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		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{short description|Mathematical function that describes the probability of occurrence of different possible outcomes in an experiment}}&lt;br /&gt;
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{{other uses|Distribution (disambiguation){{!}}Distribution}}&lt;br /&gt;
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{{Off topic|date=August 2020|Random variable|talksectionheading}}&lt;br /&gt;
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In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].&amp;lt;ref name=&amp;quot;:02&amp;quot;&amp;gt;{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}&amp;lt;/ref&amp;gt; It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In Probability Theory and Statistics, a Probability Distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;概率论 Probability Theory&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;统计学 Statistics&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率分布 Probability Distribution&amp;lt;/font&amp;gt;'''是一个给出一个实验不同可能结果出现的概率的数学函数。它是根据'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 样本空间 Sample Space&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 事件概率 Probabilities of Events&amp;lt;/font&amp;gt;'''(样本空间的子集)对随机现象的数学描述。&lt;br /&gt;
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For instance, if the [[random variable]] {{mvar|X}} is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of {{mvar|X}} would take the value 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} heads}}, and 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} tails}} (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a [[Survey methodology|survey]], etc.&amp;lt;ref name=&amp;quot;ross&amp;quot;&amp;gt;{{cite book|first=Sheldon M.|last=Ross|title=A first course in probability|publisher=Pearson|year=2010}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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For instance, if the random variable X is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of X would take the value 0.5 for X=heads, and 0.5 for X=tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.&lt;br /&gt;
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例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。随机现象的例子包括未来某一天的天气状况、一个人的身高、学校中男生的比例、调查结果等等。&lt;br /&gt;
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A probability distribution is a mathematical function that has a [[sample space]] as its '''input''', and gives a '''probability''' as its output. The sample space is the [[Set (mathematics)|set]] of all possible [[Outcome (probability)|outcomes]] of a random phenomenon being observed; it may be the set of [[real numbers]] or a set of [[vector (mathematics)|vectors]], or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be {{math|{heads, tails} }}.&lt;br /&gt;
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A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be .&lt;br /&gt;
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概率分布是一个数学函数，它的输入是一个样本空间，输出是一个概率。样本空间是所观察到的随机现象的所有可能结果的集合; 它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。&lt;br /&gt;
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Probability distributions are generally divided into two classes. A '''discrete probability distribution''' is applicable to the scenarios where the set of possible outcomes is [[discrete probability distribution|discrete]] (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the [[probability mass function]]. On the other hand, '''continuous probability distributions''' are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a [[probability density function]].&amp;lt;ref name=&amp;quot;ross&amp;quot; /&amp;gt;&amp;lt;ref&amp;gt;{{cite book|author1=DeGroot, Morris H.|author2=Schervish, Mark J.|title=Probability and Statistics|publisher=Addison-Wesley|year=2002}}&amp;lt;/ref&amp;gt; The [[normal distribution]] is a commonly encountered continuous probability distribution. More complex experiments, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.&lt;br /&gt;
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Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function. The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.&lt;br /&gt;
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概率分布一般分为两类。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 离散概率分布 Discrete Probability Distribution&amp;lt;/font&amp;gt;'''适用于一组可能的结果是离散的情况，如抛硬币或掷骰子。这里的概率被编码为结果概率的离散列表，称为概率质量函数。另一方面，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 连续概率分布 Continuous Probability Distribution&amp;lt;/font&amp;gt;'''适用于一组可以在一个连续的范围内取值的结果的情况(例如:实数)，例如某一天的温度。在这种情况下，概率通常由概率密度函数描述。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 正态分布 Normal Distribution&amp;lt;/font&amp;gt;'''是一种常见的连续概率分布。更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called [[Univariate distribution|univariate]], while a distribution whose sample space is a [[vector space]] of dimension 2 or more is called [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a [[joint probability distribution]]) gives the probabilities of a [[random vector]] – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the [[binomial distribution]], the [[hypergeometric distribution]], and the [[normal distribution]]. The [[multivariate normal distribution]] is a commonly encountered multivariate distribution.&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.&lt;br /&gt;
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一个一维的样本空间(例如实数、标签列表、有序标签或二进制)的概率分布被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 单变量 Univariate&amp;lt;/font&amp;gt;'''，而样本空间为二维或更多向量空间的分布被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 多变量 Multivariate&amp;lt;/font&amp;gt;'''。单变量分布给出了单个随机变量取不同替代值的概率; 联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。重要的和常见的单变量概率分布包括二项分布、超几何分布和正态分布。多变量正态分布是一种常见的联合分布。&lt;br /&gt;
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==Introduction 简介==&lt;br /&gt;
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[[File:Dice Distribution (bar).svg|thumb|250px|right|&lt;br /&gt;
图1:The [[probability mass function]] (pmf) ''p''(''S'') specifies the probability distribution for the sum ''S'' of counts from two [[dice]].  For example, the figure shows that ''p''(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as ''P''(''S'' &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.&lt;br /&gt;
概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布。例如，图中显示 p (11) = 2/36 = 1/18。Pmf 允许计算事件的概率，如 p (s &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6，以及分布中的所有其他概率。&lt;br /&gt;
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The [[probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice.  For example, the figure shows that p(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as P(S &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]&lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between '''discrete''' and '''continuous''' [[random variable]]s. In the discrete case, it is sufficient to specify a [[probability mass function]] &amp;lt;math&amp;gt;p&amp;lt;/math&amp;gt; assigning a probability to each possible outcome: for example, when throwing a fair [[Dice|die]], each of the six values 1 to 6 has the probability 1/6. The probability of an [[Event (probability theory)|event]] is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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为了定义最简单的概率分布，有必要区分离散和连续的随机变量。在离散情况下，指定一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;概率质量函数 Probability Mass Function&amp;lt;/font&amp;gt;''' P就足够了，它为每个可能的结果赋予一个概率: 例如，当投掷一个骰子时，6个值中的每一个的概率为1/6。然后将事件的概率定义为满足事件的结果的概率之和; 例如，事件”骰子掷出偶数值”的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2.&amp;lt;/math&amp;gt;&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs ''exactly'' 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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相比之下，当一个随机变量从一个连续体中取值时，那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率。例如，一个给定的物体重量正好是500克的概率为零，因为随着我们测量仪器精度的提高，正好测量500克的概率趋向于零。然而，在质量控制方面，人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98% ，而这一要求对测量仪器的准确性不太敏感。&lt;br /&gt;
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Continuous probability distributions can be described in several ways. The [[probability density function]] describes the [[infinitesimal]] probability of any given value, and the probability that the outcome lies in a given interval can be computed by [[Integration (mathematics)|integrating]] the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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连续概率分布可以用几种方法来描述。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;概率密度函数 Probability Density Function&amp;lt;/font&amp;gt;'''描述了任意给定值的无穷小概率，并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。可能值位于某一固定区间的概率可以与和收敛于积分的方式有关，因此，连续概率是基于积分的定义。&lt;br /&gt;
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[[File:Combined Cumulative Distribution Graphs.png|thumb|455x455px|&lt;br /&gt;
图2:On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve.&lt;br /&gt;
左边是概率密度函数。右边是累积分布函数，它是概率密度曲线下面的区域。]]&lt;br /&gt;
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The [[cumulative distribution function]] describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the [[antiderivative]] of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the [[probability density function]] from minus infinity &amp;lt;math&amp;gt;\infty&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; as described by the picture to the right.&amp;lt;ref&amp;gt;{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=Springer|others=Dekking, Michel, 1946-|isbn=978-1-85233-896-1|location=London|oclc=262680588}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity \infty to x as described by the picture to the right.&lt;br /&gt;
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累积分布函数指标描述了随机变量不大于给定值的概率; 结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算。累积分布函数是概率密度函数的反导函数，前提是后者存在。正如右边图片所描述的那样，累积分布函数是从负无穷到 x 的概率密度函数下面的区域。&lt;br /&gt;
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[[File:Standard deviation diagram.svg|right|thumb|250px|&lt;br /&gt;
图3:The [[probability density function]] (pdf) of the [[normal distribution]], also called Gaussian or &amp;quot;bell curve&amp;quot;, the most important continuous random distribution.  As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.&lt;br /&gt;
正态分布的[概率密度函数(pdf) ，也称为高斯或钟形曲线，是最重要的连续随机分布。如图所示，值间隔的概率对应于曲线下面积。]]&lt;br /&gt;
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== Terminology 术语==&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.&amp;lt;ref name=&amp;quot;:02&amp;quot; /&amp;gt;&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X &amp;gt; a, X &amp;lt; b or a union thereof.&lt;br /&gt;
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一些关键的，广泛用于以概率分布为主题的文献中的概念和术语，列出如下。&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Functions for discrete variables 离散变量函数&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
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Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.&lt;br /&gt;
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用于统计建模的著名离散概率分布包括泊松分佈、伯努利分布、二项分布、几何分布和负二项分布。&lt;br /&gt;
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*'''Probability function''': describes the probability distribution of a discrete random variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率函数 Probability Function&amp;lt;/font&amp;gt;'''：描述离散随机变量的概率分布。&lt;br /&gt;
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*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率质量函数（pmf Probability Mass Function&amp;lt;/font&amp;gt;'''：给出离散随机变量等于某个值的概率的函数。&lt;br /&gt;
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*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 频率分布 Frequency Distribution&amp;lt;/font&amp;gt;'''：显示样本中各种结果的频率的表格。&lt;br /&gt;
&lt;br /&gt;
For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
&lt;br /&gt;
对于一个离散的随机变量 x，设 u0，u1，... 是它在非零概率情况下可以取的值。表示&lt;br /&gt;
&lt;br /&gt;
*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 相对频率分布 Relative Frequency Distribution&amp;lt;/font&amp;gt;'''：一种频率分布，其中每个值均已被样本中的多个结果（即样本大小）除（归一化）。&lt;br /&gt;
&lt;br /&gt;
*'''Discrete probability distribution function''': general term to indicate the way the total probability of 1 is distributed over '''all''' various possible outcomes (i.e. over entire population) for discrete random variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;离散概率分布函数 Discrete Probability Distribution Function&amp;lt;/font&amp;gt;'''：通用术语，表示总概率1在离散随机变量的所有各种可能结果（即整个人群）中的分布方式。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for a discrete random variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 累积分布函数 Cumulative distribution function&amp;lt;/font&amp;gt;'''：该函数评估离散随机变量X取小于或等于x的值的概率。&lt;br /&gt;
&lt;br /&gt;
*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 分类分布 Categorical Distribution&amp;lt;/font&amp;gt;'''：适用于具有有限值集的离散随机变量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Functions for continuous variables 连续变量函数&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* '''[[Probability density function]] (pdf):''' function whose value at any given sample (or point) in the [[sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率密度函数（pdf）Probability Density function&amp;lt;/font&amp;gt;'''：可以将其在样本空间中任意给定样本（或点）上的值（随机变量可能获得的一组值）的值解释为提供随机变量值将具有的相对可能性的函数等于那个样本。&lt;br /&gt;
&lt;br /&gt;
It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as&lt;br /&gt;
&lt;br /&gt;
由此可见，除了 u0、 u1、 ... 之外，x 取任何值的概率为零，因此可以将 x 写为&lt;br /&gt;
&lt;br /&gt;
* '''Continuous probability distribution function''': most often reserved for continuous random variables.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;连续概率分布函数 Continuous Probability Distribution Function&amp;lt;/font&amp;gt;'''：最常保留的连续随机变量。&lt;br /&gt;
&lt;br /&gt;
* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for continuous variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 累积分布函数 Cumulative distribution function&amp;lt;/font&amp;gt;'''：评估连续变量X取小于或等于x的值的概率的函数。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Basic terms 基本术语===&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Mode (statistics)|'''Mode''']]: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.&lt;br /&gt;
模式：对于离散随机变量，该值具有最高概率；对于连续随机变量，是概率密度函数具有局部峰值的位置。&lt;br /&gt;
&lt;br /&gt;
* [[Support (mathematics)|'''Support''']]: set of values that can be assumed, with non-zero probability, by the random variable.&lt;br /&gt;
支持：可以由随机变量以非零概率假定的一组值。对于随机变量X，有时表示为R_ {X}&lt;br /&gt;
&lt;br /&gt;
* '''Tail''':&amp;lt;ref name='tail'&amp;gt;More information and examples can be found in the articles [[Heavy-tailed distribution]], [[Long-tailed distribution]], [[fat-tailed distribution]]&amp;lt;/ref&amp;gt; the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form &amp;lt;math&amp;gt;X &amp;gt; a&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;X &amp;lt; b&amp;lt;/math&amp;gt; or a union thereof.&lt;br /&gt;
尾巴：如果pmf或pdf相对较低，则靠近随机变量边界的区域。通常形式为X&amp;gt; a，X &amp;lt;b或它们的并集。&lt;br /&gt;
&lt;br /&gt;
*'''Head''':&amp;lt;ref name='tail' /&amp;gt; the region where the pmf or pdf is relatively high. Usually has the form &amp;lt;math&amp;gt;a &amp;lt; X &amp;lt; b&amp;lt;/math&amp;gt;.&lt;br /&gt;
头部：pmf或pdf较高的区域。通常具有a &amp;lt;X &amp;lt;b的形式&lt;br /&gt;
&lt;br /&gt;
* '''[[Expected value]]''' or '''mean''': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.&lt;br /&gt;
期望值或均值：可能值的加权平均值，以其概率作为权重；或其连续类似物。&lt;br /&gt;
&lt;br /&gt;
A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a cumulative density function that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.&lt;br /&gt;
&lt;br /&gt;
一个连续的概率分布是一个支持不可数集的概率分布，比如实数行中的一个区间。它们是唯一拥有属性的一个累积密度函数，可以用来计算支持的每个子集的概率。有许多连续概率分布的例子: 正态、均匀、卡方等等。&lt;br /&gt;
&lt;br /&gt;
*'''[[Median]]''': the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.&lt;br /&gt;
中位数：这样的值，即一组值小于中位数，而该组大于中位数，每一个的概率不大于二分之一。&lt;br /&gt;
&lt;br /&gt;
*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.&lt;br /&gt;
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。&lt;br /&gt;
&lt;br /&gt;
* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.&lt;br /&gt;
标准偏差：方差的平方根，因此是色散的另一种度量。&lt;br /&gt;
&lt;br /&gt;
* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.&lt;br /&gt;
对称性：某些分布的一种属性，其中特定值左侧（通常是中位数）的分布部分是其右侧部分的镜像。&lt;br /&gt;
&lt;br /&gt;
*'''[[Skewness]]''': a measure of the extent to which a pmf or pdf &amp;quot;leans&amp;quot; to one side of its mean. The third [[standardized moment]] of the distribution.&lt;br /&gt;
偏度：衡量pmf或pdf在其均值的一侧“倾斜”的程度。分布的第三个标准化时刻。&lt;br /&gt;
&lt;br /&gt;
*'''[[Kurtosis]]''': a measure of the &amp;quot;fatness&amp;quot; of the tails of a pmf or pdf. The fourth standardized moment of the distribution.&lt;br /&gt;
峰度：pmf或pdf尾部“脂肪”的量度。分布的第四个标准化时刻。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
=='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Discrete probability distribution 离散概率分布&amp;lt;/font&amp;gt;'''==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Discrete probability distrib.svg|right|thumb|&lt;br /&gt;
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Discrete probability distribution.svg|right|thumb|&lt;br /&gt;
图5:The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...离散概率分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Normal probability distribution.svg|right|thumb|&lt;br /&gt;
图6:... of a continuous probability distribution, ...连续概率分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Mixed probability distribution.svg|right|thumb|&lt;br /&gt;
图7:... of a distribution which has both a continuous part and a discrete part.既有连续部分又有离散部分]]&lt;br /&gt;
&lt;br /&gt;
 &lt;br /&gt;
A '''discrete probability distribution''' is a probability distribution that can take on a countable number of values.&amp;lt;ref&amp;gt;{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer|isbn=9780387878591|location=New York|pages=51|oclc=710149819}}&amp;lt;/ref&amp;gt; For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if &amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.&lt;br /&gt;
&lt;br /&gt;
离散概率分布是可以具有可数数量的值的概率分布。在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1。例如，如果&amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2，概率之和为1/2 + 1/4 + 1/8 + ... = 1。&lt;br /&gt;
&lt;br /&gt;
Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt; Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.&lt;br /&gt;
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
When a [[Sample (statistics)|sample]] (a set of observations) is drawn from a larger population, the sample points have an [[empirical distribution function|empirical distribution]] that is discrete and that provides information about the population distribution.&lt;br /&gt;
当从更大的总体中抽取一个样本（一组观察值）时，这些采样点的经验分布是离散的，并且提供了有关总体分布的信息。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Measure theoretic formulation 测量理论公式===&lt;br /&gt;
&lt;br /&gt;
A [[measurable function]] &amp;lt;math&amp;gt;X \colon A \to B &amp;lt;/math&amp;gt; between a [[probability space]] &amp;lt;math&amp;gt;(A, \mathcal A, P)&amp;lt;/math&amp;gt; and a [[measurable space]] &amp;lt;math&amp;gt;(B, \mathcal B) &amp;lt;/math&amp;gt; is called a '''discrete random variable''' provided that its image is a countable set. In this case measurability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; means that the pre-images of singleton sets are measurable, i.e., &amp;lt;math&amp;gt;X^{-1}(\{b\}) \in \mathcal A&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;b \in B&amp;lt;/math&amp;gt;.&lt;br /&gt;
The latter requirement induces a [[probability mass function]] &amp;lt;math&amp;gt;f_X \colon X(A) \to \mathbb R&amp;lt;/math&amp;gt; via &amp;lt;math&amp;gt; f_X(b):=P(X^{-1}(\{b\}))&amp;lt;/math&amp;gt;.  Since the pre-images of disjoint sets&lt;br /&gt;
are disjoint,&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
This recovers the definition given above.&lt;br /&gt;
&lt;br /&gt;
一个可测量的函数 &amp;lt;math&amp;gt;X \colon A \to B &amp;lt;/math&amp;gt; 在一个概率空间中 &amp;lt;math&amp;gt;(A, \mathcal A, P)&amp;lt;/math&amp;gt; and 和一个可测量空间 &amp;lt;math&amp;gt;(B, \mathcal B) &amp;lt;/math&amp;gt; 被叫做离散随机变量。该图像是一个可数的集合。在这种情况下&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;的测量意味着单例集的原像是可测量的 i.e., &amp;lt;math&amp;gt;X^{-1}(\{b\}) \in \mathcal A&amp;lt;/math&amp;gt; 对于所有的&amp;lt;math&amp;gt;b \in B&amp;lt;/math&amp;gt;.&lt;br /&gt;
后者需要包括概率质量函数 &amp;lt;math&amp;gt;f_X \colon X(A) \to \mathbb R&amp;lt;/math&amp;gt; via &amp;lt;math&amp;gt; f_X(b):=P(X^{-1}(\{b\}))&amp;lt;/math&amp;gt;.  由于不相交集的原像不相交&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
这包含了上面所提到的定义&lt;br /&gt;
&lt;br /&gt;
==='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 累积分布函数 Cumulative distribution function&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it &amp;quot;jumps&amp;quot; to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.&lt;br /&gt;
&lt;br /&gt;
与上述等效，可以将离散随机变量定义为其累积分布函数（cdf）仅因跳跃不连续性而增加的随机变量，也就是说，其cdf仅在“跳跃”至较高值时才增加，并且在那些跳跃点间是常数。但是请注意，cdf跳转的点可能会形成密集的实数集。发生跳变的点恰好是随机变量可能取的值。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Delta-function representation 三角函数表示===&lt;br /&gt;
&lt;br /&gt;
Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions.  This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.&amp;lt;ref&amp;gt;{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此，离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数，该函数实质上统一了对连续分布和离散分布的处理。当处理涉及连续和离散部分的概率分布时，这特别有用。&lt;br /&gt;
&lt;br /&gt;
===Indicator-function representation 指标功能表示===&lt;br /&gt;
&lt;br /&gt;
For a discrete random variable ''X'', let ''u''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;, ''u''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
对于离散随机变量X，令u0，u1，...是它可以以非零概率获取的值。表示&lt;br /&gt;
:&amp;lt;math&amp;gt;\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这些是不相交的集合，对于这样的集合：&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此，X取u0，u1，...以外的任何值的概率为零，因此可以将X写入为&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
except on a set of probability zero, where &amp;lt;math&amp;gt;1_A&amp;lt;/math&amp;gt; is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.&lt;br /&gt;
除了概率为零的集合外，其中1_ {A}是A的指标函数。这可以用作离散随机变量的替代定义。&lt;br /&gt;
&lt;br /&gt;
==Continuous probability distribution 连续概率分布==&lt;br /&gt;
&lt;br /&gt;
{{See also|Probability density function}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.&amp;lt;ref&amp;gt;{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}&amp;lt;/ref&amp;gt; They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].&lt;br /&gt;
连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]。连续概率分布有很多示例：正态分布，均匀分布，卡方分布和其他分布。&lt;br /&gt;
&lt;br /&gt;
A random variable &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; has a continuous probability distribution if there is a function &amp;lt;math&amp;gt;f: \mathbb{R} \rightarrow [0, \infty)&amp;lt;/math&amp;gt; such that for each interval &amp;lt;math&amp;gt;I \subset \mathbb{R}&amp;lt;/math&amp;gt; the probability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; belonging to &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt; is given by the integral of &amp;lt;math&amp;gt;f&amp;lt;/math&amp;gt; over &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}&amp;lt;/ref&amp;gt; For example, if &amp;lt;math&amp;gt;I = [a, b]&amp;lt;/math&amp;gt; then we would have:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一个随机变量&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; 有一个连续的概率分布，如果这有一个函数&amp;lt;math&amp;gt;f: \mathbb{R} \rightarrow [0, \infty)&amp;lt;/math&amp;gt; 对于每一个区间&amp;lt;math&amp;gt;I \subset \mathbb{R}&amp;lt;/math&amp;gt;  &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;的概率 &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;是&amp;lt;math&amp;gt;f&amp;lt;/math&amp;gt; 在 &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;上的积分.&amp;lt;ref&amp;gt;Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}&amp;lt;/ref&amp;gt; 例如, 如果&amp;lt;math&amp;gt;I = [a, b]&amp;lt;/math&amp;gt; 可得到:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In particular, the probability for &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; to take any single value &amp;lt;math&amp;gt;a&amp;lt;/math&amp;gt; (that is &amp;lt;math&amp;gt;a \le X \le a&amp;lt;/math&amp;gt;) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;F(x) = \operatorname{P}\left[-\infty &amp;lt; X \le x\right] = \int_{-\infty}^x f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
特别是，X取任何单个值a（即a≤X≤a）的概率为零，因为上下限一致的积分始终等于零。满足上述条件的变量称为连续随机变量。其累积密度函数定义为&lt;br /&gt;
&lt;br /&gt;
which, by this definition, has the properties:&lt;br /&gt;
根据定义有以下一些性质&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is non-decreasing;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;0 \le F(x) \le 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;\lim_{x \rightarrow -\infty} F(x) = 0&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\lim_{x \rightarrow \infty} F(x) = 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;P(a \le X &amp;lt; b) = F(b) - F(a)&amp;lt;/math&amp;gt;; and&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is continuous (due to the [[Riemann integral]] properties).&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
It is also possible to think in the opposite direction, which allows more flexibility. Say &amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is a function that satisfies all but the last of the properties above, then &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; represents the cumulative density function for some random variable: a discrete random variable if &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; is a step function, and a continuous random variable otherwise.&amp;lt;ref&amp;gt;See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.&amp;lt;/ref&amp;gt; This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].&lt;br /&gt;
也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; is such an absolutely continuous random variable, then it has a [[probability density function]] &amp;lt;math&amp;gt;f(x)&amp;lt;/math&amp;gt;, and its probability of falling into a Lebesgue-measurable set &amp;lt;math&amp;gt;A \subset \mathbb{R}&amp;lt;/math&amp;gt; is:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; is the Lebesgue measure.&lt;br /&gt;
&lt;br /&gt;
对于实线的更多任意子集，通常有必要对上述定义进行概括。在这些情况下，连续概率分布定义为具有绝对连续的累积分布函数的概率分布。等效地，就Lebesgue测度而言，它是实数上的概率分布，它是绝对连续的。这样的分布可以用它们的概率密度函数表示。如果X是这样一个绝对连续的随机变量，则它具有概率密度函数f（x），并且落入Lebesgue可测量集合A⊂R的概率为：&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这里，mu是Lebesgue度量&lt;br /&gt;
&lt;br /&gt;
Note on terminology: some authors use the term &amp;quot;continuous distribution&amp;quot; to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;\mu\{x\}\,=\,0&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;\,x&amp;lt;/math&amp;gt;. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].&lt;br /&gt;
&lt;br /&gt;
关于术语的注释：一些作者使用术语“连续分布”来表示其累积分布函数是连续的而不是绝对连续的分布。这些分布是所有x的μ{x} = 0的μ分布。该定义包括上面定义的（绝对）连续分布，但也包括奇异分布，既不是绝对连续也不是离散的，也不是它们的混合。没有密度。 Cantor分布给出了一个示例。&lt;br /&gt;
&lt;br /&gt;
== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; [[Andrey Kolmogorov|Kolmogorov]] definition 柯尔莫哥洛夫的定义&amp;lt;/font&amp;gt;'''==&lt;br /&gt;
&lt;br /&gt;
{{Main|Probability space|Probability measure}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; from a [[probability space]] &amp;lt;math&amp;gt;(\Omega, \mathcal{F}, \mathbb{P})&amp;lt;/math&amp;gt; to a [[measurable space]] &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt;. Given that probabilities of events of the form &amp;lt;math&amp;gt;\{\omega\in\Omega\mid X(\omega)\in A\}&amp;lt;/math&amp;gt; satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] &amp;lt;math&amp;gt;X_*\mathbb{P}&amp;lt;/math&amp;gt; of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; , which is a [[probability measure]] on &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt; satisfying &amp;lt;math&amp;gt;X_*\mathbb{P} = \mathbb{P}X^{-1}&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
在概率论的度量理论形式化中，将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）。给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P，它是满足&amp;lt;math&amp;gt;X_*\mathbb{P} = \mathbb{P}X^{-1}&amp;lt;/math&amp;gt;的概率量度&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Random number generation 随机数生成==&lt;br /&gt;
{{Main|Pseudo-random number sampling}}&lt;br /&gt;
&lt;br /&gt;
Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Citation|last1=Dekking|first1=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
大多数算法基于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 伪随机数生成器 Pseudorandom Number Generator&amp;lt;/font&amp;gt;'''，该伪随机数生成器生成在半开间隔[0,1）中均匀分布的数字X。然后，通过某种算法对这些随机变量X进行转换，以创建具有所需概率分布的新随机变量。利用这种统一的伪随机源，可以生成任何随机变量的实现。&lt;br /&gt;
&lt;br /&gt;
For example, suppose &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt; has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some &amp;lt;math&amp;gt;0 &amp;lt; p &amp;lt; 1&amp;lt;/math&amp;gt;, we define&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
so that&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\textrm{P}(X=1) = \textrm{P}(U&amp;lt;p) = p,&lt;br /&gt;
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
例如，假设U具有介于0和1之间的均匀分布。为某些对象构造一个随机的Bernoulli变量0 &amp;lt;p &amp;lt;1我们定义&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\textrm{P}(X=1) = \textrm{P}(U&amp;lt;p) = p,&lt;br /&gt;
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
This random variable X has a Bernoulli distribution with parameter &amp;lt;math&amp;gt;p&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:0&amp;quot;/&amp;gt; Note that this is a transformation of discrete random variable.&lt;br /&gt;
该随机变量X具有参数的伯努利分布p。请注意，这是离散随机变量的变换。&lt;br /&gt;
&lt;br /&gt;
For a distribution function &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; of a continuous random variable, a continuous random variable must be constructed. &amp;lt;math&amp;gt;F^{inv}&amp;lt;/math&amp;gt;, an inverse function of &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt;, relates to the uniform variable &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{U\leq F(x)} = {F^{inv}(U)\leq x}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
F或连续随机变量的分布函数F，必须构造连续随机变量。 &amp;lt;math&amp;gt;F^{inv}&amp;lt;/math&amp;gt;，F的反函数，涉及均匀变量U：&lt;br /&gt;
&amp;lt;math&amp;gt;{U\leq F(x)} = {F^{inv}(U)\leq x}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For example, suppose a random variable that has an exponential distribution &amp;lt;math&amp;gt;F(x) = 1 - e^{-\lambda x}&amp;lt;/math&amp;gt; must be constructed.&lt;br /&gt;
例如，假设必须构造一个具有指数分布&amp;lt;math&amp;gt;F(x) = 1 - e^{-\lambda x}&amp;lt;/math&amp;gt; 的随机变量。&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
F(x) = u &amp;amp;\Leftrightarrow 1-e^{-\lambda x} = u \\ &amp;amp;\Leftrightarrow e^{-\lambda x } = 1-u \\&amp;amp;\Leftrightarrow -\lambda x = \ln(1-u) \\ &amp;amp;\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)&lt;br /&gt;
\end{align}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
所以&amp;lt;math&amp;gt;F^{inv}(u) = \frac{-1}{\lambda}\ln(1-u)&amp;lt;/math&amp;gt; 并且如果 &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt; 有一个&amp;lt;math&amp;gt;U(0,1)&amp;lt;/math&amp;gt; 分布, 然后随机变量 &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; 被定义为 &amp;lt;math&amp;gt;X = F^{inv}(U) = \frac{-1}{\lambda} \ln(1-U)&amp;lt;/math&amp;gt;. 这里有一个指数分布 &amp;lt;math&amp;gt;\lambda&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.&lt;br /&gt;
统计模拟（蒙特卡洛方法）中经常遇到的一个问题是生成以给定方式分布的伪随机数。&lt;br /&gt;
&lt;br /&gt;
== Common probability distributions and their applications 共同概率分布及其应用==&lt;br /&gt;
{{Main list|List of probability distributions}}&lt;br /&gt;
&lt;br /&gt;
The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.&lt;br /&gt;
&lt;br /&gt;
概率分布的概念及其描述的随机变量是概率论和统计学科学的数学基础。人口中几乎可以测量的任何值都存在价差或可变性（例如人的身高，金属的耐用性，销售增长，交通流量等）;几乎所有测量均存在一定的固有误差；在物理学中，从气体的动力学特性到基本粒子的量子力学描述，很多过程都用概率论来描述。由于这些以及许多其他原因，简单数字通常不足以描述数量，而概率分布通常更合适。&lt;br /&gt;
&lt;br /&gt;
The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see [[list of probability distributions]], which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)&lt;br /&gt;
以下是一些最常见的概率分布列表，按与之相关的过程类型进行分组。有关更完整的列表，请参见概率分布列表，该列表按要考虑的结果的性质（离散，连续，多元等）进行分组。&lt;br /&gt;
&lt;br /&gt;
All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a [[mixture distribution]].&lt;br /&gt;
下面所有的单变量分布都达到了峰值。也就是说，假设值聚集在单个点周围。实际上，实际观察到的量可能会聚集在多个值附近。可以使用混合物分布对此类数量进行建模。&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Linear growth (e.g. errors, offsets) 线性增长 &amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
* [[Normal distribution]] (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 正态分布（高斯分布 Normal Distribution&amp;lt;/font&amp;gt;'''，对于单个这样的数量；最常用的连续分布&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Exponential growth (e.g. prices, incomes, populations) 指数增长&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;对数正态分布 Log-normal Distribution&amp;lt;/font&amp;gt;'''，对于单个此类数量的对数正态分布&lt;br /&gt;
&lt;br /&gt;
* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;帕累托分布 Pareto Distribution&amp;lt;/font&amp;gt;'''，对于单个这样的数量，其对数呈指数分布；原型幂律分布&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Uniformly distributed quantities 数量均匀分布&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 离散均匀分布 Discrete Uniform Distributed&amp;lt;/font&amp;gt;'''，用于有限的一组值（例如，公平死亡的结果）&lt;br /&gt;
&lt;br /&gt;
* [[Continuous uniform distribution]], for continuously distributed values&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;连续均匀分布 Continuous Uniform Distributed&amp;lt;/font&amp;gt;'''，用于连续分布的值&lt;br /&gt;
&lt;br /&gt;
=== Bernoulli trials (yes/no events, with a given probability) 伯努利试验（是/否事件，具有给定的概率）===&lt;br /&gt;
&lt;br /&gt;
* Basic distributions 基本分布:&lt;br /&gt;
** [[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 伯努利分布 Bernoulli Distribution&amp;lt;/font&amp;gt;'''，用于单个伯努利试验的结果（例如成功/失败，是/否）&lt;br /&gt;
&lt;br /&gt;
** [[Binomial distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed total number of [[Independent (statistics)|independent]] occurrences&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项式分布 Binomial Distribution &amp;lt;/font&amp;gt;'''，对于给定固定总数的独立“出现次数”（例如，成功，赞成票等）的数量&lt;br /&gt;
&lt;br /&gt;
** [[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;负二项分布 Negative Binomial Distribution&amp;lt;/font&amp;gt;'''，用于二项式观察，但是关注的数量是在给定成功次数之前发生的失败次数&lt;br /&gt;
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** [[Geometric distribution]], for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the [[negative binomial distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;几何分布 Geometric Distribution&amp;lt;/font&amp;gt;'''，用于二项式观测，但是关注的数量是首次成功之前的失败数量；负二项式分布的特殊情况&lt;br /&gt;
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* Related to sampling schemes over a finite population 与有限人口抽样方案有关:&lt;br /&gt;
&lt;br /&gt;
** [[Hypergeometric distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using [[sampling without replacement]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 超几何分布 Hypergeometric Distribution&amp;lt;/font&amp;gt;'''，对于“肯定出现”的数量（例如成功，赞成票等），给定了一定的总出现数量，使用采样而无需替换&lt;br /&gt;
&lt;br /&gt;
** [[Beta-binomial distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a [[Pólya urn model]] (in some sense, the &amp;quot;opposite&amp;quot; of [[sampling without replacement]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 贝塔二项式分布 Beta-binomial Distribution&amp;lt;/font&amp;gt;'''，对于给定的总发生次数为“阳性”的次数（例如，成功，赞成票等），使用Pólyaurn模型进行采样（在某种意义上，为“替代”而不进行替换） ）&lt;br /&gt;
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=== Categorical outcomes (events with ''K'' possible outcomes, with a given probability for each outcome) 分类结果（具有K个可能结果的事件，每个结果具有给定的概率）===&lt;br /&gt;
&lt;br /&gt;
* [[Categorical distribution]], for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the [[Bernoulli distribution]]&lt;br /&gt;
针对单个分类结果的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分类分布 Categorical Distribution&amp;lt;/font&amp;gt;'''（例如，调查中的是/否/也许）；伯努利分布的一般化&lt;br /&gt;
&lt;br /&gt;
* [[Multinomial distribution]], for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the [[binomial distribution]]&lt;br /&gt;
给定总结果的固定数量，针对每种类别结果的数量的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;多项式分布 Multinomial Distribution&amp;lt;/font&amp;gt;'''；二项式分布的一般化&lt;br /&gt;
&lt;br /&gt;
* [[Multivariate hypergeometric distribution]], similar to the [[multinomial distribution]], but using [[sampling without replacement]]; a generalization of the [[hypergeometric distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;多元超几何分布 Multivariate Distribution&amp;lt;/font&amp;gt;'''，类似于多项式分布，但使用采样而不进行替换；超几何分布的一般化&lt;br /&gt;
&lt;br /&gt;
=== Poisson process (events that occur independently with a given rate) 泊松过程（以给定速率独立发生的事件）===&lt;br /&gt;
&lt;br /&gt;
* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 泊松分布 Poisson Distribution&amp;lt;/font&amp;gt;'''，用于给定时间段内泊松型事件的发生次数&lt;br /&gt;
&lt;br /&gt;
* [[Exponential distribution]], for the time before the next Poisson-type event occurs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;指数分布 Exponential Distribution&amp;lt;/font&amp;gt;'''，在下一次泊松型事件发生之前的时间&lt;br /&gt;
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* [[Gamma distribution]], for the time before the next k Poisson-type events occur&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;伽马分布 Gamma Distribution&amp;lt;/font&amp;gt;'''，在接下来的k个泊松型事件发生之前的时间&lt;br /&gt;
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=== Absolute values of vectors with normally distributed components 具有正态分布分量的向量的绝对值===&lt;br /&gt;
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* [[Rayleigh distribution]], for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;瑞利分布 Rayleigh Distribution&amp;lt;/font&amp;gt;'''，用于具有高斯分布正交分量的矢量幅度分布。在具有高斯实部和虚部的RF信号中发现瑞利分布。&lt;br /&gt;
&lt;br /&gt;
* [[Rice distribution]], a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in [[Rician fading]] of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;莱斯分布 Rice Distribution&amp;lt;/font&amp;gt;'''，是在背景信号分量稳定的情况下瑞利分布的概括。由于多径传播而在无线电信号的Rician衰落中发现，并且在非零NMR信号中出现噪声破坏的MR图像中也发现了这种情况。&lt;br /&gt;
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=== Normally distributed quantities operated with sum of squares (for hypothesis testing) 以平方和运算的正态分布量（用于假设检验）===&lt;br /&gt;
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* [[Chi-squared distribution]], the distribution of a sum of squared [[standard normal]] variables; useful e.g. for inference regarding the [[sample variance]] of normally distributed samples (see [[chi-squared test]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;卡方分布 Chi-squared Distribution&amp;lt;/font&amp;gt;'''，标准正态变量平方和的分布；有用的关于正态分布样本的样本方差的推论（请参见卡方检验）&lt;br /&gt;
&lt;br /&gt;
* [[Student's t distribution]], the distribution of the ratio of a [[standard normal]] variable and the square root of a scaled [[Chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;学生t分布 Student‘s t Distribution&amp;lt;/font&amp;gt;'''，标准正态变量与缩放的卡方变量的平方根之比的分布；有助于推断方差未知的正态分布样本的平均值（请参阅学生的t检验）&lt;br /&gt;
&lt;br /&gt;
* [[F-distribution]], the distribution of the ratio of two scaled [[Chi squared distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;F-分布 F-Distribution&amp;lt;/font&amp;gt;'''，两个比例卡方变量的比例分布；有用的用于涉及比较方差或涉及R平方（相关系数平方）的推论&lt;br /&gt;
&lt;br /&gt;
=== As a conjugate prior distributions in Bayesian inference 作为贝叶斯推断中的共轭先验分布===&lt;br /&gt;
{{Main|Conjugate prior}}&lt;br /&gt;
&lt;br /&gt;
* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Beta分布 Beta Distribution&amp;lt;/font&amp;gt;'''，具有单个概率（0到1之间的实数）；与伯努利分布和二项式分布共轭&lt;br /&gt;
&lt;br /&gt;
* [[Gamma distribution]], for a non-negative scaling parameter; conjugate to the rate parameter of a [[Poisson distribution]] or [[exponential distribution]], the [[Precision (statistics)|precision]] (inverse [[variance]]) of a [[normal distribution]], etc.&lt;br /&gt;
伽玛分布，用于非负比例缩放参数；与泊松分布或指数分布的速率参数，正态分布的精度（逆方差）等共轭。&lt;br /&gt;
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* [[Dirichlet distribution]], for a vector of probabilities that must sum to 1; conjugate to the [[categorical distribution]] and [[multinomial distribution]]; generalization of the [[beta distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Dirichlet分布 Dirichlet Distribution&amp;lt;/font&amp;gt;'''，对于必须为1的概率向量；与分类分布和多项式分布共轭； beta分布的一般化&lt;br /&gt;
&lt;br /&gt;
*[[Wishart distribution]], for a symmetric [[non-negative definite]] matrix; conjugate to the inverse of the [[covariance matrix]] of a [[multivariate normal distribution]]; generalization of the [[gamma distribution]]&amp;lt;ref&amp;gt;{{Cite book|title=Pattern recognition and machine learning|last=Bishop, Christopher M.|date=2006|publisher=Springer|isbn=0-387-31073-8|location=New York|oclc=71008143}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Wishart分布 Wishart Distribution&amp;lt;/font&amp;gt;'''，用于对称非负定矩阵；与多元正态分布的协方差矩阵的逆共轭；伽玛分布的一般化&lt;br /&gt;
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=== Some specialized applications of probability distributions 概率分布的一些特殊应用===&lt;br /&gt;
&lt;br /&gt;
* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.&lt;br /&gt;
在自然语言处理中使用的高速缓存语言模型和其他统计语言模型通过概率分布来为特定单词和单词序列的出现分配概率。&lt;br /&gt;
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* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by &amp;lt;math&amp;gt;P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 &amp;lt;/math&amp;gt;, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.&amp;lt;ref&amp;gt;{{Cite book|title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|publisher=|others=Thoman, John W., Jr., 1960-|year=|isbn=978-1-68015-835-9|location=[Mill Valley, California]|pages=403–406|oclc=927509011}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
在量子力学中，在给定点处找到粒子的概率密度与该点处粒子波函数大小的平方成正比（请参阅博恩法则）。因此，粒子位置的概率分布函数描述为&amp;lt;math&amp;gt;P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 &amp;lt;/math&amp;gt;，粒子位置的概率x在第一个维度中的间隔为a≤x≤b，在第三个维度中的间隔类似。这是量子力学的关键原理。&lt;br /&gt;
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* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution.&amp;lt;ref&amp;gt;{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.|last2=Chen|first2=Z.|last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review|doi=10.1109/drpt.2008.4523658|s2cid=18669309}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
潮流研究中的概率潮流解释了作为概率分布的输入变量的不确定性，并以概率分布的形式提供了潮流计算。&lt;br /&gt;
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* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.&amp;lt;ref&amp;gt;{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity, Rajib|isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
根据先前的频率分布（例如热带气旋，冰雹，事件之间的时间等）预测自然现象的发生。&lt;br /&gt;
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==See also 另请参见==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Copula (statistics)]]&lt;br /&gt;
Copula（统计数据）&lt;br /&gt;
* [[Empirical probability]]&lt;br /&gt;
经验概率&lt;br /&gt;
* [[Histogram]]&lt;br /&gt;
直方图&lt;br /&gt;
* [[Likelihood function]]&lt;br /&gt;
似然函数&lt;br /&gt;
* [[Kirkwood approximation]]&lt;br /&gt;
柯克伍德近似&lt;br /&gt;
* [[Moment-generating function]]&lt;br /&gt;
瞬时产生功能&lt;br /&gt;
* [[Riemann–Stieltjes integral#Application to probability theory|Riemann–Stieltjes integral application to probability theory]]&lt;br /&gt;
Riemann–Stieltjes积分＃在概率论中的应用&lt;br /&gt;
*[[Pairwise independence]]&lt;br /&gt;
成对独立&lt;br /&gt;
&lt;br /&gt;
=== Lists 清单===&lt;br /&gt;
&lt;br /&gt;
* [[List of probability distributions]]&lt;br /&gt;
概率分布的清单&lt;br /&gt;
* [[List of statistical topics]]&lt;br /&gt;
统计学话题的清淡&lt;br /&gt;
=== Probability distributions 概率分布 ===&lt;br /&gt;
&lt;br /&gt;
* [[Conditional probability distribution]]&lt;br /&gt;
条件概率分布&lt;br /&gt;
* [[Joint probability distribution]]&lt;br /&gt;
联合概率分布&lt;br /&gt;
* [[Quasiprobability distribution]]&lt;br /&gt;
拟概率分布&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
=== Citations ===&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
&lt;br /&gt;
=== Sources ===&lt;br /&gt;
{{refbegin|}}&lt;br /&gt;
* {{cite journal|doi=10.1016/j.ejmp.2014.05.002|pmid=25059432|title=Data distributions in magnetic resonance images: A review|url=|journal=[[Physica Medica]]|volume=30|issue=7|pages=725–741|year=2014|last1=den Dekker|first1=A. J.|last2=Sijbers|first2=J.}}&lt;br /&gt;
* {{cite book|last=Vapnik|first=Vladimir Naumovich|year=1998|title=Statistical Learning Theory|publisher=John Wiley and Sons}}&lt;br /&gt;
{{refend}}&lt;br /&gt;
&lt;br /&gt;
==External links==&lt;br /&gt;
{{commons|Probability distribution|Probability distribution}}&lt;br /&gt;
*{{springer|title=Probability distribution|id=p/p074900}}&lt;br /&gt;
*[http://threeplusone.com/FieldGuide.pdf Field Guide to Continuous Probability Distributions],  Gavin E. Crooks.&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{short description|Mathematical function that describes the probability of occurrence of different possible outcomes in an experiment}}&lt;br /&gt;
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{{other uses|Distribution (disambiguation){{!}}Distribution}}&lt;br /&gt;
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{{Off topic|date=August 2020|Random variable|talksectionheading}}&lt;br /&gt;
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In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].&amp;lt;ref name=&amp;quot;:02&amp;quot;&amp;gt;{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}&amp;lt;/ref&amp;gt; It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In Probability Theory and Statistics, a Probability Distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;概率论 Probability Theory&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;统计学 Statistics&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率分布 Probability Distribution&amp;lt;/font&amp;gt;'''是一个给出一个实验不同可能结果出现的概率的数学函数。它是根据'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 样本空间 Sample Space&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 事件概率 Probabilities of Events&amp;lt;/font&amp;gt;'''(样本空间的子集)对随机现象的数学描述。&lt;br /&gt;
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For instance, if the [[random variable]] {{mvar|X}} is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of {{mvar|X}} would take the value 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} heads}}, and 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} tails}} (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a [[Survey methodology|survey]], etc.&amp;lt;ref name=&amp;quot;ross&amp;quot;&amp;gt;{{cite book|first=Sheldon M.|last=Ross|title=A first course in probability|publisher=Pearson|year=2010}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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For instance, if the random variable X is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of X would take the value 0.5 for X=heads, and 0.5 for X=tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.&lt;br /&gt;
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例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。随机现象的例子包括未来某一天的天气状况、一个人的身高、学校中男生的比例、调查结果等等。&lt;br /&gt;
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A probability distribution is a mathematical function that has a [[sample space]] as its '''input''', and gives a '''probability''' as its output. The sample space is the [[Set (mathematics)|set]] of all possible [[Outcome (probability)|outcomes]] of a random phenomenon being observed; it may be the set of [[real numbers]] or a set of [[vector (mathematics)|vectors]], or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be {{math|{heads, tails} }}.&lt;br /&gt;
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A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be .&lt;br /&gt;
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概率分布是一个数学函数，它的输入是一个样本空间，输出是一个概率。样本空间是所观察到的随机现象的所有可能结果的集合; 它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。&lt;br /&gt;
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Probability distributions are generally divided into two classes. A '''discrete probability distribution''' is applicable to the scenarios where the set of possible outcomes is [[discrete probability distribution|discrete]] (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the [[probability mass function]]. On the other hand, '''continuous probability distributions''' are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a [[probability density function]].&amp;lt;ref name=&amp;quot;ross&amp;quot; /&amp;gt;&amp;lt;ref&amp;gt;{{cite book|author1=DeGroot, Morris H.|author2=Schervish, Mark J.|title=Probability and Statistics|publisher=Addison-Wesley|year=2002}}&amp;lt;/ref&amp;gt; The [[normal distribution]] is a commonly encountered continuous probability distribution. More complex experiments, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.&lt;br /&gt;
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Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function. The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.&lt;br /&gt;
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概率分布一般分为两类。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 离散概率分布 Discrete Probability Distribution&amp;lt;/font&amp;gt;'''适用于一组可能的结果是离散的情况，如抛硬币或掷骰子。这里的概率被编码为结果概率的离散列表，称为概率质量函数。另一方面，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 连续概率分布 Continuous Probability Distribution&amp;lt;/font&amp;gt;'''适用于一组可以在一个连续的范围内取值的结果的情况(例如:实数)，例如某一天的温度。在这种情况下，概率通常由概率密度函数描述。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 正态分布 Normal Distribution&amp;lt;/font&amp;gt;'''是一种常见的连续概率分布。更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called [[Univariate distribution|univariate]], while a distribution whose sample space is a [[vector space]] of dimension 2 or more is called [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a [[joint probability distribution]]) gives the probabilities of a [[random vector]] – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the [[binomial distribution]], the [[hypergeometric distribution]], and the [[normal distribution]]. The [[multivariate normal distribution]] is a commonly encountered multivariate distribution.&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.&lt;br /&gt;
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一个一维的样本空间(例如实数、标签列表、有序标签或二进制)的概率分布被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 单变量 Univariate&amp;lt;/font&amp;gt;'''，而样本空间为二维或更多向量空间的分布被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 多变量 Multivariate&amp;lt;/font&amp;gt;'''。单变量分布给出了单个随机变量取不同替代值的概率; 联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。重要的和常见的单变量概率分布包括二项分布、超几何分布和正态分布。多变量正态分布是一种常见的联合分布。&lt;br /&gt;
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==Introduction 简介==&lt;br /&gt;
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[[File:Dice Distribution (bar).svg|thumb|250px|right|&lt;br /&gt;
图1:The [[probability mass function]] (pmf) ''p''(''S'') specifies the probability distribution for the sum ''S'' of counts from two [[dice]].  For example, the figure shows that ''p''(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as ''P''(''S'' &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.&lt;br /&gt;
概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布。例如，图中显示 p (11) = 2/36 = 1/18。Pmf 允许计算事件的概率，如 p (s &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6，以及分布中的所有其他概率。&lt;br /&gt;
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The [[probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice.  For example, the figure shows that p(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as P(S &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]&lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between '''discrete''' and '''continuous''' [[random variable]]s. In the discrete case, it is sufficient to specify a [[probability mass function]] &amp;lt;math&amp;gt;p&amp;lt;/math&amp;gt; assigning a probability to each possible outcome: for example, when throwing a fair [[Dice|die]], each of the six values 1 to 6 has the probability 1/6. The probability of an [[Event (probability theory)|event]] is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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为了定义最简单的概率分布，有必要区分离散和连续的随机变量。在离散情况下，指定一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;概率质量函数 Probability Mass Function&amp;lt;/font&amp;gt;'''p 就足够了，它为每个可能的结果赋予一个概率: 例如，当投掷一个骰子时，6个值中的每一个的概率为1/6。然后将事件的概率定义为满足事件的结果的概率之和; 例如，事件”骰子掷出偶数值”的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2.&amp;lt;/math&amp;gt;&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs ''exactly'' 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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相比之下，当一个随机变量从一个连续体中取值时，那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率。例如，一个给定的物体重量正好是500克的概率为零，因为随着我们测量仪器精度的提高，正好测量500克的概率趋向于零。然而，在质量控制方面，人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98% ，而这一要求对测量仪器的准确性不太敏感。&lt;br /&gt;
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Continuous probability distributions can be described in several ways. The [[probability density function]] describes the [[infinitesimal]] probability of any given value, and the probability that the outcome lies in a given interval can be computed by [[Integration (mathematics)|integrating]] the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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连续概率分布可以用几种方法来描述。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;概率密度函数 Probability Density Function&amp;lt;/font&amp;gt;'''描述了任意给定值的无穷小概率，并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。可能值位于某一固定区间的概率可以与和收敛于积分的方式有关，因此，连续概率是基于积分的定义。&lt;br /&gt;
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[[File:Combined Cumulative Distribution Graphs.png|thumb|455x455px|&lt;br /&gt;
图2:On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve.&lt;br /&gt;
左边是概率密度函数。右边是累积分布函数，它是概率密度曲线下面的区域。]]&lt;br /&gt;
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The [[cumulative distribution function]] describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the [[antiderivative]] of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the [[probability density function]] from minus infinity &amp;lt;math&amp;gt;\infty&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; as described by the picture to the right.&amp;lt;ref&amp;gt;{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=Springer|others=Dekking, Michel, 1946-|isbn=978-1-85233-896-1|location=London|oclc=262680588}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity \infty to x as described by the picture to the right.&lt;br /&gt;
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累积分布函数指标描述了随机变量不大于给定值的概率; 结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算。累积分布函数是概率密度函数的反导函数，前提是后者存在。正如右边图片所描述的那样，累积分布函数是从负无穷到 x 的概率密度函数下面的区域。&lt;br /&gt;
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[[File:Standard deviation diagram.svg|right|thumb|250px|&lt;br /&gt;
图3:The [[probability density function]] (pdf) of the [[normal distribution]], also called Gaussian or &amp;quot;bell curve&amp;quot;, the most important continuous random distribution.  As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.&lt;br /&gt;
正态分布的[概率密度函数(pdf) ，也称为高斯或钟形曲线，是最重要的连续随机分布。如图所示，值间隔的概率对应于曲线下面积。]]&lt;br /&gt;
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== Terminology 术语==&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.&amp;lt;ref name=&amp;quot;:02&amp;quot; /&amp;gt;&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X &amp;gt; a, X &amp;lt; b or a union thereof.&lt;br /&gt;
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一些关键的，广泛用于以概率分布为主题的文献中的概念和术语，列出如下。&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Functions for discrete variables 离散变量函数&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
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Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.&lt;br /&gt;
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用于统计建模的著名离散概率分布包括泊松分佈、伯努利分布、二项分布、几何分布和负二项分布。&lt;br /&gt;
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*'''Probability function''': describes the probability distribution of a discrete random variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率函数 Probability Function&amp;lt;/font&amp;gt;'''：描述离散随机变量的概率分布。&lt;br /&gt;
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*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率质量函数（pmf Probability Mass Function&amp;lt;/font&amp;gt;'''：给出离散随机变量等于某个值的概率的函数。&lt;br /&gt;
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*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 频率分布 Frequency Distribution&amp;lt;/font&amp;gt;'''：显示样本中各种结果的频率的表格。&lt;br /&gt;
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For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
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对于一个离散的随机变量 x，设 u0，u1，... 是它在非零概率情况下可以取的值。表示&lt;br /&gt;
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*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 相对频率分布 Relative Frequency Distribution&amp;lt;/font&amp;gt;'''：一种频率分布，其中每个值均已被样本中的多个结果（即样本大小）除（归一化）。&lt;br /&gt;
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*'''Discrete probability distribution function''': general term to indicate the way the total probability of 1 is distributed over '''all''' various possible outcomes (i.e. over entire population) for discrete random variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;离散概率分布函数 Discrete Probability Distribution Function&amp;lt;/font&amp;gt;'''：通用术语，表示总概率1在离散随机变量的所有各种可能结果（即整个人群）中的分布方式。&lt;br /&gt;
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*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for a discrete random variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 累积分布函数 Cumulative distribution function&amp;lt;/font&amp;gt;'''：该函数评估离散随机变量X取小于或等于x的值的概率。&lt;br /&gt;
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*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 分类分布 Categorical Distribution&amp;lt;/font&amp;gt;'''：适用于具有有限值集的离散随机变量。&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Functions for continuous variables 连续变量函数&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
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* '''[[Probability density function]] (pdf):''' function whose value at any given sample (or point) in the [[sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率密度函数（pdf）Probability Density function&amp;lt;/font&amp;gt;'''：可以将其在样本空间中任意给定样本（或点）上的值（随机变量可能获得的一组值）的值解释为提供随机变量值将具有的相对可能性的函数等于那个样本。&lt;br /&gt;
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It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as&lt;br /&gt;
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由此可见，除了 u0、 u1、 ... 之外，x 取任何值的概率为零，因此可以将 x 写为&lt;br /&gt;
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* '''Continuous probability distribution function''': most often reserved for continuous random variables.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;连续概率分布函数 Continuous Probability Distribution Function&amp;lt;/font&amp;gt;'''：最常保留的连续随机变量。&lt;br /&gt;
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* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for continuous variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 累积分布函数 Cumulative distribution function&amp;lt;/font&amp;gt;'''：评估连续变量X取小于或等于x的值的概率的函数。&lt;br /&gt;
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===Basic terms 基本术语===&lt;br /&gt;
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* [[Mode (statistics)|'''Mode''']]: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.&lt;br /&gt;
模式：对于离散随机变量，该值具有最高概率；对于连续随机变量，是概率密度函数具有局部峰值的位置。&lt;br /&gt;
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* [[Support (mathematics)|'''Support''']]: set of values that can be assumed, with non-zero probability, by the random variable.&lt;br /&gt;
支持：可以由随机变量以非零概率假定的一组值。对于随机变量X，有时表示为R_ {X}&lt;br /&gt;
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* '''Tail''':&amp;lt;ref name='tail'&amp;gt;More information and examples can be found in the articles [[Heavy-tailed distribution]], [[Long-tailed distribution]], [[fat-tailed distribution]]&amp;lt;/ref&amp;gt; the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form &amp;lt;math&amp;gt;X &amp;gt; a&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;X &amp;lt; b&amp;lt;/math&amp;gt; or a union thereof.&lt;br /&gt;
尾巴：如果pmf或pdf相对较低，则靠近随机变量边界的区域。通常形式为X&amp;gt; a，X &amp;lt;b或它们的并集。&lt;br /&gt;
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*'''Head''':&amp;lt;ref name='tail' /&amp;gt; the region where the pmf or pdf is relatively high. Usually has the form &amp;lt;math&amp;gt;a &amp;lt; X &amp;lt; b&amp;lt;/math&amp;gt;.&lt;br /&gt;
头部：pmf或pdf较高的区域。通常具有a &amp;lt;X &amp;lt;b的形式&lt;br /&gt;
&lt;br /&gt;
* '''[[Expected value]]''' or '''mean''': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.&lt;br /&gt;
期望值或均值：可能值的加权平均值，以其概率作为权重；或其连续类似物。&lt;br /&gt;
&lt;br /&gt;
A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a cumulative density function that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.&lt;br /&gt;
&lt;br /&gt;
一个连续的概率分布是一个支持不可数集的概率分布，比如实数行中的一个区间。它们是唯一拥有属性的一个累积密度函数，可以用来计算支持的每个子集的概率。有许多连续概率分布的例子: 正态、均匀、卡方等等。&lt;br /&gt;
&lt;br /&gt;
*'''[[Median]]''': the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.&lt;br /&gt;
中位数：这样的值，即一组值小于中位数，而该组大于中位数，每一个的概率不大于二分之一。&lt;br /&gt;
&lt;br /&gt;
*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.&lt;br /&gt;
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。&lt;br /&gt;
&lt;br /&gt;
* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.&lt;br /&gt;
标准偏差：方差的平方根，因此是色散的另一种度量。&lt;br /&gt;
&lt;br /&gt;
* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.&lt;br /&gt;
对称性：某些分布的一种属性，其中特定值左侧（通常是中位数）的分布部分是其右侧部分的镜像。&lt;br /&gt;
&lt;br /&gt;
*'''[[Skewness]]''': a measure of the extent to which a pmf or pdf &amp;quot;leans&amp;quot; to one side of its mean. The third [[standardized moment]] of the distribution.&lt;br /&gt;
偏度：衡量pmf或pdf在其均值的一侧“倾斜”的程度。分布的第三个标准化时刻。&lt;br /&gt;
&lt;br /&gt;
*'''[[Kurtosis]]''': a measure of the &amp;quot;fatness&amp;quot; of the tails of a pmf or pdf. The fourth standardized moment of the distribution.&lt;br /&gt;
峰度：pmf或pdf尾部“脂肪”的量度。分布的第四个标准化时刻。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
=='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Discrete probability distribution 离散概率分布&amp;lt;/font&amp;gt;'''==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Discrete probability distrib.svg|right|thumb|&lt;br /&gt;
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Discrete probability distribution.svg|right|thumb|&lt;br /&gt;
图5:The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...离散概率分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Normal probability distribution.svg|right|thumb|&lt;br /&gt;
图6:... of a continuous probability distribution, ...连续概率分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Mixed probability distribution.svg|right|thumb|&lt;br /&gt;
图7:... of a distribution which has both a continuous part and a discrete part.既有连续部分又有离散部分]]&lt;br /&gt;
&lt;br /&gt;
 &lt;br /&gt;
A '''discrete probability distribution''' is a probability distribution that can take on a countable number of values.&amp;lt;ref&amp;gt;{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer|isbn=9780387878591|location=New York|pages=51|oclc=710149819}}&amp;lt;/ref&amp;gt; For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if &amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.&lt;br /&gt;
&lt;br /&gt;
离散概率分布是可以具有可数数量的值的概率分布。在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1。例如，如果&amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2，概率之和为1/2 + 1/4 + 1/8 + ... = 1。&lt;br /&gt;
&lt;br /&gt;
Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt; Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.&lt;br /&gt;
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
When a [[Sample (statistics)|sample]] (a set of observations) is drawn from a larger population, the sample points have an [[empirical distribution function|empirical distribution]] that is discrete and that provides information about the population distribution.&lt;br /&gt;
当从更大的总体中抽取一个样本（一组观察值）时，这些采样点的经验分布是离散的，并且提供了有关总体分布的信息。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Measure theoretic formulation 测量理论公式===&lt;br /&gt;
&lt;br /&gt;
A [[measurable function]] &amp;lt;math&amp;gt;X \colon A \to B &amp;lt;/math&amp;gt; between a [[probability space]] &amp;lt;math&amp;gt;(A, \mathcal A, P)&amp;lt;/math&amp;gt; and a [[measurable space]] &amp;lt;math&amp;gt;(B, \mathcal B) &amp;lt;/math&amp;gt; is called a '''discrete random variable''' provided that its image is a countable set. In this case measurability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; means that the pre-images of singleton sets are measurable, i.e., &amp;lt;math&amp;gt;X^{-1}(\{b\}) \in \mathcal A&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;b \in B&amp;lt;/math&amp;gt;.&lt;br /&gt;
The latter requirement induces a [[probability mass function]] &amp;lt;math&amp;gt;f_X \colon X(A) \to \mathbb R&amp;lt;/math&amp;gt; via &amp;lt;math&amp;gt; f_X(b):=P(X^{-1}(\{b\}))&amp;lt;/math&amp;gt;.  Since the pre-images of disjoint sets&lt;br /&gt;
are disjoint,&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
This recovers the definition given above.&lt;br /&gt;
&lt;br /&gt;
一个可测量的函数 &amp;lt;math&amp;gt;X \colon A \to B &amp;lt;/math&amp;gt; 在一个概率空间中 &amp;lt;math&amp;gt;(A, \mathcal A, P)&amp;lt;/math&amp;gt; and 和一个可测量空间 &amp;lt;math&amp;gt;(B, \mathcal B) &amp;lt;/math&amp;gt; 被叫做离散随机变量。该图像是一个可数的集合。在这种情况下&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;的测量意味着单例集的原像是可测量的 i.e., &amp;lt;math&amp;gt;X^{-1}(\{b\}) \in \mathcal A&amp;lt;/math&amp;gt; 对于所有的&amp;lt;math&amp;gt;b \in B&amp;lt;/math&amp;gt;.&lt;br /&gt;
后者需要包括概率质量函数 &amp;lt;math&amp;gt;f_X \colon X(A) \to \mathbb R&amp;lt;/math&amp;gt; via &amp;lt;math&amp;gt; f_X(b):=P(X^{-1}(\{b\}))&amp;lt;/math&amp;gt;.  由于不相交集的原像不相交&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
这包含了上面所提到的定义&lt;br /&gt;
&lt;br /&gt;
==='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 累积分布函数 Cumulative distribution function&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it &amp;quot;jumps&amp;quot; to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.&lt;br /&gt;
&lt;br /&gt;
与上述等效，可以将离散随机变量定义为其累积分布函数（cdf）仅因跳跃不连续性而增加的随机变量，也就是说，其cdf仅在“跳跃”至较高值时才增加，并且在那些跳跃点间是常数。但是请注意，cdf跳转的点可能会形成密集的实数集。发生跳变的点恰好是随机变量可能取的值。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Delta-function representation 三角函数表示===&lt;br /&gt;
&lt;br /&gt;
Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions.  This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.&amp;lt;ref&amp;gt;{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此，离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数，该函数实质上统一了对连续分布和离散分布的处理。当处理涉及连续和离散部分的概率分布时，这特别有用。&lt;br /&gt;
&lt;br /&gt;
===Indicator-function representation 指标功能表示===&lt;br /&gt;
&lt;br /&gt;
For a discrete random variable ''X'', let ''u''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;, ''u''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
对于离散随机变量X，令u0，u1，...是它可以以非零概率获取的值。表示&lt;br /&gt;
:&amp;lt;math&amp;gt;\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这些是不相交的集合，对于这样的集合：&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此，X取u0，u1，...以外的任何值的概率为零，因此可以将X写入为&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
except on a set of probability zero, where &amp;lt;math&amp;gt;1_A&amp;lt;/math&amp;gt; is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.&lt;br /&gt;
除了概率为零的集合外，其中1_ {A}是A的指标函数。这可以用作离散随机变量的替代定义。&lt;br /&gt;
&lt;br /&gt;
==Continuous probability distribution 连续概率分布==&lt;br /&gt;
&lt;br /&gt;
{{See also|Probability density function}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.&amp;lt;ref&amp;gt;{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}&amp;lt;/ref&amp;gt; They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].&lt;br /&gt;
连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]。连续概率分布有很多示例：正态分布，均匀分布，卡方分布和其他分布。&lt;br /&gt;
&lt;br /&gt;
A random variable &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; has a continuous probability distribution if there is a function &amp;lt;math&amp;gt;f: \mathbb{R} \rightarrow [0, \infty)&amp;lt;/math&amp;gt; such that for each interval &amp;lt;math&amp;gt;I \subset \mathbb{R}&amp;lt;/math&amp;gt; the probability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; belonging to &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt; is given by the integral of &amp;lt;math&amp;gt;f&amp;lt;/math&amp;gt; over &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}&amp;lt;/ref&amp;gt; For example, if &amp;lt;math&amp;gt;I = [a, b]&amp;lt;/math&amp;gt; then we would have:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一个随机变量&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; 有一个连续的概率分布，如果这有一个函数&amp;lt;math&amp;gt;f: \mathbb{R} \rightarrow [0, \infty)&amp;lt;/math&amp;gt; 对于每一个区间&amp;lt;math&amp;gt;I \subset \mathbb{R}&amp;lt;/math&amp;gt;  &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;的概率 &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;是&amp;lt;math&amp;gt;f&amp;lt;/math&amp;gt; 在 &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;上的积分.&amp;lt;ref&amp;gt;Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}&amp;lt;/ref&amp;gt; 例如, 如果&amp;lt;math&amp;gt;I = [a, b]&amp;lt;/math&amp;gt; 可得到:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In particular, the probability for &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; to take any single value &amp;lt;math&amp;gt;a&amp;lt;/math&amp;gt; (that is &amp;lt;math&amp;gt;a \le X \le a&amp;lt;/math&amp;gt;) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;F(x) = \operatorname{P}\left[-\infty &amp;lt; X \le x\right] = \int_{-\infty}^x f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
特别是，X取任何单个值a（即a≤X≤a）的概率为零，因为上下限一致的积分始终等于零。满足上述条件的变量称为连续随机变量。其累积密度函数定义为&lt;br /&gt;
&lt;br /&gt;
which, by this definition, has the properties:&lt;br /&gt;
根据定义有以下一些性质&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is non-decreasing;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;0 \le F(x) \le 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;\lim_{x \rightarrow -\infty} F(x) = 0&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\lim_{x \rightarrow \infty} F(x) = 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;P(a \le X &amp;lt; b) = F(b) - F(a)&amp;lt;/math&amp;gt;; and&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is continuous (due to the [[Riemann integral]] properties).&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
It is also possible to think in the opposite direction, which allows more flexibility. Say &amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is a function that satisfies all but the last of the properties above, then &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; represents the cumulative density function for some random variable: a discrete random variable if &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; is a step function, and a continuous random variable otherwise.&amp;lt;ref&amp;gt;See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.&amp;lt;/ref&amp;gt; This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].&lt;br /&gt;
也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; is such an absolutely continuous random variable, then it has a [[probability density function]] &amp;lt;math&amp;gt;f(x)&amp;lt;/math&amp;gt;, and its probability of falling into a Lebesgue-measurable set &amp;lt;math&amp;gt;A \subset \mathbb{R}&amp;lt;/math&amp;gt; is:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; is the Lebesgue measure.&lt;br /&gt;
&lt;br /&gt;
对于实线的更多任意子集，通常有必要对上述定义进行概括。在这些情况下，连续概率分布定义为具有绝对连续的累积分布函数的概率分布。等效地，就Lebesgue测度而言，它是实数上的概率分布，它是绝对连续的。这样的分布可以用它们的概率密度函数表示。如果X是这样一个绝对连续的随机变量，则它具有概率密度函数f（x），并且落入Lebesgue可测量集合A⊂R的概率为：&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这里，mu是Lebesgue度量&lt;br /&gt;
&lt;br /&gt;
Note on terminology: some authors use the term &amp;quot;continuous distribution&amp;quot; to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;\mu\{x\}\,=\,0&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;\,x&amp;lt;/math&amp;gt;. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].&lt;br /&gt;
&lt;br /&gt;
关于术语的注释：一些作者使用术语“连续分布”来表示其累积分布函数是连续的而不是绝对连续的分布。这些分布是所有x的μ{x} = 0的μ分布。该定义包括上面定义的（绝对）连续分布，但也包括奇异分布，既不是绝对连续也不是离散的，也不是它们的混合。没有密度。 Cantor分布给出了一个示例。&lt;br /&gt;
&lt;br /&gt;
== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; [[Andrey Kolmogorov|Kolmogorov]] definition 柯尔莫哥洛夫的定义&amp;lt;/font&amp;gt;'''==&lt;br /&gt;
&lt;br /&gt;
{{Main|Probability space|Probability measure}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; from a [[probability space]] &amp;lt;math&amp;gt;(\Omega, \mathcal{F}, \mathbb{P})&amp;lt;/math&amp;gt; to a [[measurable space]] &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt;. Given that probabilities of events of the form &amp;lt;math&amp;gt;\{\omega\in\Omega\mid X(\omega)\in A\}&amp;lt;/math&amp;gt; satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] &amp;lt;math&amp;gt;X_*\mathbb{P}&amp;lt;/math&amp;gt; of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; , which is a [[probability measure]] on &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt; satisfying &amp;lt;math&amp;gt;X_*\mathbb{P} = \mathbb{P}X^{-1}&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
在概率论的度量理论形式化中，将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）。给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P，它是满足&amp;lt;math&amp;gt;X_*\mathbb{P} = \mathbb{P}X^{-1}&amp;lt;/math&amp;gt;的概率量度&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Random number generation 随机数生成==&lt;br /&gt;
{{Main|Pseudo-random number sampling}}&lt;br /&gt;
&lt;br /&gt;
Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Citation|last1=Dekking|first1=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
大多数算法基于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 伪随机数生成器 Pseudorandom Number Generator&amp;lt;/font&amp;gt;'''，该伪随机数生成器生成在半开间隔[0,1）中均匀分布的数字X。然后，通过某种算法对这些随机变量X进行转换，以创建具有所需概率分布的新随机变量。利用这种统一的伪随机源，可以生成任何随机变量的实现。&lt;br /&gt;
&lt;br /&gt;
For example, suppose &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt; has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some &amp;lt;math&amp;gt;0 &amp;lt; p &amp;lt; 1&amp;lt;/math&amp;gt;, we define&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
so that&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\textrm{P}(X=1) = \textrm{P}(U&amp;lt;p) = p,&lt;br /&gt;
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
例如，假设U具有介于0和1之间的均匀分布。为某些对象构造一个随机的Bernoulli变量0 &amp;lt;p &amp;lt;1我们定义&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\textrm{P}(X=1) = \textrm{P}(U&amp;lt;p) = p,&lt;br /&gt;
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
This random variable X has a Bernoulli distribution with parameter &amp;lt;math&amp;gt;p&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:0&amp;quot;/&amp;gt; Note that this is a transformation of discrete random variable.&lt;br /&gt;
该随机变量X具有参数的伯努利分布p。请注意，这是离散随机变量的变换。&lt;br /&gt;
&lt;br /&gt;
For a distribution function &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; of a continuous random variable, a continuous random variable must be constructed. &amp;lt;math&amp;gt;F^{inv}&amp;lt;/math&amp;gt;, an inverse function of &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt;, relates to the uniform variable &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{U\leq F(x)} = {F^{inv}(U)\leq x}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
F或连续随机变量的分布函数F，必须构造连续随机变量。 &amp;lt;math&amp;gt;F^{inv}&amp;lt;/math&amp;gt;，F的反函数，涉及均匀变量U：&lt;br /&gt;
&amp;lt;math&amp;gt;{U\leq F(x)} = {F^{inv}(U)\leq x}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For example, suppose a random variable that has an exponential distribution &amp;lt;math&amp;gt;F(x) = 1 - e^{-\lambda x}&amp;lt;/math&amp;gt; must be constructed.&lt;br /&gt;
例如，假设必须构造一个具有指数分布&amp;lt;math&amp;gt;F(x) = 1 - e^{-\lambda x}&amp;lt;/math&amp;gt; 的随机变量。&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
F(x) = u &amp;amp;\Leftrightarrow 1-e^{-\lambda x} = u \\ &amp;amp;\Leftrightarrow e^{-\lambda x } = 1-u \\&amp;amp;\Leftrightarrow -\lambda x = \ln(1-u) \\ &amp;amp;\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)&lt;br /&gt;
\end{align}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
所以&amp;lt;math&amp;gt;F^{inv}(u) = \frac{-1}{\lambda}\ln(1-u)&amp;lt;/math&amp;gt; 并且如果 &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt; 有一个&amp;lt;math&amp;gt;U(0,1)&amp;lt;/math&amp;gt; 分布, 然后随机变量 &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; 被定义为 &amp;lt;math&amp;gt;X = F^{inv}(U) = \frac{-1}{\lambda} \ln(1-U)&amp;lt;/math&amp;gt;. 这里有一个指数分布 &amp;lt;math&amp;gt;\lambda&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.&lt;br /&gt;
统计模拟（蒙特卡洛方法）中经常遇到的一个问题是生成以给定方式分布的伪随机数。&lt;br /&gt;
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== Common probability distributions and their applications 共同概率分布及其应用==&lt;br /&gt;
{{Main list|List of probability distributions}}&lt;br /&gt;
&lt;br /&gt;
The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.&lt;br /&gt;
&lt;br /&gt;
概率分布的概念及其描述的随机变量是概率论和统计学科学的数学基础。人口中几乎可以测量的任何值都存在价差或可变性（例如人的身高，金属的耐用性，销售增长，交通流量等）;几乎所有测量均存在一定的固有误差；在物理学中，从气体的动力学特性到基本粒子的量子力学描述，很多过程都用概率论来描述。由于这些以及许多其他原因，简单数字通常不足以描述数量，而概率分布通常更合适。&lt;br /&gt;
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The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see [[list of probability distributions]], which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)&lt;br /&gt;
以下是一些最常见的概率分布列表，按与之相关的过程类型进行分组。有关更完整的列表，请参见概率分布列表，该列表按要考虑的结果的性质（离散，连续，多元等）进行分组。&lt;br /&gt;
&lt;br /&gt;
All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a [[mixture distribution]].&lt;br /&gt;
下面所有的单变量分布都达到了峰值。也就是说，假设值聚集在单个点周围。实际上，实际观察到的量可能会聚集在多个值附近。可以使用混合物分布对此类数量进行建模。&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Linear growth (e.g. errors, offsets) 线性增长 &amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
* [[Normal distribution]] (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 正态分布（高斯分布 Normal Distribution&amp;lt;/font&amp;gt;'''，对于单个这样的数量；最常用的连续分布&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Exponential growth (e.g. prices, incomes, populations) 指数增长&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;对数正态分布 Log-normal Distribution&amp;lt;/font&amp;gt;'''，对于单个此类数量的对数正态分布&lt;br /&gt;
&lt;br /&gt;
* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;帕累托分布 Pareto Distribution&amp;lt;/font&amp;gt;'''，对于单个这样的数量，其对数呈指数分布；原型幂律分布&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Uniformly distributed quantities 数量均匀分布&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 离散均匀分布 Discrete Uniform Distributed&amp;lt;/font&amp;gt;'''，用于有限的一组值（例如，公平死亡的结果）&lt;br /&gt;
&lt;br /&gt;
* [[Continuous uniform distribution]], for continuously distributed values&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;连续均匀分布 Continuous Uniform Distributed&amp;lt;/font&amp;gt;'''，用于连续分布的值&lt;br /&gt;
&lt;br /&gt;
=== Bernoulli trials (yes/no events, with a given probability) 伯努利试验（是/否事件，具有给定的概率）===&lt;br /&gt;
&lt;br /&gt;
* Basic distributions 基本分布:&lt;br /&gt;
** [[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 伯努利分布 Bernoulli Distribution&amp;lt;/font&amp;gt;'''，用于单个伯努利试验的结果（例如成功/失败，是/否）&lt;br /&gt;
&lt;br /&gt;
** [[Binomial distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed total number of [[Independent (statistics)|independent]] occurrences&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项式分布 Binomial Distribution &amp;lt;/font&amp;gt;'''，对于给定固定总数的独立“出现次数”（例如，成功，赞成票等）的数量&lt;br /&gt;
&lt;br /&gt;
** [[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;负二项分布 Negative Binomial Distribution&amp;lt;/font&amp;gt;'''，用于二项式观察，但是关注的数量是在给定成功次数之前发生的失败次数&lt;br /&gt;
&lt;br /&gt;
** [[Geometric distribution]], for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the [[negative binomial distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;几何分布 Geometric Distribution&amp;lt;/font&amp;gt;'''，用于二项式观测，但是关注的数量是首次成功之前的失败数量；负二项式分布的特殊情况&lt;br /&gt;
&lt;br /&gt;
* Related to sampling schemes over a finite population 与有限人口抽样方案有关:&lt;br /&gt;
&lt;br /&gt;
** [[Hypergeometric distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using [[sampling without replacement]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 超几何分布 Hypergeometric Distribution&amp;lt;/font&amp;gt;'''，对于“肯定出现”的数量（例如成功，赞成票等），给定了一定的总出现数量，使用采样而无需替换&lt;br /&gt;
&lt;br /&gt;
** [[Beta-binomial distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a [[Pólya urn model]] (in some sense, the &amp;quot;opposite&amp;quot; of [[sampling without replacement]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 贝塔二项式分布 Beta-binomial Distribution&amp;lt;/font&amp;gt;'''，对于给定的总发生次数为“阳性”的次数（例如，成功，赞成票等），使用Pólyaurn模型进行采样（在某种意义上，为“替代”而不进行替换） ）&lt;br /&gt;
&lt;br /&gt;
=== Categorical outcomes (events with ''K'' possible outcomes, with a given probability for each outcome) 分类结果（具有K个可能结果的事件，每个结果具有给定的概率）===&lt;br /&gt;
&lt;br /&gt;
* [[Categorical distribution]], for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the [[Bernoulli distribution]]&lt;br /&gt;
针对单个分类结果的分类分布（例如，调查中的是/否/也许）；伯努利分布的一般化&lt;br /&gt;
&lt;br /&gt;
* [[Multinomial distribution]], for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the [[binomial distribution]]&lt;br /&gt;
给定总结果的固定数量，针对每种类别结果的数量的多项式分布；二项式分布的一般化&lt;br /&gt;
&lt;br /&gt;
* [[Multivariate hypergeometric distribution]], similar to the [[multinomial distribution]], but using [[sampling without replacement]]; a generalization of the [[hypergeometric distribution]]&lt;br /&gt;
多元超几何分布，类似于多项式分布，但使用采样而不进行替换；超几何分布的一般化&lt;br /&gt;
&lt;br /&gt;
=== Poisson process (events that occur independently with a given rate) 泊松过程（以给定速率独立发生的事件）===&lt;br /&gt;
&lt;br /&gt;
* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 泊松分布 Poisson Distribution&amp;lt;/font&amp;gt;'''，用于给定时间段内泊松型事件的发生次数&lt;br /&gt;
&lt;br /&gt;
* [[Exponential distribution]], for the time before the next Poisson-type event occurs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;指数分布 Exponential Distribution&amp;lt;/font&amp;gt;'''，在下一次泊松型事件发生之前的时间&lt;br /&gt;
&lt;br /&gt;
* [[Gamma distribution]], for the time before the next k Poisson-type events occur&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;伽马分布 Gamma Distribution&amp;lt;/font&amp;gt;'''，在接下来的k个泊松型事件发生之前的时间&lt;br /&gt;
&lt;br /&gt;
=== Absolute values of vectors with normally distributed components 具有正态分布分量的向量的绝对值===&lt;br /&gt;
&lt;br /&gt;
* [[Rayleigh distribution]], for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;瑞利分布 Rayleigh Distribution&amp;lt;/font&amp;gt;'''，用于具有高斯分布正交分量的矢量幅度分布。在具有高斯实部和虚部的RF信号中发现瑞利分布。&lt;br /&gt;
&lt;br /&gt;
* [[Rice distribution]], a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in [[Rician fading]] of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;莱斯分布 Rice Distribution&amp;lt;/font&amp;gt;'''，是在背景信号分量稳定的情况下瑞利分布的概括。由于多径传播而在无线电信号的Rician衰落中发现，并且在非零NMR信号中出现噪声破坏的MR图像中也发现了这种情况。&lt;br /&gt;
&lt;br /&gt;
=== Normally distributed quantities operated with sum of squares (for hypothesis testing) 以平方和运算的正态分布量（用于假设检验）===&lt;br /&gt;
&lt;br /&gt;
* [[Chi-squared distribution]], the distribution of a sum of squared [[standard normal]] variables; useful e.g. for inference regarding the [[sample variance]] of normally distributed samples (see [[chi-squared test]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;卡方分布 Chi-squared Distribution&amp;lt;/font&amp;gt;'''，标准正态变量平方和的分布；有用的关于正态分布样本的样本方差的推论（请参见卡方检验）&lt;br /&gt;
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* [[Student's t distribution]], the distribution of the ratio of a [[standard normal]] variable and the square root of a scaled [[Chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;学生t分布 Student‘s t Distribution&amp;lt;/font&amp;gt;'''，标准正态变量与缩放的卡方变量的平方根之比的分布；有助于推断方差未知的正态分布样本的平均值（请参阅学生的t检验）&lt;br /&gt;
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* [[F-distribution]], the distribution of the ratio of two scaled [[Chi squared distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;F-分布 F-Distribution&amp;lt;/font&amp;gt;'''，两个比例卡方变量的比例分布；有用的用于涉及比较方差或涉及R平方（相关系数平方）的推论&lt;br /&gt;
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=== As a conjugate prior distributions in Bayesian inference 作为贝叶斯推断中的共轭先验分布===&lt;br /&gt;
{{Main|Conjugate prior}}&lt;br /&gt;
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* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Beta分布 Beta Distribution&amp;lt;/font&amp;gt;'''，具有单个概率（0到1之间的实数）；与伯努利分布和二项式分布共轭&lt;br /&gt;
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* [[Gamma distribution]], for a non-negative scaling parameter; conjugate to the rate parameter of a [[Poisson distribution]] or [[exponential distribution]], the [[Precision (statistics)|precision]] (inverse [[variance]]) of a [[normal distribution]], etc.&lt;br /&gt;
伽玛分布，用于非负比例缩放参数；与泊松分布或指数分布的速率参数，正态分布的精度（逆方差）等共轭。&lt;br /&gt;
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* [[Dirichlet distribution]], for a vector of probabilities that must sum to 1; conjugate to the [[categorical distribution]] and [[multinomial distribution]]; generalization of the [[beta distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Dirichlet分布 Dirichlet Distribution&amp;lt;/font&amp;gt;'''，对于必须为1的概率向量；与分类分布和多项式分布共轭； beta分布的一般化&lt;br /&gt;
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*[[Wishart distribution]], for a symmetric [[non-negative definite]] matrix; conjugate to the inverse of the [[covariance matrix]] of a [[multivariate normal distribution]]; generalization of the [[gamma distribution]]&amp;lt;ref&amp;gt;{{Cite book|title=Pattern recognition and machine learning|last=Bishop, Christopher M.|date=2006|publisher=Springer|isbn=0-387-31073-8|location=New York|oclc=71008143}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Wishart分布 Wishart Distribution&amp;lt;/font&amp;gt;'''，用于对称非负定矩阵；与多元正态分布的协方差矩阵的逆共轭；伽玛分布的一般化&lt;br /&gt;
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=== Some specialized applications of probability distributions 概率分布的一些特殊应用===&lt;br /&gt;
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* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.&lt;br /&gt;
在自然语言处理中使用的高速缓存语言模型和其他统计语言模型通过概率分布来为特定单词和单词序列的出现分配概率。&lt;br /&gt;
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* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by &amp;lt;math&amp;gt;P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 &amp;lt;/math&amp;gt;, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.&amp;lt;ref&amp;gt;{{Cite book|title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|publisher=|others=Thoman, John W., Jr., 1960-|year=|isbn=978-1-68015-835-9|location=[Mill Valley, California]|pages=403–406|oclc=927509011}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
在量子力学中，在给定点处找到粒子的概率密度与该点处粒子波函数大小的平方成正比（请参阅博恩法则）。因此，粒子位置的概率分布函数描述为&amp;lt;math&amp;gt;P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 &amp;lt;/math&amp;gt;，粒子位置的概率x在第一个维度中的间隔为a≤x≤b，在第三个维度中的间隔类似。这是量子力学的关键原理。&lt;br /&gt;
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* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution.&amp;lt;ref&amp;gt;{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.|last2=Chen|first2=Z.|last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review|doi=10.1109/drpt.2008.4523658|s2cid=18669309}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
潮流研究中的概率潮流解释了作为概率分布的输入变量的不确定性，并以概率分布的形式提供了潮流计算。&lt;br /&gt;
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* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.&amp;lt;ref&amp;gt;{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity, Rajib|isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
根据先前的频率分布（例如热带气旋，冰雹，事件之间的时间等）预测自然现象的发生。&lt;br /&gt;
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==See also 另请参见==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Copula (statistics)]]&lt;br /&gt;
Copula（统计数据）&lt;br /&gt;
* [[Empirical probability]]&lt;br /&gt;
经验概率&lt;br /&gt;
* [[Histogram]]&lt;br /&gt;
直方图&lt;br /&gt;
* [[Likelihood function]]&lt;br /&gt;
似然函数&lt;br /&gt;
* [[Kirkwood approximation]]&lt;br /&gt;
柯克伍德近似&lt;br /&gt;
* [[Moment-generating function]]&lt;br /&gt;
瞬时产生功能&lt;br /&gt;
* [[Riemann–Stieltjes integral#Application to probability theory|Riemann–Stieltjes integral application to probability theory]]&lt;br /&gt;
Riemann–Stieltjes积分＃在概率论中的应用&lt;br /&gt;
*[[Pairwise independence]]&lt;br /&gt;
成对独立&lt;br /&gt;
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=== Lists 清单===&lt;br /&gt;
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* [[List of probability distributions]]&lt;br /&gt;
概率分布的清单&lt;br /&gt;
* [[List of statistical topics]]&lt;br /&gt;
统计学话题的清淡&lt;br /&gt;
=== Probability distributions 概率分布 ===&lt;br /&gt;
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* [[Conditional probability distribution]]&lt;br /&gt;
条件概率分布&lt;br /&gt;
* [[Joint probability distribution]]&lt;br /&gt;
联合概率分布&lt;br /&gt;
* [[Quasiprobability distribution]]&lt;br /&gt;
拟概率分布&lt;br /&gt;
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== References ==&lt;br /&gt;
=== Citations ===&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
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=== Sources ===&lt;br /&gt;
{{refbegin|}}&lt;br /&gt;
* {{cite journal|doi=10.1016/j.ejmp.2014.05.002|pmid=25059432|title=Data distributions in magnetic resonance images: A review|url=|journal=[[Physica Medica]]|volume=30|issue=7|pages=725–741|year=2014|last1=den Dekker|first1=A. J.|last2=Sijbers|first2=J.}}&lt;br /&gt;
* {{cite book|last=Vapnik|first=Vladimir Naumovich|year=1998|title=Statistical Learning Theory|publisher=John Wiley and Sons}}&lt;br /&gt;
{{refend}}&lt;br /&gt;
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==External links==&lt;br /&gt;
{{commons|Probability distribution|Probability distribution}}&lt;br /&gt;
*{{springer|title=Probability distribution|id=p/p074900}}&lt;br /&gt;
*[http://threeplusone.com/FieldGuide.pdf Field Guide to Continuous Probability Distributions],  Gavin E. Crooks.&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
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		<title>概率分布</title>
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		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{short description|Mathematical function that describes the probability of occurrence of different possible outcomes in an experiment}}&lt;br /&gt;
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{{other uses|Distribution (disambiguation){{!}}Distribution}}&lt;br /&gt;
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{{Off topic|date=August 2020|Random variable|talksectionheading}}&lt;br /&gt;
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In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].&amp;lt;ref name=&amp;quot;:02&amp;quot;&amp;gt;{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}&amp;lt;/ref&amp;gt; It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In Probability Theory and Statistics, a Probability Distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;概率论 Probability Theory&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;统计学 Statistics&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率分布 Probability Distribution&amp;lt;/font&amp;gt;'''是一个给出一个实验不同可能结果出现的概率的数学函数。它是根据'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 样本空间 Sample Space&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 事件概率 Probabilities of Events&amp;lt;/font&amp;gt;'''(样本空间的子集)对随机现象的数学描述。&lt;br /&gt;
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For instance, if the [[random variable]] {{mvar|X}} is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of {{mvar|X}} would take the value 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} heads}}, and 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} tails}} (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a [[Survey methodology|survey]], etc.&amp;lt;ref name=&amp;quot;ross&amp;quot;&amp;gt;{{cite book|first=Sheldon M.|last=Ross|title=A first course in probability|publisher=Pearson|year=2010}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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For instance, if the random variable X is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of X would take the value 0.5 for X=heads, and 0.5 for X=tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.&lt;br /&gt;
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例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。随机现象的例子包括未来某一天的天气状况、一个人的身高、学校中男生的比例、调查结果等等。&lt;br /&gt;
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A probability distribution is a mathematical function that has a [[sample space]] as its '''input''', and gives a '''probability''' as its output. The sample space is the [[Set (mathematics)|set]] of all possible [[Outcome (probability)|outcomes]] of a random phenomenon being observed; it may be the set of [[real numbers]] or a set of [[vector (mathematics)|vectors]], or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be {{math|{heads, tails} }}.&lt;br /&gt;
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A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be .&lt;br /&gt;
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概率分布是一个数学函数，它的输入是一个样本空间，输出是一个概率。样本空间是所观察到的随机现象的所有可能结果的集合; 它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。&lt;br /&gt;
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Probability distributions are generally divided into two classes. A '''discrete probability distribution''' is applicable to the scenarios where the set of possible outcomes is [[discrete probability distribution|discrete]] (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the [[probability mass function]]. On the other hand, '''continuous probability distributions''' are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a [[probability density function]].&amp;lt;ref name=&amp;quot;ross&amp;quot; /&amp;gt;&amp;lt;ref&amp;gt;{{cite book|author1=DeGroot, Morris H.|author2=Schervish, Mark J.|title=Probability and Statistics|publisher=Addison-Wesley|year=2002}}&amp;lt;/ref&amp;gt; The [[normal distribution]] is a commonly encountered continuous probability distribution. More complex experiments, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.&lt;br /&gt;
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Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function. The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.&lt;br /&gt;
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概率分布一般分为两类。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 离散概率分布 Discrete Probability Distribution&amp;lt;/font&amp;gt;'''适用于一组可能的结果是离散的情况，如抛硬币或掷骰子。这里的概率被编码为结果概率的离散列表，称为概率质量函数。另一方面，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 连续概率分布 Continuous Probability Distribution&amp;lt;/font&amp;gt;'''适用于一组可以在一个连续的范围内取值的结果的情况(例如:实数)，例如某一天的温度。在这种情况下，概率通常由概率密度函数描述。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 正态分布 Normal Distribution&amp;lt;/font&amp;gt;'''是一种常见的连续概率分布。更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called [[Univariate distribution|univariate]], while a distribution whose sample space is a [[vector space]] of dimension 2 or more is called [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a [[joint probability distribution]]) gives the probabilities of a [[random vector]] – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the [[binomial distribution]], the [[hypergeometric distribution]], and the [[normal distribution]]. The [[multivariate normal distribution]] is a commonly encountered multivariate distribution.&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.&lt;br /&gt;
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一个一维的样本空间(例如实数、标签列表、有序标签或二进制)的概率分布被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 单变量 Univariate&amp;lt;/font&amp;gt;'''，而样本空间为二维或更多向量空间的分布被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 多变量 Multivariate&amp;lt;/font&amp;gt;'''。单变量分布给出了单个随机变量取不同替代值的概率; 联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。重要的和常见的单变量概率分布包括二项分布、超几何分布和正态分布。多变量正态分布是一种常见的联合分布。&lt;br /&gt;
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==Introduction 简介==&lt;br /&gt;
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[[File:Dice Distribution (bar).svg|thumb|250px|right|&lt;br /&gt;
图1:The [[probability mass function]] (pmf) ''p''(''S'') specifies the probability distribution for the sum ''S'' of counts from two [[dice]].  For example, the figure shows that ''p''(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as ''P''(''S'' &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.&lt;br /&gt;
概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布。例如，图中显示 p (11) = 2/36 = 1/18。Pmf 允许计算事件的概率，如 p (s &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6，以及分布中的所有其他概率。&lt;br /&gt;
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The [[probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice.  For example, the figure shows that p(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as P(S &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]&lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between '''discrete''' and '''continuous''' [[random variable]]s. In the discrete case, it is sufficient to specify a [[probability mass function]] &amp;lt;math&amp;gt;p&amp;lt;/math&amp;gt; assigning a probability to each possible outcome: for example, when throwing a fair [[Dice|die]], each of the six values 1 to 6 has the probability 1/6. The probability of an [[Event (probability theory)|event]] is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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为了定义最简单的概率分布，有必要区分离散和连续的随机变量。在离散情况下，指定一个概率质量函数 p 就足够了，它为每个可能的结果赋予一个概率: 例如，当投掷一个骰子时，6个值中的每一个的概率为1/6。然后将事件的概率定义为满足事件的结果的概率之和; 例如，事件”骰子掷出偶数值”的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2.&amp;lt;/math&amp;gt;&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs ''exactly'' 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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相比之下，当一个随机变量从一个连续体中取值时，那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率。例如，一个给定的物体重量正好是500克的概率为零，因为随着我们测量仪器精度的提高，正好测量500克的概率趋向于零。然而，在质量控制方面，人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98% ，而这一要求对测量仪器的准确性不太敏感。&lt;br /&gt;
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Continuous probability distributions can be described in several ways. The [[probability density function]] describes the [[infinitesimal]] probability of any given value, and the probability that the outcome lies in a given interval can be computed by [[Integration (mathematics)|integrating]] the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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连续概率分布可以用几种方法来描述。概率密度函数描述了任意给定值的无穷小概率，并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。可能值位于某一固定区间的概率可以与和收敛于积分的方式有关，因此，连续概率是基于积分的定义。&lt;br /&gt;
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[[File:Combined Cumulative Distribution Graphs.png|thumb|455x455px|&lt;br /&gt;
图2:On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve.&lt;br /&gt;
左边是概率密度函数。右边是累积分布函数，它是概率密度曲线下面的区域。]]&lt;br /&gt;
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The [[cumulative distribution function]] describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the [[antiderivative]] of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the [[probability density function]] from minus infinity &amp;lt;math&amp;gt;\infty&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; as described by the picture to the right.&amp;lt;ref&amp;gt;{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=Springer|others=Dekking, Michel, 1946-|isbn=978-1-85233-896-1|location=London|oclc=262680588}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity \infty to x as described by the picture to the right.&lt;br /&gt;
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累积分布函数指标描述了随机变量不大于给定值的概率; 结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算。累积分布函数是概率密度函数的反导函数，前提是后者存在。正如右边图片所描述的那样，累积分布函数是从负无穷到 x 的概率密度函数下面的区域。&lt;br /&gt;
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[[File:Standard deviation diagram.svg|right|thumb|250px|&lt;br /&gt;
图3:The [[probability density function]] (pdf) of the [[normal distribution]], also called Gaussian or &amp;quot;bell curve&amp;quot;, the most important continuous random distribution.  As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.&lt;br /&gt;
正态分布的[概率密度函数(pdf) ，也称为高斯或钟形曲线，是最重要的连续随机分布。如图所示，值间隔的概率对应于曲线下面积。]]&lt;br /&gt;
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== Terminology 术语==&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.&amp;lt;ref name=&amp;quot;:02&amp;quot; /&amp;gt;&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X &amp;gt; a, X &amp;lt; b or a union thereof.&lt;br /&gt;
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一些关键的，广泛用于以概率分布为主题的文献中的概念和术语，列出如下。&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Functions for discrete variables 离散变量函数&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
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Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.&lt;br /&gt;
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用于统计建模的著名离散概率分布包括泊松分佈、伯努利分布、二项分布、几何分布和负二项分布。&lt;br /&gt;
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*'''Probability function''': describes the probability distribution of a discrete random variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率函数 Probability Function&amp;lt;/font&amp;gt;'''：描述离散随机变量的概率分布。&lt;br /&gt;
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*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率质量函数（pmf Probability Mass Function&amp;lt;/font&amp;gt;'''：给出离散随机变量等于某个值的概率的函数。&lt;br /&gt;
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*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 频率分布 Frequency Distribution&amp;lt;/font&amp;gt;'''：显示样本中各种结果的频率的表格。&lt;br /&gt;
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For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
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对于一个离散的随机变量 x，设 u0，u1，... 是它在非零概率情况下可以取的值。表示&lt;br /&gt;
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*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 相对频率分布 Relative Frequency Distribution&amp;lt;/font&amp;gt;'''：一种频率分布，其中每个值均已被样本中的多个结果（即样本大小）除（归一化）。&lt;br /&gt;
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*'''Discrete probability distribution function''': general term to indicate the way the total probability of 1 is distributed over '''all''' various possible outcomes (i.e. over entire population) for discrete random variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;离散概率分布函数 Discrete Probability Distribution Function&amp;lt;/font&amp;gt;'''：通用术语，表示总概率1在离散随机变量的所有各种可能结果（即整个人群）中的分布方式。&lt;br /&gt;
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*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for a discrete random variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 累积分布函数 Cumulative distribution function&amp;lt;/font&amp;gt;'''：该函数评估离散随机变量X取小于或等于x的值的概率。&lt;br /&gt;
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*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 分类分布 Categorical Distribution&amp;lt;/font&amp;gt;'''：适用于具有有限值集的离散随机变量。&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Functions for continuous variables 连续变量函数&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
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* '''[[Probability density function]] (pdf):''' function whose value at any given sample (or point) in the [[sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 概率密度函数（pdf）Probability Density function&amp;lt;/font&amp;gt;'''：可以将其在样本空间中任意给定样本（或点）上的值（随机变量可能获得的一组值）的值解释为提供随机变量值将具有的相对可能性的函数等于那个样本。&lt;br /&gt;
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It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as&lt;br /&gt;
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由此可见，除了 u0、 u1、 ... 之外，x 取任何值的概率为零，因此可以将 x 写为&lt;br /&gt;
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* '''Continuous probability distribution function''': most often reserved for continuous random variables.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;连续概率分布函数 Continuous Probability Distribution Function&amp;lt;/font&amp;gt;'''：最常保留的连续随机变量。&lt;br /&gt;
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* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for continuous variable.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 累积分布函数 Cumulative distribution function&amp;lt;/font&amp;gt;'''：评估连续变量X取小于或等于x的值的概率的函数。&lt;br /&gt;
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===Basic terms 基本术语===&lt;br /&gt;
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* [[Mode (statistics)|'''Mode''']]: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.&lt;br /&gt;
模式：对于离散随机变量，该值具有最高概率；对于连续随机变量，是概率密度函数具有局部峰值的位置。&lt;br /&gt;
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* [[Support (mathematics)|'''Support''']]: set of values that can be assumed, with non-zero probability, by the random variable.&lt;br /&gt;
支持：可以由随机变量以非零概率假定的一组值。对于随机变量X，有时表示为R_ {X}&lt;br /&gt;
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* '''Tail''':&amp;lt;ref name='tail'&amp;gt;More information and examples can be found in the articles [[Heavy-tailed distribution]], [[Long-tailed distribution]], [[fat-tailed distribution]]&amp;lt;/ref&amp;gt; the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form &amp;lt;math&amp;gt;X &amp;gt; a&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;X &amp;lt; b&amp;lt;/math&amp;gt; or a union thereof.&lt;br /&gt;
尾巴：如果pmf或pdf相对较低，则靠近随机变量边界的区域。通常形式为X&amp;gt; a，X &amp;lt;b或它们的并集。&lt;br /&gt;
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*'''Head''':&amp;lt;ref name='tail' /&amp;gt; the region where the pmf or pdf is relatively high. Usually has the form &amp;lt;math&amp;gt;a &amp;lt; X &amp;lt; b&amp;lt;/math&amp;gt;.&lt;br /&gt;
头部：pmf或pdf较高的区域。通常具有a &amp;lt;X &amp;lt;b的形式&lt;br /&gt;
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* '''[[Expected value]]''' or '''mean''': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.&lt;br /&gt;
期望值或均值：可能值的加权平均值，以其概率作为权重；或其连续类似物。&lt;br /&gt;
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A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a cumulative density function that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.&lt;br /&gt;
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一个连续的概率分布是一个支持不可数集的概率分布，比如实数行中的一个区间。它们是唯一拥有属性的一个累积密度函数，可以用来计算支持的每个子集的概率。有许多连续概率分布的例子: 正态、均匀、卡方等等。&lt;br /&gt;
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*'''[[Median]]''': the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.&lt;br /&gt;
中位数：这样的值，即一组值小于中位数，而该组大于中位数，每一个的概率不大于二分之一。&lt;br /&gt;
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*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.&lt;br /&gt;
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。&lt;br /&gt;
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* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.&lt;br /&gt;
标准偏差：方差的平方根，因此是色散的另一种度量。&lt;br /&gt;
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* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.&lt;br /&gt;
对称性：某些分布的一种属性，其中特定值左侧（通常是中位数）的分布部分是其右侧部分的镜像。&lt;br /&gt;
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*'''[[Skewness]]''': a measure of the extent to which a pmf or pdf &amp;quot;leans&amp;quot; to one side of its mean. The third [[standardized moment]] of the distribution.&lt;br /&gt;
偏度：衡量pmf或pdf在其均值的一侧“倾斜”的程度。分布的第三个标准化时刻。&lt;br /&gt;
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*'''[[Kurtosis]]''': a measure of the &amp;quot;fatness&amp;quot; of the tails of a pmf or pdf. The fourth standardized moment of the distribution.&lt;br /&gt;
峰度：pmf或pdf尾部“脂肪”的量度。分布的第四个标准化时刻。&lt;br /&gt;
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=='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Discrete probability distribution 离散概率分布&amp;lt;/font&amp;gt;'''==&lt;br /&gt;
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[[File:Discrete probability distrib.svg|right|thumb|&lt;br /&gt;
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]&lt;br /&gt;
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[[File:Discrete probability distribution.svg|right|thumb|&lt;br /&gt;
图5:The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...离散概率分布]]&lt;br /&gt;
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[[File:Normal probability distribution.svg|right|thumb|&lt;br /&gt;
图6:... of a continuous probability distribution, ...连续概率分布]]&lt;br /&gt;
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[[File:Mixed probability distribution.svg|right|thumb|&lt;br /&gt;
图7:... of a distribution which has both a continuous part and a discrete part.既有连续部分又有离散部分]]&lt;br /&gt;
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A '''discrete probability distribution''' is a probability distribution that can take on a countable number of values.&amp;lt;ref&amp;gt;{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer|isbn=9780387878591|location=New York|pages=51|oclc=710149819}}&amp;lt;/ref&amp;gt; For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if &amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.&lt;br /&gt;
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离散概率分布是可以具有可数数量的值的概率分布。在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1。例如，如果&amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2，概率之和为1/2 + 1/4 + 1/8 + ... = 1。&lt;br /&gt;
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Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt; Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.&lt;br /&gt;
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。&lt;br /&gt;
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When a [[Sample (statistics)|sample]] (a set of observations) is drawn from a larger population, the sample points have an [[empirical distribution function|empirical distribution]] that is discrete and that provides information about the population distribution.&lt;br /&gt;
当从更大的总体中抽取一个样本（一组观察值）时，这些采样点的经验分布是离散的，并且提供了有关总体分布的信息。&lt;br /&gt;
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===Measure theoretic formulation 测量理论公式===&lt;br /&gt;
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A [[measurable function]] &amp;lt;math&amp;gt;X \colon A \to B &amp;lt;/math&amp;gt; between a [[probability space]] &amp;lt;math&amp;gt;(A, \mathcal A, P)&amp;lt;/math&amp;gt; and a [[measurable space]] &amp;lt;math&amp;gt;(B, \mathcal B) &amp;lt;/math&amp;gt; is called a '''discrete random variable''' provided that its image is a countable set. In this case measurability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; means that the pre-images of singleton sets are measurable, i.e., &amp;lt;math&amp;gt;X^{-1}(\{b\}) \in \mathcal A&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;b \in B&amp;lt;/math&amp;gt;.&lt;br /&gt;
The latter requirement induces a [[probability mass function]] &amp;lt;math&amp;gt;f_X \colon X(A) \to \mathbb R&amp;lt;/math&amp;gt; via &amp;lt;math&amp;gt; f_X(b):=P(X^{-1}(\{b\}))&amp;lt;/math&amp;gt;.  Since the pre-images of disjoint sets&lt;br /&gt;
are disjoint,&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
This recovers the definition given above.&lt;br /&gt;
&lt;br /&gt;
一个可测量的函数 &amp;lt;math&amp;gt;X \colon A \to B &amp;lt;/math&amp;gt; 在一个概率空间中 &amp;lt;math&amp;gt;(A, \mathcal A, P)&amp;lt;/math&amp;gt; and 和一个可测量空间 &amp;lt;math&amp;gt;(B, \mathcal B) &amp;lt;/math&amp;gt; 被叫做离散随机变量。该图像是一个可数的集合。在这种情况下&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;的测量意味着单例集的原像是可测量的 i.e., &amp;lt;math&amp;gt;X^{-1}(\{b\}) \in \mathcal A&amp;lt;/math&amp;gt; 对于所有的&amp;lt;math&amp;gt;b \in B&amp;lt;/math&amp;gt;.&lt;br /&gt;
后者需要包括概率质量函数 &amp;lt;math&amp;gt;f_X \colon X(A) \to \mathbb R&amp;lt;/math&amp;gt; via &amp;lt;math&amp;gt; f_X(b):=P(X^{-1}(\{b\}))&amp;lt;/math&amp;gt;.  由于不相交集的原像不相交&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
这包含了上面所提到的定义&lt;br /&gt;
&lt;br /&gt;
==='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 累积分布函数 Cumulative distribution function&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it &amp;quot;jumps&amp;quot; to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.&lt;br /&gt;
&lt;br /&gt;
与上述等效，可以将离散随机变量定义为其累积分布函数（cdf）仅因跳跃不连续性而增加的随机变量，也就是说，其cdf仅在“跳跃”至较高值时才增加，并且在那些跳跃点间是常数。但是请注意，cdf跳转的点可能会形成密集的实数集。发生跳变的点恰好是随机变量可能取的值。&lt;br /&gt;
&lt;br /&gt;
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===Delta-function representation 三角函数表示===&lt;br /&gt;
&lt;br /&gt;
Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions.  This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.&amp;lt;ref&amp;gt;{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此，离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数，该函数实质上统一了对连续分布和离散分布的处理。当处理涉及连续和离散部分的概率分布时，这特别有用。&lt;br /&gt;
&lt;br /&gt;
===Indicator-function representation 指标功能表示===&lt;br /&gt;
&lt;br /&gt;
For a discrete random variable ''X'', let ''u''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;, ''u''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
对于离散随机变量X，令u0，u1，...是它可以以非零概率获取的值。表示&lt;br /&gt;
:&amp;lt;math&amp;gt;\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这些是不相交的集合，对于这样的集合：&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此，X取u0，u1，...以外的任何值的概率为零，因此可以将X写入为&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
except on a set of probability zero, where &amp;lt;math&amp;gt;1_A&amp;lt;/math&amp;gt; is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.&lt;br /&gt;
除了概率为零的集合外，其中1_ {A}是A的指标函数。这可以用作离散随机变量的替代定义。&lt;br /&gt;
&lt;br /&gt;
==Continuous probability distribution 连续概率分布==&lt;br /&gt;
&lt;br /&gt;
{{See also|Probability density function}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.&amp;lt;ref&amp;gt;{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}&amp;lt;/ref&amp;gt; They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].&lt;br /&gt;
连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]。连续概率分布有很多示例：正态分布，均匀分布，卡方分布和其他分布。&lt;br /&gt;
&lt;br /&gt;
A random variable &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; has a continuous probability distribution if there is a function &amp;lt;math&amp;gt;f: \mathbb{R} \rightarrow [0, \infty)&amp;lt;/math&amp;gt; such that for each interval &amp;lt;math&amp;gt;I \subset \mathbb{R}&amp;lt;/math&amp;gt; the probability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; belonging to &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt; is given by the integral of &amp;lt;math&amp;gt;f&amp;lt;/math&amp;gt; over &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}&amp;lt;/ref&amp;gt; For example, if &amp;lt;math&amp;gt;I = [a, b]&amp;lt;/math&amp;gt; then we would have:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一个随机变量&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; 有一个连续的概率分布，如果这有一个函数&amp;lt;math&amp;gt;f: \mathbb{R} \rightarrow [0, \infty)&amp;lt;/math&amp;gt; 对于每一个区间&amp;lt;math&amp;gt;I \subset \mathbb{R}&amp;lt;/math&amp;gt;  &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;的概率 &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;是&amp;lt;math&amp;gt;f&amp;lt;/math&amp;gt; 在 &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;上的积分.&amp;lt;ref&amp;gt;Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}&amp;lt;/ref&amp;gt; 例如, 如果&amp;lt;math&amp;gt;I = [a, b]&amp;lt;/math&amp;gt; 可得到:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In particular, the probability for &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; to take any single value &amp;lt;math&amp;gt;a&amp;lt;/math&amp;gt; (that is &amp;lt;math&amp;gt;a \le X \le a&amp;lt;/math&amp;gt;) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;F(x) = \operatorname{P}\left[-\infty &amp;lt; X \le x\right] = \int_{-\infty}^x f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
特别是，X取任何单个值a（即a≤X≤a）的概率为零，因为上下限一致的积分始终等于零。满足上述条件的变量称为连续随机变量。其累积密度函数定义为&lt;br /&gt;
&lt;br /&gt;
which, by this definition, has the properties:&lt;br /&gt;
根据定义有以下一些性质&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is non-decreasing;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;0 \le F(x) \le 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;\lim_{x \rightarrow -\infty} F(x) = 0&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\lim_{x \rightarrow \infty} F(x) = 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;P(a \le X &amp;lt; b) = F(b) - F(a)&amp;lt;/math&amp;gt;; and&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is continuous (due to the [[Riemann integral]] properties).&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
It is also possible to think in the opposite direction, which allows more flexibility. Say &amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is a function that satisfies all but the last of the properties above, then &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; represents the cumulative density function for some random variable: a discrete random variable if &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; is a step function, and a continuous random variable otherwise.&amp;lt;ref&amp;gt;See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.&amp;lt;/ref&amp;gt; This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].&lt;br /&gt;
也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; is such an absolutely continuous random variable, then it has a [[probability density function]] &amp;lt;math&amp;gt;f(x)&amp;lt;/math&amp;gt;, and its probability of falling into a Lebesgue-measurable set &amp;lt;math&amp;gt;A \subset \mathbb{R}&amp;lt;/math&amp;gt; is:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; is the Lebesgue measure.&lt;br /&gt;
&lt;br /&gt;
对于实线的更多任意子集，通常有必要对上述定义进行概括。在这些情况下，连续概率分布定义为具有绝对连续的累积分布函数的概率分布。等效地，就Lebesgue测度而言，它是实数上的概率分布，它是绝对连续的。这样的分布可以用它们的概率密度函数表示。如果X是这样一个绝对连续的随机变量，则它具有概率密度函数f（x），并且落入Lebesgue可测量集合A⊂R的概率为：&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这里，mu是Lebesgue度量&lt;br /&gt;
&lt;br /&gt;
Note on terminology: some authors use the term &amp;quot;continuous distribution&amp;quot; to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;\mu\{x\}\,=\,0&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;\,x&amp;lt;/math&amp;gt;. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].&lt;br /&gt;
&lt;br /&gt;
关于术语的注释：一些作者使用术语“连续分布”来表示其累积分布函数是连续的而不是绝对连续的分布。这些分布是所有x的μ{x} = 0的μ分布。该定义包括上面定义的（绝对）连续分布，但也包括奇异分布，既不是绝对连续也不是离散的，也不是它们的混合。没有密度。 Cantor分布给出了一个示例。&lt;br /&gt;
&lt;br /&gt;
== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; [[Andrey Kolmogorov|Kolmogorov]] definition 柯尔莫哥洛夫的定义&amp;lt;/font&amp;gt;'''==&lt;br /&gt;
&lt;br /&gt;
{{Main|Probability space|Probability measure}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; from a [[probability space]] &amp;lt;math&amp;gt;(\Omega, \mathcal{F}, \mathbb{P})&amp;lt;/math&amp;gt; to a [[measurable space]] &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt;. Given that probabilities of events of the form &amp;lt;math&amp;gt;\{\omega\in\Omega\mid X(\omega)\in A\}&amp;lt;/math&amp;gt; satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] &amp;lt;math&amp;gt;X_*\mathbb{P}&amp;lt;/math&amp;gt; of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; , which is a [[probability measure]] on &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt; satisfying &amp;lt;math&amp;gt;X_*\mathbb{P} = \mathbb{P}X^{-1}&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
在概率论的度量理论形式化中，将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）。给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P，它是满足&amp;lt;math&amp;gt;X_*\mathbb{P} = \mathbb{P}X^{-1}&amp;lt;/math&amp;gt;的概率量度&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Random number generation 随机数生成==&lt;br /&gt;
{{Main|Pseudo-random number sampling}}&lt;br /&gt;
&lt;br /&gt;
Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Citation|last1=Dekking|first1=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
大多数算法基于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 伪随机数生成器 Pseudorandom Number Generator&amp;lt;/font&amp;gt;'''，该伪随机数生成器生成在半开间隔[0,1）中均匀分布的数字X。然后，通过某种算法对这些随机变量X进行转换，以创建具有所需概率分布的新随机变量。利用这种统一的伪随机源，可以生成任何随机变量的实现。&lt;br /&gt;
&lt;br /&gt;
For example, suppose &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt; has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some &amp;lt;math&amp;gt;0 &amp;lt; p &amp;lt; 1&amp;lt;/math&amp;gt;, we define&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
so that&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\textrm{P}(X=1) = \textrm{P}(U&amp;lt;p) = p,&lt;br /&gt;
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
例如，假设U具有介于0和1之间的均匀分布。为某些对象构造一个随机的Bernoulli变量0 &amp;lt;p &amp;lt;1我们定义&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\textrm{P}(X=1) = \textrm{P}(U&amp;lt;p) = p,&lt;br /&gt;
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
This random variable X has a Bernoulli distribution with parameter &amp;lt;math&amp;gt;p&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:0&amp;quot;/&amp;gt; Note that this is a transformation of discrete random variable.&lt;br /&gt;
该随机变量X具有参数的伯努利分布p。请注意，这是离散随机变量的变换。&lt;br /&gt;
&lt;br /&gt;
For a distribution function &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; of a continuous random variable, a continuous random variable must be constructed. &amp;lt;math&amp;gt;F^{inv}&amp;lt;/math&amp;gt;, an inverse function of &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt;, relates to the uniform variable &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{U\leq F(x)} = {F^{inv}(U)\leq x}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
F或连续随机变量的分布函数F，必须构造连续随机变量。 &amp;lt;math&amp;gt;F^{inv}&amp;lt;/math&amp;gt;，F的反函数，涉及均匀变量U：&lt;br /&gt;
&amp;lt;math&amp;gt;{U\leq F(x)} = {F^{inv}(U)\leq x}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For example, suppose a random variable that has an exponential distribution &amp;lt;math&amp;gt;F(x) = 1 - e^{-\lambda x}&amp;lt;/math&amp;gt; must be constructed.&lt;br /&gt;
例如，假设必须构造一个具有指数分布&amp;lt;math&amp;gt;F(x) = 1 - e^{-\lambda x}&amp;lt;/math&amp;gt; 的随机变量。&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
F(x) = u &amp;amp;\Leftrightarrow 1-e^{-\lambda x} = u \\ &amp;amp;\Leftrightarrow e^{-\lambda x } = 1-u \\&amp;amp;\Leftrightarrow -\lambda x = \ln(1-u) \\ &amp;amp;\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)&lt;br /&gt;
\end{align}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
所以&amp;lt;math&amp;gt;F^{inv}(u) = \frac{-1}{\lambda}\ln(1-u)&amp;lt;/math&amp;gt; 并且如果 &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt; 有一个&amp;lt;math&amp;gt;U(0,1)&amp;lt;/math&amp;gt; 分布, 然后随机变量 &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; 被定义为 &amp;lt;math&amp;gt;X = F^{inv}(U) = \frac{-1}{\lambda} \ln(1-U)&amp;lt;/math&amp;gt;. 这里有一个指数分布 &amp;lt;math&amp;gt;\lambda&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.&lt;br /&gt;
统计模拟（蒙特卡洛方法）中经常遇到的一个问题是生成以给定方式分布的伪随机数。&lt;br /&gt;
&lt;br /&gt;
== Common probability distributions and their applications 共同概率分布及其应用==&lt;br /&gt;
{{Main list|List of probability distributions}}&lt;br /&gt;
&lt;br /&gt;
The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.&lt;br /&gt;
&lt;br /&gt;
概率分布的概念及其描述的随机变量是概率论和统计学科学的数学基础。人口中几乎可以测量的任何值都存在价差或可变性（例如人的身高，金属的耐用性，销售增长，交通流量等）;几乎所有测量均存在一定的固有误差；在物理学中，从气体的动力学特性到基本粒子的量子力学描述，很多过程都用概率论来描述。由于这些以及许多其他原因，简单数字通常不足以描述数量，而概率分布通常更合适。&lt;br /&gt;
&lt;br /&gt;
The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see [[list of probability distributions]], which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)&lt;br /&gt;
以下是一些最常见的概率分布列表，按与之相关的过程类型进行分组。有关更完整的列表，请参见概率分布列表，该列表按要考虑的结果的性质（离散，连续，多元等）进行分组。&lt;br /&gt;
&lt;br /&gt;
All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a [[mixture distribution]].&lt;br /&gt;
下面所有的单变量分布都达到了峰值。也就是说，假设值聚集在单个点周围。实际上，实际观察到的量可能会聚集在多个值附近。可以使用混合物分布对此类数量进行建模。&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Linear growth (e.g. errors, offsets) 线性增长 &amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
* [[Normal distribution]] (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 正态分布（高斯分布 Normal Distribution&amp;lt;/font&amp;gt;'''，对于单个这样的数量；最常用的连续分布&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Exponential growth (e.g. prices, incomes, populations) 指数增长&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;对数正态分布 Log-normal Distribution&amp;lt;/font&amp;gt;'''，对于单个此类数量的对数正态分布&lt;br /&gt;
&lt;br /&gt;
* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;帕累托分布 Pareto Distribution&amp;lt;/font&amp;gt;'''，对于单个这样的数量，其对数呈指数分布；原型幂律分布&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Uniformly distributed quantities 数量均匀分布&amp;lt;/font&amp;gt;'''===&lt;br /&gt;
&lt;br /&gt;
* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 离散均匀分布 Discrete Uniform Distributed&amp;lt;/font&amp;gt;'''，用于有限的一组值（例如，公平死亡的结果）&lt;br /&gt;
&lt;br /&gt;
* [[Continuous uniform distribution]], for continuously distributed values&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;连续均匀分布 Continuous Uniform Distributed&amp;lt;/font&amp;gt;'''，用于连续分布的值&lt;br /&gt;
&lt;br /&gt;
=== Bernoulli trials (yes/no events, with a given probability) 伯努利试验（是/否事件，具有给定的概率）===&lt;br /&gt;
&lt;br /&gt;
* Basic distributions 基本分布:&lt;br /&gt;
** [[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 伯努利分布 Bernoulli Distribution&amp;lt;/font&amp;gt;'''，用于单个伯努利试验的结果（例如成功/失败，是/否）&lt;br /&gt;
&lt;br /&gt;
** [[Binomial distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed total number of [[Independent (statistics)|independent]] occurrences&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项式分布 Binomial Distribution &amp;lt;/font&amp;gt;'''，对于给定固定总数的独立“出现次数”（例如，成功，赞成票等）的数量&lt;br /&gt;
&lt;br /&gt;
** [[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;负二项分布 Negative Binomial Distribution&amp;lt;/font&amp;gt;'''，用于二项式观察，但是关注的数量是在给定成功次数之前发生的失败次数&lt;br /&gt;
&lt;br /&gt;
** [[Geometric distribution]], for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the [[negative binomial distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;几何分布 Geometric Distribution&amp;lt;/font&amp;gt;'''，用于二项式观测，但是关注的数量是首次成功之前的失败数量；负二项式分布的特殊情况&lt;br /&gt;
&lt;br /&gt;
* Related to sampling schemes over a finite population 与有限人口抽样方案有关:&lt;br /&gt;
&lt;br /&gt;
** [[Hypergeometric distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using [[sampling without replacement]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 超几何分布 Hypergeometric Distribution&amp;lt;/font&amp;gt;'''，对于“肯定出现”的数量（例如成功，赞成票等），给定了一定的总出现数量，使用采样而无需替换&lt;br /&gt;
&lt;br /&gt;
** [[Beta-binomial distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a [[Pólya urn model]] (in some sense, the &amp;quot;opposite&amp;quot; of [[sampling without replacement]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 贝塔二项式分布 Beta-binomial Distribution&amp;lt;/font&amp;gt;'''，对于给定的总发生次数为“阳性”的次数（例如，成功，赞成票等），使用Pólyaurn模型进行采样（在某种意义上，为“替代”而不进行替换） ）&lt;br /&gt;
&lt;br /&gt;
=== Categorical outcomes (events with ''K'' possible outcomes, with a given probability for each outcome) 分类结果（具有K个可能结果的事件，每个结果具有给定的概率）===&lt;br /&gt;
&lt;br /&gt;
* [[Categorical distribution]], for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the [[Bernoulli distribution]]&lt;br /&gt;
针对单个分类结果的分类分布（例如，调查中的是/否/也许）；伯努利分布的一般化&lt;br /&gt;
&lt;br /&gt;
* [[Multinomial distribution]], for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the [[binomial distribution]]&lt;br /&gt;
给定总结果的固定数量，针对每种类别结果的数量的多项式分布；二项式分布的一般化&lt;br /&gt;
&lt;br /&gt;
* [[Multivariate hypergeometric distribution]], similar to the [[multinomial distribution]], but using [[sampling without replacement]]; a generalization of the [[hypergeometric distribution]]&lt;br /&gt;
多元超几何分布，类似于多项式分布，但使用采样而不进行替换；超几何分布的一般化&lt;br /&gt;
&lt;br /&gt;
=== Poisson process (events that occur independently with a given rate) 泊松过程（以给定速率独立发生的事件）===&lt;br /&gt;
&lt;br /&gt;
* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 泊松分布 Poisson Distribution&amp;lt;/font&amp;gt;'''，用于给定时间段内泊松型事件的发生次数&lt;br /&gt;
&lt;br /&gt;
* [[Exponential distribution]], for the time before the next Poisson-type event occurs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;指数分布 Exponential Distribution&amp;lt;/font&amp;gt;'''，在下一次泊松型事件发生之前的时间&lt;br /&gt;
&lt;br /&gt;
* [[Gamma distribution]], for the time before the next k Poisson-type events occur&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;伽马分布 Gamma Distribution&amp;lt;/font&amp;gt;'''，在接下来的k个泊松型事件发生之前的时间&lt;br /&gt;
&lt;br /&gt;
=== Absolute values of vectors with normally distributed components 具有正态分布分量的向量的绝对值===&lt;br /&gt;
&lt;br /&gt;
* [[Rayleigh distribution]], for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;瑞利分布 Rayleigh Distribution&amp;lt;/font&amp;gt;'''，用于具有高斯分布正交分量的矢量幅度分布。在具有高斯实部和虚部的RF信号中发现瑞利分布。&lt;br /&gt;
&lt;br /&gt;
* [[Rice distribution]], a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in [[Rician fading]] of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;莱斯分布 Rice Distribution&amp;lt;/font&amp;gt;'''，是在背景信号分量稳定的情况下瑞利分布的概括。由于多径传播而在无线电信号的Rician衰落中发现，并且在非零NMR信号中出现噪声破坏的MR图像中也发现了这种情况。&lt;br /&gt;
&lt;br /&gt;
=== Normally distributed quantities operated with sum of squares (for hypothesis testing) 以平方和运算的正态分布量（用于假设检验）===&lt;br /&gt;
&lt;br /&gt;
* [[Chi-squared distribution]], the distribution of a sum of squared [[standard normal]] variables; useful e.g. for inference regarding the [[sample variance]] of normally distributed samples (see [[chi-squared test]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;卡方分布 Chi-squared Distribution&amp;lt;/font&amp;gt;'''，标准正态变量平方和的分布；有用的关于正态分布样本的样本方差的推论（请参见卡方检验）&lt;br /&gt;
&lt;br /&gt;
* [[Student's t distribution]], the distribution of the ratio of a [[standard normal]] variable and the square root of a scaled [[Chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;学生t分布 Student‘s t Distribution&amp;lt;/font&amp;gt;'''，标准正态变量与缩放的卡方变量的平方根之比的分布；有助于推断方差未知的正态分布样本的平均值（请参阅学生的t检验）&lt;br /&gt;
&lt;br /&gt;
* [[F-distribution]], the distribution of the ratio of two scaled [[Chi squared distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;F-分布 F-Distribution&amp;lt;/font&amp;gt;'''，两个比例卡方变量的比例分布；有用的用于涉及比较方差或涉及R平方（相关系数平方）的推论&lt;br /&gt;
&lt;br /&gt;
=== As a conjugate prior distributions in Bayesian inference 作为贝叶斯推断中的共轭先验分布===&lt;br /&gt;
{{Main|Conjugate prior}}&lt;br /&gt;
&lt;br /&gt;
* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Beta分布 Beta Distribution&amp;lt;/font&amp;gt;'''，具有单个概率（0到1之间的实数）；与伯努利分布和二项式分布共轭&lt;br /&gt;
&lt;br /&gt;
* [[Gamma distribution]], for a non-negative scaling parameter; conjugate to the rate parameter of a [[Poisson distribution]] or [[exponential distribution]], the [[Precision (statistics)|precision]] (inverse [[variance]]) of a [[normal distribution]], etc.&lt;br /&gt;
伽玛分布，用于非负比例缩放参数；与泊松分布或指数分布的速率参数，正态分布的精度（逆方差）等共轭。&lt;br /&gt;
&lt;br /&gt;
* [[Dirichlet distribution]], for a vector of probabilities that must sum to 1; conjugate to the [[categorical distribution]] and [[multinomial distribution]]; generalization of the [[beta distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Dirichlet分布 Dirichlet Distribution&amp;lt;/font&amp;gt;'''，对于必须为1的概率向量；与分类分布和多项式分布共轭； beta分布的一般化&lt;br /&gt;
&lt;br /&gt;
*[[Wishart distribution]], for a symmetric [[non-negative definite]] matrix; conjugate to the inverse of the [[covariance matrix]] of a [[multivariate normal distribution]]; generalization of the [[gamma distribution]]&amp;lt;ref&amp;gt;{{Cite book|title=Pattern recognition and machine learning|last=Bishop, Christopher M.|date=2006|publisher=Springer|isbn=0-387-31073-8|location=New York|oclc=71008143}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Wishart分布 Wishart Distribution&amp;lt;/font&amp;gt;'''，用于对称非负定矩阵；与多元正态分布的协方差矩阵的逆共轭；伽玛分布的一般化&lt;br /&gt;
&lt;br /&gt;
=== Some specialized applications of probability distributions 概率分布的一些特殊应用===&lt;br /&gt;
&lt;br /&gt;
* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.&lt;br /&gt;
在自然语言处理中使用的高速缓存语言模型和其他统计语言模型通过概率分布来为特定单词和单词序列的出现分配概率。&lt;br /&gt;
&lt;br /&gt;
* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by &amp;lt;math&amp;gt;P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 &amp;lt;/math&amp;gt;, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.&amp;lt;ref&amp;gt;{{Cite book|title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|publisher=|others=Thoman, John W., Jr., 1960-|year=|isbn=978-1-68015-835-9|location=[Mill Valley, California]|pages=403–406|oclc=927509011}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
在量子力学中，在给定点处找到粒子的概率密度与该点处粒子波函数大小的平方成正比（请参阅博恩法则）。因此，粒子位置的概率分布函数描述为&amp;lt;math&amp;gt;P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 &amp;lt;/math&amp;gt;，粒子位置的概率x在第一个维度中的间隔为a≤x≤b，在第三个维度中的间隔类似。这是量子力学的关键原理。&lt;br /&gt;
&lt;br /&gt;
* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution.&amp;lt;ref&amp;gt;{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.|last2=Chen|first2=Z.|last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review|doi=10.1109/drpt.2008.4523658|s2cid=18669309}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
潮流研究中的概率潮流解释了作为概率分布的输入变量的不确定性，并以概率分布的形式提供了潮流计算。&lt;br /&gt;
&lt;br /&gt;
* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.&amp;lt;ref&amp;gt;{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity, Rajib|isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
根据先前的频率分布（例如热带气旋，冰雹，事件之间的时间等）预测自然现象的发生。&lt;br /&gt;
&lt;br /&gt;
==See also 另请参见==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Copula (statistics)]]&lt;br /&gt;
Copula（统计数据）&lt;br /&gt;
* [[Empirical probability]]&lt;br /&gt;
经验概率&lt;br /&gt;
* [[Histogram]]&lt;br /&gt;
直方图&lt;br /&gt;
* [[Likelihood function]]&lt;br /&gt;
似然函数&lt;br /&gt;
* [[Kirkwood approximation]]&lt;br /&gt;
柯克伍德近似&lt;br /&gt;
* [[Moment-generating function]]&lt;br /&gt;
瞬时产生功能&lt;br /&gt;
* [[Riemann–Stieltjes integral#Application to probability theory|Riemann–Stieltjes integral application to probability theory]]&lt;br /&gt;
Riemann–Stieltjes积分＃在概率论中的应用&lt;br /&gt;
*[[Pairwise independence]]&lt;br /&gt;
成对独立&lt;br /&gt;
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=== Lists 清单===&lt;br /&gt;
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* [[List of probability distributions]]&lt;br /&gt;
概率分布的清单&lt;br /&gt;
* [[List of statistical topics]]&lt;br /&gt;
统计学话题的清淡&lt;br /&gt;
=== Probability distributions 概率分布 ===&lt;br /&gt;
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* [[Conditional probability distribution]]&lt;br /&gt;
条件概率分布&lt;br /&gt;
* [[Joint probability distribution]]&lt;br /&gt;
联合概率分布&lt;br /&gt;
* [[Quasiprobability distribution]]&lt;br /&gt;
拟概率分布&lt;br /&gt;
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== References ==&lt;br /&gt;
=== Citations ===&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
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=== Sources ===&lt;br /&gt;
{{refbegin|}}&lt;br /&gt;
* {{cite journal|doi=10.1016/j.ejmp.2014.05.002|pmid=25059432|title=Data distributions in magnetic resonance images: A review|url=|journal=[[Physica Medica]]|volume=30|issue=7|pages=725–741|year=2014|last1=den Dekker|first1=A. J.|last2=Sijbers|first2=J.}}&lt;br /&gt;
* {{cite book|last=Vapnik|first=Vladimir Naumovich|year=1998|title=Statistical Learning Theory|publisher=John Wiley and Sons}}&lt;br /&gt;
{{refend}}&lt;br /&gt;
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==External links==&lt;br /&gt;
{{commons|Probability distribution|Probability distribution}}&lt;br /&gt;
*{{springer|title=Probability distribution|id=p/p074900}}&lt;br /&gt;
*[http://threeplusone.com/FieldGuide.pdf Field Guide to Continuous Probability Distributions],  Gavin E. Crooks.&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
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		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{short description|Mathematical function that describes the probability of occurrence of different possible outcomes in an experiment}}&lt;br /&gt;
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{{other uses|Distribution (disambiguation){{!}}Distribution}}&lt;br /&gt;
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{{Off topic|date=August 2020|Random variable|talksectionheading}}&lt;br /&gt;
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In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].&amp;lt;ref name=&amp;quot;:02&amp;quot;&amp;gt;{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}&amp;lt;/ref&amp;gt; It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).&lt;br /&gt;
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在概率论和统计学中，概率分布是一个给出一个实验不同可能结果出现的概率的数学函数。它是根据样本空间和事件概率(样本空间的子集)对随机现象的数学描述。&lt;br /&gt;
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For instance, if the [[random variable]] {{mvar|X}} is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of {{mvar|X}} would take the value 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} heads}}, and 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} tails}} (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a [[Survey methodology|survey]], etc.&amp;lt;ref name=&amp;quot;ross&amp;quot;&amp;gt;{{cite book|first=Sheldon M.|last=Ross|title=A first course in probability|publisher=Pearson|year=2010}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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For instance, if the random variable X is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of X would take the value 0.5 for X=heads, and 0.5 for X=tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.&lt;br /&gt;
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例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。随机现象的例子包括未来某一天的天气状况、一个人的身高、学校中男生的比例、调查结果等等。&lt;br /&gt;
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A probability distribution is a mathematical function that has a [[sample space]] as its '''input''', and gives a '''probability''' as its output. The sample space is the [[Set (mathematics)|set]] of all possible [[Outcome (probability)|outcomes]] of a random phenomenon being observed; it may be the set of [[real numbers]] or a set of [[vector (mathematics)|vectors]], or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be {{math|{heads, tails} }}.&lt;br /&gt;
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A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be .&lt;br /&gt;
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概率分布是一个数学函数，它的输入是一个样本空间，输出是一个概率。样本空间是所观察到的随机现象的所有可能结果的集合; 它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。&lt;br /&gt;
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Probability distributions are generally divided into two classes. A '''discrete probability distribution''' is applicable to the scenarios where the set of possible outcomes is [[discrete probability distribution|discrete]] (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the [[probability mass function]]. On the other hand, '''continuous probability distributions''' are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a [[probability density function]].&amp;lt;ref name=&amp;quot;ross&amp;quot; /&amp;gt;&amp;lt;ref&amp;gt;{{cite book|author1=DeGroot, Morris H.|author2=Schervish, Mark J.|title=Probability and Statistics|publisher=Addison-Wesley|year=2002}}&amp;lt;/ref&amp;gt; The [[normal distribution]] is a commonly encountered continuous probability distribution. More complex experiments, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.&lt;br /&gt;
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Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function. The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.&lt;br /&gt;
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概率分布一般分为两类。离散概率分布适用于一组可能的结果是离散的情况，如抛硬币或掷骰子。这里的概率被编码为结果概率的离散列表，称为概率质量函数。另一方面，连续概率分布适用于一组可以在一个连续的范围内取值的结果的情况(例如:实数)，例如某一天的温度。在这种情况下，概率通常由概率密度函数描述。正态分布是一种常见的连续概率分布。更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called [[Univariate distribution|univariate]], while a distribution whose sample space is a [[vector space]] of dimension 2 or more is called [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a [[joint probability distribution]]) gives the probabilities of a [[random vector]] – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the [[binomial distribution]], the [[hypergeometric distribution]], and the [[normal distribution]]. The [[multivariate normal distribution]] is a commonly encountered multivariate distribution.&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.&lt;br /&gt;
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一个一维的样本空间(例如实数、标签列表、有序标签或二进制)的概率分布被称为单变量，而样本空间为二维或更多向量空间的分布被称为多变量。单变量分布给出了单个随机变量取不同替代值的概率; 联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。重要的和常见的单变量概率分布包括二项分布、超几何分布和正态分布。多变量正态分布是一种常见的联合分布。&lt;br /&gt;
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==Introduction 简介==&lt;br /&gt;
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[[File:Dice Distribution (bar).svg|thumb|250px|right|&lt;br /&gt;
图1:The [[probability mass function]] (pmf) ''p''(''S'') specifies the probability distribution for the sum ''S'' of counts from two [[dice]].  For example, the figure shows that ''p''(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as ''P''(''S'' &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.&lt;br /&gt;
概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布。例如，图中显示 p (11) = 2/36 = 1/18。Pmf 允许计算事件的概率，如 p (s &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6，以及分布中的所有其他概率。&lt;br /&gt;
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The [[probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice.  For example, the figure shows that p(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as P(S &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]&lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between '''discrete''' and '''continuous''' [[random variable]]s. In the discrete case, it is sufficient to specify a [[probability mass function]] &amp;lt;math&amp;gt;p&amp;lt;/math&amp;gt; assigning a probability to each possible outcome: for example, when throwing a fair [[Dice|die]], each of the six values 1 to 6 has the probability 1/6. The probability of an [[Event (probability theory)|event]] is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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为了定义最简单的概率分布，有必要区分离散和连续的随机变量。在离散情况下，指定一个概率质量函数 p 就足够了，它为每个可能的结果赋予一个概率: 例如，当投掷一个骰子时，6个值中的每一个的概率为1/6。然后将事件的概率定义为满足事件的结果的概率之和; 例如，事件”骰子掷出偶数值”的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2.&amp;lt;/math&amp;gt;&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs ''exactly'' 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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相比之下，当一个随机变量从一个连续体中取值时，那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率。例如，一个给定的物体重量正好是500克的概率为零，因为随着我们测量仪器精度的提高，正好测量500克的概率趋向于零。然而，在质量控制方面，人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98% ，而这一要求对测量仪器的准确性不太敏感。&lt;br /&gt;
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Continuous probability distributions can be described in several ways. The [[probability density function]] describes the [[infinitesimal]] probability of any given value, and the probability that the outcome lies in a given interval can be computed by [[Integration (mathematics)|integrating]] the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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连续概率分布可以用几种方法来描述。概率密度函数描述了任意给定值的无穷小概率，并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。可能值位于某一固定区间的概率可以与和收敛于积分的方式有关，因此，连续概率是基于积分的定义。&lt;br /&gt;
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[[File:Combined Cumulative Distribution Graphs.png|thumb|455x455px|&lt;br /&gt;
图2:On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve.&lt;br /&gt;
左边是概率密度函数。右边是累积分布函数，它是概率密度曲线下面的区域。]]&lt;br /&gt;
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The [[cumulative distribution function]] describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the [[antiderivative]] of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the [[probability density function]] from minus infinity &amp;lt;math&amp;gt;\infty&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; as described by the picture to the right.&amp;lt;ref&amp;gt;{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=Springer|others=Dekking, Michel, 1946-|isbn=978-1-85233-896-1|location=London|oclc=262680588}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity \infty to x as described by the picture to the right.&lt;br /&gt;
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累积分布函数指标描述了随机变量不大于给定值的概率; 结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算。累积分布函数是概率密度函数的反导函数，前提是后者存在。正如右边图片所描述的那样，累积分布函数是从负无穷到 x 的概率密度函数下面的区域。&lt;br /&gt;
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[[File:Standard deviation diagram.svg|right|thumb|250px|&lt;br /&gt;
图3:The [[probability density function]] (pdf) of the [[normal distribution]], also called Gaussian or &amp;quot;bell curve&amp;quot;, the most important continuous random distribution.  As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.&lt;br /&gt;
正态分布的[概率密度函数(pdf) ，也称为高斯或钟形曲线，是最重要的连续随机分布。如图所示，值间隔的概率对应于曲线下面积。]]&lt;br /&gt;
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== Terminology 术语==&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.&amp;lt;ref name=&amp;quot;:02&amp;quot; /&amp;gt;&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X &amp;gt; a, X &amp;lt; b or a union thereof.&lt;br /&gt;
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一些关键的，广泛用于以概率分布为主题的文献中的概念和术语，列出如下。&lt;br /&gt;
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=== Functions for discrete variables 离散变量的函数===&lt;br /&gt;
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Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.&lt;br /&gt;
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用于统计建模的著名离散概率分布包括泊松分佈、伯努利分布、二项分布、几何分布和负二项分布。&lt;br /&gt;
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*'''Probability function''': describes the probability distribution of a discrete random variable.&lt;br /&gt;
概率函数：描述离散随机变量的概率分布。&lt;br /&gt;
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*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.&lt;br /&gt;
概率质量函数（pmf）：给出离散随机变量等于某个值的概率的函数。&lt;br /&gt;
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*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.&lt;br /&gt;
频率分布：显示样本中各种结果的频率的表格。&lt;br /&gt;
&lt;br /&gt;
For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
&lt;br /&gt;
对于一个离散的随机变量 x，设 u0，u1，... 是它在非零概率情况下可以取的值。表示&lt;br /&gt;
&lt;br /&gt;
*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.&lt;br /&gt;
相对频率分布：一种频率分布，其中每个值均已被样本中的多个结果（即样本大小）除（归一化）。&lt;br /&gt;
&lt;br /&gt;
*'''Discrete probability distribution function''': general term to indicate the way the total probability of 1 is distributed over '''all''' various possible outcomes (i.e. over entire population) for discrete random variable.&lt;br /&gt;
离散概率分布函数：通用术语，表示总概率1在离散随机变量的所有各种可能结果（即整个人群）中的分布方式。&lt;br /&gt;
&lt;br /&gt;
\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots&lt;br /&gt;
&lt;br /&gt;
Omega _ i = x ^ {-1}(u _ i) = { Omega: x (Omega) = u _ i } ，i = 0,1,2，点&lt;br /&gt;
&lt;br /&gt;
*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for a discrete random variable.&lt;br /&gt;
累积分布函数：该函数评估离散随机变量X取小于或等于x的值的概率。&lt;br /&gt;
&lt;br /&gt;
*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.&lt;br /&gt;
分类分布：适用于具有有限值集的离散随机变量。&lt;br /&gt;
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&lt;br /&gt;
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=== Functions for continuous variables 连续变量函数===&lt;br /&gt;
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&lt;br /&gt;
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* '''[[Probability density function]] (pdf):''' function whose value at any given sample (or point) in the [[sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.&lt;br /&gt;
概率密度函数（pdf）：可以将其在样本空间中任意给定样本（或点）上的值（随机变量可能获得的一组值）的值解释为提供随机变量值将具有的相对可能性的函数等于那个样本。&lt;br /&gt;
&lt;br /&gt;
It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as&lt;br /&gt;
&lt;br /&gt;
由此可见，除了 u0、 u1、 ... 之外，x 取任何值的概率为零，因此可以将 x 写为&lt;br /&gt;
&lt;br /&gt;
* '''Continuous probability distribution function''': most often reserved for continuous random variables.&lt;br /&gt;
连续概率分布函数：最常保留的连续随机变量。&lt;br /&gt;
&lt;br /&gt;
* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for continuous variable.&lt;br /&gt;
累积分布函数：评估连续变量X取小于或等于x的值的概率的函数。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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===Basic terms 基本术语===&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Mode (statistics)|'''Mode''']]: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.&lt;br /&gt;
模式：对于离散随机变量，该值具有最高概率；对于连续随机变量，是概率密度函数具有局部峰值的位置。&lt;br /&gt;
&lt;br /&gt;
* [[Support (mathematics)|'''Support''']]: set of values that can be assumed, with non-zero probability, by the random variable.&lt;br /&gt;
支持：可以由随机变量以非零概率假定的一组值。对于随机变量X，有时表示为R_ {X}&lt;br /&gt;
&lt;br /&gt;
* '''Tail''':&amp;lt;ref name='tail'&amp;gt;More information and examples can be found in the articles [[Heavy-tailed distribution]], [[Long-tailed distribution]], [[fat-tailed distribution]]&amp;lt;/ref&amp;gt; the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form &amp;lt;math&amp;gt;X &amp;gt; a&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;X &amp;lt; b&amp;lt;/math&amp;gt; or a union thereof.&lt;br /&gt;
尾巴：如果pmf或pdf相对较低，则靠近随机变量边界的区域。通常形式为X&amp;gt; a，X &amp;lt;b或它们的并集。&lt;br /&gt;
&lt;br /&gt;
*'''Head''':&amp;lt;ref name='tail' /&amp;gt; the region where the pmf or pdf is relatively high. Usually has the form &amp;lt;math&amp;gt;a &amp;lt; X &amp;lt; b&amp;lt;/math&amp;gt;.&lt;br /&gt;
头部：pmf或pdf较高的区域。通常具有a &amp;lt;X &amp;lt;b的形式&lt;br /&gt;
&lt;br /&gt;
* '''[[Expected value]]''' or '''mean''': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.&lt;br /&gt;
期望值或均值：可能值的加权平均值，以其概率作为权重；或其连续类似物。&lt;br /&gt;
&lt;br /&gt;
A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a cumulative density function that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.&lt;br /&gt;
&lt;br /&gt;
一个连续的概率分布是一个支持不可数集的概率分布，比如实数行中的一个区间。它们是唯一拥有属性的一个累积密度函数，可以用来计算支持的每个子集的概率。有许多连续概率分布的例子: 正态、均匀、卡方等等。&lt;br /&gt;
&lt;br /&gt;
*'''[[Median]]''': the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.&lt;br /&gt;
中位数：这样的值，即一组值小于中位数，而该组大于中位数，每一个的概率不大于二分之一。&lt;br /&gt;
&lt;br /&gt;
*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.&lt;br /&gt;
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。&lt;br /&gt;
&lt;br /&gt;
* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.&lt;br /&gt;
标准偏差：方差的平方根，因此是色散的另一种度量。&lt;br /&gt;
&lt;br /&gt;
* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.&lt;br /&gt;
对称性：某些分布的一种属性，其中特定值左侧（通常是中位数）的分布部分是其右侧部分的镜像。&lt;br /&gt;
&lt;br /&gt;
*'''[[Skewness]]''': a measure of the extent to which a pmf or pdf &amp;quot;leans&amp;quot; to one side of its mean. The third [[standardized moment]] of the distribution.&lt;br /&gt;
偏度：衡量pmf或pdf在其均值的一侧“倾斜”的程度。分布的第三个标准化时刻。&lt;br /&gt;
&lt;br /&gt;
*'''[[Kurtosis]]''': a measure of the &amp;quot;fatness&amp;quot; of the tails of a pmf or pdf. The fourth standardized moment of the distribution.&lt;br /&gt;
峰度：pmf或pdf尾部“脂肪”的量度。分布的第四个标准化时刻。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Discrete probability distribution 离散概率分布==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Discrete probability distrib.svg|right|thumb|&lt;br /&gt;
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Discrete probability distribution.svg|right|thumb|&lt;br /&gt;
图5:The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...离散概率分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Normal probability distribution.svg|right|thumb|&lt;br /&gt;
图6:... of a continuous probability distribution, ...连续概率分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Mixed probability distribution.svg|right|thumb|&lt;br /&gt;
图7:... of a distribution which has both a continuous part and a discrete part.既有连续部分又有离散部分]]&lt;br /&gt;
&lt;br /&gt;
 &lt;br /&gt;
A '''discrete probability distribution''' is a probability distribution that can take on a countable number of values.&amp;lt;ref&amp;gt;{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer|isbn=9780387878591|location=New York|pages=51|oclc=710149819}}&amp;lt;/ref&amp;gt; For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if &amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.&lt;br /&gt;
&lt;br /&gt;
离散概率分布是可以具有可数数量的值的概率分布。在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1。例如，如果&amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2，概率之和为1/2 + 1/4 + 1/8 + ... = 1。&lt;br /&gt;
&lt;br /&gt;
Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt; Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.&lt;br /&gt;
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
When a [[Sample (statistics)|sample]] (a set of observations) is drawn from a larger population, the sample points have an [[empirical distribution function|empirical distribution]] that is discrete and that provides information about the population distribution.&lt;br /&gt;
当从更大的总体中抽取一个样本（一组观察值）时，这些采样点的经验分布是离散的，并且提供了有关总体分布的信息。&lt;br /&gt;
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&lt;br /&gt;
===Measure theoretic formulation 测量理论公式===&lt;br /&gt;
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A [[measurable function]] &amp;lt;math&amp;gt;X \colon A \to B &amp;lt;/math&amp;gt; between a [[probability space]] &amp;lt;math&amp;gt;(A, \mathcal A, P)&amp;lt;/math&amp;gt; and a [[measurable space]] &amp;lt;math&amp;gt;(B, \mathcal B) &amp;lt;/math&amp;gt; is called a '''discrete random variable''' provided that its image is a countable set. In this case measurability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; means that the pre-images of singleton sets are measurable, i.e., &amp;lt;math&amp;gt;X^{-1}(\{b\}) \in \mathcal A&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;b \in B&amp;lt;/math&amp;gt;.&lt;br /&gt;
The latter requirement induces a [[probability mass function]] &amp;lt;math&amp;gt;f_X \colon X(A) \to \mathbb R&amp;lt;/math&amp;gt; via &amp;lt;math&amp;gt; f_X(b):=P(X^{-1}(\{b\}))&amp;lt;/math&amp;gt;.  Since the pre-images of disjoint sets&lt;br /&gt;
are disjoint,&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
This recovers the definition given above.&lt;br /&gt;
&lt;br /&gt;
一个可测量的函数 &amp;lt;math&amp;gt;X \colon A \to B &amp;lt;/math&amp;gt; 在一个概率空间中 &amp;lt;math&amp;gt;(A, \mathcal A, P)&amp;lt;/math&amp;gt; and 和一个可测量空间 &amp;lt;math&amp;gt;(B, \mathcal B) &amp;lt;/math&amp;gt; 被叫做离散随机变量。该图像是一个可数的集合。在这种情况下&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;的测量意味着单例集的原像是可测量的 i.e., &amp;lt;math&amp;gt;X^{-1}(\{b\}) \in \mathcal A&amp;lt;/math&amp;gt; 对于所有的&amp;lt;math&amp;gt;b \in B&amp;lt;/math&amp;gt;.&lt;br /&gt;
后者需要包括概率质量函数 &amp;lt;math&amp;gt;f_X \colon X(A) \to \mathbb R&amp;lt;/math&amp;gt; via &amp;lt;math&amp;gt; f_X(b):=P(X^{-1}(\{b\}))&amp;lt;/math&amp;gt;.  由于不相交集的原像不相交&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
这包含了上面所提到的定义&lt;br /&gt;
&lt;br /&gt;
===Cumulative distribution function===&lt;br /&gt;
&lt;br /&gt;
Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it &amp;quot;jumps&amp;quot; to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.&lt;br /&gt;
&lt;br /&gt;
与上述等效，可以将离散随机变量定义为其累积分布函数（cdf）仅因跳跃不连续性而增加的随机变量，也就是说，其cdf仅在“跳跃”至较高值时才增加，并且在那些跳跃点间是常数。但是请注意，cdf跳转的点可能会形成密集的实数集。发生跳变的点恰好是随机变量可能取的值。&lt;br /&gt;
&lt;br /&gt;
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===Delta-function representation 三角函数表示===&lt;br /&gt;
&lt;br /&gt;
Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions.  This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.&amp;lt;ref&amp;gt;{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此，离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数，该函数实质上统一了对连续分布和离散分布的处理。当处理涉及连续和离散部分的概率分布时，这特别有用。&lt;br /&gt;
&lt;br /&gt;
===Indicator-function representation 指标功能表示===&lt;br /&gt;
&lt;br /&gt;
For a discrete random variable ''X'', let ''u''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;, ''u''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
对于离散随机变量X，令u0，u1，...是它可以以非零概率获取的值。表示&lt;br /&gt;
:&amp;lt;math&amp;gt;\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这些是不相交的集合，对于这样的集合：&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此，X取u0，u1，...以外的任何值的概率为零，因此可以将X写入为&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
except on a set of probability zero, where &amp;lt;math&amp;gt;1_A&amp;lt;/math&amp;gt; is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.&lt;br /&gt;
除了概率为零的集合外，其中1_ {A}是A的指标函数。这可以用作离散随机变量的替代定义。&lt;br /&gt;
&lt;br /&gt;
==Continuous probability distribution 连续概率分布==&lt;br /&gt;
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{{See also|Probability density function}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.&amp;lt;ref&amp;gt;{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}&amp;lt;/ref&amp;gt; They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].&lt;br /&gt;
连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]。连续概率分布有很多示例：正态分布，均匀分布，卡方分布和其他分布。&lt;br /&gt;
&lt;br /&gt;
A random variable &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; has a continuous probability distribution if there is a function &amp;lt;math&amp;gt;f: \mathbb{R} \rightarrow [0, \infty)&amp;lt;/math&amp;gt; such that for each interval &amp;lt;math&amp;gt;I \subset \mathbb{R}&amp;lt;/math&amp;gt; the probability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; belonging to &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt; is given by the integral of &amp;lt;math&amp;gt;f&amp;lt;/math&amp;gt; over &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}&amp;lt;/ref&amp;gt; For example, if &amp;lt;math&amp;gt;I = [a, b]&amp;lt;/math&amp;gt; then we would have:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一个随机变量&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; 有一个连续的概率分布，如果这有一个函数&amp;lt;math&amp;gt;f: \mathbb{R} \rightarrow [0, \infty)&amp;lt;/math&amp;gt; 对于每一个区间&amp;lt;math&amp;gt;I \subset \mathbb{R}&amp;lt;/math&amp;gt;  &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;的概率 &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;是&amp;lt;math&amp;gt;f&amp;lt;/math&amp;gt; 在 &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;上的积分.&amp;lt;ref&amp;gt;Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}&amp;lt;/ref&amp;gt; 例如, 如果&amp;lt;math&amp;gt;I = [a, b]&amp;lt;/math&amp;gt; 可得到:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In particular, the probability for &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; to take any single value &amp;lt;math&amp;gt;a&amp;lt;/math&amp;gt; (that is &amp;lt;math&amp;gt;a \le X \le a&amp;lt;/math&amp;gt;) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;F(x) = \operatorname{P}\left[-\infty &amp;lt; X \le x\right] = \int_{-\infty}^x f(x) \, dx&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
特别是，X取任何单个值a（即a≤X≤a）的概率为零，因为上下限一致的积分始终等于零。满足上述条件的变量称为连续随机变量。其累积密度函数定义为&lt;br /&gt;
&lt;br /&gt;
which, by this definition, has the properties:&lt;br /&gt;
根据定义有以下一些性质&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is non-decreasing;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;0 \le F(x) \le 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;\lim_{x \rightarrow -\infty} F(x) = 0&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\lim_{x \rightarrow \infty} F(x) = 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;P(a \le X &amp;lt; b) = F(b) - F(a)&amp;lt;/math&amp;gt;; and&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is continuous (due to the [[Riemann integral]] properties).&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
It is also possible to think in the opposite direction, which allows more flexibility. Say &amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is a function that satisfies all but the last of the properties above, then &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; represents the cumulative density function for some random variable: a discrete random variable if &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; is a step function, and a continuous random variable otherwise.&amp;lt;ref&amp;gt;See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.&amp;lt;/ref&amp;gt; This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].&lt;br /&gt;
也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; is such an absolutely continuous random variable, then it has a [[probability density function]] &amp;lt;math&amp;gt;f(x)&amp;lt;/math&amp;gt;, and its probability of falling into a Lebesgue-measurable set &amp;lt;math&amp;gt;A \subset \mathbb{R}&amp;lt;/math&amp;gt; is:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; is the Lebesgue measure.&lt;br /&gt;
&lt;br /&gt;
对于实线的更多任意子集，通常有必要对上述定义进行概括。在这些情况下，连续概率分布定义为具有绝对连续的累积分布函数的概率分布。等效地，就Lebesgue测度而言，它是实数上的概率分布，它是绝对连续的。这样的分布可以用它们的概率密度函数表示。如果X是这样一个绝对连续的随机变量，则它具有概率密度函数f（x），并且落入Lebesgue可测量集合A⊂R的概率为：&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这里，mu是Lebesgue度量&lt;br /&gt;
&lt;br /&gt;
Note on terminology: some authors use the term &amp;quot;continuous distribution&amp;quot; to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;\mu\{x\}\,=\,0&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;\,x&amp;lt;/math&amp;gt;. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].&lt;br /&gt;
&lt;br /&gt;
关于术语的注释：一些作者使用术语“连续分布”来表示其累积分布函数是连续的而不是绝对连续的分布。这些分布是所有x的μ{x} = 0的μ分布。该定义包括上面定义的（绝对）连续分布，但也包括奇异分布，既不是绝对连续也不是离散的，也不是它们的混合。没有密度。 Cantor分布给出了一个示例。&lt;br /&gt;
&lt;br /&gt;
== [[Andrey Kolmogorov|Kolmogorov]] definition 柯尔莫哥洛夫的定义==&lt;br /&gt;
&lt;br /&gt;
{{Main|Probability space|Probability measure}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; from a [[probability space]] &amp;lt;math&amp;gt;(\Omega, \mathcal{F}, \mathbb{P})&amp;lt;/math&amp;gt; to a [[measurable space]] &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt;. Given that probabilities of events of the form &amp;lt;math&amp;gt;\{\omega\in\Omega\mid X(\omega)\in A\}&amp;lt;/math&amp;gt; satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] &amp;lt;math&amp;gt;X_*\mathbb{P}&amp;lt;/math&amp;gt; of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; , which is a [[probability measure]] on &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt; satisfying &amp;lt;math&amp;gt;X_*\mathbb{P} = \mathbb{P}X^{-1}&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
在概率论的度量理论形式化中，将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）。给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P，它是满足&amp;lt;math&amp;gt;X_*\mathbb{P} = \mathbb{P}X^{-1}&amp;lt;/math&amp;gt;的概率量度&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Random number generation 随机数生成==&lt;br /&gt;
{{Main|Pseudo-random number sampling}}&lt;br /&gt;
&lt;br /&gt;
Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Citation|last1=Dekking|first1=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
大多数算法基于伪随机数生成器，该伪随机数生成器生成在半开间隔[0,1）中均匀分布的数字X。然后，通过某种算法对这些随机变量X进行转换，以创建具有所需概率分布的新随机变量。利用这种统一的伪随机源，可以生成任何随机变量的实现。&lt;br /&gt;
&lt;br /&gt;
For example, suppose &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt; has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some &amp;lt;math&amp;gt;0 &amp;lt; p &amp;lt; 1&amp;lt;/math&amp;gt;, we define&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
so that&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\textrm{P}(X=1) = \textrm{P}(U&amp;lt;p) = p,&lt;br /&gt;
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
例如，假设U具有介于0和1之间的均匀分布。为某些对象构造一个随机的Bernoulli变量0 &amp;lt;p &amp;lt;1我们定义&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
因此&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\textrm{P}(X=1) = \textrm{P}(U&amp;lt;p) = p,&lt;br /&gt;
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
This random variable X has a Bernoulli distribution with parameter &amp;lt;math&amp;gt;p&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:0&amp;quot;/&amp;gt; Note that this is a transformation of discrete random variable.&lt;br /&gt;
该随机变量X具有参数的伯努利分布p。请注意，这是离散随机变量的变换。&lt;br /&gt;
&lt;br /&gt;
For a distribution function &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; of a continuous random variable, a continuous random variable must be constructed. &amp;lt;math&amp;gt;F^{inv}&amp;lt;/math&amp;gt;, an inverse function of &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt;, relates to the uniform variable &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{U\leq F(x)} = {F^{inv}(U)\leq x}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
F或连续随机变量的分布函数F，必须构造连续随机变量。 &amp;lt;math&amp;gt;F^{inv}&amp;lt;/math&amp;gt;，F的反函数，涉及均匀变量U：&lt;br /&gt;
&amp;lt;math&amp;gt;{U\leq F(x)} = {F^{inv}(U)\leq x}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For example, suppose a random variable that has an exponential distribution &amp;lt;math&amp;gt;F(x) = 1 - e^{-\lambda x}&amp;lt;/math&amp;gt; must be constructed.&lt;br /&gt;
例如，假设必须构造一个具有指数分布&amp;lt;math&amp;gt;F(x) = 1 - e^{-\lambda x}&amp;lt;/math&amp;gt; 的随机变量。&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
F(x) = u &amp;amp;\Leftrightarrow 1-e^{-\lambda x} = u \\ &amp;amp;\Leftrightarrow e^{-\lambda x } = 1-u \\&amp;amp;\Leftrightarrow -\lambda x = \ln(1-u) \\ &amp;amp;\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)&lt;br /&gt;
\end{align}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
所以&amp;lt;math&amp;gt;F^{inv}(u) = \frac{-1}{\lambda}\ln(1-u)&amp;lt;/math&amp;gt; 并且如果 &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt; 有一个&amp;lt;math&amp;gt;U(0,1)&amp;lt;/math&amp;gt; 分布, 然后随机变量 &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; 被定义为 &amp;lt;math&amp;gt;X = F^{inv}(U) = \frac{-1}{\lambda} \ln(1-U)&amp;lt;/math&amp;gt;. 这里有一个指数分布 &amp;lt;math&amp;gt;\lambda&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.&lt;br /&gt;
统计模拟（蒙特卡洛方法）中经常遇到的一个问题是生成以给定方式分布的伪随机数。&lt;br /&gt;
&lt;br /&gt;
== Common probability distributions and their applications 共同概率分布及其应用==&lt;br /&gt;
{{Main list|List of probability distributions}}&lt;br /&gt;
&lt;br /&gt;
The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.&lt;br /&gt;
&lt;br /&gt;
概率分布的概念及其描述的随机变量是概率论和统计学科学的数学基础。人口中几乎可以测量的任何值都存在价差或可变性（例如人的身高，金属的耐用性，销售增长，交通流量等）;几乎所有测量均存在一定的固有误差；在物理学中，从气体的动力学特性到基本粒子的量子力学描述，很多过程都用概率论来描述。由于这些以及许多其他原因，简单数字通常不足以描述数量，而概率分布通常更合适。&lt;br /&gt;
&lt;br /&gt;
The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see [[list of probability distributions]], which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)&lt;br /&gt;
以下是一些最常见的概率分布列表，按与之相关的过程类型进行分组。有关更完整的列表，请参见概率分布列表，该列表按要考虑的结果的性质（离散，连续，多元等）进行分组。&lt;br /&gt;
&lt;br /&gt;
All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a [[mixture distribution]].&lt;br /&gt;
下面所有的单变量分布都达到了峰值。也就是说，假设值聚集在单个点周围。实际上，实际观察到的量可能会聚集在多个值附近。可以使用混合物分布对此类数量进行建模。&lt;br /&gt;
&lt;br /&gt;
=== Linear growth (e.g. errors, offsets) 线性增长===&lt;br /&gt;
&lt;br /&gt;
* [[Normal distribution]] (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution&lt;br /&gt;
正态分布（高斯分布），对于单个这样的数量；最常用的连续分布&lt;br /&gt;
&lt;br /&gt;
=== Exponential growth (e.g. prices, incomes, populations) 指数增长===&lt;br /&gt;
&lt;br /&gt;
* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed&lt;br /&gt;
对数正态分布，对于单个此类数量的对数正态分布&lt;br /&gt;
&lt;br /&gt;
* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution&lt;br /&gt;
帕累托分布，对于单个这样的数量，其对数呈指数分布；原型幂律分布&lt;br /&gt;
&lt;br /&gt;
=== Uniformly distributed quantities 数量均匀分布===&lt;br /&gt;
&lt;br /&gt;
* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)&lt;br /&gt;
离散均匀分布，用于有限的一组值（例如，公平死亡的结果）&lt;br /&gt;
&lt;br /&gt;
* [[Continuous uniform distribution]], for continuously distributed values&lt;br /&gt;
连续均匀分布，用于连续分布的值&lt;br /&gt;
&lt;br /&gt;
=== Bernoulli trials (yes/no events, with a given probability) 伯努利试验（是/否事件，具有给定的概率）===&lt;br /&gt;
&lt;br /&gt;
* Basic distributions 基本分布:&lt;br /&gt;
** [[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)&lt;br /&gt;
伯努利分布，用于单个伯努利试验的结果（例如成功/失败，是/否）&lt;br /&gt;
&lt;br /&gt;
** [[Binomial distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed total number of [[Independent (statistics)|independent]] occurrences&lt;br /&gt;
二项式分布，对于给定固定总数的独立“出现次数”（例如，成功，赞成票等）的数量&lt;br /&gt;
&lt;br /&gt;
** [[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs&lt;br /&gt;
负二项分布，用于二项式观察，但是关注的数量是在给定成功次数之前发生的失败次数&lt;br /&gt;
&lt;br /&gt;
** [[Geometric distribution]], for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the [[negative binomial distribution]]&lt;br /&gt;
几何分布，用于二项式观测，但是关注的数量是首次成功之前的失败数量；负二项式分布的特殊情况&lt;br /&gt;
&lt;br /&gt;
* Related to sampling schemes over a finite population 与有限人口抽样方案有关:&lt;br /&gt;
&lt;br /&gt;
** [[Hypergeometric distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using [[sampling without replacement]]&lt;br /&gt;
超几何分布，对于“肯定出现”的数量（例如成功，赞成票等），给定了一定的总出现数量，使用采样而无需替换&lt;br /&gt;
&lt;br /&gt;
** [[Beta-binomial distribution]], for the number of &amp;quot;positive occurrences&amp;quot; (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a [[Pólya urn model]] (in some sense, the &amp;quot;opposite&amp;quot; of [[sampling without replacement]])&lt;br /&gt;
贝塔二项式分布，对于给定的总发生次数为“阳性”的次数（例如，成功，赞成票等），使用Pólyaurn模型进行采样（在某种意义上，为“替代”而不进行替换） ）&lt;br /&gt;
&lt;br /&gt;
=== Categorical outcomes (events with ''K'' possible outcomes, with a given probability for each outcome) 分类结果（具有K个可能结果的事件，每个结果具有给定的概率）===&lt;br /&gt;
&lt;br /&gt;
* [[Categorical distribution]], for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the [[Bernoulli distribution]]&lt;br /&gt;
针对单个分类结果的分类分布（例如，调查中的是/否/也许）；伯努利分布的一般化&lt;br /&gt;
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* [[Multinomial distribution]], for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the [[binomial distribution]]&lt;br /&gt;
给定总结果的固定数量，针对每种类别结果的数量的多项式分布；二项式分布的一般化&lt;br /&gt;
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* [[Multivariate hypergeometric distribution]], similar to the [[multinomial distribution]], but using [[sampling without replacement]]; a generalization of the [[hypergeometric distribution]]&lt;br /&gt;
多元超几何分布，类似于多项式分布，但使用采样而不进行替换；超几何分布的一般化&lt;br /&gt;
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=== Poisson process (events that occur independently with a given rate) 泊松过程（以给定速率独立发生的事件）===&lt;br /&gt;
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* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time&lt;br /&gt;
泊松分布，用于给定时间段内泊松型事件的发生次数&lt;br /&gt;
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* [[Exponential distribution]], for the time before the next Poisson-type event occurs&lt;br /&gt;
指数分布，在下一次泊松型事件发生之前的时间&lt;br /&gt;
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* [[Gamma distribution]], for the time before the next k Poisson-type events occur&lt;br /&gt;
伽马分布，在接下来的k个泊松型事件发生之前的时间&lt;br /&gt;
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=== Absolute values of vectors with normally distributed components 具有正态分布分量的向量的绝对值===&lt;br /&gt;
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* [[Rayleigh distribution]], for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.&lt;br /&gt;
瑞利分布，用于具有高斯分布正交分量的矢量幅度分布。在具有高斯实部和虚部的RF信号中发现瑞利分布。&lt;br /&gt;
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* [[Rice distribution]], a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in [[Rician fading]] of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.&lt;br /&gt;
莱斯分布，是在背景信号分量稳定的情况下瑞利分布的概括。由于多径传播而在无线电信号的Rician衰落中发现，并且在非零NMR信号中出现噪声破坏的MR图像中也发现了这种情况。&lt;br /&gt;
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=== Normally distributed quantities operated with sum of squares (for hypothesis testing) 以平方和运算的正态分布量（用于假设检验）===&lt;br /&gt;
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* [[Chi-squared distribution]], the distribution of a sum of squared [[standard normal]] variables; useful e.g. for inference regarding the [[sample variance]] of normally distributed samples (see [[chi-squared test]])&lt;br /&gt;
卡方分布，标准正态变量平方和的分布；有用的关于正态分布样本的样本方差的推论（请参见卡方检验）&lt;br /&gt;
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* [[Student's t distribution]], the distribution of the ratio of a [[standard normal]] variable and the square root of a scaled [[Chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])&lt;br /&gt;
学生t分布，标准正态变量与缩放的卡方变量的平方根之比的分布；有助于推断方差未知的正态分布样本的平均值（请参阅学生的t检验）&lt;br /&gt;
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* [[F-distribution]], the distribution of the ratio of two scaled [[Chi squared distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])&lt;br /&gt;
F-分布，两个比例卡方变量的比例分布；有用的用于涉及比较方差或涉及R平方（相关系数平方）的推论&lt;br /&gt;
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=== As a conjugate prior distributions in Bayesian inference 作为贝叶斯推断中的共轭先验分布===&lt;br /&gt;
{{Main|Conjugate prior}}&lt;br /&gt;
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* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]&lt;br /&gt;
Beta分布，具有单个概率（0到1之间的实数）；与伯努利分布和二项式分布共轭&lt;br /&gt;
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* [[Gamma distribution]], for a non-negative scaling parameter; conjugate to the rate parameter of a [[Poisson distribution]] or [[exponential distribution]], the [[Precision (statistics)|precision]] (inverse [[variance]]) of a [[normal distribution]], etc.&lt;br /&gt;
伽玛分布，用于非负比例缩放参数；与泊松分布或指数分布的速率参数，正态分布的精度（逆方差）等共轭。&lt;br /&gt;
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* [[Dirichlet distribution]], for a vector of probabilities that must sum to 1; conjugate to the [[categorical distribution]] and [[multinomial distribution]]; generalization of the [[beta distribution]]&lt;br /&gt;
Dirichlet分布，对于必须为1的概率向量；与分类分布和多项式分布共轭； beta分布的一般化&lt;br /&gt;
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*[[Wishart distribution]], for a symmetric [[non-negative definite]] matrix; conjugate to the inverse of the [[covariance matrix]] of a [[multivariate normal distribution]]; generalization of the [[gamma distribution]]&amp;lt;ref&amp;gt;{{Cite book|title=Pattern recognition and machine learning|last=Bishop, Christopher M.|date=2006|publisher=Springer|isbn=0-387-31073-8|location=New York|oclc=71008143}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
Wishart分布，用于对称非负定矩阵；与多元正态分布的协方差矩阵的逆共轭；伽玛分布的一般化&lt;br /&gt;
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=== Some specialized applications of probability distributions 概率分布的一些特殊应用===&lt;br /&gt;
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* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.&lt;br /&gt;
在自然语言处理中使用的高速缓存语言模型和其他统计语言模型通过概率分布来为特定单词和单词序列的出现分配概率。&lt;br /&gt;
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* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by &amp;lt;math&amp;gt;P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 &amp;lt;/math&amp;gt;, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.&amp;lt;ref&amp;gt;{{Cite book|title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|publisher=|others=Thoman, John W., Jr., 1960-|year=|isbn=978-1-68015-835-9|location=[Mill Valley, California]|pages=403–406|oclc=927509011}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
在量子力学中，在给定点处找到粒子的概率密度与该点处粒子波函数大小的平方成正比（请参阅博恩法则）。因此，粒子位置的概率分布函数描述为&amp;lt;math&amp;gt;P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 &amp;lt;/math&amp;gt;，粒子位置的概率x在第一个维度中的间隔为a≤x≤b，在第三个维度中的间隔类似。这是量子力学的关键原理。&lt;br /&gt;
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* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution.&amp;lt;ref&amp;gt;{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.|last2=Chen|first2=Z.|last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review|doi=10.1109/drpt.2008.4523658|s2cid=18669309}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
潮流研究中的概率潮流解释了作为概率分布的输入变量的不确定性，并以概率分布的形式提供了潮流计算。&lt;br /&gt;
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* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.&amp;lt;ref&amp;gt;{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity, Rajib|isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
根据先前的频率分布（例如热带气旋，冰雹，事件之间的时间等）预测自然现象的发生。&lt;br /&gt;
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==See also 另请参见==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Copula (statistics)]]&lt;br /&gt;
Copula（统计数据）&lt;br /&gt;
* [[Empirical probability]]&lt;br /&gt;
经验概率&lt;br /&gt;
* [[Histogram]]&lt;br /&gt;
直方图&lt;br /&gt;
* [[Likelihood function]]&lt;br /&gt;
似然函数&lt;br /&gt;
* [[Kirkwood approximation]]&lt;br /&gt;
柯克伍德近似&lt;br /&gt;
* [[Moment-generating function]]&lt;br /&gt;
瞬时产生功能&lt;br /&gt;
* [[Riemann–Stieltjes integral#Application to probability theory|Riemann–Stieltjes integral application to probability theory]]&lt;br /&gt;
Riemann–Stieltjes积分＃在概率论中的应用&lt;br /&gt;
*[[Pairwise independence]]&lt;br /&gt;
成对独立&lt;br /&gt;
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=== Lists 清单===&lt;br /&gt;
&lt;br /&gt;
* [[List of probability distributions]]&lt;br /&gt;
概率分布的清单&lt;br /&gt;
* [[List of statistical topics]]&lt;br /&gt;
统计学话题的清淡&lt;br /&gt;
=== Probability distributions 概率分布 ===&lt;br /&gt;
&lt;br /&gt;
* [[Conditional probability distribution]]&lt;br /&gt;
条件概率分布&lt;br /&gt;
* [[Joint probability distribution]]&lt;br /&gt;
联合概率分布&lt;br /&gt;
* [[Quasiprobability distribution]]&lt;br /&gt;
拟概率分布&lt;br /&gt;
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== References ==&lt;br /&gt;
=== Citations ===&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
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=== Sources ===&lt;br /&gt;
{{refbegin|}}&lt;br /&gt;
* {{cite journal|doi=10.1016/j.ejmp.2014.05.002|pmid=25059432|title=Data distributions in magnetic resonance images: A review|url=|journal=[[Physica Medica]]|volume=30|issue=7|pages=725–741|year=2014|last1=den Dekker|first1=A. J.|last2=Sijbers|first2=J.}}&lt;br /&gt;
* {{cite book|last=Vapnik|first=Vladimir Naumovich|year=1998|title=Statistical Learning Theory|publisher=John Wiley and Sons}}&lt;br /&gt;
{{refend}}&lt;br /&gt;
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==External links==&lt;br /&gt;
{{commons|Probability distribution|Probability distribution}}&lt;br /&gt;
*{{springer|title=Probability distribution|id=p/p074900}}&lt;br /&gt;
*[http://threeplusone.com/FieldGuide.pdf Field Guide to Continuous Probability Distributions],  Gavin E. Crooks.&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
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		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{short description|Mathematical function that describes the probability of occurrence of different possible outcomes in an experiment}}&lt;br /&gt;
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{{other uses|Distribution (disambiguation){{!}}Distribution}}&lt;br /&gt;
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{{Off topic|date=August 2020|Random variable|talksectionheading}}&lt;br /&gt;
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In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].&amp;lt;ref name=&amp;quot;:02&amp;quot;&amp;gt;{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}&amp;lt;/ref&amp;gt; It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).&lt;br /&gt;
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在概率论和统计学中，概率分布是一个给出一个实验不同可能结果出现的概率的数学函数。它是根据样本空间和事件概率(样本空间的子集)对随机现象的数学描述。&lt;br /&gt;
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For instance, if the [[random variable]] {{mvar|X}} is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of {{mvar|X}} would take the value 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} heads}}, and 0.5 for {{math|''X''&amp;amp;nbsp;{{=}} tails}} (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a [[Survey methodology|survey]], etc.&amp;lt;ref name=&amp;quot;ross&amp;quot;&amp;gt;{{cite book|first=Sheldon M.|last=Ross|title=A first course in probability|publisher=Pearson|year=2010}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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For instance, if the random variable X is used to denote the outcome of a coin toss (&amp;quot;the experiment&amp;quot;), then the probability distribution of X would take the value 0.5 for X=heads, and 0.5 for X=tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.&lt;br /&gt;
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例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。随机现象的例子包括未来某一天的天气状况、一个人的身高、学校中男生的比例、调查结果等等。&lt;br /&gt;
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A probability distribution is a mathematical function that has a [[sample space]] as its '''input''', and gives a '''probability''' as its output. The sample space is the [[Set (mathematics)|set]] of all possible [[Outcome (probability)|outcomes]] of a random phenomenon being observed; it may be the set of [[real numbers]] or a set of [[vector (mathematics)|vectors]], or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be {{math|{heads, tails} }}.&lt;br /&gt;
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A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values.  For example, the sample space of a coin flip would be .&lt;br /&gt;
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概率分布是一个数学函数，它的输入是一个样本空间，输出是一个概率。样本空间是所观察到的随机现象的所有可能结果的集合; 它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。&lt;br /&gt;
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Probability distributions are generally divided into two classes. A '''discrete probability distribution''' is applicable to the scenarios where the set of possible outcomes is [[discrete probability distribution|discrete]] (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the [[probability mass function]]. On the other hand, '''continuous probability distributions''' are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a [[probability density function]].&amp;lt;ref name=&amp;quot;ross&amp;quot; /&amp;gt;&amp;lt;ref&amp;gt;{{cite book|author1=DeGroot, Morris H.|author2=Schervish, Mark J.|title=Probability and Statistics|publisher=Addison-Wesley|year=2002}}&amp;lt;/ref&amp;gt; The [[normal distribution]] is a commonly encountered continuous probability distribution. More complex experiments, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.&lt;br /&gt;
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Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function. The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.&lt;br /&gt;
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概率分布一般分为两类。离散概率分布适用于一组可能的结果是离散的情况，如抛硬币或掷骰子。这里的概率被编码为结果概率的离散列表，称为概率质量函数。另一方面，连续概率分布适用于一组可以在一个连续的范围内取值的结果的情况(例如:实数)，例如某一天的温度。在这种情况下，概率通常由概率密度函数描述。正态分布是一种常见的连续概率分布。更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called [[Univariate distribution|univariate]], while a distribution whose sample space is a [[vector space]] of dimension 2 or more is called [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a [[joint probability distribution]]) gives the probabilities of a [[random vector]] – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the [[binomial distribution]], the [[hypergeometric distribution]], and the [[normal distribution]]. The [[multivariate normal distribution]] is a commonly encountered multivariate distribution.&lt;br /&gt;
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A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.&lt;br /&gt;
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一个一维的样本空间(例如实数、标签列表、有序标签或二进制)的概率分布被称为单变量，而样本空间为二维或更多向量空间的分布被称为多变量。单变量分布给出了单个随机变量取不同替代值的概率; 联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。重要的和常见的单变量概率分布包括二项分布、超几何分布和正态分布。多变量正态分布是一种常见的联合分布。&lt;br /&gt;
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==Introduction 简介==&lt;br /&gt;
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[[File:Dice Distribution (bar).svg|thumb|250px|right|&lt;br /&gt;
图1:The [[probability mass function]] (pmf) ''p''(''S'') specifies the probability distribution for the sum ''S'' of counts from two [[dice]].  For example, the figure shows that ''p''(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as ''P''(''S'' &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.&lt;br /&gt;
概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布。例如，图中显示 p (11) = 2/36 = 1/18。Pmf 允许计算事件的概率，如 p (s &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6，以及分布中的所有其他概率。&lt;br /&gt;
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The [[probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice.  For example, the figure shows that p(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as P(S &amp;gt; 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]&lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between '''discrete''' and '''continuous''' [[random variable]]s. In the discrete case, it is sufficient to specify a [[probability mass function]] &amp;lt;math&amp;gt;p&amp;lt;/math&amp;gt; assigning a probability to each possible outcome: for example, when throwing a fair [[Dice|die]], each of the six values 1 to 6 has the probability 1/6. The probability of an [[Event (probability theory)|event]] is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event &amp;quot;the dice rolls an even value&amp;quot; is &lt;br /&gt;
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为了定义最简单的概率分布，有必要区分离散和连续的随机变量。在离散情况下，指定一个概率质量函数 p 就足够了，它为每个可能的结果赋予一个概率: 例如，当投掷一个骰子时，6个值中的每一个的概率为1/6。然后将事件的概率定义为满足事件的结果的概率之和; 例如，事件”骰子掷出偶数值”的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2.&amp;lt;/math&amp;gt;&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs ''exactly'' 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500&amp;amp;nbsp;g is zero, because the probability of measuring exactly 500&amp;amp;nbsp;g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a &amp;quot;500&amp;amp;nbsp;g&amp;quot; package containing between 490&amp;amp;nbsp;g and 510&amp;amp;nbsp;g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.&lt;br /&gt;
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相比之下，当一个随机变量从一个连续体中取值时，那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率。例如，一个给定的物体重量正好是500克的概率为零，因为随着我们测量仪器精度的提高，正好测量500克的概率趋向于零。然而，在质量控制方面，人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98% ，而这一要求对测量仪器的准确性不太敏感。&lt;br /&gt;
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Continuous probability distributions can be described in several ways. The [[probability density function]] describes the [[infinitesimal]] probability of any given value, and the probability that the outcome lies in a given interval can be computed by [[Integration (mathematics)|integrating]] the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral. &lt;br /&gt;
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连续概率分布可以用几种方法来描述。概率密度函数描述了任意给定值的无穷小概率，并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。可能值位于某一固定区间的概率可以与和收敛于积分的方式有关，因此，连续概率是基于积分的定义。&lt;br /&gt;
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[[File:Combined Cumulative Distribution Graphs.png|thumb|455x455px|&lt;br /&gt;
图2:On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve.&lt;br /&gt;
左边是概率密度函数。右边是累积分布函数，它是概率密度曲线下面的区域。]]&lt;br /&gt;
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The [[cumulative distribution function]] describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the [[antiderivative]] of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the [[probability density function]] from minus infinity &amp;lt;math&amp;gt;\infty&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; as described by the picture to the right.&amp;lt;ref&amp;gt;{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=Springer|others=Dekking, Michel, 1946-|isbn=978-1-85233-896-1|location=London|oclc=262680588}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity \infty to x as described by the picture to the right.&lt;br /&gt;
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累积分布函数指标描述了随机变量不大于给定值的概率; 结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算。累积分布函数是概率密度函数的反导函数，前提是后者存在。正如右边图片所描述的那样，累积分布函数是从负无穷到 x 的概率密度函数下面的区域。&lt;br /&gt;
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[[File:Standard deviation diagram.svg|right|thumb|250px|&lt;br /&gt;
图3:The [[probability density function]] (pdf) of the [[normal distribution]], also called Gaussian or &amp;quot;bell curve&amp;quot;, the most important continuous random distribution.  As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.&lt;br /&gt;
正态分布的[概率密度函数(pdf) ，也称为高斯或钟形曲线，是最重要的连续随机分布。如图所示，值间隔的概率对应于曲线下面积。]]&lt;br /&gt;
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== Terminology 术语==&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.&amp;lt;ref name=&amp;quot;:02&amp;quot; /&amp;gt;&lt;br /&gt;
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Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X &amp;gt; a, X &amp;lt; b or a union thereof.&lt;br /&gt;
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一些关键的，广泛用于以概率分布为主题的文献中的概念和术语，列出如下。&lt;br /&gt;
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=== Functions for discrete variables 离散变量的函数===&lt;br /&gt;
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Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.&lt;br /&gt;
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用于统计建模的著名离散概率分布包括泊松分佈、伯努利分布、二项分布、几何分布和负二项分布。&lt;br /&gt;
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*'''Probability function''': describes the probability distribution of a discrete random variable.&lt;br /&gt;
概率函数：描述离散随机变量的概率分布。&lt;br /&gt;
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*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.&lt;br /&gt;
概率质量函数（pmf）：给出离散随机变量等于某个值的概率的函数。&lt;br /&gt;
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*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.&lt;br /&gt;
频率分布：显示样本中各种结果的频率的表格。&lt;br /&gt;
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For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
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对于一个离散的随机变量 x，设 u0，u1，... 是它在非零概率情况下可以取的值。表示&lt;br /&gt;
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*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.&lt;br /&gt;
相对频率分布：一种频率分布，其中每个值均已被样本中的多个结果（即样本大小）除（归一化）。&lt;br /&gt;
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*'''Discrete probability distribution function''': general term to indicate the way the total probability of 1 is distributed over '''all''' various possible outcomes (i.e. over entire population) for discrete random variable.&lt;br /&gt;
离散概率分布函数：通用术语，表示总概率1在离散随机变量的所有各种可能结果（即整个人群）中的分布方式。&lt;br /&gt;
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\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots&lt;br /&gt;
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Omega _ i = x ^ {-1}(u _ i) = { Omega: x (Omega) = u _ i } ，i = 0,1,2，点&lt;br /&gt;
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*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for a discrete random variable.&lt;br /&gt;
累积分布函数：该函数评估离散随机变量X取小于或等于x的值的概率。&lt;br /&gt;
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*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.&lt;br /&gt;
分类分布：适用于具有有限值集的离散随机变量。&lt;br /&gt;
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=== Functions for continuous variables 连续变量函数===&lt;br /&gt;
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* '''[[Probability density function]] (pdf):''' function whose value at any given sample (or point) in the [[sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.&lt;br /&gt;
概率密度函数（pdf）：可以将其在样本空间中任意给定样本（或点）上的值（随机变量可能获得的一组值）的值解释为提供随机变量值将具有的相对可能性的函数等于那个样本。&lt;br /&gt;
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It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as&lt;br /&gt;
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由此可见，除了 u0、 u1、 ... 之外，x 取任何值的概率为零，因此可以将 x 写为&lt;br /&gt;
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* '''Continuous probability distribution function''': most often reserved for continuous random variables.&lt;br /&gt;
连续概率分布函数：最常保留的连续随机变量。&lt;br /&gt;
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* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; will take a value less than or equal to &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; for continuous variable.&lt;br /&gt;
累积分布函数：评估连续变量X取小于或等于x的值的概率的函数。&lt;br /&gt;
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===Basic terms 基本术语===&lt;br /&gt;
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* [[Mode (statistics)|'''Mode''']]: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.&lt;br /&gt;
模式：对于离散随机变量，该值具有最高概率；对于连续随机变量，是概率密度函数具有局部峰值的位置。&lt;br /&gt;
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* [[Support (mathematics)|'''Support''']]: set of values that can be assumed, with non-zero probability, by the random variable.&lt;br /&gt;
支持：可以由随机变量以非零概率假定的一组值。对于随机变量X，有时表示为R_ {X}&lt;br /&gt;
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* '''Tail''':&amp;lt;ref name='tail'&amp;gt;More information and examples can be found in the articles [[Heavy-tailed distribution]], [[Long-tailed distribution]], [[fat-tailed distribution]]&amp;lt;/ref&amp;gt; the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form &amp;lt;math&amp;gt;X &amp;gt; a&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;X &amp;lt; b&amp;lt;/math&amp;gt; or a union thereof.&lt;br /&gt;
尾巴：如果pmf或pdf相对较低，则靠近随机变量边界的区域。通常形式为X&amp;gt; a，X &amp;lt;b或它们的并集。&lt;br /&gt;
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*'''Head''':&amp;lt;ref name='tail' /&amp;gt; the region where the pmf or pdf is relatively high. Usually has the form &amp;lt;math&amp;gt;a &amp;lt; X &amp;lt; b&amp;lt;/math&amp;gt;.&lt;br /&gt;
头部：pmf或pdf较高的区域。通常具有a &amp;lt;X &amp;lt;b的形式&lt;br /&gt;
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* '''[[Expected value]]''' or '''mean''': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.&lt;br /&gt;
期望值或均值：可能值的加权平均值，以其概率作为权重；或其连续类似物。&lt;br /&gt;
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A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a cumulative density function that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.&lt;br /&gt;
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一个连续的概率分布是一个支持不可数集的概率分布，比如实数行中的一个区间。它们是唯一拥有属性的一个累积密度函数，可以用来计算支持的每个子集的概率。有许多连续概率分布的例子: 正态、均匀、卡方等等。&lt;br /&gt;
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*'''[[Median]]''': the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.&lt;br /&gt;
中位数：这样的值，即一组值小于中位数，而该组大于中位数，每一个的概率不大于二分之一。&lt;br /&gt;
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*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.&lt;br /&gt;
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。&lt;br /&gt;
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* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.&lt;br /&gt;
标准偏差：方差的平方根，因此是色散的另一种度量。&lt;br /&gt;
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* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.&lt;br /&gt;
对称性：某些分布的一种属性，其中特定值左侧（通常是中位数）的分布部分是其右侧部分的镜像。&lt;br /&gt;
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*'''[[Skewness]]''': a measure of the extent to which a pmf or pdf &amp;quot;leans&amp;quot; to one side of its mean. The third [[standardized moment]] of the distribution.&lt;br /&gt;
偏度：衡量pmf或pdf在其均值的一侧“倾斜”的程度。分布的第三个标准化时刻。&lt;br /&gt;
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*'''[[Kurtosis]]''': a measure of the &amp;quot;fatness&amp;quot; of the tails of a pmf or pdf. The fourth standardized moment of the distribution.&lt;br /&gt;
峰度：pmf或pdf尾部“脂肪”的量度。分布的第四个标准化时刻。&lt;br /&gt;
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==Discrete probability distribution 离散概率分布==&lt;br /&gt;
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[[File:Discrete probability distrib.svg|right|thumb|&lt;br /&gt;
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]&lt;br /&gt;
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图5:The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...离散概率分布]]&lt;br /&gt;
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[[File:Normal probability distribution.svg|right|thumb|&lt;br /&gt;
图6:... of a continuous probability distribution, ...连续概率分布]]&lt;br /&gt;
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[[File:Mixed probability distribution.svg|right|thumb|&lt;br /&gt;
图7:... of a distribution which has both a continuous part and a discrete part.既有连续部分又有离散部分]]&lt;br /&gt;
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A '''discrete probability distribution''' is a probability distribution that can take on a countable number of values.&amp;lt;ref&amp;gt;{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer|isbn=9780387878591|location=New York|pages=51|oclc=710149819}}&amp;lt;/ref&amp;gt; For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if &amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.&lt;br /&gt;
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离散概率分布是可以具有可数数量的值的概率分布。在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1。例如，如果&amp;lt;math&amp;gt;\operatorname{P}(X=n) = \tfrac{1}{2^n}&amp;lt;/math&amp;gt; for ''n'' = 1, 2，概率之和为1/2 + 1/4 + 1/8 + ... = 1。&lt;br /&gt;
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Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt; Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.&lt;br /&gt;
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。&lt;br /&gt;
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It is also possible to think in the opposite direction, which allows more flexibility. Say F(x) is a function that satisfies all but the last of the properties above, then F represents the cumulative density function for some random variable: a discrete random variable if F is a step function, and a continuous random variable otherwise. This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the Cantor distribution.&lt;br /&gt;
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也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
When a [[Sample (statistics)|sample]] (a set of observations) is drawn from a larger population, the sample points have an [[empirical distribution function|empirical distribution]] that is discrete and that provides information about the population distribution.&lt;br /&gt;
当从更大的总体中抽取一个样本（一组观察值）时，这些采样点的经验分布是离散的，并且提供了有关总体分布的信息。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Measure theoretic formulation 测量理论公式===&lt;br /&gt;
&lt;br /&gt;
A [[measurable function]] &amp;lt;math&amp;gt;X \colon A \to B &amp;lt;/math&amp;gt; between a [[probability space]] &amp;lt;math&amp;gt;(A, \mathcal A, P)&amp;lt;/math&amp;gt; and a [[measurable space]] &amp;lt;math&amp;gt;(B, \mathcal B) &amp;lt;/math&amp;gt; is called a '''discrete random variable''' provided that its image is a countable set. In this case measurability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; means that the pre-images of singleton sets are measurable, i.e., &amp;lt;math&amp;gt;X^{-1}(\{b\}) \in \mathcal A&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;b \in B&amp;lt;/math&amp;gt;.&lt;br /&gt;
The latter requirement induces a [[probability mass function]] &amp;lt;math&amp;gt;f_X \colon X(A) \to \mathbb R&amp;lt;/math&amp;gt; via &amp;lt;math&amp;gt; f_X(b):=P(X^{-1}(\{b\}))&amp;lt;/math&amp;gt;.  Since the pre-images of disjoint sets&lt;br /&gt;
are disjoint,&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
This recovers the definition given above.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Cumulative distribution function===&lt;br /&gt;
&lt;br /&gt;
Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it &amp;quot;jumps&amp;quot; to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.&lt;br /&gt;
&lt;br /&gt;
In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function X from a probability space (\Omega, \mathcal{F}, \mathbb{P}) to a measurable space (\mathcal{X},\mathcal{A}). Given that probabilities of events of the form \{\omega\in\Omega\mid X(\omega)\in A\} satisfy Kolmogorov's probability axioms, the probability distribution of X is the pushforward measure X_*\mathbb{P} of X , which is a probability measure on (\mathcal{X},\mathcal{A}) satisfying X_*\mathbb{P} = \mathbb{P}X^{-1}.&lt;br /&gt;
&lt;br /&gt;
在概率论的度量理论形式化中，一个随机变量被定义为一个可测函数 x，从一个概率空间(Omega，mathcal { f } ，mathbb { p })到一个可测空间(mathcal { x } ，mathcal { a })。考虑到 a }中 Omega 中 x (Omega)中 ω 的事件概率满足 Kolmogorov 的概率公理，x 的概率分布是 x 的 x * mathbb { p }的前推测度，它是关于(cal { x } ，mathcal { a })满足 x * mathbb { p } = mathbb { p } x ^ {-1}的机率量测。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Delta-function representation===&lt;br /&gt;
&lt;br /&gt;
Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions.  This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.&amp;lt;ref&amp;gt;{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Indicator-function representation===&lt;br /&gt;
&lt;br /&gt;
Most algorithms are based on a pseudorandom number generator that produces numbers X that are uniformly distributed in the half-open interval [0,1). These random variates X are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.&lt;br /&gt;
&lt;br /&gt;
大多数算法都是基于一个伪随机数生成器，它产生的数字 x 在半开区间内均匀分布。这些随机变量 x 然后通过一些算法转换，创建一个新的具有所需概率分布的随机变量。有了这种均匀的伪随机性，任何随机变量都可以被实现。&lt;br /&gt;
&lt;br /&gt;
For a discrete random variable ''X'', let ''u''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;, ''u''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ... be the values it can take with non-zero probability. Denote&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
For example, suppose U has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some 0 &amp;lt; p &amp;lt; 1, we define&lt;br /&gt;
&lt;br /&gt;
例如，假设 u 在0和1之间有一个均匀分布。为了构造一个0 &amp;lt; p &amp;lt; 1的随机贝努利变量，我们定义了&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&lt;br /&gt;
{ displaystyle x = { begin { cases }1，&amp;amp; { mbox { if } u &amp;lt; p 0，&amp;amp; { mbox { if } u geq p end { cases }}&lt;br /&gt;
&lt;br /&gt;
These are [[disjoint set]]s, and for such sets&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
so that&lt;br /&gt;
&lt;br /&gt;
所以&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
It follows that the probability that ''X'' takes any value except for ''u''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;, ''u''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ... is zero, and thus one can write ''X'' as&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\textrm{P}(X=1) = \textrm{P}(U&amp;lt;p) = p,&lt;br /&gt;
&lt;br /&gt;
&amp;lt; math &amp;gt; textrm { p }(x = 1) = textrm { p }(u &amp;lt; p) = p,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
文本{ p }(x = 0) = 文本{ p }(u geq p) = 1-p. &amp;lt;/math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
This random variable X has a Bernoulli distribution with parameter p.&lt;br /&gt;
&lt;br /&gt;
这个随机变量 x 有一个带参数 p 的伯努利分布。&lt;br /&gt;
&lt;br /&gt;
except on a set of probability zero, where &amp;lt;math&amp;gt;1_A&amp;lt;/math&amp;gt; is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Continuous probability distribution==&lt;br /&gt;
&lt;br /&gt;
{{See also|Probability density function}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.&amp;lt;ref&amp;gt;{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}&amp;lt;/ref&amp;gt; They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A random variable &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; has a continuous probability distribution if there is a function &amp;lt;math&amp;gt;f: \mathbb{R} \rightarrow [0, \infty)&amp;lt;/math&amp;gt; such that for each interval &amp;lt;math&amp;gt;I \subset \mathbb{R}&amp;lt;/math&amp;gt; the probability of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; belonging to &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt; is given by the integral of &amp;lt;math&amp;gt;f&amp;lt;/math&amp;gt; over &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}&amp;lt;/ref&amp;gt; For example, if &amp;lt;math&amp;gt;I = [a, b]&amp;lt;/math&amp;gt; then we would have:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In particular, the probability for &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; to take any single value &amp;lt;math&amp;gt;a&amp;lt;/math&amp;gt; (that is &amp;lt;math&amp;gt;a \le X \le a&amp;lt;/math&amp;gt;) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
F(x) = \operatorname{P}\left[-\infty &amp;lt; X \le x\right] = \int_{-\infty}^x f(x) \, dx&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
which, by this definition, has the properties:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is non-decreasing;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;0 \le F(x) \le 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;\lim_{x \rightarrow -\infty} F(x) = 0&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\lim_{x \rightarrow \infty} F(x) = 1&amp;lt;/math&amp;gt;;&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;P(a \le X &amp;lt; b) = F(b) - F(a)&amp;lt;/math&amp;gt;; and&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;li style=&amp;quot;margin: 0.7rem 0;&amp;quot;&amp;gt;&amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is continuous (due to the [[Riemann integral]] properties).&amp;lt;/li&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/ul&amp;gt;&lt;br /&gt;
&lt;br /&gt;
It is also possible to think in the opposite direction, which allows more flexibility. Say &amp;lt;math&amp;gt;F(x)&amp;lt;/math&amp;gt; is a function that satisfies all but the last of the properties above, then &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; represents the cumulative density function for some random variable: a discrete random variable if &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; is a step function, and a continuous random variable otherwise.&amp;lt;ref&amp;gt;See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.&amp;lt;/ref&amp;gt; This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; is such an absolutely continuous random variable, then it has a [[probability density function]] &amp;lt;math&amp;gt;f(x)&amp;lt;/math&amp;gt;, and its probability of falling into a Lebesgue-measurable set &amp;lt;math&amp;gt;A \subset \mathbb{R}&amp;lt;/math&amp;gt; is:&lt;br /&gt;
&lt;br /&gt;
: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
    \operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu&lt;br /&gt;
&lt;br /&gt;
  &amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; is the Lebesgue measure.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Note on terminology: some authors use the term &amp;quot;continuous distribution&amp;quot; to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;\mu\{x\}\,=\,0&amp;lt;/math&amp;gt; for all &amp;lt;math&amp;gt;\,x&amp;lt;/math&amp;gt;. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== [[Andrey Kolmogorov|Kolmogorov]] definition ==&lt;br /&gt;
&lt;br /&gt;
{{Main|Probability space|Probability measure}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; from a [[probability space]] &amp;lt;math&amp;gt;(\Omega, \mathcal{F}, \mathbb{P})&amp;lt;/math&amp;gt; to a [[measurable space]] &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt;. Given that probabilities of events of the form &amp;lt;math&amp;gt;\{\omega\in\Omega\mid X(\omega)\in A\}&amp;lt;/math&amp;gt; satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] &amp;lt;math&amp;gt;X_*\mathbb{P}&amp;lt;/math&amp;gt; of &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt; , which is a [[probability measure]] on &amp;lt;math&amp;gt;(\mathcal{X},\mathcal{A})&amp;lt;/math&amp;gt; satisfying &amp;lt;math&amp;gt;X_*\mathbb{P} = \mathbb{P}X^{-1}&amp;lt;/math&amp;gt;.&amp;lt;ref&amp;gt;{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Random number generation==&lt;br /&gt;
&lt;br /&gt;
{{Main|Pseudo-random number sampling}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Citation|last=Dekking|first=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
For example, suppose &amp;lt;math&amp;gt;U&amp;lt;/math&amp;gt; has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some &amp;lt;math&amp;gt;0 &amp;lt; p &amp;lt; 1&amp;lt;/math&amp;gt;, we define&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;{\displaystyle X ={\begin{cases}1,&amp;amp;{\mbox{if }}U&amp;lt;p\\0,&amp;amp;{\mbox{if }}U\geq p\end{cases}}}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Category:Mathematical and quantitative methods (economics)&lt;br /&gt;
&lt;br /&gt;
类别: 数学和定量方法(经济学)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
so that&lt;br /&gt;
&lt;br /&gt;
it:Variabile casuale#Distribuzione di probabilità&lt;br /&gt;
&lt;br /&gt;
它: Variabile casuale # distributione di 盖率 à&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Probability distribution]]. Its edit history can be viewed at [[概率分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%9B%BE%E8%B7%9D%E7%A6%BB&amp;diff=13618</id>
		<title>图距离</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%9B%BE%E8%B7%9D%E7%A6%BB&amp;diff=13618"/>
		<updated>2020-09-02T13:54:14Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Redirect|Geodesic distance|distances on the surface of a sphere|Great-circle distance|distances on the surface of the Earth|Geodesics on an ellipsoid|geodesics in differential geometry|Geodesic}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In the [[mathematics|mathematical]] field of [[graph theory]], the '''distance''' between two [[vertex (graph theory)|vertices]] in a [[Graph (discrete mathematics)|graph]] is the number of edges in a [[shortest path problem|shortest path]] (also called a '''graph geodesic''') connecting them.  This is also known as the '''geodesic distance'''.&amp;lt;ref&amp;gt;{{cite journal |last=Bouttier  |first=Jérémie |author2=Di Francesco,P. |author3=Guitter, E. |date=July 2003  &lt;br /&gt;
&lt;br /&gt;
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them.  This is also known as the geodesic distance.&amp;lt;ref&amp;gt;{{cite journal |last=Bouttier  |first=Jérémie |author2=Di Francesco,P. |author3=Guitter, E. |date=July 2003  &lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''的数学领域中，图中两个顶点之间的距离是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;最短路径 Shortest Path&amp;lt;/font&amp;gt;'''(也称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图测地线 Graph Geodesic&amp;lt;/font&amp;gt;''')中连接它们的边的数目。这也被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;测地距离 Geodesic Distance&amp;lt;/font&amp;gt;'''。3 = Guitter，e. | date = July 2003&lt;br /&gt;
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|title=Geodesic distance in planar graphs |journal= Nuclear Physics B|volume=663 |issue=3 |pages=535–567 |quote=By distance we mean here geodesic distance along the graph, namely the length of any shortest path between say two given faces|doi=10.1016/S0550-3213(03)00355-9|arxiv=cond-mat/0303272 }}&amp;lt;/ref&amp;gt; Notice that there may be more than one shortest path between two vertices.&amp;lt;ref&amp;gt;&lt;br /&gt;
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|title=Geodesic distance in planar graphs |journal= Nuclear Physics B|volume=663 |issue=3 |pages=535–567 |quote=By distance we mean here geodesic distance along the graph, namely the length of any shortest path between say two given faces  |doi=10.1016/S0550-3213(03)00355-9|arxiv=cond-mat/0303272 }}&amp;lt;/ref&amp;gt; &lt;br /&gt;
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Notice that there may be more than one shortest path between two vertices.&amp;lt;ref&amp;gt;&lt;br /&gt;
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注意，在两个顶点之间可能有一个以上的最短路径&lt;br /&gt;
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{{cite web |url=http://mathworld.wolfram.com/GraphGeodesic.html |title=Graph Geodesic |accessdate= 2008-04-23&lt;br /&gt;
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{{cite web |url=http://mathworld.wolfram.com/GraphGeodesic.html |title=Graph Geodesic |accessdate= 2008-04-23&lt;br /&gt;
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{ cite web | url =  http://mathworld.wolfram.com/graphgeodesic.html  | title = Graph Geodesic | accessdate = 2008-04-23&lt;br /&gt;
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|last=Weisstein |first=Eric W. |authorlink=Eric W. Weisstein |work=MathWorld--A Wolfram Web Resource &lt;br /&gt;
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|last=Weisstein |first=Eric W. |authorlink=Eric W. Weisstein |work=MathWorld--A Wolfram Web Resource &lt;br /&gt;
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|publisher= Wolfram Research &lt;br /&gt;
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|publisher= Wolfram Research &lt;br /&gt;
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|quote=The length of the graph geodesic between these points d(u,v) is called the graph distance between u and v }}&lt;br /&gt;
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|quote=The length of the graph geodesic between these points d(u,v) is called the graph distance between u and v }}&lt;br /&gt;
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&amp;lt;/ref&amp;gt;  If there is no path connecting the two vertices, i.e., if they belong to different [[connected component (graph theory)|connected component]]s, then conventionally the distance is defined as infinite.&lt;br /&gt;
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&amp;lt;/ref&amp;gt;  If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite.&lt;br /&gt;
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如果两个顶点间没有路径连接，也就是说，如果它们属于不同的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;连通分支 Connected Components&amp;lt;/font&amp;gt;'''，那么按照惯例，距离被定义为无穷大。&lt;br /&gt;
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In the case of a [[directed graph]] the distance &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; between two vertices &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is defined as the length of a shortest directed path from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; consisting of arcs, provided at least one such path exists.&amp;lt;ref&amp;gt;F. Harary, Graph Theory, Addison-Wesley, 1969, p.199.&amp;lt;/ref&amp;gt; Notice that, in contrast with the case of undirected graphs, &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; does not necessarily coincide with &amp;lt;math&amp;gt;d(v,u)&amp;lt;/math&amp;gt;, and it might be the case that one is defined while the other is not.&lt;br /&gt;
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In the case of a directed graph the distance &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; between two vertices &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is defined as the length of a shortest directed path from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; does not necessarily coincide with &amp;lt;math&amp;gt;d(v,u)&amp;lt;/math&amp;gt;, and it might be the case that one is defined while the other is not.&lt;br /&gt;
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在有向图的情况下，两个顶点之间 &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;的距离&amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt;被定义为从&amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; 到 &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;由弧组成的最短有向路径的长度，前提是至少存在一条这样的路径。请注意，与无向图的情况不同， &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; 不一定与&amp;lt;math&amp;gt;d(v,u)&amp;lt;/math&amp;gt;一致，而且可能一个是被定义的，而另一个不是。&lt;br /&gt;
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==Related concepts 相关概念==&lt;br /&gt;
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A [[metric space]] defined over a set of points in terms of distances in a graph defined over the set is called a '''graph metric'''.&lt;br /&gt;
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A metric space defined over a set of points in terms of distances in a graph defined over the set is called a graph metric.&lt;br /&gt;
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一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度量空间 Metric Space&amp;lt;/font&amp;gt;'''定义在一个集合上的点集合上，该集合上的图的距离称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图度量 Graph Metric&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is [[connected (graph theory)|connected]].&lt;br /&gt;
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The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is connected.&lt;br /&gt;
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当且仅当图是连通的时，(无向图的)顶点集和距离函数构成度量空间。&lt;br /&gt;
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The '''eccentricity''' &amp;lt;math&amp;gt;\epsilon(v)&amp;lt;/math&amp;gt; of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is the greatest distance between &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; and any other vertex; in symbols that is &amp;lt;math&amp;gt;\epsilon(v) = \max_{u \in V}d(v,u)&amp;lt;/math&amp;gt;. It can be thought of as how far a node is from the node most distant from it in the graph.&lt;br /&gt;
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The eccentricity &amp;lt;math&amp;gt;\epsilon(v)&amp;lt;/math&amp;gt; of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is the greatest distance between &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; and any other vertex; in symbols that is &amp;lt;math&amp;gt;\epsilon(v) = \max_{u \in V}d(v,u)&amp;lt;/math&amp;gt;. It can be thought of as how far a node is from the node most distant from it in the graph.&lt;br /&gt;
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一个顶点的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;偏心率 Eccentricity&amp;lt;/font&amp;gt;'''是它与其他顶点之间最大的距离。它可以被认为是一个节点距离图中离它最远的节点有多远。&lt;br /&gt;
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The '''radius''' &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt; of a graph is the minimum eccentricity of any vertex or, in symbols, &amp;lt;math&amp;gt;r = \min_{v \in V} \epsilon(v) = \min_{v \in V}\max_{u \in V}d(v,u)&amp;lt;/math&amp;gt;.&lt;br /&gt;
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The radius &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt; of a graph is the minimum eccentricity of any vertex or, in symbols, &amp;lt;math&amp;gt;r = \min_{v \in V} \epsilon(v) = \min_{v \in V}\max_{u \in V}d(v,u)&amp;lt;/math&amp;gt;.&lt;br /&gt;
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图的半径&amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;是任何顶点的最小偏心率，或者，在符号中，&amp;lt;math&amp;gt;r = \min_{v \in V} \epsilon(v) = \min_{v \in V}\max_{u \in V}d(v,u)&amp;lt;/math&amp;gt; 。&lt;br /&gt;
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The '''diameter''' &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; of a graph is the maximum eccentricity of any vertex in the graph.  That is, &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; is the greatest distance between any pair of vertices or, alternatively, &amp;lt;math&amp;gt;d = \max_{v \in V}\epsilon(v)&amp;lt;/math&amp;gt;. To find the diameter of a graph, first find the [[Shortest path problem|shortest path]] between each pair of [[vertex (graph theory)|vertices]]. The greatest length of any of these paths is the diameter of the graph. &lt;br /&gt;
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The diameter &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; of a graph is the maximum eccentricity of any vertex in the graph.  That is, &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; is the greatest distance between any pair of vertices or, alternatively, &amp;lt;math&amp;gt;d = \max_{v \in V}\epsilon(v)&amp;lt;/math&amp;gt;. To find the diameter of a graph, first find the shortest path between each pair of vertices. The greatest length of any of these paths is the diameter of the graph. &lt;br /&gt;
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图的直径 &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt;是图中任何顶点的最大偏心率。也就是说， &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; 是任何一对顶点之间最大的距离，或者，&amp;lt;math&amp;gt;d = \max_{v \in V}\epsilon(v)&amp;lt;/math&amp;gt;。要找到图的直径，首先要找到每对顶点之间的最短路径。这些路径的最大长度是图的直径。&lt;br /&gt;
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A '''central vertex''' in a graph of radius &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt; is one whose eccentricity is &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;&amp;amp;mdash;that is, a vertex that achieves the radius or, equivalently, a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;\epsilon(v) = r&amp;lt;/math&amp;gt;.&lt;br /&gt;
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A central vertex in a graph of radius &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt; is one whose eccentricity is &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;&amp;amp;mdash;that is, a vertex that achieves the radius or, equivalently, a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;\epsilon(v) = r&amp;lt;/math&amp;gt;.&lt;br /&gt;
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半径&amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;图中的中心顶点的偏心率是 &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;&amp;amp;mdash，也就是说，达到半径的顶点，或者等效于一个顶点&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;，这样 &amp;lt;math&amp;gt;\epsilon(v) = r&amp;lt;/math&amp;gt; 。&lt;br /&gt;
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A '''peripheral vertex''' in a graph of diameter &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; is one that is distance &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; from some other vertex&amp;amp;mdash;that is, a vertex that achieves the diameter. Formally, &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is peripheral if &amp;lt;math&amp;gt;\epsilon(v) = d&amp;lt;/math&amp;gt;.&lt;br /&gt;
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A peripheral vertex in a graph of diameter &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; is one that is distance &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; from some other vertex&amp;amp;mdash;that is, a vertex that achieves the diameter. Formally, &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is peripheral if &amp;lt;math&amp;gt;\epsilon(v) = d&amp;lt;/math&amp;gt;.&lt;br /&gt;
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直径&amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt;图中的边缘顶点与其他顶点之间的距离为&amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt;，即达到直径的顶点。形式上，如果&amp;lt;math&amp;gt;\epsilon(v) = d&amp;lt;/math&amp;gt;，&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;是次要的。&lt;br /&gt;
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A '''pseudo-peripheral vertex''' &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; has the property that for any vertex &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;, if &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is as far away from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; as possible, then &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; is as far away from &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; as possible.  Formally, a vertex ''u'' is pseudo-peripheral, &lt;br /&gt;
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A pseudo-peripheral vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; has the property that for any vertex &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;, if &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is as far away from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; as possible, then &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; is as far away from &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; as possible.  Formally, a vertex u is pseudo-peripheral, &lt;br /&gt;
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一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;伪周边顶点 Pseudo-Peripheral Vertex&amp;lt;/font&amp;gt;'''具有这样的属性: 对于任何顶点 &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;，如果&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;离&amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;越远，那么 &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;离 &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;越远。形式上，一个顶点 u 是伪外围的,&lt;br /&gt;
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if for each vertex ''v'' with &amp;lt;math&amp;gt;d(u,v) = \epsilon(u)&amp;lt;/math&amp;gt; holds &amp;lt;math&amp;gt;\epsilon(u)=\epsilon(v)&amp;lt;/math&amp;gt;.&lt;br /&gt;
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if for each vertex v with &amp;lt;math&amp;gt;d(u,v) = \epsilon(u)&amp;lt;/math&amp;gt; holds &amp;lt;math&amp;gt;\epsilon(u)=\epsilon(v)&amp;lt;/math&amp;gt;.&lt;br /&gt;
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如果对于每个顶点 v 都有&amp;lt;math&amp;gt;d(u,v) = \epsilon(u)&amp;lt;/math&amp;gt;保持&amp;lt;math&amp;gt;\epsilon(u)=\epsilon(v)&amp;lt;/math&amp;gt;。&lt;br /&gt;
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The [[partition of a set|partition]] of a graph's vertices into subsets by their distances from a given starting vertex is called the [[level structure]] of the graph.&lt;br /&gt;
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The partition of a graph's vertices into subsets by their distances from a given starting vertex is called the level structure of the graph.&lt;br /&gt;
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图的顶点按照它们与给定顶点之间的距离划分成子集的过程称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图的层次结构 Level Structure of The Graph&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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A graph such that for every pair of vertices there is a unique shortest path connecting them is called a '''geodetic graph'''. For example, all [[Tree (graph theory)|trees]] are geodetic.&amp;lt;ref&amp;gt;[[Øystein Ore]], Theory of graphs [3rd ed., 1967], Colloquium Publications, American Mathematical Society,  	p. 104&amp;lt;/ref&amp;gt;&lt;br /&gt;
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A graph such that for every pair of vertices there is a unique shortest path connecting them is called a geodetic graph. For example, all trees are geodetic.&lt;br /&gt;
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对于每一对顶点，有一条唯一的最短路径连接它们，这样的图称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;测地图 Geodetic Graph&amp;lt;/font&amp;gt;'''。例如，所有的树都是测地图。&lt;br /&gt;
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==Algorithm for finding pseudo-peripheral vertices 寻找伪周边的算法==&lt;br /&gt;
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Often peripheral [[sparse matrix]] algorithms need a starting vertex with a high eccentricity. A peripheral vertex would be perfect, but is often hard to calculate. In most circumstances a pseudo-peripheral vertex can be used.  A pseudo-peripheral vertex can easily be found with the following algorithm:&lt;br /&gt;
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Often peripheral sparse matrix algorithms need a starting vertex with a high eccentricity. A peripheral vertex would be perfect, but is often hard to calculate. In most circumstances a pseudo-peripheral vertex can be used.  A pseudo-peripheral vertex can easily be found with the following algorithm:&lt;br /&gt;
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通常周边稀疏矩阵算法需要一个高偏心率的起始点。一个外围的顶点会是完美的，但往往是难以计算。在大多数情况下，可以使用伪周边顶点。通过以下算法，可以很容易地找到伪周边顶点:&lt;br /&gt;
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# Choose a vertex &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;.&lt;br /&gt;
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Choose a vertex &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;.&lt;br /&gt;
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选择顶点&amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;。&lt;br /&gt;
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# Among all the vertices that are as far from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; as possible, let &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; be one with minimal [[degree (graph theory)|degree]].&lt;br /&gt;
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Among all the vertices that are as far from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; as possible, let &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; be one with minimal degree.&lt;br /&gt;
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在所有尽可能远离&amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;的顶点中，让&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;是一个最小度的顶点。&lt;br /&gt;
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# If &amp;lt;math&amp;gt;\epsilon(v) &amp;gt; \epsilon(u)&amp;lt;/math&amp;gt; then set &amp;lt;math&amp;gt;u=v&amp;lt;/math&amp;gt; and repeat with step 2, else &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; is a pseudo-peripheral vertex.&lt;br /&gt;
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If &amp;lt;math&amp;gt;\epsilon(v) &amp;gt; \epsilon(u)&amp;lt;/math&amp;gt; then set &amp;lt;math&amp;gt;u=v&amp;lt;/math&amp;gt; and repeat with step 2, else &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; is a pseudo-peripheral vertex.&lt;br /&gt;
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如果&amp;lt;math&amp;gt;\epsilon(v) &amp;gt; \epsilon(u)&amp;lt;/math&amp;gt;然后设置 &amp;lt;math&amp;gt;u=v&amp;lt;/math&amp;gt;并重复步骤2，否则 &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;是一个伪周边顶点。&lt;br /&gt;
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==See also 另请参见==&lt;br /&gt;
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* [[Distance matrix]]&lt;br /&gt;
距离矩阵&lt;br /&gt;
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* [[Resistance distance]]&lt;br /&gt;
电阻距离&lt;br /&gt;
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* [[Betweenness centrality]]&lt;br /&gt;
介数中心性&lt;br /&gt;
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* [[Centrality]]&lt;br /&gt;
中心性&lt;br /&gt;
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* [[Closeness (graph theory)|Closeness]]&lt;br /&gt;
封闭性&lt;br /&gt;
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* [[Degree diameter problem]] for [[Graph (discrete mathematics)|graph]]s and [[digraph (mathematics)|digraph]]s&lt;br /&gt;
图和有向图的度数直径问题&lt;br /&gt;
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* [[Metric graph]]&lt;br /&gt;
指标图&lt;br /&gt;
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==Notes==&lt;br /&gt;
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{{reflist}}&lt;br /&gt;
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[[Category:Graph theory]]&lt;br /&gt;
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Category:Graph theory&lt;br /&gt;
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范畴: 图论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Distance (graph theory)]]. Its edit history can be viewed at [[图距离/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%9B%BE%E8%B7%9D%E7%A6%BB&amp;diff=13617</id>
		<title>图距离</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%9B%BE%E8%B7%9D%E7%A6%BB&amp;diff=13617"/>
		<updated>2020-09-02T13:43:39Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{Redirect|Geodesic distance|distances on the surface of a sphere|Great-circle distance|distances on the surface of the Earth|Geodesics on an ellipsoid|geodesics in differential geometry|Geodesic}}&lt;br /&gt;
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In the [[mathematics|mathematical]] field of [[graph theory]], the '''distance''' between two [[vertex (graph theory)|vertices]] in a [[Graph (discrete mathematics)|graph]] is the number of edges in a [[shortest path problem|shortest path]] (also called a '''graph geodesic''') connecting them.  This is also known as the '''geodesic distance'''.&amp;lt;ref&amp;gt;{{cite journal |last=Bouttier  |first=Jérémie |author2=Di Francesco,P. |author3=Guitter, E. |date=July 2003  &lt;br /&gt;
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In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them.  This is also known as the geodesic distance.&amp;lt;ref&amp;gt;{{cite journal |last=Bouttier  |first=Jérémie |author2=Di Francesco,P. |author3=Guitter, E. |date=July 2003  &lt;br /&gt;
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在图论的数学领域中，图中两个顶点之间的距离是最短路径(也称为图测地线)中连接它们的边的数目。这也被称为测地距离。3 = Guitter，e. | date = July 2003&lt;br /&gt;
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|title=Geodesic distance in planar graphs |journal= Nuclear Physics B|volume=663 |issue=3 |pages=535–567 |quote=By distance we mean here geodesic distance along the graph, namely the length of any shortest path between say two given faces|doi=10.1016/S0550-3213(03)00355-9|arxiv=cond-mat/0303272 }}&amp;lt;/ref&amp;gt; Notice that there may be more than one shortest path between two vertices.&amp;lt;ref&amp;gt;&lt;br /&gt;
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|title=Geodesic distance in planar graphs |journal= Nuclear Physics B|volume=663 |issue=3 |pages=535–567 |quote=By distance we mean here geodesic distance along the graph, namely the length of any shortest path between say two given faces  |doi=10.1016/S0550-3213(03)00355-9|arxiv=cond-mat/0303272 }}&amp;lt;/ref&amp;gt; &lt;br /&gt;
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Notice that there may be more than one shortest path between two vertices.&amp;lt;ref&amp;gt;&lt;br /&gt;
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注意，在两个顶点之间可能有一个以上的最短路径&lt;br /&gt;
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{{cite web |url=http://mathworld.wolfram.com/GraphGeodesic.html |title=Graph Geodesic |accessdate= 2008-04-23&lt;br /&gt;
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{{cite web |url=http://mathworld.wolfram.com/GraphGeodesic.html |title=Graph Geodesic |accessdate= 2008-04-23&lt;br /&gt;
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{ cite web | url =  http://mathworld.wolfram.com/graphgeodesic.html  | title = Graph Geodesic | accessdate = 2008-04-23&lt;br /&gt;
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|last=Weisstein |first=Eric W. |authorlink=Eric W. Weisstein |work=MathWorld--A Wolfram Web Resource &lt;br /&gt;
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|last=Weisstein |first=Eric W. |authorlink=Eric W. Weisstein |work=MathWorld--A Wolfram Web Resource &lt;br /&gt;
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|publisher= Wolfram Research &lt;br /&gt;
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|publisher= Wolfram Research &lt;br /&gt;
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|quote=The length of the graph geodesic between these points d(u,v) is called the graph distance between u and v }}&lt;br /&gt;
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|quote=The length of the graph geodesic between these points d(u,v) is called the graph distance between u and v }}&lt;br /&gt;
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&amp;lt;/ref&amp;gt;  If there is no path connecting the two vertices, i.e., if they belong to different [[connected component (graph theory)|connected component]]s, then conventionally the distance is defined as infinite.&lt;br /&gt;
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&amp;lt;/ref&amp;gt;  If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite.&lt;br /&gt;
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如果两个顶点间没有路径连接，也就是说，如果它们属于不同的连通分支，那么按照惯例，距离被定义为无穷大。&lt;br /&gt;
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In the case of a [[directed graph]] the distance &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; between two vertices &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is defined as the length of a shortest directed path from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; consisting of arcs, provided at least one such path exists.&amp;lt;ref&amp;gt;F. Harary, Graph Theory, Addison-Wesley, 1969, p.199.&amp;lt;/ref&amp;gt; Notice that, in contrast with the case of undirected graphs, &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; does not necessarily coincide with &amp;lt;math&amp;gt;d(v,u)&amp;lt;/math&amp;gt;, and it might be the case that one is defined while the other is not.&lt;br /&gt;
&lt;br /&gt;
In the case of a directed graph the distance &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; between two vertices &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is defined as the length of a shortest directed path from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; does not necessarily coincide with &amp;lt;math&amp;gt;d(v,u)&amp;lt;/math&amp;gt;, and it might be the case that one is defined while the other is not.&lt;br /&gt;
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在有向图的情况下，两个顶点之间 &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;的距离&amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt;被定义为从&amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; 到 &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;由弧组成的最短有向路径的长度，前提是至少存在一条这样的路径。请注意，与无向图的情况不同， &amp;lt;math&amp;gt;d(u,v)&amp;lt;/math&amp;gt; 不一定与&amp;lt;math&amp;gt;d(v,u)&amp;lt;/math&amp;gt;一致，而且可能一个是被定义的，而另一个不是。&lt;br /&gt;
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==Related concepts 相关概念==&lt;br /&gt;
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A [[metric space]] defined over a set of points in terms of distances in a graph defined over the set is called a '''graph metric'''.&lt;br /&gt;
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A metric space defined over a set of points in terms of distances in a graph defined over the set is called a graph metric.&lt;br /&gt;
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一个度量空间定义在一个集合上的点集合上，该集合上的图的距离称为图度量。&lt;br /&gt;
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The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is [[connected (graph theory)|connected]].&lt;br /&gt;
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The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is connected.&lt;br /&gt;
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当且仅当图是连通的时，(无向图的)顶点集和距离函数构成度量空间。&lt;br /&gt;
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The '''eccentricity''' &amp;lt;math&amp;gt;\epsilon(v)&amp;lt;/math&amp;gt; of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is the greatest distance between &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; and any other vertex; in symbols that is &amp;lt;math&amp;gt;\epsilon(v) = \max_{u \in V}d(v,u)&amp;lt;/math&amp;gt;. It can be thought of as how far a node is from the node most distant from it in the graph.&lt;br /&gt;
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The eccentricity &amp;lt;math&amp;gt;\epsilon(v)&amp;lt;/math&amp;gt; of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is the greatest distance between &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; and any other vertex; in symbols that is &amp;lt;math&amp;gt;\epsilon(v) = \max_{u \in V}d(v,u)&amp;lt;/math&amp;gt;. It can be thought of as how far a node is from the node most distant from it in the graph.&lt;br /&gt;
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一个顶点的偏心率是它与其他顶点之间最大的距离。它可以被认为是一个节点距离图中离它最远的节点有多远。&lt;br /&gt;
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The '''radius''' &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt; of a graph is the minimum eccentricity of any vertex or, in symbols, &amp;lt;math&amp;gt;r = \min_{v \in V} \epsilon(v) = \min_{v \in V}\max_{u \in V}d(v,u)&amp;lt;/math&amp;gt;.&lt;br /&gt;
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The radius &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt; of a graph is the minimum eccentricity of any vertex or, in symbols, &amp;lt;math&amp;gt;r = \min_{v \in V} \epsilon(v) = \min_{v \in V}\max_{u \in V}d(v,u)&amp;lt;/math&amp;gt;.&lt;br /&gt;
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图的半径&amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;是任何顶点的最小偏心率，或者，在符号中，&amp;lt;math&amp;gt;r = \min_{v \in V} \epsilon(v) = \min_{v \in V}\max_{u \in V}d(v,u)&amp;lt;/math&amp;gt; 。&lt;br /&gt;
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The '''diameter''' &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; of a graph is the maximum eccentricity of any vertex in the graph.  That is, &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; is the greatest distance between any pair of vertices or, alternatively, &amp;lt;math&amp;gt;d = \max_{v \in V}\epsilon(v)&amp;lt;/math&amp;gt;. To find the diameter of a graph, first find the [[Shortest path problem|shortest path]] between each pair of [[vertex (graph theory)|vertices]]. The greatest length of any of these paths is the diameter of the graph. &lt;br /&gt;
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The diameter &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; of a graph is the maximum eccentricity of any vertex in the graph.  That is, &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; is the greatest distance between any pair of vertices or, alternatively, &amp;lt;math&amp;gt;d = \max_{v \in V}\epsilon(v)&amp;lt;/math&amp;gt;. To find the diameter of a graph, first find the shortest path between each pair of vertices. The greatest length of any of these paths is the diameter of the graph. &lt;br /&gt;
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图的直径 &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt;是图中任何顶点的最大偏心率。也就是说， &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; 是任何一对顶点之间最大的距离，或者，&amp;lt;math&amp;gt;d = \max_{v \in V}\epsilon(v)&amp;lt;/math&amp;gt;。要找到图的直径，首先要找到每对顶点之间的最短路径。这些路径的最大长度是图的直径。&lt;br /&gt;
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A '''central vertex''' in a graph of radius &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt; is one whose eccentricity is &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;&amp;amp;mdash;that is, a vertex that achieves the radius or, equivalently, a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;\epsilon(v) = r&amp;lt;/math&amp;gt;.&lt;br /&gt;
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A central vertex in a graph of radius &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt; is one whose eccentricity is &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;&amp;amp;mdash;that is, a vertex that achieves the radius or, equivalently, a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;\epsilon(v) = r&amp;lt;/math&amp;gt;.&lt;br /&gt;
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半径&amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;图中的中心顶点的偏心率是 &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;&amp;amp;mdash，也就是说，达到半径的顶点，或者等效于一个顶点&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;，这样 &amp;lt;math&amp;gt;\epsilon(v) = r&amp;lt;/math&amp;gt; 。&lt;br /&gt;
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A '''peripheral vertex''' in a graph of diameter &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; is one that is distance &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; from some other vertex&amp;amp;mdash;that is, a vertex that achieves the diameter. Formally, &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is peripheral if &amp;lt;math&amp;gt;\epsilon(v) = d&amp;lt;/math&amp;gt;.&lt;br /&gt;
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A peripheral vertex in a graph of diameter &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; is one that is distance &amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt; from some other vertex&amp;amp;mdash;that is, a vertex that achieves the diameter. Formally, &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is peripheral if &amp;lt;math&amp;gt;\epsilon(v) = d&amp;lt;/math&amp;gt;.&lt;br /&gt;
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直径&amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt;图中的边缘顶点与其他顶点之间的距离为&amp;lt;math&amp;gt;d&amp;lt;/math&amp;gt;，即达到直径的顶点。形式上，如果&amp;lt;math&amp;gt;\epsilon(v) = d&amp;lt;/math&amp;gt;，&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;是次要的。&lt;br /&gt;
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A '''pseudo-peripheral vertex''' &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; has the property that for any vertex &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;, if &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is as far away from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; as possible, then &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; is as far away from &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; as possible.  Formally, a vertex ''u'' is pseudo-peripheral, &lt;br /&gt;
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A pseudo-peripheral vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; has the property that for any vertex &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;, if &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is as far away from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; as possible, then &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; is as far away from &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; as possible.  Formally, a vertex u is pseudo-peripheral, &lt;br /&gt;
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一个伪周边顶点具有这样的属性: 对于任何顶点 &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;，如果&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;离&amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;越远，那么 &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;离 &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;越远。形式上，一个顶点 u 是伪外围的,&lt;br /&gt;
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if for each vertex ''v'' with &amp;lt;math&amp;gt;d(u,v) = \epsilon(u)&amp;lt;/math&amp;gt; holds &amp;lt;math&amp;gt;\epsilon(u)=\epsilon(v)&amp;lt;/math&amp;gt;.&lt;br /&gt;
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if for each vertex v with &amp;lt;math&amp;gt;d(u,v) = \epsilon(u)&amp;lt;/math&amp;gt; holds &amp;lt;math&amp;gt;\epsilon(u)=\epsilon(v)&amp;lt;/math&amp;gt;.&lt;br /&gt;
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如果对于每个顶点 v 都有&amp;lt;math&amp;gt;d(u,v) = \epsilon(u)&amp;lt;/math&amp;gt;保持&amp;lt;math&amp;gt;\epsilon(u)=\epsilon(v)&amp;lt;/math&amp;gt;。&lt;br /&gt;
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The [[partition of a set|partition]] of a graph's vertices into subsets by their distances from a given starting vertex is called the [[level structure]] of the graph.&lt;br /&gt;
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The partition of a graph's vertices into subsets by their distances from a given starting vertex is called the level structure of the graph.&lt;br /&gt;
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图的顶点按照它们与给定顶点之间的距离划分成子集的过程称为图的层次结构。&lt;br /&gt;
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A graph such that for every pair of vertices there is a unique shortest path connecting them is called a '''geodetic graph'''. For example, all [[Tree (graph theory)|trees]] are geodetic.&amp;lt;ref&amp;gt;[[Øystein Ore]], Theory of graphs [3rd ed., 1967], Colloquium Publications, American Mathematical Society,  	p. 104&amp;lt;/ref&amp;gt;&lt;br /&gt;
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A graph such that for every pair of vertices there is a unique shortest path connecting them is called a geodetic graph. For example, all trees are geodetic.&lt;br /&gt;
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对于每一对顶点，有一条唯一的最短路径连接它们，这样的图称为大地图。例如，所有的树都是大地图。&lt;br /&gt;
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==Algorithm for finding pseudo-peripheral vertices 寻找伪周边的算法==&lt;br /&gt;
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Often peripheral [[sparse matrix]] algorithms need a starting vertex with a high eccentricity. A peripheral vertex would be perfect, but is often hard to calculate. In most circumstances a pseudo-peripheral vertex can be used.  A pseudo-peripheral vertex can easily be found with the following algorithm:&lt;br /&gt;
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Often peripheral sparse matrix algorithms need a starting vertex with a high eccentricity. A peripheral vertex would be perfect, but is often hard to calculate. In most circumstances a pseudo-peripheral vertex can be used.  A pseudo-peripheral vertex can easily be found with the following algorithm:&lt;br /&gt;
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通常周边稀疏矩阵算法需要一个高偏心率的起始点。一个外围的顶点会是完美的，但往往是难以计算。在大多数情况下，可以使用伪周边顶点。通过以下算法，可以很容易地找到伪周边顶点:&lt;br /&gt;
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# Choose a vertex &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;.&lt;br /&gt;
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 Choose a vertex &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;.&lt;br /&gt;
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选择顶点&amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;。&lt;br /&gt;
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# Among all the vertices that are as far from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; as possible, let &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; be one with minimal [[degree (graph theory)|degree]].&lt;br /&gt;
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 Among all the vertices that are as far from &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; as possible, let &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; be one with minimal degree.&lt;br /&gt;
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在所有尽可能远离&amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;的顶点中，让&amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt;是一个最小度的顶点。&lt;br /&gt;
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# If &amp;lt;math&amp;gt;\epsilon(v) &amp;gt; \epsilon(u)&amp;lt;/math&amp;gt; then set &amp;lt;math&amp;gt;u=v&amp;lt;/math&amp;gt; and repeat with step 2, else &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; is a pseudo-peripheral vertex.&lt;br /&gt;
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 If &amp;lt;math&amp;gt;\epsilon(v) &amp;gt; \epsilon(u)&amp;lt;/math&amp;gt; then set &amp;lt;math&amp;gt;u=v&amp;lt;/math&amp;gt; and repeat with step 2, else &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt; is a pseudo-peripheral vertex.&lt;br /&gt;
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如果&amp;lt;math&amp;gt;\epsilon(v) &amp;gt; \epsilon(u)&amp;lt;/math&amp;gt;然后设置 &amp;lt;math&amp;gt;u=v&amp;lt;/math&amp;gt;并重复步骤2，否则 &amp;lt;math&amp;gt;u&amp;lt;/math&amp;gt;是一个伪周边顶点。&lt;br /&gt;
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==See also 另请参见==&lt;br /&gt;
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* [[Distance matrix]]&lt;br /&gt;
距离矩阵&lt;br /&gt;
&lt;br /&gt;
* [[Resistance distance]]&lt;br /&gt;
电阻距离&lt;br /&gt;
&lt;br /&gt;
* [[Betweenness centrality]]&lt;br /&gt;
介数中心性&lt;br /&gt;
&lt;br /&gt;
* [[Centrality]]&lt;br /&gt;
中心性&lt;br /&gt;
&lt;br /&gt;
* [[Closeness (graph theory)|Closeness]]&lt;br /&gt;
封闭性&lt;br /&gt;
&lt;br /&gt;
* [[Degree diameter problem]] for [[Graph (discrete mathematics)|graph]]s and [[digraph (mathematics)|digraph]]s&lt;br /&gt;
图和有向图的度数直径问题&lt;br /&gt;
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* [[Metric graph]]&lt;br /&gt;
指标图&lt;br /&gt;
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&lt;br /&gt;
==Notes==&lt;br /&gt;
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{{reflist}}&lt;br /&gt;
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[[Category:Graph theory]]&lt;br /&gt;
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Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Distance (graph theory)]]. Its edit history can be viewed at [[图距离/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E8%B7%AF%E5%BE%84_path&amp;diff=13614</id>
		<title>路径 path</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E8%B7%AF%E5%BE%84_path&amp;diff=13614"/>
		<updated>2020-09-02T12:00:54Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{for|the family of graphs known as paths|Path graph}}&lt;br /&gt;
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[[File:Snake-in-the-box and Hamiltonian path.svg|thumb|right|&lt;br /&gt;
图1：A three-dimensional [[hypercube graph]] showing a [[Hamiltonian path]] in red, and a [[Snake-in-the-box|longest induced path]] in bold black. 一个三维的[超立方体图表显示一个红色的哈密顿图，和一个黑色的最长诱导路径]]&lt;br /&gt;
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In [[graph theory]], a '''path''' in a [[Graph (discrete mathematics)|graph]] is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] which joins a sequence of [[Vertex (graph theory)|vertices]] which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A '''directed path''' (sometimes called '''dipath'''&amp;lt;ref&amp;gt;Graph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991, [https://books.google.com/books?id=idigH5CTGWAC&amp;amp;pg=PA205 p.205]&amp;lt;/ref&amp;gt;) in a [[directed graph]] is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.&lt;br /&gt;
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In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''中，图中的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;路径 Path&amp;lt;/font&amp;gt;'''是一个有限或无限的边序列，这些边连接着一系列顶点，这些顶点在大多数定义中都是不同的(因为顶点是不同的，所以边也是不同的)。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向图 Directed Graph&amp;lt;/font&amp;gt;'''中的有向路径(有时称为dipath)是一个有限或无限的边序列，它连接一系列不同的顶点，但附加的限制是所有的边都朝向同一个方向。&lt;br /&gt;
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Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced [[algorithm]]ic topics concerning paths in graphs.&lt;br /&gt;
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Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs.&lt;br /&gt;
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路径是图论的基本概念，在大多数图论文本的导论部分都有描述。邦迪和默蒂(1976) ，吉本斯(1985) ，或迪斯特尔(2005)。柯尔特(1990)等人著作中，涵盖了更多关于图中路径的高级算法的话题。&lt;br /&gt;
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== Definitions 定义==&lt;br /&gt;
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=== Walk, trail, path 步道 轨迹 路径===&lt;br /&gt;
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[[File:Trail but not path.svg|200px|thumb|right &lt;br /&gt;
图2：Trail from A to E，but not path 从A到E的轨迹而非路径]]&lt;br /&gt;
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* A '''walk''' is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] which joins a sequence of [[Vertex (graph theory)|vertices]].{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;步道 Walk&amp;lt;/font&amp;gt;'''是连接一系列顶点的有限或无限的边序列&lt;br /&gt;
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Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a graph. A finite walk is a sequence of edges {{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} for which there is a sequence of vertices {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} such that {{nowrap begin}}''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = {''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;}{{nowrap end}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} is the ''vertex sequence'' of the walk. This walk is  ''closed'' if {{nowrap begin}}''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; = ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;{{nowrap end}}, and ''open'' else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.&lt;br /&gt;
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Let  be a graph. A finite walk is a sequence of edges  for which there is a sequence of vertices  such that ϕ(e&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;) = {v&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;, v&amp;lt;sub&amp;gt;i + 1&amp;lt;/sub&amp;gt;} for .  is the vertex sequence of the walk. This walk is  closed if v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; = v&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;, and open else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or ray) has a first vertex but no last vertex.&lt;br /&gt;
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让我们做一个图{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} 。有限步道是一系列的边{{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}，其顶点序列{{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}。 {{nowrap begin}}''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = {''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;}{{nowrap end}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} 是行走的顶点序列。如果 {{nowrap begin}}''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; = ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;{{nowrap end}} ，则此步道关闭，否则就打开。一个无限步道是一系列的边，它们的类型与这里描述的相同，但是没有第一个顶点或最后一个顶点，而一个半无限步道(或光线)有第一个顶点，但是没有最后一个顶点。&lt;br /&gt;
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* A '''trail''' is a walk in which all edges are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;轨迹 Trail&amp;lt;/font&amp;gt;'''是所有的边缘都清晰可见的步道&lt;br /&gt;
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* A '''path''' is a trail in which all vertices (and therefore also all edges) are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;路径 Path&amp;lt;/font&amp;gt;'''是一条所有顶点(因此也包括所有边)都是不同的轨迹。&lt;br /&gt;
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If {{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} is a finite walk with vertex sequence {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} then ''w'' is said to be a ''walk from'' ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''to'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Similarly for a trail or a path. If there is a finite walk between two ''distinct'' vertices then there is also a finite trail and a finite path between them.&lt;br /&gt;
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If is a finite walk with vertex sequence  then w is said to be a walk from v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; to v&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;. Similarly for a trail or a path. If there is a finite walk between two distinct vertices then there is also a finite trail and a finite path between them.&lt;br /&gt;
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如果 {{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}是有顶点序列 {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}的有限步道，那么 w 就是从  ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''to'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;的步行。同样，对于一条轨迹或者一条路径也是如此。如果在两个不同的顶点之间有一个有限步道，那么在它们之间也有一个有限的轨迹和一个有限的路径。&lt;br /&gt;
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Some authors do not require that all vertices of a path be distinct and instead use the term '''simple path''' to refer to such a path.&lt;br /&gt;
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Some authors do not require that all vertices of a path be distinct and instead use the term simple path to refer to such a path.&lt;br /&gt;
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有些作者并不要求路径的所有顶点都是不同的，而是使用术语“简单路径”来指代这样的路径。&lt;br /&gt;
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A [[weighted graph]] associates a value (''weight'') with every edge in the graph. The ''weight of a walk'' (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight.&lt;br /&gt;
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A weighted graph associates a value (weight) with every edge in the graph. The weight of a walk (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.&lt;br /&gt;
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一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;加权图 Weighted Graph&amp;lt;/font&amp;gt;'''将一个值(权)与图中的每条边相关联。一个加权图中的步道(或轨迹或路径)的权是所有边的权之和。有时，“成本”或“长度”这两个词用来代替“权重”。&lt;br /&gt;
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=== Directed walk, trail, path 有向的步道，轨迹，路径===&lt;br /&gt;
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* A '''directed walk''' is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] directed in the same direction which joins a sequence of [[Vertex (graph theory)|vertices]].{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向步道 Directed Walk&amp;lt;/font&amp;gt;'''是沿相同方向定向的边的有限或无限序列，该边连接一系列顶点。&lt;br /&gt;
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Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a directed graph. A finite directed walk is a sequence of edges {{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} for which there is a sequence of vertices {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} such that {{nowrap|1=''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = (''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;)}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} is the ''vertex sequence'' of the directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.&lt;br /&gt;
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Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a directed graph. A finite directed walk is a sequence of edges  for which there is a sequence of vertices  such that  for .  is the vertex sequence of the directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or ray) has a first vertex but no last vertex.&lt;br /&gt;
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一个有向图{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}}。有限有向步道是一系列的边 {{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}，对于这些边，存在一系列的顶点 {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}。{{nowrap|1=''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = (''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;)}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}是有向步道的顶点序列。无限有向步道是一个边序列，其类型与本文描述的相同，但没有第一个顶点或最后一个顶点，而半无限有向步道(或射线)有第一个顶点，但没有最后一个顶点。&lt;br /&gt;
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* A '''directed trail''' is a directed walk in which all edges are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向轨迹 Directed Trail&amp;lt;/font&amp;gt;'''是指所有边缘都清晰可见的有向轨迹。&lt;br /&gt;
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* A '''directed path''' is a directed trail in which all vertices are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向路径 Directed Path&amp;lt;/font&amp;gt;'''是指所有顶点都是不同的有向路径。&lt;br /&gt;
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If {{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} is a finite directed walk with vertex sequence {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} then ''w'' is said to be a ''walk from'' ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''to'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Similarly for a directed trail or a path. If there is a finite directed walk between two ''distinct'' vertices then there is also a finite directed trail and a finite directed path between them.&lt;br /&gt;
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If  is a finite directed walk with vertex sequence  then w is said to be a walk from v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; to v&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;. Similarly for a directed trail or a path. If there is a finite directed walk between two distinct vertices then there is also a finite directed trail and a finite directed path between them.&lt;br /&gt;
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如果{{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}是一个顶点序列有限的{{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}有向步道，那么 w 就是从 ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''到'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;的步行。同样，对于有向轨迹或路径也是如此。如果在两个不同的顶点之间存在有限有向步道，那么在它们之间也存在有限有向轨迹和有限有向路径。&lt;br /&gt;
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Some authors do not require that all vertices of a directed path be distinct and instead use the term '''simple directed path''' to refer to such a directed path.&lt;br /&gt;
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Some authors do not require that all vertices of a directed path be distinct and instead use the term simple directed path to refer to such a directed path.&lt;br /&gt;
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有些作者并不要求有向路径的所有顶点都是不同的，而是用“简单有向路径”这个术语来表示这样一个有向路。&lt;br /&gt;
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A [[Weighted graph|weighted directed graph]] associates a value (''weight'') with every edge in the directed graph. The ''weight of a directed walk'' (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight.&lt;br /&gt;
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A weighted directed graph associates a value (weight) with every edge in the directed graph. The weight of a directed walk (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.&lt;br /&gt;
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一个加权有向图将一个值(权)与有向图中的每条边相关联。加权有向图中有向步道(或路径)的权是有向边权重的和。有时，“成本”或“长度”这两个词用来代替“权重”。&lt;br /&gt;
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== Examples 例子==&lt;br /&gt;
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* A graph is [[Connectivity (graph theory)|connected]] if there are paths containing each pair of vertices.&lt;br /&gt;
如果路径包含每一对顶点，则图是连通的。&lt;br /&gt;
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* A directed graph is [[Strongly-connected digraph|strongly connected]] if there are oppositely oriented directed paths containing each pair of vertices.&lt;br /&gt;
如果一个有向图中存在包含每对顶点的相对有向路径，那么这个有向图就是强连通的。&lt;br /&gt;
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* A path such that no graph edges connect two nonconsecutive path vertices is called an [[induced path]].&lt;br /&gt;
没有图的边连接两个不连续的路径顶点的路径称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;诱导路径 Induced Path&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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* A path that includes every vertex of the graph is known as a [[Hamiltonian path]].&lt;br /&gt;
包含图的每个顶点的路径称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;哈密顿路径 Hamiltonian Path&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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* Two paths are ''vertex-independent'' (alternatively, ''internally vertex-disjoint'') if they do not have any internal vertex in common. Similarly, two paths are ''edge-independent'' (or ''edge-disjoint'') if they do not have any internal edge in common.  Two internally vertex-disjoint paths are edge-disjoint, but the converse is not necessarily true.&lt;br /&gt;
如果两条路径没有任何共同的内部顶点，则它们与顶点无关(或者，内部顶点不相交)。类似地，如果两条路径没有任何共同的内边，则它们是边独立的(或边不相交)。两条内部顶点不相交的路径是边不相交的，但反过来也不一定是正确的。&lt;br /&gt;
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* The [[Distance (graph theory)|distance]] between two vertices in a graph is the length of a shortest path between them, if one exists, and otherwise the distance is infinity.&lt;br /&gt;
图中两个顶点之间的距离是它们之间最短路径的长度（如果存在），否则该距离是无穷大。&lt;br /&gt;
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* The diameter of a connected graph is the largest distance (defined above) between pairs of vertices of the graph.&lt;br /&gt;
连接图的直径是图的顶点对之间的最大距离（如上定义）。&lt;br /&gt;
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== Finding paths 路径寻找==&lt;br /&gt;
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Several algorithms exist to find [[Shortest path problem|shortest]] and [[Longest path problem|longest]] paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.&lt;br /&gt;
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Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.&lt;br /&gt;
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图的最短路径和最长路径的求解有几种算法，但一个重要区别是前者的计算比后者简单得多。&lt;br /&gt;
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[[Dijkstra's algorithm]] produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the [[Bellman–Ford algorithm]] can be applied to directed graphs with negative edge weights.  The [[Floyd–Warshall algorithm]] can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.&lt;br /&gt;
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Dijkstra's algorithm produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the Bellman–Ford algorithm can be applied to directed graphs with negative edge weights.  The Floyd–Warshall algorithm can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Dijkstra 算法&amp;lt;/font&amp;gt;'''产生了一个从源顶点到无向或有向图中每个顶点的最短路径列表，而'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Bellman-Ford 算法&amp;lt;/font&amp;gt;''' 可以应用于具有负边权重的有向图。利用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Floyd-Warshall 算法&amp;lt;/font&amp;gt;'''可以求出加权有向图中所有顶点对之间的最短路径。&lt;br /&gt;
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== See also 另请参见==&lt;br /&gt;
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* [[Glossary of graph theory]]&lt;br /&gt;
图论词汇&lt;br /&gt;
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* [[Path graph]]&lt;br /&gt;
路径图&lt;br /&gt;
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* [[Polygonal chain]]&lt;br /&gt;
多边形链&lt;br /&gt;
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* [[Shortest path problem]]&lt;br /&gt;
最短路径问题&lt;br /&gt;
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* [[Longest path problem]]&lt;br /&gt;
最长路径问题&lt;br /&gt;
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* [[Dijkstra's algorithm]]&lt;br /&gt;
Dijkstra算法&lt;br /&gt;
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* [[Bellman–Ford algorithm]]&lt;br /&gt;
Bellman–Ford算法&lt;br /&gt;
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* [[Floyd–Warshall algorithm]]&lt;br /&gt;
Floyd–Warshall算法&lt;br /&gt;
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* [[Self-avoiding walk]]&lt;br /&gt;
自我避免的步道&lt;br /&gt;
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*[[Shortest-path graph]]&lt;br /&gt;
最短路径图&lt;br /&gt;
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== References 引用==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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* {{cite book |last1=Bender |first1=Edward A. |last2=Williamson |first2=S. Gill |date=2010 |title=Lists, Decisions and Graphs. With an Introduction to Probability |url=https://books.google.fr/books?id=vaXv_yhefG8C |location= |publisher= |page= |isbn= |author-link= }}&lt;br /&gt;
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* {{cite book |last1=Bondy |first1=J. A. |last2=Murty |first2=U. S. R. |date=1976 |title=Graph Theory with Applications |url=https://archive.org/details/graphtheorywitha0000bond/page/12 |location= |publisher=North Holland |page=[https://archive.org/details/graphtheorywitha0000bond/page/12 12-21] |isbn=0-444-19451-7 |author-link= }}&lt;br /&gt;
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* {{cite book |last=Diestel |first=Reinhard |date=2005 |title=Graph Theory |url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ |location= |publisher=Springer-Verlag |page=6-9 |isbn=3-540-26182-6 |author-link= }}&lt;br /&gt;
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* {{cite book |last=Gibbons |first=A. |date=1985 |title=Algorithmic Graph Theory |url= |location= |publisher=Cambridge University Press |page=5-6 |isbn=0-521-28881-9 |author-link= }}&lt;br /&gt;
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* {{cite book |last1=Korte |first1=Bernhard |last2=Lovász |first2=László |last3=Prömel |first3=Hans Jürgen |last4=Schrijver |first4=Alexander |date=1990 |title=Paths, Flows, and VLSI-Layout |url=https://archive.org/details/pathsflowsvlsila0000unse |location= |publisher=Springer-Verlag |page= |isbn=0-387-52685-4 |author-link= |url-access=registration }}&lt;br /&gt;
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[[Category:Graph theory objects]]&lt;br /&gt;
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Category:Graph theory objects&lt;br /&gt;
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范畴: 图论对象&lt;br /&gt;
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[[Category:Graph connectivity]]&lt;br /&gt;
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Category:Graph connectivity&lt;br /&gt;
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类别: 图形连接性&lt;br /&gt;
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{{interwiki extra|qid=Q917421}}&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Path (graph theory)]]. Its edit history can be viewed at [[路径/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E8%B7%AF%E5%BE%84_path&amp;diff=13613</id>
		<title>路径 path</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E8%B7%AF%E5%BE%84_path&amp;diff=13613"/>
		<updated>2020-09-02T11:59:32Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{for|the family of graphs known as paths|Path graph}}&lt;br /&gt;
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[[File:Snake-in-the-box and Hamiltonian path.svg|thumb|right|&lt;br /&gt;
图1：A three-dimensional [[hypercube graph]] showing a [[Hamiltonian path]] in red, and a [[Snake-in-the-box|longest induced path]] in bold black. 一个三维的[超立方体图表显示一个红色的哈密顿图，和一个黑色的最长诱导路径]]&lt;br /&gt;
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In [[graph theory]], a '''path''' in a [[Graph (discrete mathematics)|graph]] is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] which joins a sequence of [[Vertex (graph theory)|vertices]] which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A '''directed path''' (sometimes called '''dipath'''&amp;lt;ref&amp;gt;Graph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991, [https://books.google.com/books?id=idigH5CTGWAC&amp;amp;pg=PA205 p.205]&amp;lt;/ref&amp;gt;) in a [[directed graph]] is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.&lt;br /&gt;
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In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''中，图中的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;路径 Path&amp;lt;/font&amp;gt;'''是一个有限或无限的边序列，这些边连接着一系列顶点，这些顶点在大多数定义中都是不同的(因为顶点是不同的，所以边也是不同的)。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向图 Directed Graph&amp;lt;/font&amp;gt;'''中的有向路径(有时称为dipath)是一个有限或无限的边序列，它连接一系列不同的顶点，但附加的限制是所有的边都朝向同一个方向。&lt;br /&gt;
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Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced [[algorithm]]ic topics concerning paths in graphs.&lt;br /&gt;
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Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs.&lt;br /&gt;
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路径是图论的基本概念，在大多数图论文本的导论部分都有描述。邦迪和默蒂(1976) ，吉本斯(1985) ，或迪斯特尔(2005)。柯尔特(1990)等人著作中，涵盖了更多关于图中路径的高级算法的话题。&lt;br /&gt;
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== Definitions 定义==&lt;br /&gt;
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=== Walk, trail, path 步道 轨迹 路径===&lt;br /&gt;
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[[File:Trail but not path.svg|200px|thumb|right &lt;br /&gt;
图2：Trail from A to E，but not path 从A到E的轨迹而非路径]]&lt;br /&gt;
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* A '''walk''' is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] which joins a sequence of [[Vertex (graph theory)|vertices]].{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;步道 Walk&amp;lt;/font&amp;gt;'''是连接一系列顶点的有限或无限的边序列&lt;br /&gt;
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Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a graph. A finite walk is a sequence of edges {{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} for which there is a sequence of vertices {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} such that {{nowrap begin}}''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = {''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;}{{nowrap end}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} is the ''vertex sequence'' of the walk. This walk is  ''closed'' if {{nowrap begin}}''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; = ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;{{nowrap end}}, and ''open'' else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.&lt;br /&gt;
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 Let  be a graph. A finite walk is a sequence of edges  for which there is a sequence of vertices  such that ϕ(e&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;) = {v&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;, v&amp;lt;sub&amp;gt;i + 1&amp;lt;/sub&amp;gt;} for .  is the vertex sequence of the walk. This walk is  closed if v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; = v&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;, and open else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or ray) has a first vertex but no last vertex.&lt;br /&gt;
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让我们做一个图{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} 。有限步道是一系列的边{{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}，其顶点序列{{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}。 {{nowrap begin}}''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = {''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;}{{nowrap end}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} 是行走的顶点序列。如果 {{nowrap begin}}''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; = ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;{{nowrap end}} ，则此步道关闭，否则就打开。一个无限步道是一系列的边，它们的类型与这里描述的相同，但是没有第一个顶点或最后一个顶点，而一个半无限步道(或光线)有第一个顶点，但是没有最后一个顶点。&lt;br /&gt;
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* A '''trail''' is a walk in which all edges are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;轨迹 Trail&amp;lt;/font&amp;gt;'''是所有的边缘都清晰可见的步道&lt;br /&gt;
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* A '''path''' is a trail in which all vertices (and therefore also all edges) are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;路径 Path&amp;lt;/font&amp;gt;'''是一条所有顶点(因此也包括所有边)都是不同的轨迹。&lt;br /&gt;
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If {{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} is a finite walk with vertex sequence {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} then ''w'' is said to be a ''walk from'' ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''to'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Similarly for a trail or a path. If there is a finite walk between two ''distinct'' vertices then there is also a finite trail and a finite path between them.&lt;br /&gt;
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If is a finite walk with vertex sequence  then w is said to be a walk from v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; to v&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;. Similarly for a trail or a path. If there is a finite walk between two distinct vertices then there is also a finite trail and a finite path between them.&lt;br /&gt;
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如果 {{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}是有顶点序列 {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}的有限步道，那么 w 就是从  ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''to'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;的步行。同样，对于一条轨迹或者一条路径也是如此。如果在两个不同的顶点之间有一个有限步道，那么在它们之间也有一个有限的轨迹和一个有限的路径。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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Some authors do not require that all vertices of a path be distinct and instead use the term '''simple path''' to refer to such a path.&lt;br /&gt;
&lt;br /&gt;
Some authors do not require that all vertices of a path be distinct and instead use the term simple path to refer to such a path.&lt;br /&gt;
&lt;br /&gt;
有些作者并不要求路径的所有顶点都是不同的，而是使用术语“简单路径”来指代这样的路径。&lt;br /&gt;
&lt;br /&gt;
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A [[weighted graph]] associates a value (''weight'') with every edge in the graph. The ''weight of a walk'' (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight.&lt;br /&gt;
&lt;br /&gt;
A weighted graph associates a value (weight) with every edge in the graph. The weight of a walk (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.&lt;br /&gt;
&lt;br /&gt;
一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;加权图 Weighted Graph&amp;lt;/font&amp;gt;'''将一个值(权)与图中的每条边相关联。一个加权图中的步道(或轨迹或路径)的权是所有边的权之和。有时，“成本”或“长度”这两个词用来代替“权重”。&lt;br /&gt;
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=== Directed walk, trail, path 有向的步道，轨迹，路径===&lt;br /&gt;
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* A '''directed walk''' is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] directed in the same direction which joins a sequence of [[Vertex (graph theory)|vertices]].{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向步道 Directed Walk&amp;lt;/font&amp;gt;'''是沿相同方向定向的边的有限或无限序列，该边连接一系列顶点。&lt;br /&gt;
&lt;br /&gt;
Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a directed graph. A finite directed walk is a sequence of edges {{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} for which there is a sequence of vertices {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} such that {{nowrap|1=''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = (''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;)}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} is the ''vertex sequence'' of the directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.&lt;br /&gt;
&lt;br /&gt;
Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a directed graph. A finite directed walk is a sequence of edges  for which there is a sequence of vertices  such that  for .  is the vertex sequence of the directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or ray) has a first vertex but no last vertex.&lt;br /&gt;
&lt;br /&gt;
一个有向图{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}}。有限有向步道是一系列的边 {{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}，对于这些边，存在一系列的顶点 {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}。{{nowrap|1=''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = (''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;)}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}是有向步道的顶点序列。无限有向步道是一个边序列，其类型与本文描述的相同，但没有第一个顶点或最后一个顶点，而半无限有向步道(或射线)有第一个顶点，但没有最后一个顶点。&lt;br /&gt;
&lt;br /&gt;
* A '''directed trail''' is a directed walk in which all edges are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向轨迹 Directed Trail&amp;lt;/font&amp;gt;'''是指所有边缘都清晰可见的有向轨迹。&lt;br /&gt;
&lt;br /&gt;
* A '''directed path''' is a directed trail in which all vertices are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向路径 Directed Path&amp;lt;/font&amp;gt;'''是指所有顶点都是不同的有向路径。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
If {{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} is a finite directed walk with vertex sequence {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} then ''w'' is said to be a ''walk from'' ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''to'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Similarly for a directed trail or a path. If there is a finite directed walk between two ''distinct'' vertices then there is also a finite directed trail and a finite directed path between them.&lt;br /&gt;
&lt;br /&gt;
If  is a finite directed walk with vertex sequence  then w is said to be a walk from v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; to v&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;. Similarly for a directed trail or a path. If there is a finite directed walk between two distinct vertices then there is also a finite directed trail and a finite directed path between them.&lt;br /&gt;
&lt;br /&gt;
如果{{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}是一个顶点序列有限的{{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}有向步道，那么 w 就是从 ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''到'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;的步行。同样，对于有向轨迹或路径也是如此。如果在两个不同的顶点之间存在有限有向步道，那么在它们之间也存在有限有向轨迹和有限有向路径。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Some authors do not require that all vertices of a directed path be distinct and instead use the term '''simple directed path''' to refer to such a directed path.&lt;br /&gt;
&lt;br /&gt;
Some authors do not require that all vertices of a directed path be distinct and instead use the term simple directed path to refer to such a directed path.&lt;br /&gt;
&lt;br /&gt;
有些作者并不要求有向路径的所有顶点都是不同的，而是用“简单有向路径”这个术语来表示这样一个有向路。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A [[Weighted graph|weighted directed graph]] associates a value (''weight'') with every edge in the directed graph. The ''weight of a directed walk'' (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight.&lt;br /&gt;
&lt;br /&gt;
A weighted directed graph associates a value (weight) with every edge in the directed graph. The weight of a directed walk (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.&lt;br /&gt;
&lt;br /&gt;
一个加权有向图将一个值(权)与有向图中的每条边相关联。加权有向图中有向步道(或路径)的权是有向边权重的和。有时，“成本”或“长度”这两个词用来代替“权重”。&lt;br /&gt;
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== Examples 例子==&lt;br /&gt;
&lt;br /&gt;
* A graph is [[Connectivity (graph theory)|connected]] if there are paths containing each pair of vertices.&lt;br /&gt;
如果路径包含每一对顶点，则图是连通的。&lt;br /&gt;
&lt;br /&gt;
* A directed graph is [[Strongly-connected digraph|strongly connected]] if there are oppositely oriented directed paths containing each pair of vertices.&lt;br /&gt;
如果一个有向图中存在包含每对顶点的相对有向路径，那么这个有向图就是强连通的。&lt;br /&gt;
&lt;br /&gt;
* A path such that no graph edges connect two nonconsecutive path vertices is called an [[induced path]].&lt;br /&gt;
没有图的边连接两个不连续的路径顶点的路径称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;诱导路径 Induced Path&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
&lt;br /&gt;
* A path that includes every vertex of the graph is known as a [[Hamiltonian path]].&lt;br /&gt;
包含图的每个顶点的路径称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;哈密顿路径 Hamiltonian Path&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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* Two paths are ''vertex-independent'' (alternatively, ''internally vertex-disjoint'') if they do not have any internal vertex in common. Similarly, two paths are ''edge-independent'' (or ''edge-disjoint'') if they do not have any internal edge in common.  Two internally vertex-disjoint paths are edge-disjoint, but the converse is not necessarily true.&lt;br /&gt;
如果两条路径没有任何共同的内部顶点，则它们与顶点无关(或者，内部顶点不相交)。类似地，如果两条路径没有任何共同的内边，则它们是边独立的(或边不相交)。两条内部顶点不相交的路径是边不相交的，但反过来也不一定是正确的。&lt;br /&gt;
&lt;br /&gt;
* The [[Distance (graph theory)|distance]] between two vertices in a graph is the length of a shortest path between them, if one exists, and otherwise the distance is infinity.&lt;br /&gt;
图中两个顶点之间的距离是它们之间最短路径的长度（如果存在），否则该距离是无穷大。&lt;br /&gt;
&lt;br /&gt;
* The diameter of a connected graph is the largest distance (defined above) between pairs of vertices of the graph.&lt;br /&gt;
连接图的直径是图的顶点对之间的最大距离（如上定义）。&lt;br /&gt;
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== Finding paths 路径寻找==&lt;br /&gt;
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Several algorithms exist to find [[Shortest path problem|shortest]] and [[Longest path problem|longest]] paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.&lt;br /&gt;
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Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.&lt;br /&gt;
&lt;br /&gt;
图的最短路径和最长路径的求解有几种算法，但一个重要区别是前者的计算比后者简单得多。&lt;br /&gt;
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[[Dijkstra's algorithm]] produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the [[Bellman–Ford algorithm]] can be applied to directed graphs with negative edge weights.  The [[Floyd–Warshall algorithm]] can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.&lt;br /&gt;
&lt;br /&gt;
Dijkstra's algorithm produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the Bellman–Ford algorithm can be applied to directed graphs with negative edge weights.  The Floyd–Warshall algorithm can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Dijkstra 算法&amp;lt;/font&amp;gt;'''产生了一个从源顶点到无向或有向图中每个顶点的最短路径列表，而'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Bellman-Ford 算法&amp;lt;/font&amp;gt;''' 可以应用于具有负边权重的有向图。利用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; Floyd-Warshall 算法&amp;lt;/font&amp;gt;'''可以求出加权有向图中所有顶点对之间的最短路径。&lt;br /&gt;
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== See also 另请参见==&lt;br /&gt;
&lt;br /&gt;
* [[Glossary of graph theory]]&lt;br /&gt;
图论词汇&lt;br /&gt;
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* [[Path graph]]&lt;br /&gt;
路径图&lt;br /&gt;
&lt;br /&gt;
* [[Polygonal chain]]&lt;br /&gt;
多边形链&lt;br /&gt;
&lt;br /&gt;
* [[Shortest path problem]]&lt;br /&gt;
最短路径问题&lt;br /&gt;
&lt;br /&gt;
* [[Longest path problem]]&lt;br /&gt;
最长路径问题&lt;br /&gt;
&lt;br /&gt;
* [[Dijkstra's algorithm]]&lt;br /&gt;
Dijkstra算法&lt;br /&gt;
&lt;br /&gt;
* [[Bellman–Ford algorithm]]&lt;br /&gt;
Bellman–Ford算法&lt;br /&gt;
&lt;br /&gt;
* [[Floyd–Warshall algorithm]]&lt;br /&gt;
Floyd–Warshall算法&lt;br /&gt;
&lt;br /&gt;
* [[Self-avoiding walk]]&lt;br /&gt;
自我避免的步道&lt;br /&gt;
&lt;br /&gt;
*[[Shortest-path graph]]&lt;br /&gt;
最短路径图&lt;br /&gt;
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== References 引用==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
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* {{cite book |last1=Bender |first1=Edward A. |last2=Williamson |first2=S. Gill |date=2010 |title=Lists, Decisions and Graphs. With an Introduction to Probability |url=https://books.google.fr/books?id=vaXv_yhefG8C |location= |publisher= |page= |isbn= |author-link= }}&lt;br /&gt;
&lt;br /&gt;
* {{cite book |last1=Bondy |first1=J. A. |last2=Murty |first2=U. S. R. |date=1976 |title=Graph Theory with Applications |url=https://archive.org/details/graphtheorywitha0000bond/page/12 |location= |publisher=North Holland |page=[https://archive.org/details/graphtheorywitha0000bond/page/12 12-21] |isbn=0-444-19451-7 |author-link= }}&lt;br /&gt;
&lt;br /&gt;
* {{cite book |last=Diestel |first=Reinhard |date=2005 |title=Graph Theory |url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ |location= |publisher=Springer-Verlag |page=6-9 |isbn=3-540-26182-6 |author-link= }}&lt;br /&gt;
&lt;br /&gt;
* {{cite book |last=Gibbons |first=A. |date=1985 |title=Algorithmic Graph Theory |url= |location= |publisher=Cambridge University Press |page=5-6 |isbn=0-521-28881-9 |author-link= }}&lt;br /&gt;
&lt;br /&gt;
* {{cite book |last1=Korte |first1=Bernhard |last2=Lovász |first2=László |last3=Prömel |first3=Hans Jürgen |last4=Schrijver |first4=Alexander |date=1990 |title=Paths, Flows, and VLSI-Layout |url=https://archive.org/details/pathsflowsvlsila0000unse |location= |publisher=Springer-Verlag |page= |isbn=0-387-52685-4 |author-link= |url-access=registration }}&lt;br /&gt;
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[[Category:Graph theory objects]]&lt;br /&gt;
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Category:Graph theory objects&lt;br /&gt;
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范畴: 图论对象&lt;br /&gt;
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[[Category:Graph connectivity]]&lt;br /&gt;
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Category:Graph connectivity&lt;br /&gt;
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类别: 图形连接性&lt;br /&gt;
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{{interwiki extra|qid=Q917421}}&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Path (graph theory)]]. Its edit history can be viewed at [[路径/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%9B%BE%E7%9A%84%E7%BB%98%E5%88%B6_Graph_drawing&amp;diff=13612</id>
		<title>图的绘制 Graph drawing</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%9B%BE%E7%9A%84%E7%BB%98%E5%88%B6_Graph_drawing&amp;diff=13612"/>
		<updated>2020-09-02T11:54:42Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{about|the general subject of graph drawing|the annual research symposium|International Symposium on Graph Drawing}}&lt;br /&gt;
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[[File:WorldWideWebAroundWikipedia.png|thumb|&lt;br /&gt;
图1：Graphic representation of a minute fraction of the [[WWW]], demonstrating [[hyperlink]]s. 以图的形式展示万维网的一小部分]]&lt;br /&gt;
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'''Graph drawing''' is an area of [[mathematics]] and [[computer science]] combining methods from [[geometric graph theory]] and [[information visualization]] to derive two-dimensional depictions of [[graph (discrete mathematics)|graph]]s arising from applications such as [[social network analysis]], [[cartography]], [[linguistics]], and [[bioinformatics]].&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, pp. vii–viii; {{harvtxt|Herman|Melançon|Marshall|2000}}, Section 1.1, &amp;quot;Typical Application Areas&amp;quot;.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图的绘制 Graph Drawing&amp;lt;/font&amp;gt;'''是 '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;数学 Mathematics&amp;lt;/font&amp;gt;''' 和 '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;计算机科学 Computer Science&amp;lt;/font&amp;gt;'''的一个交叉领域，它结合了'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;几何图形理论 Geometric Graph Theory&amp;lt;/font&amp;gt;'''和 '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;信息可视化 Information Visualization&amp;lt;/font&amp;gt;'''的方法，对来自社会网络分析、制图学、语言学和生物信息学等各种应用场景进行二维描述。&lt;br /&gt;
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A drawing of a graph or '''network diagram''' is a pictorial representation of the [[vertex (graph theory)|vertices]] and [[edge (graph theory)|edges]] of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.&amp;lt;ref name=&amp;quot;dett6&amp;quot;&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p. 6.&amp;lt;/ref&amp;gt;  In the abstract, all that matters is which pairs of vertices are connected by edges.  In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and [[aesthetics]].&amp;lt;ref name=&amp;quot;dett-viii&amp;quot;/&amp;gt; The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's mental map.&amp;lt;ref&amp;gt;{{harvtxt|Misue|Eades|Lai|Sugiyama|1995}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.  In the abstract, all that matters is which pairs of vertices are connected by edges.  In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics.The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's mental map.&lt;br /&gt;
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图形或'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络图 Network Diagram&amp;lt;/font&amp;gt;'''的绘制是通过从图形抽象出的顶点和边表示的。这个图不应与图形本身混淆: 非常不同的图可以对应于同一个图形。抽象地说，重要的是哪一对顶点是由边连在一起。然而，在具体的绘图中，这些顶点和边的排列影响了它的可理解性、可用性、制造成本和美学。如果图形通过添加和删除边缘(动态绘制图形)而随时间变化，并且目标是保存用户的心理映射，那么问题会变得更糟。&lt;br /&gt;
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==Graphical conventions 画图公约==&lt;br /&gt;
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[[File:4node-digraph-natural.svg|thumb|upright=0.5|&lt;br /&gt;
图2：[[Directed graph]] with arrowheads showing edge directions 带箭头显示边方向的有向图]]&lt;br /&gt;
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Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as [[line segment]]s, [[Polygonal chain|polylines]], or curves in the [[Euclidean plane]].&amp;lt;ref name=&amp;quot;dett-viii&amp;quot;&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p. viii.&amp;lt;/ref&amp;gt; Node–link diagrams can be traced back to the 13th century work of [[Ramon Llull]], who drew diagrams of this type for [[complete graph]]s in order to analyze all pairwise combinations among sets of metaphysical concepts.&amp;lt;ref&amp;gt;{{citation|contribution=Two thousand years of combinatorics|first=Donald E.|last=Knuth|authorlink=Donald Knuth|pages=7–37|title=Combinatorics: Ancient and Modern|publisher=Oxford University Press|year=2013|editor1-first=Robin|editor1-last=Wilson|editor2-first=John J.|editor2-last=Watkins}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as line segments, polylines, or curves in the Euclidean plane. Node–link diagrams can be traced back to the 13th century work of Ramon Llull, who drew diagrams of this type for complete graphs in order to analyze all pairwise combinations among sets of metaphysical concepts.&lt;br /&gt;
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图经常绘制成节点链接图，其中顶点可表示为圆盘、框或文本标签，边表示为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;欧几里德平面 Euclidean Plane&amp;lt;/font&amp;gt;'''上的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;线段 Line Segments&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;折线 Polylines&amp;lt;/font&amp;gt;'''或曲线。节点链接图可以追溯到13世纪雷蒙 · 卢尔的作品，他为得到完全图，绘制了这种类型的图，以便分析形而上学概念集合之间的所有成对组合。&lt;br /&gt;
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In the case of [[directed graph]]s, [[Arrow (symbol)|arrowheads]] form a commonly used graphical convention to show their [[Orientability|orientation]];&amp;lt;ref name=&amp;quot;dett6&amp;quot;/&amp;gt; however, user studies have shown that other conventions such as tapering provide this information more effectively.&amp;lt;ref&amp;gt;{{harvtxt|Holten|van Wijk|2009}}; {{harvtxt|Holten|Isenberg|van Wijk|Fekete|2011}}.&amp;lt;/ref&amp;gt; [[Upward planar drawing]] uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary.&amp;lt;ref&amp;gt;{{harvtxt|Garg|Tamassia|1995}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In the case of directed graphs, arrowheads form a commonly used graphical convention to show their orientation; Upward planar drawing uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向图 Directed Graphs&amp;lt;/font&amp;gt;'''的情况下，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;箭头 Arrowheads&amp;lt;/font&amp;gt;'''约定俗成的来显示它们的方向; '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;向上平面绘图 Upward Planar Drawing&amp;lt;/font&amp;gt;'''使用以下约定：每个边都从较低的顶点定向到较高的顶点，因此不需要箭头。&lt;br /&gt;
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Alternative conventions to node–link diagrams include adjacency representations such as [[circle packing theorem|circle packings]], in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; [[intersection graph|intersection representations]] in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other; confluent drawings, in which edges are represented as smooth curves within mathematical [[train track (mathematics)|train tracks]]; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines;&amp;lt;ref name=&amp;quot;Longabaugh 2012&amp;quot;&amp;gt;{{harvtxt|Longabaugh|2012}}.&amp;lt;/ref&amp;gt; and visualizations of the [[adjacency matrix]] of the graph.&lt;br /&gt;
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Alternative conventions to node–link diagrams include adjacency representations such as circle packings, in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; intersection representations in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other; confluent drawings, in which edges are represented as smooth curves within mathematical train tracks; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines; and visualizations of the adjacency matrix of the graph.&lt;br /&gt;
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节点链接图可被其他画图方式替代，包括邻接表示法，如'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;圆填充 Circle Packings&amp;lt;/font&amp;gt;'''，其中顶点表示为平面上不相交的区域，边表示为区域之间的邻接; '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;交点表示 Intersection representations&amp;lt;/font&amp;gt;'''，其中顶点由非相交的几何对象表示，而边线由其交点表示；可见性表示，其中顶点由平面中的区域表示，边由彼此无障碍视线的区域表示; 汇流图，其中边表示为数学轨迹上的光滑曲线; 织物，其中顶点表示为水平线，边表示为垂直线; 以及图形'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;邻接矩阵 Adjiacency Matrix&amp;lt;/font&amp;gt;'''的可视化结点。&lt;br /&gt;
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==Quality measures 质量度量==&lt;br /&gt;
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Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Section 2.1.2, Aesthetics, pp. 14–16; {{harvtxt|Purchase|Cohen|James|1997}}.&amp;lt;/ref&amp;gt; In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.&lt;br /&gt;
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Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability. In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.&lt;br /&gt;
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为了找到客观评价图形美观性和可用性的方法，已经定义了许多不同的图形质量度量方法。除了指导针对同一图形的不同布局方法之间的选择之外，某些布局方法还尝试直接优化这些度量。&lt;br /&gt;
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[[File:4node-digraph-embed.svg|upright=0.5|thumb|&lt;br /&gt;
图3：[[Planar graph]] drawn without overlapping edges 没有重叠边的平面图]]&lt;br /&gt;
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*The [[Crossing number (graph theory)|crossing number]] of a drawing is the number of pairs of edges that cross each other.  If the graph is [[planar graph|planar]], then it is often convenient to draw it without any edge intersections; that is, in this case, a graph drawing represents a [[graph embedding]]. However, nonplanar graphs frequently arise in applications, so graph drawing algorithms must generally allow for edge crossings.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p 14.&amp;lt;/ref&amp;gt;&lt;br /&gt;
图的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;交叉数 Crossing Number&amp;lt;/font&amp;gt;'''是相互交叉的边对的数目。如果图形是平面的，那么通常可以方便地画出没有任何边缘相交的图形;也就是说，在本例中，图形绘制表示'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图形嵌入 Graph Embedding&amp;lt;/font&amp;gt;'''。然而，应用中经常出现非平面图，因此图形绘制算法通常必须考虑边缘交叉。&lt;br /&gt;
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*The [[Area (graph drawing)|area]] of a drawing is the size of its smallest [[bounding box]], relative to the closest distance between any two vertices. Drawings with smaller area are generally preferable to those with larger area, because they allow the features of the drawing to be shown at greater size and therefore more legibly. The [[aspect ratio]] of the bounding box may also be important.&lt;br /&gt;
一个图形的面积是它的最小'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;边界盒 Bounding Box&amp;lt;/font&amp;gt;'''的大小，相对于任意两个顶点之间的最近距离。面积小的画通常比面积大的画更可取，因为它们可以让图画的特征以更大的尺寸显示，因此更清晰。边界框的纵横比也很重要。&lt;br /&gt;
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*Symmetry display is the problem of finding [[Graph automorphism|symmetry group]]s within a given graph, and finding a drawing that displays as much of the symmetry as possible. Some layout methods automatically lead to symmetric drawings; alternatively, some drawing methods start by finding symmetries in the input graph and using them to construct a drawing.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p. 16.&amp;lt;/ref&amp;gt;&lt;br /&gt;
对称显示的问题是在给定的图形中找到'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;对称组 Symmetry Groups&amp;lt;/font&amp;gt;'''，并找到尽可能多地显示对称的绘图。一些布局方法自动导致对称图形;另外，一些绘图方法从查找输入图形中的对称性开始，并使用它们来构造绘图。&lt;br /&gt;
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*It is important that edges have shapes that are as simple as possible, to make it easier for the eye to follow them. In polyline drawings, the complexity of an edge may be measured by its [[Bend minimization|number of bends]], and many methods aim to provide drawings with few total bends or few bends per edge. Similarly for spline curves the complexity of an edge may be measured by the number of control points on the edge.&lt;br /&gt;
重要的是，边缘的形状要尽可能简单，以使眼睛更容易跟踪它们。在折线图中，一条边的复杂性可以通过它的弯曲数来衡量，许多方法的目的是提供很少的总弯曲或每条边很少弯曲的图。类似地，对于样条曲线，边的复杂性可以通过边上控制点的数量来度量。&lt;br /&gt;
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*Several commonly used quality measures concern lengths of edges: it is generally desirable to minimize the total length of the edges as well as the maximum length of any edge. Additionally, it may be preferable for the lengths of edges to be uniform rather than highly varied.&lt;br /&gt;
几种常用的质量度量方法与边的长度有关:通常需要最小化边的总长度以及任何边的最大长度。此外，最好使边缘的长度一致而不是变化很大。&lt;br /&gt;
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*[[Angular resolution (graph drawing)|Angular resolution]] is a measure of the sharpest angles in a graph drawing. If a graph has vertices with high [[degree (graph theory)|degree]] then it necessarily will have small angular resolution, but the angular resolution can be bounded below by a function of the degree.&amp;lt;ref name=&amp;quot;ps09&amp;quot;&amp;gt;{{harvtxt|Pach|Sharir|2009}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;角度分辨率 Angular Resolution&amp;lt;/font&amp;gt;'''是图形绘制中最锐角的度量。如果一个图的顶点高度高，那么它的角分辨率就会很小，但是角分辨率可以由角的函数来限定。&lt;br /&gt;
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*The [[slope number]] of a graph is the minimum number of distinct edge slopes needed in a drawing with straight line segment edges (allowing crossings). [[Cubic graph]]s have slope number at most four, but graphs of degree five may have unbounded slope number; it remains open whether the slope number of degree-4 graphs is bounded.&amp;lt;ref name=&amp;quot;ps09&amp;quot;/&amp;gt;&lt;br /&gt;
图的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;斜率数 Slope Number&amp;lt;/font&amp;gt;'''是在具有直线段边(允许交叉)的图中所需的明显边缘斜率的最小值。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;三次图 Cubic Graphs&amp;lt;/font&amp;gt;'''的斜率数最多为4，五次图的斜率数可能是无界的;4度图的斜率数是否有界仍然是开放的。&lt;br /&gt;
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==Layout methods 布局方法==&lt;br /&gt;
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[[File:Social Network Analysis Visualization.png|thumb|right|&lt;br /&gt;
图4：A force-based network visualization. 基于力的网络可视化 &amp;lt;ref&amp;gt;Published in {{Cite journal | volume = 10| issue = 3| last = Grandjean| first = Martin| title = La connaissance est un réseau| journal =Les Cahiers du Numérique| accessdate = 2014-10-15| date = 2014| pages = 37–54| url = http://www.cairn.info/resume.php?ID_ARTICLE=LCN_103_0037| doi=10.3166/lcn.10.3.37-54}}&amp;lt;/ref&amp;gt;]]&lt;br /&gt;
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There are many different graph layout strategies:&lt;br /&gt;
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There are many different graph layout strategies:&lt;br /&gt;
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有许多不同的图形布局策略:&lt;br /&gt;
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*In [[force-based layout]] systems, the graph drawing software modifies an initial vertex placement by continuously moving the vertices according to a system of forces based on physical metaphors related to systems of [[spring (device)|springs]] or [[molecular mechanics]]. Typically, these systems combine attractive forces between adjacent vertices with repulsive forces between all pairs of vertices, in order to seek a layout in which edge lengths are small while vertices are well-separated. These systems may perform [[gradient descent]] based minimization of an [[energy function]], or they may translate the forces directly into velocities or accelerations for the moving vertices.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Section 2.7, &amp;quot;The Force-Directed Approach&amp;quot;, pp. 29–30, and Chapter 10, &amp;quot;Force-Directed Methods&amp;quot;, pp. 303–326.&amp;lt;/ref&amp;gt;&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'基于力的布局系统 Force-Based Layout&amp;lt;/font&amp;gt;''中，图形绘制软件通过根据基于弹簧系统或分子力学的物理隐喻的力系统不断移动顶点来修改初始顶点的位置。通常，这些系统将相邻顶点之间的吸引力与所有对顶点之间的排斥力结合起来，以寻求一种边长较小而顶点分离良好的布局。这些系统可以执行'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;梯度下降 Gradient Descent&amp;lt;/font&amp;gt;'''最小化的一个能量函数，或者他们可以把力直接转化为节点的速度或加速度&lt;br /&gt;
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*[[Spectral layout]] methods use as coordinates the [[eigenvector]]s of a [[Matrix (mathematics)|matrix]] such as the [[Discrete Laplace operator|Laplacian]] derived from the [[adjacency matrix]] of the graph.&amp;lt;ref&amp;gt;{{harvtxt|Beckman|1994}}; {{harvtxt|Koren|2005}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;频谱布局方法 Spectral Layout&amp;lt;/font&amp;gt;'''使用从图的邻接矩阵派生的矩阵（例如Laplacian）的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;特征向量 Eigenvector&amp;lt;/font&amp;gt;'''作为坐标。&lt;br /&gt;
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*Orthogonal layout methods, which allow the edges of the graph to run horizontally or vertically, parallel to the coordinate axes of the layout. These methods were originally designed for [[VLSI]] and [[Printed circuit board|PCB]] layout problems but they have also been adapted for graph drawing. They typically involve a multiphase approach in which an input graph is planarized by replacing crossing points by vertices, a topological embedding of the planarized graph is found, edge orientations are chosen to minimize bends, vertices are placed consistently with these orientations, and finally a layout compaction stage reduces the area of the drawing.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Chapter 5, &amp;quot;Flow and Orthogonal Drawings&amp;quot;, pp. 137–170; {{harv|Eiglsperger|Fekete|Klau|2001}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
正交布局方法，允许图形的边缘水平或垂直运行，平行于布局的坐标轴。这些方法最初是为超大规模集成电路和PCB布局问题而设计的，但它们也适用于图形绘制。它们通常涉及到一种多阶段的方法，通过用顶点替换交叉点来平面化输入图，找到一个平面化图的拓扑嵌入，选择边的方向来最小化弯曲，顶点的放置始终与这些方向一致，最后一个布局压缩阶段减少&lt;br /&gt;
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*Tree layout algorithms these show a rooted [[tree structure|tree]]-like formation, suitable for [[tree (graph theory)|trees]]. Often, in a technique called &amp;quot;balloon layout&amp;quot;, the children of each node in the tree are drawn on a circle surrounding the node, with the radii of these circles diminishing at lower levels in the tree so that these circles do not overlap.&amp;lt;ref&amp;gt;{{harvtxt|Herman|Melançon|Marshall|2000}}, Section 2.2, &amp;quot;Traditional Layout – An Overview&amp;quot;.&amp;lt;/ref&amp;gt;&lt;br /&gt;
这些树布局算法显示了一种类似树的根状结构，适合树。通常，在一种称为气球布局的技术中，树中每个节点的子节点被画在围绕该节点的圆上，这些圆的半径在树的较低层次上递减，以便这些圆不会重叠。&lt;br /&gt;
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*[[Layered graph drawing]] methods (often called Sugiyama-style drawing) are best suited for [[directed acyclic graph]]s or graphs that are nearly acyclic, such as the graphs of dependencies between modules or functions in a software system. In these methods, the nodes of the graph are arranged into horizontal layers using methods such as the [[Coffman–Graham algorithm]], in such a way that most edges go downwards from one layer to the next; after this step, the nodes within each layer are arranged in order to minimize crossings.&amp;lt;ref&amp;gt;{{harvtxt|Sugiyama|Tagawa|Toda|1981}}; {{harvtxt|Bastert|Matuszewski|2001}}; {{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Chapter 9, &amp;quot;Layered Drawings of Digraphs&amp;quot;, pp. 265–302.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分层图绘制方法 Layered Graph Drawing&amp;lt;/font&amp;gt;'''(通常称为杉山式图)最适合于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向无环图 Directed Acyclic Graphs&amp;lt;/font&amp;gt;'''或接近无环的图，例如软件系统中模块或函数之间的依赖关系图。在这些方法中，图的节点使用Coffman Graham算法等方法被安排到水平层中，以这样的方式，大多数边从一层向下到下一层;在这一步之后，将每一层内的节点进行排列，以最小化交叉。&lt;br /&gt;
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[[File:Goldner-Harary-linear.svg|thumb|&lt;br /&gt;
图5：Arc diagram 弧线图]]&lt;br /&gt;
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*[[Arc diagram]]s, a layout style dating back to the 1960s,&amp;lt;ref&amp;gt;{{harvtxt|Saaty|1964}}.&amp;lt;/ref&amp;gt; place vertices on a line; edges may be drawn as semicircles above or below the line, or as smooth curves linked together from multiple semicircles.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;弧图 Arc Diagram&amp;lt;/font&amp;gt;'''，一种布局风格可以追溯到20世纪60年代，将顶点放在一条直线上;边缘可以画成在直线上方或下方的半圆，也可以画成由多个半圆连接在一起的光滑曲线。&lt;br /&gt;
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*[[Circular layout]] methods place the vertices of the graph on a circle, choosing carefully the ordering of the vertices around the circle to reduce crossings and place adjacent vertices close to each other. Edges may be drawn either as chords of the circle or as arcs inside or outside of the circle. In some cases, multiple circles may be used.&amp;lt;ref&amp;gt;{{harvtxt|Doğrusöz|Madden|Madden|1997}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;圆形布局方法 Circular Layout&amp;lt;/font&amp;gt;'''将图形的顶点放置在一个圆上，仔细选择圆周上顶点的顺序，以减少交叉，并将相邻的顶点放置在彼此接近的位置。边可以画成圆的弦，也可以画成圆内外的弧线。在某些情况下，可以使用多个圆。&lt;br /&gt;
&lt;br /&gt;
*[[Dominance drawing]] places vertices in such a way that one vertex is upwards, rightwards, or both of another if and only if it is [[reachability|reachable]] from the other vertex. In this way, the layout style makes the reachability relation of the graph visually apparent.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Section 4.7, &amp;quot;Dominance Drawings&amp;quot;, pp. 112–127.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;优势图 Dominance Drawing&amp;lt;/font&amp;gt;'''当且仅当一个顶点可以从另一个顶点到达时，优势画顶点的方式是:一个顶点向上，或向右，或两个顶点同时向上。这样，布局风格使得图的可达性关系在视觉上变得明显。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Application-specific graph drawings 专用图==&lt;br /&gt;
&lt;br /&gt;
Graphs and graph drawings arising in other areas of application include&lt;br /&gt;
&lt;br /&gt;
Graphs and graph drawings arising in other areas of application include&lt;br /&gt;
&lt;br /&gt;
在其他应用领域出现的图和图绘制包括&lt;br /&gt;
&lt;br /&gt;
* [[Sociogram]]s, drawings of a [[social network]], as often offered by [[social network analysis software]]&amp;lt;ref&amp;gt;{{Harvard citation text|Scott|2000}}; {{Harvard citation text|Brandes|Freeman|Wagner|2014}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会图 Sociogram&amp;lt;/font&amp;gt;'''，社会网络图，通常由社会网络分析软件提供&lt;br /&gt;
&lt;br /&gt;
* [[Hasse diagram]]s, a type of graph drawing specialized to [[partial order]]s&amp;lt;ref&amp;gt;{{Harvard citation text|Di Battista|Eades|Tamassia|Tollis|1994}}, pp. 15–16, and Chapter 6, &amp;quot;Flow and Upward Planarity&amp;quot;, pp. 171–214; {{Harvard citation text|Freese|2004}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;哈塞图 Hasse Diagram&amp;lt;/font&amp;gt;'''，一种专门用于部分顺序的图形绘制&lt;br /&gt;
&lt;br /&gt;
* [[Dessin d'enfant]]s, a type of graph drawing used in [[algebraic geometry]]{{sfnp|Zapponi|2003}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'Dessin d'enfants&amp;lt;/font&amp;gt;''，代数几何中使用的一种图形画法&lt;br /&gt;
&lt;br /&gt;
* [[State diagram]]s, graphical representations of [[finite-state machine]]s{{sfnp|Anderson|Head|2006}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;状态图 State Diagram&amp;lt;/font&amp;gt;'''，有限状态机的图形表示&lt;br /&gt;
&lt;br /&gt;
* [[Computer network diagram]]s, depictions of the nodes and connections in a [[computer network]]{{sfnp|Di Battista|Rimondini|2014}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;计算机网络图 Computer Network Diagram&amp;lt;/font&amp;gt;'''，描述计算机网络中的节点和连接&lt;br /&gt;
&lt;br /&gt;
* [[Flowchart]]s and [[DRAKON|drakon-charts]], drawings in which the nodes represent the steps of an [[algorithm]] and the edges represent [[control flow]] between steps.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;流程图 Flowchart&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;drakon图&amp;lt;/font&amp;gt;'''，其中节点表示算法的步骤，边表示步骤之间的控制流的绘图。&lt;br /&gt;
&lt;br /&gt;
* [[Data-flow diagram]]s, drawings in which the nodes represent the components of an [[information system]] and the edges represent the movement of information from one component to another.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;数据流程图 Data-Flow Diagram&amp;lt;/font&amp;gt;'''，其中节点表示信息系统的组件，边缘表示信息从一个组件到另一个组件的移动。&lt;br /&gt;
&lt;br /&gt;
* [[Bioinformatics]] including [[phylogenetic tree]]s, [[protein–protein interaction]] networks, and [[metabolic pathway]]s.{{sfnp|Bachmaier|Brandes|Schreiber|2014}}&lt;br /&gt;
生物信息学包括系统发育树，蛋白质，蛋白质相互作用网络，和代谢途径。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In addition, the [[placement (electronic design automation)|placement]] and [[Routing (electronic design automation)|routing]] steps of [[electronic design automation]] (EDA) are similar in many ways to graph drawing, as is the problem of [[greedy embedding]] in [[distributed computing]], and the graph drawing literature includes several results borrowed from the EDA literature. However, these problems also differ in several important ways: for instance, in EDA, area minimization and signal length are more important than aesthetics, and the routing problem in EDA may have more than two terminals per net while the analogous problem in graph drawing generally only involves pairs of vertices for each edge.&lt;br /&gt;
&lt;br /&gt;
In addition, the placement and routing steps of electronic design automation (EDA) are similar in many ways to graph drawing, as is the problem of greedy embedding in distributed computing, and the graph drawing literature includes several results borrowed from the EDA literature. However, these problems also differ in several important ways: for instance, in EDA, area minimization and signal length are more important than aesthetics, and the routing problem in EDA may have more than two terminals per net while the analogous problem in graph drawing generally only involves pairs of vertices for each edge.&lt;br /&gt;
&lt;br /&gt;
此外，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;电子设计自动化（EDA）Electronic Design Automation&amp;lt;/font&amp;gt;'''的布局和布线步骤在很多方面与图形绘制相似，而在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分布式计算 Distributed Computing&amp;lt;/font&amp;gt;'''中'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;贪婪嵌入 Greedy Embedding&amp;lt;/font&amp;gt;'''的问题也是如此，并且图形绘制文献包括从EDA文献中借来的一些结果。然而，这些问题也在几个重要方面有所不同: 例如，在 EDA，面积最小化和信号长度比美观更重要，EDA中的路由问题可能每个网有两个以上的终端，而图绘制中的类似问题通常只涉及每条边的顶点对。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Software 软件==&lt;br /&gt;
&lt;br /&gt;
[[File:Gephi 0.9.1 Network Analysis and Visualization Software.png|thumb|&lt;br /&gt;
图6：A graph drawing interface 一个画图软件界面([[Gephi]] 0.9.1)]]Software, systems, and providers of systems for drawing graphs include:&lt;br /&gt;
&lt;br /&gt;
A graph drawing interface ([[Gephi 0.9.1)]]Software, systems, and providers of systems for drawing graphs include:&lt;br /&gt;
&lt;br /&gt;
软件、系统和绘制图形系统的提供者包括:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!-- DO ''not'' INCLUDE AN ENTRY HERE UNLESS IT ALREADY HAS A BLUE-LINKED ARTICLE --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!-- DO not INCLUDE AN ENTRY HERE UNLESS IT ALREADY HAS A BLUE-LINKED ARTICLE --&amp;gt;&lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
* [[BioFabric]] open-source software for visualizing large networks by drawing nodes as horizontal lines.&lt;br /&gt;
BioFabric 开源软件，用于通过将节点绘制为水平线来可视化大型网络。&lt;br /&gt;
&lt;br /&gt;
* [[Cytoscape]], open-source software for visualizing molecular interaction networks&lt;br /&gt;
Cytoscape 可视化分子相互作用网络的开源软件&lt;br /&gt;
&lt;br /&gt;
* [[Gephi]], open-source network analysis and visualization software&lt;br /&gt;
Gephi 开源网络分析和可视化软件&lt;br /&gt;
&lt;br /&gt;
* [[graph-tool]], a [[Free Software|free/libre]] [[Python (programming language)|Python]] library for analysis of graphs.&lt;br /&gt;
Graph-Tool 一个免费/免费的Python库，用于分析图形&lt;br /&gt;
&lt;br /&gt;
* [[Graphviz]], an open-source graph drawing system from [[AT&amp;amp;T Corporation]]&amp;lt;ref&amp;gt;&amp;quot;Graphviz and Dynagraph – Static and Dynamic Graph Drawing Tools&amp;quot;, by John Ellson, Emden R. Gansner, Eleftherios Koutsofios, Stephen C. North, and Gordon Woodhull, in {{harvtxt| Jünger|Mutzel|2004}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
Graphviz 来自美国电话电报公司的开放源码绘图系统&lt;br /&gt;
&lt;br /&gt;
* [[Linkurious]], a commercial network analysis and visualization software for [[graph databases]]&lt;br /&gt;
Linkurious 图形数据库的商业网络分析和可视化软件&lt;br /&gt;
&lt;br /&gt;
* [[Mathematica]], a general purpose computation tool that includes 2D and 3D graph visualization and graph analysis tools.&amp;lt;ref&amp;gt;[http://reference.wolfram.com/mathematica/ref/GraphPlot.html GraphPlot] Mathematica documentation&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[http://reference.wolfram.com/mathematica/tutorial/GraphDrawingIntroduction.html Graph drawing tutorial]&amp;lt;/ref&amp;gt;&lt;br /&gt;
Mathematica 一个通用的计算工具，包括2D和3D图形可视化和图形分析工具。&lt;br /&gt;
&lt;br /&gt;
* [[Microsoft Automatic Graph Layout]], open-source .NET library (formerly called GLEE) for laying out graphs&amp;lt;ref&amp;gt;{{harvtxt|Nachmanson|Robertson|Lee|2008}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
Microsoft Automatic Graph Layout 用于图表布局的开放源码。net库(以前称为GLEE)&lt;br /&gt;
&lt;br /&gt;
* [[NetworkX]] is a Python library for studying graphs and networks.&lt;br /&gt;
NetworkX 用于研究图和网络的Python库&lt;br /&gt;
&lt;br /&gt;
* [[Tom Sawyer Software]]&amp;lt;ref&amp;gt;{{harvtxt|Madden|Madden|Powers|Himsolt|1996}}.&amp;lt;/ref&amp;gt; Tom Sawyer Perspectives is graphics-based software for building enterprise-class graph and data visualization and analysis applications. It is a Software Development Kit (SDK) with a graphics-based design and preview environment.&lt;br /&gt;
Tom Sawyer Perspectives是一款基于图形的软件，用于构建企业级图形和数据可视化和分析应用程序。它是一个软件开发工具包(SDK)，具有基于图形的设计和预览环境&lt;br /&gt;
&lt;br /&gt;
* [[Tulip (software)]],&amp;lt;ref&amp;gt;&amp;quot;Tulip – A Huge Graph Visualization Framework&amp;quot;, by David Auber, in {{harvtxt| Jünger|Mutzel|2004}}.&amp;lt;/ref&amp;gt; an open source data visualization tool&lt;br /&gt;
Tulip (software)开源数据可视化工具&lt;br /&gt;
yEd 具有图形布局功能的图形编辑器&lt;br /&gt;
&lt;br /&gt;
* [[yEd]], a graph editor with graph layout functionality&amp;lt;ref&amp;gt;&amp;quot;yFiles – Visualization and Automatic Layout of Graphs&amp;quot;, by Roland Wiese, Markus Eiglsperger, and Michael Kaufmann, in {{harvtxt| Jünger|Mutzel|2004}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
yEd 具有图形布局功能的图形编辑器&lt;br /&gt;
&lt;br /&gt;
* [[PGF/TikZ]] 3.0 with the &amp;lt;code&amp;gt;graphdrawing&amp;lt;/code&amp;gt; package (requires [[LuaTeX]]).&amp;lt;ref&amp;gt;{{harvtxt|Tantau|2013}}; see also the older [http://www.tcs.uni-luebeck.de/downloads/mitarbeiter/tantau/2012-gd-presentation.pdf GD 2012 presentation]&amp;lt;/ref&amp;gt;&lt;br /&gt;
PGF/TikZ 带graphdrawing软件包的3.0（需要LuaTeX）&lt;br /&gt;
&lt;br /&gt;
* [[LaNet-vi]], an open-source large network visualization software&lt;br /&gt;
LaNet-vi 开源大型网络可视化软件&lt;br /&gt;
&lt;br /&gt;
* [[Edraw Max]] 2D business technical diagramming software&lt;br /&gt;
Edraw Max 2D商业技术制图软件&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&lt;br /&gt;
;Footnotes&lt;br /&gt;
&lt;br /&gt;
Footnotes&lt;br /&gt;
&lt;br /&gt;
脚注&lt;br /&gt;
&lt;br /&gt;
{{reflist|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
;General references&lt;br /&gt;
&lt;br /&gt;
General references&lt;br /&gt;
&lt;br /&gt;
一般参考资料&lt;br /&gt;
&lt;br /&gt;
{{refbegin|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
最后1个 Di Battista 最初1个 Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
作者: 彼得 · 艾德斯&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
| last 3 Tamassia | first3 Roberto | author3-link Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
4 | last 4 Tollis | first4 Ioannis g.&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Computational Geometry (journal)|Computational Geometry: Theory and Applications]]&lt;br /&gt;
&lt;br /&gt;
 | journal = Computational Geometry: Theory and Applications&lt;br /&gt;
&lt;br /&gt;
计算几何: 理论与应用&lt;br /&gt;
&lt;br /&gt;
 | pages = 235–282&lt;br /&gt;
&lt;br /&gt;
 | pages = 235–282&lt;br /&gt;
&lt;br /&gt;
第235-282页&lt;br /&gt;
&lt;br /&gt;
 | title = Algorithms for Drawing Graphs: an Annotated Bibliography&lt;br /&gt;
&lt;br /&gt;
 | title = Algorithms for Drawing Graphs: an Annotated Bibliography&lt;br /&gt;
&lt;br /&gt;
图形绘制的算法: 附有注释的参考书目&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.cs.brown.edu/people/rt/gd.html&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.cs.brown.edu/people/rt/gd.html&lt;br /&gt;
&lt;br /&gt;
Http://www.cs.brown.edu/people/rt/gd.html&lt;br /&gt;
&lt;br /&gt;
 | volume = 4&lt;br /&gt;
&lt;br /&gt;
 | volume = 4&lt;br /&gt;
&lt;br /&gt;
第四卷&lt;br /&gt;
&lt;br /&gt;
 | issue = 5&lt;br /&gt;
&lt;br /&gt;
 | issue = 5&lt;br /&gt;
&lt;br /&gt;
第五期&lt;br /&gt;
&lt;br /&gt;
 | year = 1994 | doi=10.1016/0925-7721(94)00014-x}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 1994 | doi=10.1016/0925-7721(94)00014-x}}.&lt;br /&gt;
&lt;br /&gt;
1994年 | doi 10.1016 / 0925-7721(94)00014-x }。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
最后1个 Di Battista 最初1个 Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
作者: 彼得 · 艾德斯&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
| last 3 Tamassia | first3 Roberto | author3-link Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
4 | last 4 Tollis | first4 Ioannis g.&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-0-13-301615-4&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-0-13-301615-4&lt;br /&gt;
&lt;br /&gt;
| isbn 978-0-13-301615-4&lt;br /&gt;
&lt;br /&gt;
 | publisher = [[Prentice Hall]]&lt;br /&gt;
&lt;br /&gt;
 | publisher = Prentice Hall&lt;br /&gt;
&lt;br /&gt;
出版商 Prentice Hall&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Algorithms for the Visualization of Graphs&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Algorithms for the Visualization of Graphs&lt;br /&gt;
&lt;br /&gt;
图形绘制: 图形可视化的算法&lt;br /&gt;
&lt;br /&gt;
 | year = 1998}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 1998}}.&lt;br /&gt;
&lt;br /&gt;
1998年。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/2945.841119&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/2945.841119&lt;br /&gt;
&lt;br /&gt;
10.1109 / 2945.841119&lt;br /&gt;
&lt;br /&gt;
 | last1 = Herman | first1 = Ivan&lt;br /&gt;
&lt;br /&gt;
 | last1 = Herman | first1 = Ivan&lt;br /&gt;
&lt;br /&gt;
最后1个赫尔曼最初1个伊凡&lt;br /&gt;
&lt;br /&gt;
 | last2 = Melançon | first2 = Guy&lt;br /&gt;
&lt;br /&gt;
 | last2 = Melançon | first2 = Guy&lt;br /&gt;
&lt;br /&gt;
最后2个梅兰，最初2个盖伊&lt;br /&gt;
&lt;br /&gt;
 | last3 = Marshall | first3 = M. Scott&lt;br /&gt;
&lt;br /&gt;
 | last3 = Marshall | first3 = M. Scott&lt;br /&gt;
&lt;br /&gt;
3 m · 斯科特&lt;br /&gt;
&lt;br /&gt;
 | journal = IEEE Transactions on Visualization and Computer Graphics&lt;br /&gt;
&lt;br /&gt;
 | journal = IEEE Transactions on Visualization and Computer Graphics&lt;br /&gt;
&lt;br /&gt;
可视化与计算机图形学科学杂志&lt;br /&gt;
&lt;br /&gt;
 | pages = 24–43&lt;br /&gt;
&lt;br /&gt;
 | pages = 24–43&lt;br /&gt;
&lt;br /&gt;
第24-43页&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Visualization and Navigation in Information Visualization: A Survey&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Visualization and Navigation in Information Visualization: A Survey&lt;br /&gt;
&lt;br /&gt;
信息可视化的图形可视化和导航: 一个调查&lt;br /&gt;
&lt;br /&gt;
 | issue = 1&lt;br /&gt;
&lt;br /&gt;
 | issue = 1&lt;br /&gt;
&lt;br /&gt;
第一期&lt;br /&gt;
&lt;br /&gt;
 | volume = 6&lt;br /&gt;
&lt;br /&gt;
 | volume = 6&lt;br /&gt;
&lt;br /&gt;
第六卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2000 }}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2000 }}.&lt;br /&gt;
&lt;br /&gt;
2000年开始。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Jünger | first1 = Michael&lt;br /&gt;
&lt;br /&gt;
 | last1 = Jünger | first1 = Michael&lt;br /&gt;
&lt;br /&gt;
1 j nger | first1 Michael&lt;br /&gt;
&lt;br /&gt;
 | last2 = Mutzel | first2 = Petra | author2-link = Petra Mutzel&lt;br /&gt;
&lt;br /&gt;
 | last2 = Mutzel | first2 = Petra | author2-link = Petra Mutzel&lt;br /&gt;
&lt;br /&gt;
2 Petra | author2-link Petra Mutzel&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-00881-1&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-00881-1&lt;br /&gt;
&lt;br /&gt;
[国际标准图书馆编号978-3-540-00881-1]&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing Software&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing Software&lt;br /&gt;
&lt;br /&gt;
图形绘制软件&lt;br /&gt;
&lt;br /&gt;
 | year = 2004}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2004}}.&lt;br /&gt;
&lt;br /&gt;
2004年开始。&lt;br /&gt;
&lt;br /&gt;
*{{citation|title=Drawing Graphs: Methods and Models|editor1-first=Michael|editor1-last=Kaufmann|editor2-first=Dorothea|editor2-last=Wagner|editor2-link=Dorothea Wagner|publisher=Springer-Verlag|series=[[Lecture Notes in Computer Science]]|volume=2025|year=2001|doi=10.1007/3-540-44969-8|isbn=978-3-540-42062-0}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|url=http://cs.brown.edu/~rt/gdhandbook/|year=2014|access-date=2013-08-28|archive-url=https://web.archive.org/web/20130815181243/http://cs.brown.edu/~rt/gdhandbook/|archive-date=2013-08-15|url-status=dead}}.&lt;br /&gt;
&lt;br /&gt;
{{refend}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
;Specialized subtopics&lt;br /&gt;
&lt;br /&gt;
Specialized subtopics&lt;br /&gt;
&lt;br /&gt;
专门的副主题&lt;br /&gt;
&lt;br /&gt;
{{refbegin|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
*{{citation|title=Automata Theory with Modern Applications|first1=James Andrew|last1=Anderson|first2=Thomas J.|last2=Head|publisher=Cambridge University Press|year=2006|isbn=978-0-521-84887-9|pages=38–41|url=https://books.google.com/books?id=ikS8BLdLDxIC&amp;amp;pg=PA38}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Christian|last1=Bachmaier|first2=Ulrik|last2=Brandes|first3=Falk|last3=Schreiber|contribution=Biological Networks|pages=621–651|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|year=2014}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|contribution=Layered drawings of digraphs|first1=Oliver|last1=Bastert|first2=Christian|last2=Matuszewski|title=Drawing Graphs: Methods and Models|editor1-first=Michael|editor1-last=Kaufmann|editor2-first=Dorothea|editor2-last=Wagner|editor2-link=Dorothea Wagner|publisher=Springer-Verlag|series=Lecture Notes in Computer Science|volume=2025|year=2001|pages=87–120|doi=10.1007/3-540-44969-8_5|isbn=978-3-540-42062-0}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Beckman | first = Brian&lt;br /&gt;
&lt;br /&gt;
 | last = Beckman | first = Brian&lt;br /&gt;
&lt;br /&gt;
| 最后一个 Beckman | 第一个 Brian&lt;br /&gt;
&lt;br /&gt;
 | publisher = Microsoft Research&lt;br /&gt;
&lt;br /&gt;
 | publisher = Microsoft Research&lt;br /&gt;
&lt;br /&gt;
微软研究院&lt;br /&gt;
&lt;br /&gt;
 | series = Tech. Report MSR-TR-94-04&lt;br /&gt;
&lt;br /&gt;
 | series = Tech. Report MSR-TR-94-04&lt;br /&gt;
&lt;br /&gt;
系列技术。Msr-tr-94-04报告&lt;br /&gt;
&lt;br /&gt;
 | title = Theory of Spectral Graph Layout&lt;br /&gt;
&lt;br /&gt;
 | title = Theory of Spectral Graph Layout&lt;br /&gt;
&lt;br /&gt;
光谱图布局理论&lt;br /&gt;
&lt;br /&gt;
 | url = http://research.microsoft.com/apps/pubs/default.aspx?id=69611&lt;br /&gt;
&lt;br /&gt;
 | url = http://research.microsoft.com/apps/pubs/default.aspx?id=69611&lt;br /&gt;
&lt;br /&gt;
Http://research.microsoft.com/apps/pubs/default.aspx?id=69611&lt;br /&gt;
&lt;br /&gt;
 | year = 1994}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 1994}}.&lt;br /&gt;
&lt;br /&gt;
1994年。&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Ulrik|last1=Brandes|first2=Linton C.|last2=Freeman|first3=Dorothea|last3=Wagner|author3-link=Dorothea Wagner|contribution=Social Networks|pages=805–839|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|year=2014}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Giuseppe|last1=Di Battista|first2=Massimo|last2=Rimondini|contribution=Computer Networks|pages=763–803|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|year=2014}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Doğrusöz | first1 = Uğur&lt;br /&gt;
&lt;br /&gt;
 | last1 = Doğrusöz | first1 = Uğur&lt;br /&gt;
&lt;br /&gt;
 | last1 = Doğrusöz | first1 = Uğur&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Brendan&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Brendan&lt;br /&gt;
&lt;br /&gt;
2 Madden | first2 Brendan&lt;br /&gt;
&lt;br /&gt;
 | last3 = Madden | first3 = Patrick&lt;br /&gt;
&lt;br /&gt;
 | last3 = Madden | first3 = Patrick&lt;br /&gt;
&lt;br /&gt;
最后3个马登最初3个帕特里克&lt;br /&gt;
&lt;br /&gt;
 | editor-last = North | editor-first = Stephen&lt;br /&gt;
&lt;br /&gt;
 | editor-last = North | editor-first = Stephen&lt;br /&gt;
&lt;br /&gt;
| 编辑-最后的北 | 编辑-第一斯蒂芬&lt;br /&gt;
&lt;br /&gt;
 | contribution = Circular layout in the Graph Layout toolkit&lt;br /&gt;
&lt;br /&gt;
 | contribution = Circular layout in the Graph Layout toolkit&lt;br /&gt;
&lt;br /&gt;
| 贡献图形布局工具包中的循环布局&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-62495-3_40&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-62495-3_40&lt;br /&gt;
&lt;br /&gt;
10.1007 / 3-540-62495-340&lt;br /&gt;
&lt;br /&gt;
 | pages = 92–100&lt;br /&gt;
&lt;br /&gt;
 | pages = 92–100&lt;br /&gt;
&lt;br /&gt;
第92-100页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Symposium on Graph Drawing, GD '96 Berkeley, California, USA, September 18–20, 1996, Proceedings&lt;br /&gt;
&lt;br /&gt;
 | title = Symposium on Graph Drawing, GD '96 Berkeley, California, USA, September 18–20, 1996, Proceedings&lt;br /&gt;
&lt;br /&gt;
图形绘制研讨会，美国加州伯克利，1996年9月18-20日，美国国家科学院院刊&lt;br /&gt;
&lt;br /&gt;
 | volume = 1190&lt;br /&gt;
&lt;br /&gt;
 | volume = 1190&lt;br /&gt;
&lt;br /&gt;
第1190卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1997| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
 | year = 1997| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
1997年标题链接国际图形绘制专题研讨会&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-62495-0&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-62495-0&lt;br /&gt;
&lt;br /&gt;
| isbn 978-3-540-62495-0&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
免费访问&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Eiglsperger | first1 = Markus&lt;br /&gt;
&lt;br /&gt;
 | last1 = Eiglsperger | first1 = Markus&lt;br /&gt;
&lt;br /&gt;
1 | last 1 Eiglsperger | first1 Markus&lt;br /&gt;
&lt;br /&gt;
 | last2 = Fekete | first2 = Sándor&lt;br /&gt;
&lt;br /&gt;
 | last2 = Fekete | first2 = Sándor&lt;br /&gt;
&lt;br /&gt;
 | last2 = Fekete | first2 = Sándor&lt;br /&gt;
&lt;br /&gt;
 | last3 = Klau | first3 = Gunnar&lt;br /&gt;
&lt;br /&gt;
 | last3 = Klau | first3 = Gunnar&lt;br /&gt;
&lt;br /&gt;
3 Klau | first3 Gunnar&lt;br /&gt;
&lt;br /&gt;
 | contribution = Orthogonal graph drawing&lt;br /&gt;
&lt;br /&gt;
 | contribution = Orthogonal graph drawing&lt;br /&gt;
&lt;br /&gt;
| 贡献正交图绘制&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-44969-8_6&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-44969-8_6&lt;br /&gt;
&lt;br /&gt;
10.1007 / 3-540-44969-86&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Kaufmann | editor1-first = Michael&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Kaufmann | editor1-first = Michael&lt;br /&gt;
&lt;br /&gt;
1-last Kaufmann | editor1-first Michael&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Wagner | editor2-first = Dorothea | editor2-link = Dorothea Wagner&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Wagner | editor2-first = Dorothea | editor2-link = Dorothea Wagner&lt;br /&gt;
&lt;br /&gt;
2- 最后一位瓦格纳 | 2- 第一位多萝西娅 | 2- 链接多萝西娅 · 瓦格纳&lt;br /&gt;
&lt;br /&gt;
 | pages = 121–171&lt;br /&gt;
&lt;br /&gt;
 | pages = 121–171&lt;br /&gt;
&lt;br /&gt;
第121-171页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer Berlin / Heidelberg&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer Berlin / Heidelberg&lt;br /&gt;
&lt;br /&gt;
| 出版商斯普林格 · 柏林 / 海德堡&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing Graphs&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing Graphs&lt;br /&gt;
&lt;br /&gt;
| 题目绘制图形&lt;br /&gt;
&lt;br /&gt;
 | volume = 2025&lt;br /&gt;
&lt;br /&gt;
 | volume = 2025&lt;br /&gt;
&lt;br /&gt;
2025年&lt;br /&gt;
&lt;br /&gt;
 | year = 2001| isbn = 978-3-540-42062-0&lt;br /&gt;
&lt;br /&gt;
 | year = 2001| isbn = 978-3-540-42062-0&lt;br /&gt;
&lt;br /&gt;
2001 | isbn 978-3-540-42062-0&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Freese | first = Ralph&lt;br /&gt;
&lt;br /&gt;
 | last = Freese | first = Ralph&lt;br /&gt;
&lt;br /&gt;
最后一个自由 | 第一个拉尔夫&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Eklund | editor-first = Peter&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Eklund | editor-first = Peter&lt;br /&gt;
&lt;br /&gt;
| 编辑-最后埃克伦 | 编辑-第一彼得&lt;br /&gt;
&lt;br /&gt;
 | contribution = Automated lattice drawing&lt;br /&gt;
&lt;br /&gt;
 | contribution = Automated lattice drawing&lt;br /&gt;
&lt;br /&gt;
自动点阵绘图&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-24651-0_12&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-24651-0_12&lt;br /&gt;
&lt;br /&gt;
10.1007 / 978-3-540-24651-012&lt;br /&gt;
&lt;br /&gt;
 | pages = 589–590&lt;br /&gt;
&lt;br /&gt;
 | pages = 589–590&lt;br /&gt;
&lt;br /&gt;
第589-590页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Concept Lattices: Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, February 23-26, 2004, Proceedings&lt;br /&gt;
&lt;br /&gt;
 | title = Concept Lattices: Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, February 23-26, 2004, Proceedings&lt;br /&gt;
&lt;br /&gt;
| 题目概念格: 第二届形式概念分析国际会议，2004年，悉尼，澳大利亚，2004年2月23-26日，会议录&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.math.hawaii.edu/~ralph/Preprints/latdrawing.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.math.hawaii.edu/~ralph/Preprints/latdrawing.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.math.hawaii.edu/~ralph/preprints/latdrawing.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 2961&lt;br /&gt;
&lt;br /&gt;
 | volume = 2961&lt;br /&gt;
&lt;br /&gt;
第2961卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2004| isbn = 978-3-540-21043-6&lt;br /&gt;
&lt;br /&gt;
 | year = 2004| isbn = 978-3-540-21043-6&lt;br /&gt;
&lt;br /&gt;
2004 | isbn 978-3-540-21043-6&lt;br /&gt;
&lt;br /&gt;
 | citeseerx = 10.1.1.69.6245&lt;br /&gt;
&lt;br /&gt;
 | citeseerx = 10.1.1.69.6245&lt;br /&gt;
&lt;br /&gt;
10.1.1.69.6245&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Garg | first1 = Ashim&lt;br /&gt;
&lt;br /&gt;
 | last1 = Garg | first1 = Ashim&lt;br /&gt;
&lt;br /&gt;
1 | 最后1加格 | 首先1阿希姆&lt;br /&gt;
&lt;br /&gt;
 | last2 = Tamassia | first2 = Roberto&lt;br /&gt;
&lt;br /&gt;
 | last2 = Tamassia | first2 = Roberto&lt;br /&gt;
&lt;br /&gt;
2 Tamassia | first2 Roberto&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BF01108622&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BF01108622&lt;br /&gt;
&lt;br /&gt;
| doi 10.1007 / BF01108622&lt;br /&gt;
&lt;br /&gt;
 | issue = 2&lt;br /&gt;
&lt;br /&gt;
 | issue = 2&lt;br /&gt;
&lt;br /&gt;
第二期&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Order (journal)|Order]]&lt;br /&gt;
&lt;br /&gt;
 | journal = Order&lt;br /&gt;
&lt;br /&gt;
| 日志订单&lt;br /&gt;
&lt;br /&gt;
 | mr = 1354797&lt;br /&gt;
&lt;br /&gt;
 | mr = 1354797&lt;br /&gt;
&lt;br /&gt;
1354797先生&lt;br /&gt;
&lt;br /&gt;
 | pages = 109–133&lt;br /&gt;
&lt;br /&gt;
 | pages = 109–133&lt;br /&gt;
&lt;br /&gt;
第109-133页&lt;br /&gt;
&lt;br /&gt;
 | title = Upward planarity testing&lt;br /&gt;
&lt;br /&gt;
 | title = Upward planarity testing&lt;br /&gt;
&lt;br /&gt;
| 标题向上平面性测试&lt;br /&gt;
&lt;br /&gt;
 | volume = 12&lt;br /&gt;
&lt;br /&gt;
 | volume = 12&lt;br /&gt;
&lt;br /&gt;
第12卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1995| citeseerx = 10.1.1.10.2237&lt;br /&gt;
&lt;br /&gt;
 | year = 1995| citeseerx = 10.1.1.10.2237&lt;br /&gt;
&lt;br /&gt;
1995年10.1.1.10.2237&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Holten | first1 = Danny&lt;br /&gt;
&lt;br /&gt;
 | last1 = Holten | first1 = Danny&lt;br /&gt;
&lt;br /&gt;
 | last1 = Holten | first1 = Danny&lt;br /&gt;
&lt;br /&gt;
 | last2 = Isenberg | first2 = Petra&lt;br /&gt;
&lt;br /&gt;
 | last2 = Isenberg | first2 = Petra&lt;br /&gt;
&lt;br /&gt;
2 Isenberg | first2 Petra&lt;br /&gt;
&lt;br /&gt;
 | last3 = van Wijk | first3 = Jarke J. | author3-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 | last3 = van Wijk | first3 = Jarke J. | author3-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 | last3 = van Wijk | first3 = Jarke J.| author3-link Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 | last4 = Fekete | first4 = Jean-Daniel&lt;br /&gt;
&lt;br /&gt;
 | last4 = Fekete | first4 = Jean-Daniel&lt;br /&gt;
&lt;br /&gt;
最后4个 Fekete 最早4个 Jean-Daniel&lt;br /&gt;
&lt;br /&gt;
 | contribution = An extended evaluation of the readability of tapered, animated, and textured directed-edge representations in node-link graphs&lt;br /&gt;
&lt;br /&gt;
 | contribution = An extended evaluation of the readability of tapered, animated, and textured directed-edge representations in node-link graphs&lt;br /&gt;
&lt;br /&gt;
对节点链接图中渐变的、动画的和纹理的有向边表示的可读性的扩展评估&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/PACIFICVIS.2011.5742390&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/PACIFICVIS.2011.5742390&lt;br /&gt;
&lt;br /&gt;
10.1109 / PACIFICVIS. 2011.5742390&lt;br /&gt;
&lt;br /&gt;
 | pages = 195–202&lt;br /&gt;
&lt;br /&gt;
 | pages = 195–202&lt;br /&gt;
&lt;br /&gt;
第195-202页&lt;br /&gt;
&lt;br /&gt;
 | title = IEEE Pacific Visualization Symposium (PacificVis 2011)&lt;br /&gt;
&lt;br /&gt;
 | title = IEEE Pacific Visualization Symposium (PacificVis 2011)&lt;br /&gt;
&lt;br /&gt;
美国电气和电子工程师协会太平洋可视化研讨会(PacificVis 2011)&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.lri.fr/~isenberg/publications/papers/Holten_2011_AEP.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.lri.fr/~isenberg/publications/papers/Holten_2011_AEP.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.lri.fr/~isenberg/publications/papers/holten_2011_aep.pdf&lt;br /&gt;
&lt;br /&gt;
 | year = 2011| isbn = 978-1-61284-935-5&lt;br /&gt;
&lt;br /&gt;
 | year = 2011| isbn = 978-1-61284-935-5&lt;br /&gt;
&lt;br /&gt;
| 2011年 | isbn 978-1-61284-935-5&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 |last1        = Holten&lt;br /&gt;
&lt;br /&gt;
 |last1        = Holten&lt;br /&gt;
&lt;br /&gt;
1 Holten&lt;br /&gt;
&lt;br /&gt;
 |first1       = Danny&lt;br /&gt;
&lt;br /&gt;
 |first1       = Danny&lt;br /&gt;
&lt;br /&gt;
首先是丹尼&lt;br /&gt;
&lt;br /&gt;
 |last2        = van Wijk&lt;br /&gt;
&lt;br /&gt;
 |last2        = van Wijk&lt;br /&gt;
&lt;br /&gt;
2 van Wijk&lt;br /&gt;
&lt;br /&gt;
 |first2       = Jarke J.&lt;br /&gt;
&lt;br /&gt;
 |first2       = Jarke J.&lt;br /&gt;
&lt;br /&gt;
| first2 Jarke j.&lt;br /&gt;
&lt;br /&gt;
 |author2-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 |author2-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
| author2-link Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 |contribution = A user study on visualizing directed edges in graphs&lt;br /&gt;
&lt;br /&gt;
 |contribution = A user study on visualizing directed edges in graphs&lt;br /&gt;
&lt;br /&gt;
用户研究图形中的有向边可视化&lt;br /&gt;
&lt;br /&gt;
 |doi          = 10.1145/1518701.1519054&lt;br /&gt;
&lt;br /&gt;
 |doi          = 10.1145/1518701.1519054&lt;br /&gt;
&lt;br /&gt;
10.1145 / 1518701.1519054&lt;br /&gt;
&lt;br /&gt;
 |pages        = 2299–2308&lt;br /&gt;
&lt;br /&gt;
 |pages        = 2299–2308&lt;br /&gt;
&lt;br /&gt;
2299-2308页&lt;br /&gt;
&lt;br /&gt;
 |title        = Proceedings of the 27th International Conference on Human Factors in Computing Systems (CHI '09)&lt;br /&gt;
&lt;br /&gt;
 |title        = Proceedings of the 27th International Conference on Human Factors in Computing Systems (CHI '09)&lt;br /&gt;
&lt;br /&gt;
| 第27届计算机系统中的人为因素国际会议论文集(CHI’09)&lt;br /&gt;
&lt;br /&gt;
 |url          = http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |url          = http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |year         = 2009&lt;br /&gt;
&lt;br /&gt;
 |year         = 2009&lt;br /&gt;
&lt;br /&gt;
2009年&lt;br /&gt;
&lt;br /&gt;
 |url-status      = dead&lt;br /&gt;
&lt;br /&gt;
 |url-status      = dead&lt;br /&gt;
&lt;br /&gt;
状态死机&lt;br /&gt;
&lt;br /&gt;
 |archiveurl   = https://web.archive.org/web/20111106004500/http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |archiveurl   = https://web.archive.org/web/20111106004500/http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
Https://web.archive.org/web/20111106004500/http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |archivedate  = 2011-11-06&lt;br /&gt;
&lt;br /&gt;
 |archivedate  = 2011-11-06&lt;br /&gt;
&lt;br /&gt;
2011-11-06&lt;br /&gt;
&lt;br /&gt;
|isbn       = 9781605582467&lt;br /&gt;
&lt;br /&gt;
|isbn       = 9781605582467&lt;br /&gt;
&lt;br /&gt;
9781605582467&lt;br /&gt;
&lt;br /&gt;
 |citeseerx       = 10.1.1.212.5461&lt;br /&gt;
&lt;br /&gt;
 |citeseerx       = 10.1.1.212.5461&lt;br /&gt;
&lt;br /&gt;
10.1.1.212.5461&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Koren&lt;br /&gt;
&lt;br /&gt;
 | last = Koren&lt;br /&gt;
&lt;br /&gt;
最后一个韩国&lt;br /&gt;
&lt;br /&gt;
 | first = Yehuda&lt;br /&gt;
&lt;br /&gt;
 | first = Yehuda&lt;br /&gt;
&lt;br /&gt;
首先是耶胡达&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1016/j.camwa.2004.08.015&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1016/j.camwa.2004.08.015&lt;br /&gt;
&lt;br /&gt;
10.1016 / j.camwa. 2004.08&lt;br /&gt;
&lt;br /&gt;
 | issue = 11–12&lt;br /&gt;
&lt;br /&gt;
 | issue = 11–12&lt;br /&gt;
&lt;br /&gt;
第11-12期&lt;br /&gt;
&lt;br /&gt;
 | journal = Computers &amp;amp; Mathematics with Applications&lt;br /&gt;
&lt;br /&gt;
 | journal = Computers &amp;amp; Mathematics with Applications&lt;br /&gt;
&lt;br /&gt;
计算机与数学与应用》杂志&lt;br /&gt;
&lt;br /&gt;
 | mr = 2154691&lt;br /&gt;
&lt;br /&gt;
 | mr = 2154691&lt;br /&gt;
&lt;br /&gt;
2154691先生&lt;br /&gt;
&lt;br /&gt;
 | pages = 1867–1888&lt;br /&gt;
&lt;br /&gt;
 | pages = 1867–1888&lt;br /&gt;
&lt;br /&gt;
1867-1888页&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing graphs by eigenvectors: theory and practice&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing graphs by eigenvectors: theory and practice&lt;br /&gt;
&lt;br /&gt;
按特征向量绘图: 理论与实践&lt;br /&gt;
&lt;br /&gt;
 | url = https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
Https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/dblp-journals-camwa-koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 49&lt;br /&gt;
&lt;br /&gt;
 | volume = 49&lt;br /&gt;
&lt;br /&gt;
第49卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2005&lt;br /&gt;
&lt;br /&gt;
 | year = 2005&lt;br /&gt;
&lt;br /&gt;
2005年&lt;br /&gt;
&lt;br /&gt;
 | access-date = 2011-09-17&lt;br /&gt;
&lt;br /&gt;
 | access-date = 2011-09-17&lt;br /&gt;
&lt;br /&gt;
2011-09-17&lt;br /&gt;
&lt;br /&gt;
 | archive-url = https://web.archive.org/web/20120402131402/https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | archive-url = https://web.archive.org/web/20120402131402/https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
| 档案-url  https://web.archive.org/web/20120402131402/https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/dblp-journals-camwa-koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | archive-date = 2012-04-02&lt;br /&gt;
&lt;br /&gt;
 | archive-date = 2012-04-02&lt;br /&gt;
&lt;br /&gt;
2012-04-02&lt;br /&gt;
&lt;br /&gt;
 | url-status = dead&lt;br /&gt;
&lt;br /&gt;
 | url-status = dead&lt;br /&gt;
&lt;br /&gt;
状态死机&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Longabaugh | first = William&lt;br /&gt;
&lt;br /&gt;
 | last = Longabaugh | first = William&lt;br /&gt;
&lt;br /&gt;
最后一次朗格伯格 | 第一次威廉&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1186/1471-2105-13-275&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1186/1471-2105-13-275&lt;br /&gt;
&lt;br /&gt;
10.1186 / 1471-2105-13-275&lt;br /&gt;
&lt;br /&gt;
 | journal = BMC Bioinformatics&lt;br /&gt;
&lt;br /&gt;
 | journal = BMC Bioinformatics&lt;br /&gt;
&lt;br /&gt;
生物信息学杂志&lt;br /&gt;
&lt;br /&gt;
 | pages = 275&lt;br /&gt;
&lt;br /&gt;
 | pages = 275&lt;br /&gt;
&lt;br /&gt;
第275页&lt;br /&gt;
&lt;br /&gt;
 | title = Combing the hairball with BioFabric: a new approach for visualization of large networks&lt;br /&gt;
&lt;br /&gt;
 | title = Combing the hairball with BioFabric: a new approach for visualization of large networks&lt;br /&gt;
&lt;br /&gt;
用 BioFabric 梳理毛团: 一种大型网络可视化的新方法&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.biomedcentral.com/content/pdf/1471-2105-13-275.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.biomedcentral.com/content/pdf/1471-2105-13-275.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.biomedcentral.com/content/pdf/1471-2105-13-275.pdf&lt;br /&gt;
&lt;br /&gt;
 | pmid = 23102059&lt;br /&gt;
&lt;br /&gt;
 | pmid = 23102059&lt;br /&gt;
&lt;br /&gt;
23102059&lt;br /&gt;
&lt;br /&gt;
 | volume = 13&lt;br /&gt;
&lt;br /&gt;
 | volume = 13&lt;br /&gt;
&lt;br /&gt;
第13卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2012&lt;br /&gt;
&lt;br /&gt;
 | year = 2012&lt;br /&gt;
&lt;br /&gt;
2012年&lt;br /&gt;
&lt;br /&gt;
 | pmc=3574047}}.&lt;br /&gt;
&lt;br /&gt;
 | pmc=3574047}}.&lt;br /&gt;
&lt;br /&gt;
3574047}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Madden | first1 = Brendan&lt;br /&gt;
&lt;br /&gt;
 | last1 = Madden | first1 = Brendan&lt;br /&gt;
&lt;br /&gt;
最后一个疯狂橄榄球，最初一个布兰登&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Patrick&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Patrick&lt;br /&gt;
&lt;br /&gt;
2 Madden | first2 Patrick&lt;br /&gt;
&lt;br /&gt;
 | last3 = Powers | first3 = Steve&lt;br /&gt;
&lt;br /&gt;
 | last3 = Powers | first3 = Steve&lt;br /&gt;
&lt;br /&gt;
3 power | first3 Steve&lt;br /&gt;
&lt;br /&gt;
 | last4 = Himsolt | first4 = Michael&lt;br /&gt;
&lt;br /&gt;
 | last4 = Himsolt | first4 = Michael&lt;br /&gt;
&lt;br /&gt;
最后4希望 / 最初4迈克尔&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Brandenburg | editor-first = Franz J.&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Brandenburg | editor-first = Franz J.&lt;br /&gt;
&lt;br /&gt;
最后的勃兰登堡 | 最初的弗朗茨 j。&lt;br /&gt;
&lt;br /&gt;
 | contribution = Portable graph layout and editing&lt;br /&gt;
&lt;br /&gt;
 | contribution = Portable graph layout and editing&lt;br /&gt;
&lt;br /&gt;
便携式图形布局和编辑&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BFb0021822&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BFb0021822&lt;br /&gt;
&lt;br /&gt;
10.1007 / BFb0021822&lt;br /&gt;
&lt;br /&gt;
 | pages = 385–395&lt;br /&gt;
&lt;br /&gt;
 | pages = 385–395&lt;br /&gt;
&lt;br /&gt;
第385-395页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Symposium on Graph Drawing, GD '95, Passau, Germany, September 20–22, 1995, Proceedings&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Symposium on Graph Drawing, GD '95, Passau, Germany, September 20–22, 1995, Proceedings&lt;br /&gt;
&lt;br /&gt;
图形绘制: 图形绘制研讨会，GD’95，帕绍，1995年9月20-22日，美国国家科学院院刊&lt;br /&gt;
&lt;br /&gt;
 | volume = 1027&lt;br /&gt;
&lt;br /&gt;
 | volume = 1027&lt;br /&gt;
&lt;br /&gt;
第1027卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1996| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
 | year = 1996| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
1996年标题链接国际图形绘制专题研讨会&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-60723-6&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-60723-6&lt;br /&gt;
&lt;br /&gt;
[国际标准图书编号978-3-540-60723-6]&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
免费访问&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Misue | first1= K.&lt;br /&gt;
&lt;br /&gt;
 | last1 = Misue | first1= K.&lt;br /&gt;
&lt;br /&gt;
| 最后1米苏 | 最初1 k。&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades  | first2= P.&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades  | first2= P.&lt;br /&gt;
&lt;br /&gt;
2 Eades | first2 p.&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lai  | first3 = W.&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lai  | first3 = W.&lt;br /&gt;
&lt;br /&gt;
3 Lai | first3 w.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Sugiyama | first4 = K.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Sugiyama | first4 = K.&lt;br /&gt;
&lt;br /&gt;
最后4千。&lt;br /&gt;
&lt;br /&gt;
 | title = Layout Adjustment and the Mental Map&lt;br /&gt;
&lt;br /&gt;
 | title = Layout Adjustment and the Mental Map&lt;br /&gt;
&lt;br /&gt;
| 标题布局调整和心理地图&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Visual Languages and Computing&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Visual Languages and Computing&lt;br /&gt;
&lt;br /&gt;
视觉语言与计算杂志&lt;br /&gt;
&lt;br /&gt;
 | volume = 6 | number = 2 | pages = 183–210 | year = 1995 | doi=10.1006/jvlc.1995.1010}}.&lt;br /&gt;
&lt;br /&gt;
 | volume = 6 | number = 2 | pages = 183–210 | year = 1995 | doi=10.1006/jvlc.1995.1010}}.&lt;br /&gt;
&lt;br /&gt;
183-210 | year 1995 | doi 10.1006 / jvlc. 1995.1010}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Nachmanson | first1 = Lev&lt;br /&gt;
&lt;br /&gt;
 | last1 = Nachmanson | first1 = Lev&lt;br /&gt;
&lt;br /&gt;
最后1纳克曼森最初1列弗&lt;br /&gt;
&lt;br /&gt;
 | last2 = Robertson | first2 = George&lt;br /&gt;
&lt;br /&gt;
 | last2 = Robertson | first2 = George&lt;br /&gt;
&lt;br /&gt;
2 Robertson | first2 George&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lee | first3 = Bongshin&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lee | first3 = Bongshin&lt;br /&gt;
&lt;br /&gt;
3 | last 3 Lee | first3 bongsin&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Hong | editor1-first = Seok-Hee&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Hong | editor1-first = Seok-Hee&lt;br /&gt;
&lt;br /&gt;
| 编辑1-last Hong | 编辑1-first Seok-Hee&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki&lt;br /&gt;
&lt;br /&gt;
 | editor3-last = Quan | editor3-first = Wu&lt;br /&gt;
&lt;br /&gt;
 | editor3-last = Quan | editor3-first = Wu&lt;br /&gt;
&lt;br /&gt;
| 编辑3-last Quan | 编辑3-first Wu&lt;br /&gt;
&lt;br /&gt;
 | contribution = Drawing Graphs with GLEE&lt;br /&gt;
&lt;br /&gt;
 | contribution = Drawing Graphs with GLEE&lt;br /&gt;
&lt;br /&gt;
用 GLEE 绘制图表&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-77537-9_38&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-77537-9_38&lt;br /&gt;
&lt;br /&gt;
10.1007 / 978-3-540-77537-938&lt;br /&gt;
&lt;br /&gt;
 | pages = 389–394&lt;br /&gt;
&lt;br /&gt;
 | pages = 389–394&lt;br /&gt;
&lt;br /&gt;
第389-394页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing, 15th International Symposium, GD 2007, Sydney, Australia, September 24–26, 2007, Revised Papers&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing, 15th International Symposium, GD 2007, Sydney, Australia, September 24–26, 2007, Revised Papers&lt;br /&gt;
&lt;br /&gt;
15 th International Symposium，GD 2007，Sydney，Australia，September 24-26，Revised Papers&lt;br /&gt;
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 | contribution-url = ftp://ftp.research.microsoft.com/pub/TR/TR-2007-72.pdf&lt;br /&gt;
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 | contribution-url = ftp://ftp.research.microsoft.com/pub/TR/TR-2007-72.pdf&lt;br /&gt;
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| contribution-url ftp: / / ftp.research.microsoft. com / pub / tr / tr-2007-72.pdf&lt;br /&gt;
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 | volume = 4875&lt;br /&gt;
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第4875卷&lt;br /&gt;
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 | year = 2008| title-link = International Symposium on Graph Drawing&lt;br /&gt;
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 | year = 2008| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
| 2008年 | 国际图形绘制专题研讨会链接&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-77536-2&lt;br /&gt;
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 | isbn = 978-3-540-77536-2&lt;br /&gt;
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[国际标准图书馆编号978-3-540-77536-2]&lt;br /&gt;
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 | doi-access = free&lt;br /&gt;
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免费访问&lt;br /&gt;
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*{{citation&lt;br /&gt;
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 | last1 = Pach | first1 = János | author1-link = János Pach&lt;br /&gt;
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 | last1 = Pach | first1 = János | author1-link = János Pach&lt;br /&gt;
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 | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir&lt;br /&gt;
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2 | last 2 Sharir | first2 Micha | author2-link Micha Sharir&lt;br /&gt;
&lt;br /&gt;
 | contribution = 5.5 Angular resolution and slopes&lt;br /&gt;
&lt;br /&gt;
 | contribution = 5.5 Angular resolution and slopes&lt;br /&gt;
&lt;br /&gt;
| 贡献5.5角分辨率及斜坡&lt;br /&gt;
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 | pages = 126–127&lt;br /&gt;
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 | pages = 126–127&lt;br /&gt;
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第126-127页&lt;br /&gt;
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 | publisher = [[American Mathematical Society]]&lt;br /&gt;
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 | publisher = American Mathematical Society&lt;br /&gt;
&lt;br /&gt;
美国数学学会&lt;br /&gt;
&lt;br /&gt;
 | series = Mathematical Surveys and Monographs&lt;br /&gt;
&lt;br /&gt;
 | series = Mathematical Surveys and Monographs&lt;br /&gt;
&lt;br /&gt;
| 系列数学调查和专著&lt;br /&gt;
&lt;br /&gt;
 | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures&lt;br /&gt;
&lt;br /&gt;
 | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures&lt;br /&gt;
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离散几何及其算法应用: alcal 系列讲座&lt;br /&gt;
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 | volume = 152&lt;br /&gt;
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 | volume = 152&lt;br /&gt;
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2009年开始。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
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 | last1 = Purchase&lt;br /&gt;
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 | last1 = Purchase&lt;br /&gt;
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| 最后一次购买&lt;br /&gt;
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 | first1 = H. C.&lt;br /&gt;
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 | first1 = H. C.&lt;br /&gt;
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| first1 H.c.&lt;br /&gt;
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 | author1-link = Helen Purchase&lt;br /&gt;
&lt;br /&gt;
 | author1-link = Helen Purchase&lt;br /&gt;
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| author1-link Helen Purchase&lt;br /&gt;
&lt;br /&gt;
 | last2 = Cohen&lt;br /&gt;
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2 Cohen&lt;br /&gt;
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| first2 R.f.&lt;br /&gt;
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 | last3 = James&lt;br /&gt;
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最后3名詹姆斯&lt;br /&gt;
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 | first3 = M. I.&lt;br /&gt;
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 | first3 = M. I.&lt;br /&gt;
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第一季第三集。&lt;br /&gt;
&lt;br /&gt;
 | at = Article 4&lt;br /&gt;
&lt;br /&gt;
 | at = Article 4&lt;br /&gt;
&lt;br /&gt;
| 第四条&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1145/264216.264222&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1145/264216.264222&lt;br /&gt;
&lt;br /&gt;
10.1145 / 264216.264222&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Experimental Algorithmics&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Experimental Algorithmics&lt;br /&gt;
&lt;br /&gt;
实验算法杂志&lt;br /&gt;
&lt;br /&gt;
 | title = An experimental study of the basis for graph drawing algorithms&lt;br /&gt;
&lt;br /&gt;
 | title = An experimental study of the basis for graph drawing algorithms&lt;br /&gt;
&lt;br /&gt;
关于图形绘制算法基础的实验研究&lt;br /&gt;
&lt;br /&gt;
 | url = https://secure.cs.uvic.ca/twiki/pub/Research/Chisel/ComputationalAestheticsProject/Vol2Nbr4.pdf&lt;br /&gt;
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 | url = https://secure.cs.uvic.ca/twiki/pub/Research/Chisel/ComputationalAestheticsProject/Vol2Nbr4.pdf&lt;br /&gt;
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Https://secure.cs.uvic.ca/twiki/pub/research/chisel/computationalaestheticsproject/vol2nbr4.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 2&lt;br /&gt;
&lt;br /&gt;
 | volume = 2&lt;br /&gt;
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第二卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1997&lt;br /&gt;
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 | year = 1997&lt;br /&gt;
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1997年&lt;br /&gt;
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 }}{{Dead link|date=January 2020 |bot=InternetArchiveBot |fix-attempted=yes }}.&lt;br /&gt;
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*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Saaty | first = Thomas L. | authorlink = Thomas L. Saaty&lt;br /&gt;
&lt;br /&gt;
 | last = Saaty | first = Thomas L. | authorlink = Thomas L. Saaty&lt;br /&gt;
&lt;br /&gt;
作者链接: Thomas l. Saaty&lt;br /&gt;
&lt;br /&gt;
 | journal = Proc. Natl. Acad. Sci. U.S.A.&lt;br /&gt;
&lt;br /&gt;
 | journal = Proc. Natl. Acad. Sci. U.S.A.&lt;br /&gt;
&lt;br /&gt;
华尔街日报。自然。Acad.科学。美国。&lt;br /&gt;
&lt;br /&gt;
 | pages = 688–690&lt;br /&gt;
&lt;br /&gt;
 | pages = 688–690&lt;br /&gt;
&lt;br /&gt;
第688-690页&lt;br /&gt;
&lt;br /&gt;
 | title = The minimum number of intersections in complete graphs&lt;br /&gt;
&lt;br /&gt;
 | title = The minimum number of intersections in complete graphs&lt;br /&gt;
&lt;br /&gt;
| 标题完全图中的最小交叉点数&lt;br /&gt;
&lt;br /&gt;
 | volume = 52&lt;br /&gt;
&lt;br /&gt;
 | volume = 52&lt;br /&gt;
&lt;br /&gt;
第52卷&lt;br /&gt;
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 | issue = 3 | year = 1964&lt;br /&gt;
&lt;br /&gt;
 | issue = 3 | year = 1964&lt;br /&gt;
&lt;br /&gt;
第三期1964年&lt;br /&gt;
&lt;br /&gt;
 | doi=10.1073/pnas.52.3.688| pmid = 16591215 | pmc = 300329}}.&lt;br /&gt;
&lt;br /&gt;
 | doi=10.1073/pnas.52.3.688| pmid = 16591215 | pmc = 300329}}.&lt;br /&gt;
&lt;br /&gt;
10.1073 / pnas. 52.3.688 | pmid 16591215 | pmc 300329}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|title=Social network analysis: a handbook|first=John|last=Scott|edition=2nd|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|contribution=Sociograms and Graph Theory|url=https://books.google.com/books?id=Ww3_bKcz6kgC&amp;amp;pg=PA|pages=64–69}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Kozo|last1=Sugiyama|author1-link=Kozo Sugiyama|first2=Shôjirô|last2=Tagawa|first3=Mitsuhiko|last3=Toda|title=Methods for visual understanding of hierarchical system structures|journal=[[IEEE Systems, Man &amp;amp; Cybernetics Society|IEEE Transactions on Systems, Man, and Cybernetics]]|volume=SMC-11|issue=2|pages=109–125|year=1981|mr=0611436|doi=10.1109/TSMC.1981.4308636}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Tantau | first = Till&lt;br /&gt;
&lt;br /&gt;
 | last = Tantau | first = Till&lt;br /&gt;
&lt;br /&gt;
 | last = Tantau | first = Till&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.7155/jgaa.00301&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.7155/jgaa.00301&lt;br /&gt;
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10.7155 / jgaa. 00301&lt;br /&gt;
&lt;br /&gt;
 | issue = 4&lt;br /&gt;
&lt;br /&gt;
 | issue = 4&lt;br /&gt;
&lt;br /&gt;
第四期&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Journal of Graph Algorithms and Applications]]&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Graph Algorithms and Applications&lt;br /&gt;
&lt;br /&gt;
图形算法和应用杂志&lt;br /&gt;
&lt;br /&gt;
 | pages = 495–513&lt;br /&gt;
&lt;br /&gt;
 | pages = 495–513&lt;br /&gt;
&lt;br /&gt;
第495-513页&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing in TikZ&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing in TikZ&lt;br /&gt;
&lt;br /&gt;
| title Graph Drawing in TikZ&lt;br /&gt;
&lt;br /&gt;
 | volume = 17&lt;br /&gt;
&lt;br /&gt;
 | volume = 17&lt;br /&gt;
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第17卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2013}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2013}}.&lt;br /&gt;
&lt;br /&gt;
2013年开始。&lt;br /&gt;
&lt;br /&gt;
* {{Citation | last1=Zapponi | first1=Leonardo | title=What is a Dessin d'Enfant | url=http://www.ams.org/notices/200307/what-is.pdf | date=August 2003 | journal=[[Notices of the American Mathematical Society]] | volume=50 | pages=788–789}}.&lt;br /&gt;
&lt;br /&gt;
{{refend}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==External links==&lt;br /&gt;
&lt;br /&gt;
* [http://graphx.codeplex.com GraphX library for .NET]: open-source WPF library for graph calculation and visualization. Supports many layout and edge routing algorithms.&lt;br /&gt;
&lt;br /&gt;
* [http://gdea.informatik.uni-koeln.de/ Graph drawing e-print archive]: including information on papers from all [[International Symposium on Graph Drawing|Graph Drawing symposia]].&lt;br /&gt;
&lt;br /&gt;
* {{dmoz|Science/Math/Combinatorics/Software/Graph_Drawing}} for many additional links related to graph drawing.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
{{Graph representations}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph drawing| ]]&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Graph drawing]]. Its edit history can be viewed at [[图的绘制/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E8%B7%AF%E5%BE%84_path&amp;diff=13611</id>
		<title>路径 path</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E8%B7%AF%E5%BE%84_path&amp;diff=13611"/>
		<updated>2020-09-02T11:47:08Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{for|the family of graphs known as paths|Path graph}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Snake-in-the-box and Hamiltonian path.svg|thumb|right|&lt;br /&gt;
图1：A three-dimensional [[hypercube graph]] showing a [[Hamiltonian path]] in red, and a [[Snake-in-the-box|longest induced path]] in bold black. 一个三维的[超立方体图表显示一个红色的哈密顿图，和一个黑色的最长诱导路径]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In [[graph theory]], a '''path''' in a [[Graph (discrete mathematics)|graph]] is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] which joins a sequence of [[Vertex (graph theory)|vertices]] which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A '''directed path''' (sometimes called '''dipath'''&amp;lt;ref&amp;gt;Graph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991, [https://books.google.com/books?id=idigH5CTGWAC&amp;amp;pg=PA205 p.205]&amp;lt;/ref&amp;gt;) in a [[directed graph]] is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.&lt;br /&gt;
&lt;br /&gt;
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.&lt;br /&gt;
&lt;br /&gt;
在图论中，图中的路径是一个有限或无限的边序列，这些边连接着一系列顶点，这些顶点在大多数定义中都是不同的(因为顶点是不同的，所以边也是不同的)。有向图中的有向路径(有时称为dipath)是一个有限或无限的边序列，它连接一系列不同的顶点，但附加的限制是所有的边都朝向同一个方向。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced [[algorithm]]ic topics concerning paths in graphs.&lt;br /&gt;
&lt;br /&gt;
Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs.&lt;br /&gt;
&lt;br /&gt;
路径是图论的基本概念，在大多数图论文本的导论部分都有描述。邦迪和默蒂(1976) ，吉本斯(1985) ，或迪斯特尔(2005)。柯尔特(1990)等人著作中，涵盖了更多关于图中路径的高级算法话题。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Definitions 定义==&lt;br /&gt;
&lt;br /&gt;
=== Walk, trail, path 步道 轨迹 路径===&lt;br /&gt;
&lt;br /&gt;
[[File:Trail but not path.svg|200px|thumb|right &lt;br /&gt;
图2：Trail from A to E，but not path 从A到E的轨迹而非路径]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* A '''walk''' is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] which joins a sequence of [[Vertex (graph theory)|vertices]].{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
步道是连接一系列顶点的有限或无限的边序列&lt;br /&gt;
&lt;br /&gt;
Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a graph. A finite walk is a sequence of edges {{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} for which there is a sequence of vertices {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} such that {{nowrap begin}}''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = {''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;}{{nowrap end}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} is the ''vertex sequence'' of the walk. This walk is  ''closed'' if {{nowrap begin}}''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; = ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;{{nowrap end}}, and ''open'' else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.&lt;br /&gt;
&lt;br /&gt;
 Let  be a graph. A finite walk is a sequence of edges  for which there is a sequence of vertices  such that ϕ(e&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;) = {v&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;, v&amp;lt;sub&amp;gt;i + 1&amp;lt;/sub&amp;gt;} for .  is the vertex sequence of the walk. This walk is  closed if v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; = v&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;, and open else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or ray) has a first vertex but no last vertex.&lt;br /&gt;
&lt;br /&gt;
让我们做一个图{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} 。有限步道是一系列的边{{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}，其顶点序列{{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}。 {{nowrap begin}}''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = {''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;}{{nowrap end}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} 是行走的顶点序列。如果 {{nowrap begin}}''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; = ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;{{nowrap end}} ，则此步道关闭，否则就打开。一个无限步道是一系列的边，它们的类型与这里描述的相同，但是没有第一个顶点或最后一个顶点，而一个半无限步道(或光线)有第一个顶点，但是没有最后一个顶点。&lt;br /&gt;
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* A '''trail''' is a walk in which all edges are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
轨迹是所有的边缘都清晰可见的步道&lt;br /&gt;
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* A '''path''' is a trail in which all vertices (and therefore also all edges) are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
路径是一条所有顶点(因此也包括所有边)都是不同的轨迹。&lt;br /&gt;
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If {{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} is a finite walk with vertex sequence {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} then ''w'' is said to be a ''walk from'' ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''to'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Similarly for a trail or a path. If there is a finite walk between two ''distinct'' vertices then there is also a finite trail and a finite path between them.&lt;br /&gt;
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If is a finite walk with vertex sequence  then w is said to be a walk from v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; to v&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;. Similarly for a trail or a path. If there is a finite walk between two distinct vertices then there is also a finite trail and a finite path between them.&lt;br /&gt;
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如果 {{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}是有顶点序列 {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}的有限步道，那么 w 就是从  ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''to'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;的步行。同样，对于一条轨迹或者一条路径也是如此。如果在两个不同的顶点之间有一个有限步道，那么在它们之间也有一个有限的轨迹和一个有限的路径。&lt;br /&gt;
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Some authors do not require that all vertices of a path be distinct and instead use the term '''simple path''' to refer to such a path.&lt;br /&gt;
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Some authors do not require that all vertices of a path be distinct and instead use the term simple path to refer to such a path.&lt;br /&gt;
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有些作者并不要求路径的所有顶点都是不同的，而是使用术语“简单路径”来指代这样的路径。&lt;br /&gt;
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A [[weighted graph]] associates a value (''weight'') with every edge in the graph. The ''weight of a walk'' (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight.&lt;br /&gt;
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A weighted graph associates a value (weight) with every edge in the graph. The weight of a walk (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.&lt;br /&gt;
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一个加权图将一个值(权)与图中的每条边相关联。一个加权图中的步道(或轨迹或路径)的权是所有边的权之和。有时，“成本”或“长度”这两个词用来代替“权重”。&lt;br /&gt;
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=== Directed walk, trail, path 有向的步道，轨迹，路径===&lt;br /&gt;
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* A '''directed walk''' is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] directed in the same direction which joins a sequence of [[Vertex (graph theory)|vertices]].{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
有向步道是沿相同方向定向的边的有限或无限序列，该边连接一系列顶点。&lt;br /&gt;
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Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a directed graph. A finite directed walk is a sequence of edges {{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} for which there is a sequence of vertices {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} such that {{nowrap|1=''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = (''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;)}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} is the ''vertex sequence'' of the directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.&lt;br /&gt;
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Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a directed graph. A finite directed walk is a sequence of edges  for which there is a sequence of vertices  such that  for .  is the vertex sequence of the directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or ray) has a first vertex but no last vertex.&lt;br /&gt;
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一个有向图{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}}。有限有向步道是一系列的边 {{nowrap|(''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}，对于这些边，存在一系列的顶点 {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}。{{nowrap|1=''ϕ''(''e''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;) = (''v''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;''i'' + 1&amp;lt;/sub&amp;gt;)}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}是有向步道的顶点序列。无限有向步道是一个边序列，其类型与本文描述的相同，但没有第一个顶点或最后一个顶点，而半无限有向步道(或射线)有第一个顶点，但没有最后一个顶点。&lt;br /&gt;
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* A '''directed trail''' is a directed walk in which all edges are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
有向轨迹是指所有边缘都清晰可见的有向轨迹。&lt;br /&gt;
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* A '''directed path''' is a directed trail in which all vertices are distinct.{{sfn|Bender|Williamson|2010|p=162}}&lt;br /&gt;
有向路径是指所有顶点都是不同的有向路径。&lt;br /&gt;
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If {{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}} is a finite directed walk with vertex sequence {{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}} then ''w'' is said to be a ''walk from'' ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''to'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Similarly for a directed trail or a path. If there is a finite directed walk between two ''distinct'' vertices then there is also a finite directed trail and a finite directed path between them.&lt;br /&gt;
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If  is a finite directed walk with vertex sequence  then w is said to be a walk from v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; to v&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;. Similarly for a directed trail or a path. If there is a finite directed walk between two distinct vertices then there is also a finite directed trail and a finite directed path between them.&lt;br /&gt;
&lt;br /&gt;
如果{{nowrap|1=''w'' = (''e''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''e''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''e''&amp;lt;sub&amp;gt;''n'' − 1&amp;lt;/sub&amp;gt;)}}是一个顶点序列有限的{{nowrap|(''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, ''v''&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, …, ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;)}}有向步道，那么 w 就是从 ''v''&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; ''到'' ''v''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;的步行。同样，对于有向轨迹或路径也是如此。如果在两个不同的顶点之间存在有限有向步道，那么在它们之间也存在有限有向轨迹和有限有向路径。&lt;br /&gt;
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Some authors do not require that all vertices of a directed path be distinct and instead use the term '''simple directed path''' to refer to such a directed path.&lt;br /&gt;
&lt;br /&gt;
Some authors do not require that all vertices of a directed path be distinct and instead use the term simple directed path to refer to such a directed path.&lt;br /&gt;
&lt;br /&gt;
有些作者并不要求有向路径的所有顶点都是不同的，而是用“简单有向路径”这个术语来表示这样一个有向路。&lt;br /&gt;
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A [[Weighted graph|weighted directed graph]] associates a value (''weight'') with every edge in the directed graph. The ''weight of a directed walk'' (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight.&lt;br /&gt;
&lt;br /&gt;
A weighted directed graph associates a value (weight) with every edge in the directed graph. The weight of a directed walk (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.&lt;br /&gt;
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一个加权有向图将一个值(权)与有向图中的每条边相关联。加权有向图中有向步道(或路径)的权是有向边权重的和。有时，“成本”或“长度”这两个词用来代替“权重”。&lt;br /&gt;
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== Examples 例子==&lt;br /&gt;
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* A graph is [[Connectivity (graph theory)|connected]] if there are paths containing each pair of vertices.&lt;br /&gt;
如果路径包含每一对顶点，则图是连通的。&lt;br /&gt;
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* A directed graph is [[Strongly-connected digraph|strongly connected]] if there are oppositely oriented directed paths containing each pair of vertices.&lt;br /&gt;
如果一个有向图中存在包含每对顶点的相对有向路径，那么这个有向图就是强连通的。&lt;br /&gt;
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* A path such that no graph edges connect two nonconsecutive path vertices is called an [[induced path]].&lt;br /&gt;
没有图的边连接两个不连续的路径顶点的路径称为诱导路径。&lt;br /&gt;
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* A path that includes every vertex of the graph is known as a [[Hamiltonian path]].&lt;br /&gt;
包含图的每个顶点的路径称为哈密顿路径。&lt;br /&gt;
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* Two paths are ''vertex-independent'' (alternatively, ''internally vertex-disjoint'') if they do not have any internal vertex in common. Similarly, two paths are ''edge-independent'' (or ''edge-disjoint'') if they do not have any internal edge in common.  Two internally vertex-disjoint paths are edge-disjoint, but the converse is not necessarily true.&lt;br /&gt;
如果两条路径没有任何共同的内部顶点，则它们与顶点无关(或者，内部顶点不相交)。类似地，如果两条路径没有任何共同的内边，则它们是边独立的(或边不相交)。两条内部顶点不相交的路径是边不相交的，但反过来也不一定是正确的。&lt;br /&gt;
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* The [[Distance (graph theory)|distance]] between two vertices in a graph is the length of a shortest path between them, if one exists, and otherwise the distance is infinity.&lt;br /&gt;
图中两个顶点之间的距离是它们之间最短路径的长度（如果存在），否则该距离是无穷大。&lt;br /&gt;
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* The diameter of a connected graph is the largest distance (defined above) between pairs of vertices of the graph.&lt;br /&gt;
连接图的直径是图的顶点对之间的最大距离（如上定义）。&lt;br /&gt;
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== Finding paths 路径寻找==&lt;br /&gt;
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Several algorithms exist to find [[Shortest path problem|shortest]] and [[Longest path problem|longest]] paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.&lt;br /&gt;
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Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.&lt;br /&gt;
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图的最短路径和最长路径的求解有几种算法，但一个重要区别是前者的计算比后者简单得多。&lt;br /&gt;
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[[Dijkstra's algorithm]] produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the [[Bellman–Ford algorithm]] can be applied to directed graphs with negative edge weights.  The [[Floyd–Warshall algorithm]] can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.&lt;br /&gt;
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Dijkstra's algorithm produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the Bellman–Ford algorithm can be applied to directed graphs with negative edge weights.  The Floyd–Warshall algorithm can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.&lt;br /&gt;
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Dijkstra 算法产生了一个从源顶点到无向或有向图中每个顶点的最短路径列表，而 Bellman-Ford 算法可以应用于具有负边权重的有向图。利用 Floyd-Warshall 算法可以求出加权有向图中所有顶点对之间的最短路径。&lt;br /&gt;
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== See also 另请参见==&lt;br /&gt;
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* [[Glossary of graph theory]]&lt;br /&gt;
图论词汇&lt;br /&gt;
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* [[Path graph]]&lt;br /&gt;
路径图&lt;br /&gt;
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* [[Polygonal chain]]&lt;br /&gt;
多边形链&lt;br /&gt;
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* [[Shortest path problem]]&lt;br /&gt;
最短路径问题&lt;br /&gt;
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* [[Longest path problem]]&lt;br /&gt;
最长路径问题&lt;br /&gt;
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* [[Dijkstra's algorithm]]&lt;br /&gt;
Dijkstra算法&lt;br /&gt;
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* [[Bellman–Ford algorithm]]&lt;br /&gt;
Bellman–Ford算法&lt;br /&gt;
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* [[Floyd–Warshall algorithm]]&lt;br /&gt;
Floyd–Warshall算法&lt;br /&gt;
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* [[Self-avoiding walk]]&lt;br /&gt;
自我避免的步道&lt;br /&gt;
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*[[Shortest-path graph]]&lt;br /&gt;
最短路径图&lt;br /&gt;
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== References 引用==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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* {{cite book |last1=Bender |first1=Edward A. |last2=Williamson |first2=S. Gill |date=2010 |title=Lists, Decisions and Graphs. With an Introduction to Probability |url=https://books.google.fr/books?id=vaXv_yhefG8C |location= |publisher= |page= |isbn= |author-link= }}&lt;br /&gt;
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* {{cite book |last1=Bondy |first1=J. A. |last2=Murty |first2=U. S. R. |date=1976 |title=Graph Theory with Applications |url=https://archive.org/details/graphtheorywitha0000bond/page/12 |location= |publisher=North Holland |page=[https://archive.org/details/graphtheorywitha0000bond/page/12 12-21] |isbn=0-444-19451-7 |author-link= }}&lt;br /&gt;
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* {{cite book |last=Diestel |first=Reinhard |date=2005 |title=Graph Theory |url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ |location= |publisher=Springer-Verlag |page=6-9 |isbn=3-540-26182-6 |author-link= }}&lt;br /&gt;
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* {{cite book |last=Gibbons |first=A. |date=1985 |title=Algorithmic Graph Theory |url= |location= |publisher=Cambridge University Press |page=5-6 |isbn=0-521-28881-9 |author-link= }}&lt;br /&gt;
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* {{cite book |last1=Korte |first1=Bernhard |last2=Lovász |first2=László |last3=Prömel |first3=Hans Jürgen |last4=Schrijver |first4=Alexander |date=1990 |title=Paths, Flows, and VLSI-Layout |url=https://archive.org/details/pathsflowsvlsila0000unse |location= |publisher=Springer-Verlag |page= |isbn=0-387-52685-4 |author-link= |url-access=registration }}&lt;br /&gt;
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[[Category:Graph theory objects]]&lt;br /&gt;
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Category:Graph theory objects&lt;br /&gt;
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范畴: 图论对象&lt;br /&gt;
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[[Category:Graph connectivity]]&lt;br /&gt;
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Category:Graph connectivity&lt;br /&gt;
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类别: 图形连接性&lt;br /&gt;
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{{interwiki extra|qid=Q917421}}&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Path (graph theory)]]. Its edit history can be viewed at [[路径/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
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		<title>优先链接</title>
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		<updated>2020-09-02T10:53:12Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
{{Redirect|Yule process|the type of birth process|Simple birth process}}&lt;br /&gt;
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A '''preferential attachment process''' is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not.  &amp;quot;Preferential attachment&amp;quot; is only the most recent of many names that have been given to such processes.  They are also referred to under the names &amp;quot;[[Udny Yule|Yule]] process&amp;quot;, &amp;quot;cumulative advantage&amp;quot;, &amp;quot;the rich get richer&amp;quot;, and, less correctly, the &amp;quot;[[Matthew effect (sociology)|Matthew effect]]&amp;quot;.  They are also related to [[Gibrat's law]]. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate [[power law]] distributions.&lt;br /&gt;
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A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not.  &amp;quot;Preferential attachment&amp;quot; is only the most recent of many names that have been given to such processes.  They are also referred to under the names &amp;quot;Yule process&amp;quot;, &amp;quot;cumulative advantage&amp;quot;, &amp;quot;the rich get richer&amp;quot;, and, less correctly, the &amp;quot;Matthew effect&amp;quot;.  They are also related to Gibrat's law. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;优先链接 Preferential Attachment&amp;lt;/font&amp;gt;'''过程是一类过程，其中一些数量，通常是某种形式的财富或信贷，是根据一些个人或物体已经拥有的数量来分布的，从而使那些已经富有的人比那些不富有的人得到更多。”优先链接”是该过程众多名称中最贴切其本质含义的名字。他们还被称为“Yule过程”、“优势累积”、“富人越来越富” ，以及不那么正确的“马太效应”。它们也与纪伯伦的法律有关。优先链接之所以受到科学家的关注，主要是因为它能在适当的条件下产生'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;幂律分布 Power Law Distributions&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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==Definition 定义==&lt;br /&gt;
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A preferential attachment process is a [[stochastic process|stochastic]] [[urn problem|urn process]], meaning a process in which discrete units of wealth, usually called &amp;quot;balls&amp;quot;, are added in a random or partly random fashion to a set of objects or containers, usually called &amp;quot;urns&amp;quot;.  A preferential attachment process is an urn process in which additional balls are added continuously to the system and are distributed among the urns as an increasing function of the number of balls the urns already have.  In the most commonly studied examples, the number of urns also increases continuously, although this is not a necessary condition for preferential attachment and examples have been studied with constant or even decreasing numbers of urns.&lt;br /&gt;
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A preferential attachment process is a stochastic urn process, meaning a process in which discrete units of wealth, usually called &amp;quot;balls&amp;quot;, are added in a random or partly random fashion to a set of objects or containers, usually called &amp;quot;urns&amp;quot;.  A preferential attachment process is an urn process in which additional balls are added continuously to the system and are distributed among the urns as an increasing function of the number of balls the urns already have.  In the most commonly studied examples, the number of urns also increases continuously, although this is not a necessary condition for preferential attachment and examples have been studied with constant or even decreasing numbers of urns.&lt;br /&gt;
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优先链接过程是一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机的瓮过程 Stochastic Urn Process&amp;lt;/font&amp;gt;'''，意味着在这个过程中，离散的财富单位，通常称为“球”，以随机或部分随机的方式添加到一组物体或容器，通常称为“瓮”。优先链接过程是一个瓮过程，在这个过程中，额外的球被不断地添加到系统中，并随着瓮中已有球数的增加而分布在瓮中。在最常研究的例子中，翁的数量也在不断增加，尽管这不是优先链接的必要条件，而且也有研究表示过翁数量不变甚至减少的例子。&lt;br /&gt;
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A classic example of a preferential attachment process is the growth in the number of [[species]] per [[genus]] in some higher [[taxon]] of biotic organisms.&amp;lt;ref name=YulePhilTrans&amp;gt;{{cite journal | last=Yule | first=G. U. | title=A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S | journal=[[Philosophical Transactions of the Royal Society B]] | volume=213 | pages=21–87 | year=1925 | doi=10.1098/rstb.1925.0002 | issue=402–410| doi-access=free }}&amp;lt;/ref&amp;gt;  New genera (&amp;quot;urns&amp;quot;) are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species (&amp;quot;balls&amp;quot;) are added as old ones [[speciation|speciate]] (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by [[Udny Yule|Yule]], is a ''[[linear]]'' preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.&lt;br /&gt;
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A classic example of a preferential attachment process is the growth in the number of species per genus in some higher taxon of biotic organisms.  New genera (&amp;quot;urns&amp;quot;) are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species (&amp;quot;balls&amp;quot;) are added as old ones speciate (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by Yule, is a linear preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.&lt;br /&gt;
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优先链接过程的一个典型例子是某些生物有机体的高级分类群中每个属的物种数量的增长。每当一个新出现的物种被认为与其前辈完全不同，不属于任何一个现有属时，新属(“ urns”)就被添加到分类单元中。新物种(”球”)与旧物种(即一分为二)一并加入，假设新物种与其亲本属于同一属(新属除外)，新物种加入该属的概率将与该属已有的物种数量成正比。这个过程，首先被Yule研究，是一个线性优先链接的过程，因为新物种的繁殖率与其已经拥有的数量成线性关系。&lt;br /&gt;
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Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called [[Yule distribution]].  In the most general form of the process, balls are added to the system at an overall rate of ''m'' new balls for each new urn.  Each newly created urn starts out with ''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; balls and further balls are added to urns at a rate proportional to the number ''k'' that they already have plus a constant ''a''&amp;amp;nbsp;&amp;gt;&amp;amp;nbsp;−''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;.  With these definitions, the fraction ''P''(''k'') of urns having ''k'' balls in the limit of long time is given by&amp;lt;ref&amp;gt;{{cite journal | last=Newman | first=M. E. J. | title=Power laws, Pareto distributions and Zipf's law | journal=Contemporary Physics | volume=46 | pages=323–351 | year=2005 | arxiv=cond-mat/0412004 | doi=10.1080/00107510500052444 | issue=5 | bibcode=2005ConPh..46..323N}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called Yule distribution.  In the most general form of the process, balls are added to the system at an overall rate of m new balls for each new urn.  Each newly created urn starts out with k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; balls and further balls are added to urns at a rate proportional to the number k that they already have plus a constant a&amp;amp;nbsp;&amp;gt;&amp;amp;nbsp;−k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;.  With these definitions, the fraction P(k) of urns having k balls in the limit of long time is given by&lt;br /&gt;
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已知翁的数量增加的线性优先链接过程，会按照所谓的Yule分布在翁中产生球的分布。在最一般形式的过程中，球添加到系统中的整体速度为每m个新球到一个瓮中。每个新创建的瓮都以 k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; 球开始，球被不断地添加到瓮中，其速度与瓮的数量k，加上一个常数 ''a''&amp;amp;nbsp;&amp;gt;&amp;amp;nbsp;−''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;，成正比。利用这些定义，可以给出长时极限下具有k个球的瓮的分数p(k)&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_0+a,\gamma-1)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_0+a,\gamma-1)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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for ''k''&amp;amp;nbsp;≥&amp;amp;nbsp;''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; (and zero otherwise), where B(''x'',&amp;amp;nbsp;''y'') is the Euler [[beta function]]:&lt;br /&gt;
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for k&amp;amp;nbsp;≥&amp;amp;nbsp;k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; (and zero otherwise), where B(x,&amp;amp;nbsp;y) is the Euler beta function:&lt;br /&gt;
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对于k&amp;amp;nbsp;≥&amp;amp;nbsp;k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;(否则为0) ，其中 b (x，y)是 Euler beta 函数:&lt;br /&gt;
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&amp;lt;math&amp;gt; \mathrm{B}(x,y)={\Gamma(x)\Gamma(y)\over\Gamma(x+y)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt; \mathrm{B}(x,y)={\Gamma(x)\Gamma(y)\over\Gamma(x+y)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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with Γ(''x'') being the standard [[gamma function]], and&lt;br /&gt;
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with Γ(x) being the standard gamma function, and&lt;br /&gt;
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(x)是标准伽马函数，并且&lt;br /&gt;
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&amp;lt;math&amp;gt;\gamma=2 + {k_0 + a\over m}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\gamma=2 + {k_0 + a\over m}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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The beta function behaves asymptotically as B(''x'',&amp;amp;nbsp;''y'')&amp;amp;nbsp;~&amp;amp;nbsp;''x''&amp;lt;sup&amp;gt;−''y''&amp;lt;/sup&amp;gt; for large ''x'' and fixed ''y'', which implies that for large values of ''k'' we have&lt;br /&gt;
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The beta function behaves asymptotically as B(x,&amp;amp;nbsp;y)&amp;amp;nbsp;~&amp;amp;nbsp;x&amp;lt;sup&amp;gt;−y&amp;lt;/sup&amp;gt; for large x and fixed y, which implies that for large values of k we have&lt;br /&gt;
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对于较大的x和固定的y，β函数表现为B(''x'',&amp;amp;nbsp;''y'')&amp;amp;nbsp;~&amp;amp;nbsp;''x''&amp;lt;sup&amp;gt;−''y''&amp;lt;/sup&amp;gt;，这意味着对于k的大值，我们有一个渐近的β函数&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k) \propto k^{-\gamma}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k) \propto k^{-\gamma}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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In other words, the preferential attachment process generates a &amp;quot;[[Long tail|long-tailed]]&amp;quot; distribution following a [[Pareto distribution]] or [[power law]] in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities,&amp;lt;ref name=SimonBiomet&amp;gt;{{cite journal | last=Simon | first=H. A. | title=On a class of skew distribution functions | journal=Biometrika | volume=42 | pages=425–440  | year=1955 | doi=10.1093/biomet/42.3-4.425  | issue=3–4 }}&amp;lt;/ref&amp;gt; the wealth of extremely wealthy individuals,&amp;lt;ref name=SimonBiomet /&amp;gt; the number of citations received by learned publications,&amp;lt;ref name=PriceJASIS&amp;gt;{{cite journal | last=Price | first=D. J. de S.  | title=A general theory of bibliometric and other cumulative advantage processes | journal=J. Amer. Soc. Inform. Sci. | volume=27 | pages=292–306 | year=1976 | url=http://garfield.library.upenn.edu/price/pricetheory1976.pdf | doi=10.1002/asi.4630270505 | issue=5}}&amp;lt;/ref&amp;gt; and the number of links to pages on the World Wide Web.&amp;lt;ref name=BAScience&amp;gt;{{cite journal | last=Barabási | first=A.-L. |author2=R. Albert | title=Emergence of scaling in random networks | journal=Science | volume=286 | pages=509–512 | year=1999 | arxiv=cond-mat/9910332 | doi=10.1126/science.286.5439.509 | issue=5439 | pmid=10521342| bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In other words, the preferential attachment process generates a &amp;quot;long-tailed&amp;quot; distribution following a Pareto distribution or power law in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities, the wealth of extremely wealthy individuals, and the number of links to pages on the World Wide Web.&lt;br /&gt;
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换句话说，优先链接过程在其尾部产生一个遵循'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;帕累托分布 Pareto Distribution&amp;lt;/font&amp;gt;'''或'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;幂定律 Power Law&amp;lt;/font&amp;gt;'''的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;“长尾”分布 Long-Tailed Distribution&amp;lt;/font&amp;gt;'''。这是历史上人们对优先链接感兴趣的主要原因: 经过经验的观察，物种分布和许多其他现象都遵循幂律分布，而优先链接过程是解释这种行为的主要候选机制。它还可能解释如城市规模的分布，极其富有的个人财富，以及万维网网页的链接数量等问题。&lt;br /&gt;
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The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;1) and gains new species in direct proportion to the number it already has (''a''&amp;amp;nbsp;=&amp;amp;nbsp;0), and hence ''P''(''k'')&amp;amp;nbsp;=&amp;amp;nbsp;B(''k'',&amp;amp;nbsp;''γ'')/B(''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;,&amp;amp;nbsp;''γ''&amp;amp;nbsp;−&amp;amp;nbsp;1) with ''γ''=2&amp;amp;nbsp;+&amp;amp;nbsp;1/''m''.  Similarly the Price model for scientific citations&amp;lt;ref name=PriceJASIS /&amp;gt; corresponds to the case ''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;0, ''a''&amp;amp;nbsp; =&amp;amp;nbsp;1 and the widely studied [[Barabási-Albert model]]&amp;lt;ref name=BAScience /&amp;gt; corresponds to ''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;''m'', ''a''&amp;amp;nbsp;=&amp;amp;nbsp;0.&lt;br /&gt;
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The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;1) and gains new species in direct proportion to the number it already has (a&amp;amp;nbsp;=&amp;amp;nbsp;0), and hence P(k)&amp;amp;nbsp;=&amp;amp;nbsp;B(k,&amp;amp;nbsp;γ)/B(k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;,&amp;amp;nbsp;γ&amp;amp;nbsp;−&amp;amp;nbsp;1) with γ=2&amp;amp;nbsp;+&amp;amp;nbsp;1/m.  Similarly the Price model for scientific citations corresponds to the case k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;0, a&amp;amp;nbsp; =&amp;amp;nbsp;1 and the widely studied Barabási-Albert model corresponds to k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;m, a&amp;amp;nbsp;=&amp;amp;nbsp;0.&lt;br /&gt;
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这里描述的一般模型包括作为特殊情况的许多其他特定模型。例如，在上面的种属例子中，每个属以一个单一的种(k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;1)开始，并且获得新的种(a&amp;amp;nbsp;=&amp;amp;nbsp;0) ，因此P(k)&amp;amp;nbsp;=&amp;amp;nbsp;B(k,&amp;amp;nbsp;γ)/B(k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;,&amp;amp;nbsp;γ&amp;amp;nbsp;−&amp;amp;nbsp;1)，γ=2&amp;amp;nbsp;+&amp;amp;nbsp;1/m. 。类似地，科学引文的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;价格模型 Price Model&amp;lt;/font&amp;gt;'''对应于k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;0，a&amp;amp;nbsp; =&amp;amp;nbsp;1 ，广泛研究的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Barabási-Albert 模型&amp;lt;/font&amp;gt;'''对应于 k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;m, a&amp;amp;nbsp;=&amp;amp;nbsp;0.。&lt;br /&gt;
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Preferential attachment is sometimes referred to as the [[Matthew effect]], but the two are not precisely equivalent.  The Matthew effect, first discussed by [[Robert K. Merton]],&amp;lt;ref&amp;gt;{{cite journal | last=Merton | first=Robert K. | authorlink=Robert K. Merton | title=The Matthew effect in science | journal=Science | volume=159 | pages=56–63 | year=1968 | doi=10.1126/science.159.3810.56 | pmid=17737466 | issue=3810| bibcode=1968Sci...159...56M }}&amp;lt;/ref&amp;gt; is named for a passage in the [[bible|biblical]] [[Gospel of Matthew]]: &amp;quot;For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him.&amp;quot; ([[Gospel of Matthew|Matthew]] [[s:Bible (New International Version)/Matthew#25:29|25:29]], [[New International Version]].)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.&lt;br /&gt;
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Preferential attachment is sometimes referred to as the Matthew effect, but the two are not precisely equivalent.  The Matthew effect, first discussed by Robert K. Merton, is named for a passage in the biblical Gospel of Matthew: &amp;quot;For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him.&amp;quot; (Matthew 25:29, New International Version.)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.&lt;br /&gt;
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优先链接有时被称为马太效应，但两者并不完全等同。马太效应，最早由罗伯特·金·莫顿讨论，是根据《圣经》中的一段马太福音命名的: 每个拥有的人都会得到更多。凡没有的就是他所有的，也必被夺去(马太福音25:29，新国际版)。优先链接过程不包括夺走部分。但是，由于马太效应背后的科学见解在任何情况下都是完全不同的，因此这一点可能没有意义。定性地说，它的目的不是描述一种机械的乘法效应，如优先链接，而是一种特定的人类行为。在这种行为中，人们更有可能给名人加分而不是给鲜为人知的人加分。马太效应的经典例子是两个不同的人同时做出科学发现时，其中一个是众所周知的，另一个则鲜为人知。据称，在这种情况下，人们往往更倾向于将这一发现归功于著名的科学家。因此，马太效应所要描述的现实世界现象与优先链接(虽然肯定与优先链接有关)是截然不同的。&lt;br /&gt;
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==History 历史==&lt;br /&gt;
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The first rigorous consideration of preferential attachment seems to be that of [[Udny Yule]] in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants.&amp;lt;ref name=YulePhilTrans /&amp;gt;  The process is sometimes called a &amp;quot;Yule process&amp;quot; in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.&lt;br /&gt;
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The first rigorous consideration of preferential attachment seems to be that of Udny Yule in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants.  The process is sometimes called a &amp;quot;Yule process&amp;quot; in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.&lt;br /&gt;
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1925年，那时的Udny Yule是第一个对优先链接进行严谨考量的人，他用它来解释被子植物属中物种数量的幂律分布。为了纪念他，这个过程有时被称为“Yule过程”。Yule能够证明这个过程产生了一个带有幂律尾巴的分布，但是他的证明的细节，按照今天的标准，是曲折和晦涩的，因为现代的随机过程理论工具当时还不存在，他被迫使用更加繁琐的证明方法。&lt;br /&gt;
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Most modern treatments of preferential attachment make use of the [[master equation]] method, whose use in this context was pioneered by [[Herbert A. Simon|Simon]] in 1955, in work on the distribution of sizes of cities and other phenomena.&amp;lt;ref name=SimonBiomet /&amp;gt;&lt;br /&gt;
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Most modern treatments of preferential attachment make use of the master equation method, whose use in this context was pioneered by Simon in 1955, in work on the distribution of sizes of cities and other phenomena.&lt;br /&gt;
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大多数现代优先链接的处理方法都使用了'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;主方程方法 Master Equation Method&amp;lt;/font&amp;gt;'''，这种方法在1955年由 Simon 首创，用于研究城市规模和其他现象的分布。&lt;br /&gt;
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The first application of preferential attachment to learned citations was given by [[Derek J. de Solla Price|Price]] in 1976.&amp;lt;ref name=PriceJASIS /&amp;gt;  (He referred to the process as a &amp;quot;cumulative advantage&amp;quot; process.)  His was also the first application of the process to the growth of a network, producing what would now be called a [[scale-free network]].  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including [[Lotka's law]] of scientific productivity and [[Bradford's law]] of journal use.&lt;br /&gt;
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The first application of preferential attachment to learned citations was given by Price in 1976.  (He referred to the process as a &amp;quot;cumulative advantage&amp;quot; process.)  His was also the first application of the process to the growth of a network, producing what would now be called a scale-free network.  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including Lotka's law of scientific productivity and Bradford's law of journal use.&lt;br /&gt;
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普赖斯于1976年首次将优先链接应用于学术引文。(他把这个过程称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;“优势累积” Cumulative Advantage&amp;lt;/font&amp;gt;'''过程。)他也是第一个将这一过程应用于网络发展的人，他创造了现在所谓的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Network&amp;lt;/font&amp;gt;'''。正是在网络增长的背景下，这一过程在今天得到了最频繁的研究。普莱斯还将优先链接作为许多其他现象中幂定律的可能解释，包括洛特卡的科学生产力定律和布拉德福德的期刊使用定律。&lt;br /&gt;
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The application of preferential attachment to the growth of the World Wide Web was proposed by [[BA model|Barabási and Albert]] in 1999.&amp;lt;ref name=BAScience /&amp;gt;  Barabási and Albert also coined the name &amp;quot;preferential attachment&amp;quot; by which the process is best known today and suggested that the process might apply to the growth of other networks as well. For growing networks, the precise functional form of preferential attachment can be estimated by [[maximum likelihood estimation]].&amp;lt;ref&amp;gt;{{cite journal |last1=Pham |first1=Thong |last2=Sheridan |first2=Paul |last3=Shimodaira |first3=Hidetoshi |title=PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks |journal=PLoS ONE |date=September 17, 2015 |volume=10 |issue=9 |doi=10.1371/journal.pone.0137796 |url=http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0137796 |pages=e0137796 |pmid=26378457 |pmc=4574777|bibcode=2015PLoSO..1037796P }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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The application of preferential attachment to the growth of the World Wide Web was proposed by Barabási and Albert in 1999.&lt;br /&gt;
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1999年，Barabási 和 Albert 提出了优先链接在万维网发展中的应用。&lt;br /&gt;
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==See also 另请参阅==&lt;br /&gt;
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{{colbegin}}&lt;br /&gt;
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* [[Assortative mixing]]&lt;br /&gt;
关联矩阵&lt;br /&gt;
&lt;br /&gt;
* [[BA model]]&lt;br /&gt;
BA模型&lt;br /&gt;
&lt;br /&gt;
* [[Bose–Einstein condensation: a network theory approach]]&lt;br /&gt;
玻色-爱因斯坦凝聚：一种网络理论方法&lt;br /&gt;
&lt;br /&gt;
* [[Capital accumulation]]&lt;br /&gt;
资本积累&lt;br /&gt;
&lt;br /&gt;
* [[Chinese restaurant process]]&lt;br /&gt;
中餐厅过程&lt;br /&gt;
&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
&lt;br /&gt;
* [[Double jeopardy (marketing)]]&lt;br /&gt;
双重危险（营销）&lt;br /&gt;
&lt;br /&gt;
* [[Link-centric preferential attachment]]&lt;br /&gt;
以链接为中心的优先链接&lt;br /&gt;
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* [[Matthew effect (sociology)]]&lt;br /&gt;
马太效应（社会学）&lt;br /&gt;
&lt;br /&gt;
* [[Pitman–Yor process]]&lt;br /&gt;
Pitman-Yor过程&lt;br /&gt;
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* [[Power law]]&lt;br /&gt;
幂定律&lt;br /&gt;
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* [[Price's model]]&lt;br /&gt;
价格模型&lt;br /&gt;
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* [[Proof of stake]]&lt;br /&gt;
权益证明&lt;br /&gt;
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* [[Simon model]]&lt;br /&gt;
西蒙模型&lt;br /&gt;
&lt;br /&gt;
* [[Stochastic processes]]&lt;br /&gt;
随机过程&lt;br /&gt;
&lt;br /&gt;
* [[Wealth condensation]]&lt;br /&gt;
财富集聚&lt;br /&gt;
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* [[Yule–Simon distribution]]&lt;br /&gt;
Yule-Simon 分布&lt;br /&gt;
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* [[Bibliogram]]&lt;br /&gt;
书目图&lt;br /&gt;
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{{colend}}&lt;br /&gt;
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==References 引用==&lt;br /&gt;
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&amp;lt;references/&amp;gt;&lt;br /&gt;
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{{DEFAULTSORT:Preferential Attachment}}&lt;br /&gt;
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[[Category:Stochastic processes]]&lt;br /&gt;
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Category:Stochastic processes&lt;br /&gt;
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类别: 随机过程&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Preferential attachment]]. Its edit history can be viewed at [[优先链接/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E4%BC%98%E5%85%88%E9%93%BE%E6%8E%A5&amp;diff=13597</id>
		<title>优先链接</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E4%BC%98%E5%85%88%E9%93%BE%E6%8E%A5&amp;diff=13597"/>
		<updated>2020-09-02T10:52:21Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
{{Redirect|Yule process|the type of birth process|Simple birth process}}&lt;br /&gt;
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A '''preferential attachment process''' is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not.  &amp;quot;Preferential attachment&amp;quot; is only the most recent of many names that have been given to such processes.  They are also referred to under the names &amp;quot;[[Udny Yule|Yule]] process&amp;quot;, &amp;quot;cumulative advantage&amp;quot;, &amp;quot;the rich get richer&amp;quot;, and, less correctly, the &amp;quot;[[Matthew effect (sociology)|Matthew effect]]&amp;quot;.  They are also related to [[Gibrat's law]]. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate [[power law]] distributions.&lt;br /&gt;
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A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not.  &amp;quot;Preferential attachment&amp;quot; is only the most recent of many names that have been given to such processes.  They are also referred to under the names &amp;quot;Yule process&amp;quot;, &amp;quot;cumulative advantage&amp;quot;, &amp;quot;the rich get richer&amp;quot;, and, less correctly, the &amp;quot;Matthew effect&amp;quot;.  They are also related to Gibrat's law. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;优先链接 Preferential Attachment&amp;lt;/font&amp;gt;'''过程是一类过程，其中一些数量，通常是某种形式的财富或信贷，是根据一些个人或物体已经拥有的数量来分布的，从而使那些已经富有的人比那些不富有的人得到更多。”优先链接”是该过程众多名称中最贴切其本质含义的名字。他们还被称为“Yule过程”、“优势累积”、“富人越来越富” ，以及不那么正确的“马太效应”。它们也与纪伯伦的法律有关。优先链接之所以受到科学家的关注，主要是因为它能在适当的条件下产生'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;幂律分布 Power Law Distributions&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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==Definition 定义==&lt;br /&gt;
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A preferential attachment process is a [[stochastic process|stochastic]] [[urn problem|urn process]], meaning a process in which discrete units of wealth, usually called &amp;quot;balls&amp;quot;, are added in a random or partly random fashion to a set of objects or containers, usually called &amp;quot;urns&amp;quot;.  A preferential attachment process is an urn process in which additional balls are added continuously to the system and are distributed among the urns as an increasing function of the number of balls the urns already have.  In the most commonly studied examples, the number of urns also increases continuously, although this is not a necessary condition for preferential attachment and examples have been studied with constant or even decreasing numbers of urns.&lt;br /&gt;
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A preferential attachment process is a stochastic urn process, meaning a process in which discrete units of wealth, usually called &amp;quot;balls&amp;quot;, are added in a random or partly random fashion to a set of objects or containers, usually called &amp;quot;urns&amp;quot;.  A preferential attachment process is an urn process in which additional balls are added continuously to the system and are distributed among the urns as an increasing function of the number of balls the urns already have.  In the most commonly studied examples, the number of urns also increases continuously, although this is not a necessary condition for preferential attachment and examples have been studied with constant or even decreasing numbers of urns.&lt;br /&gt;
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优先链接过程是一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机的瓮过程 Stochastic Urn Process&amp;lt;/font&amp;gt;'''，意味着在这个过程中，离散的财富单位，通常称为“球”，以随机或部分随机的方式添加到一组物体或容器，通常称为“瓮”。优先链接过程是一个瓮过程，在这个过程中，额外的球被不断地添加到系统中，并随着瓮中已有球数的增加而分布在瓮中。在最常研究的例子中，翁的数量也在不断增加，尽管这不是优先链接的必要条件，而且也有研究表示过翁数量不变甚至减少的例子。&lt;br /&gt;
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A classic example of a preferential attachment process is the growth in the number of [[species]] per [[genus]] in some higher [[taxon]] of biotic organisms.&amp;lt;ref name=YulePhilTrans&amp;gt;{{cite journal | last=Yule | first=G. U. | title=A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S | journal=[[Philosophical Transactions of the Royal Society B]] | volume=213 | pages=21–87 | year=1925 | doi=10.1098/rstb.1925.0002 | issue=402–410| doi-access=free }}&amp;lt;/ref&amp;gt;  New genera (&amp;quot;urns&amp;quot;) are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species (&amp;quot;balls&amp;quot;) are added as old ones [[speciation|speciate]] (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by [[Udny Yule|Yule]], is a ''[[linear]]'' preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.&lt;br /&gt;
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A classic example of a preferential attachment process is the growth in the number of species per genus in some higher taxon of biotic organisms.  New genera (&amp;quot;urns&amp;quot;) are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species (&amp;quot;balls&amp;quot;) are added as old ones speciate (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by Yule, is a linear preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.&lt;br /&gt;
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优先链接过程的一个典型例子是某些生物有机体的高级分类群中每个属的物种数量的增长。每当一个新出现的物种被认为与其前辈完全不同，不属于任何一个现有属时，新属(“ urns”)就被添加到分类单元中。新物种(”球”)与旧物种(即一分为二)一并加入，假设新物种与其亲本属于同一属(新属除外)，新物种加入该属的概率将与该属已有的物种数量成正比。这个过程，首先被Yule研究，是一个线性优先链接的过程，因为新物种的繁殖率与其已经拥有的数量成线性关系。&lt;br /&gt;
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Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called [[Yule distribution]].  In the most general form of the process, balls are added to the system at an overall rate of ''m'' new balls for each new urn.  Each newly created urn starts out with ''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; balls and further balls are added to urns at a rate proportional to the number ''k'' that they already have plus a constant ''a''&amp;amp;nbsp;&amp;gt;&amp;amp;nbsp;−''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;.  With these definitions, the fraction ''P''(''k'') of urns having ''k'' balls in the limit of long time is given by&amp;lt;ref&amp;gt;{{cite journal | last=Newman | first=M. E. J. | title=Power laws, Pareto distributions and Zipf's law | journal=Contemporary Physics | volume=46 | pages=323–351 | year=2005 | arxiv=cond-mat/0412004 | doi=10.1080/00107510500052444 | issue=5 | bibcode=2005ConPh..46..323N}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called Yule distribution.  In the most general form of the process, balls are added to the system at an overall rate of m new balls for each new urn.  Each newly created urn starts out with k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; balls and further balls are added to urns at a rate proportional to the number k that they already have plus a constant a&amp;amp;nbsp;&amp;gt;&amp;amp;nbsp;−k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;.  With these definitions, the fraction P(k) of urns having k balls in the limit of long time is given by&lt;br /&gt;
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已知翁的数量增加的线性优先链接过程，会按照所谓的Yule分布在翁中产生球的分布。在最一般形式的过程中，球添加到系统中的整体速度为每m个新球到一个瓮中。每个新创建的瓮都以 k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; 球开始，球被不断地添加到瓮中，其速度与瓮的数量k，加上一个常数 ''a''&amp;amp;nbsp;&amp;gt;&amp;amp;nbsp;−''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;，成正比。利用这些定义，可以给出长时极限下具有k个球的瓮的分数p(k)&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_0+a,\gamma-1)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_0+a,\gamma-1)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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for ''k''&amp;amp;nbsp;≥&amp;amp;nbsp;''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; (and zero otherwise), where B(''x'',&amp;amp;nbsp;''y'') is the Euler [[beta function]]:&lt;br /&gt;
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for k&amp;amp;nbsp;≥&amp;amp;nbsp;k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; (and zero otherwise), where B(x,&amp;amp;nbsp;y) is the Euler beta function:&lt;br /&gt;
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对于k&amp;amp;nbsp;≥&amp;amp;nbsp;k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;(否则为0) ，其中 b (x，y)是 Euler beta 函数:&lt;br /&gt;
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&amp;lt;math&amp;gt; \mathrm{B}(x,y)={\Gamma(x)\Gamma(y)\over\Gamma(x+y)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt; \mathrm{B}(x,y)={\Gamma(x)\Gamma(y)\over\Gamma(x+y)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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with Γ(''x'') being the standard [[gamma function]], and&lt;br /&gt;
&lt;br /&gt;
with Γ(x) being the standard gamma function, and&lt;br /&gt;
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(x)是标准伽马函数，并且&lt;br /&gt;
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&lt;br /&gt;
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&amp;lt;math&amp;gt;\gamma=2 + {k_0 + a\over m}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\gamma=2 + {k_0 + a\over m}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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The beta function behaves asymptotically as B(''x'',&amp;amp;nbsp;''y'')&amp;amp;nbsp;~&amp;amp;nbsp;''x''&amp;lt;sup&amp;gt;−''y''&amp;lt;/sup&amp;gt; for large ''x'' and fixed ''y'', which implies that for large values of ''k'' we have&lt;br /&gt;
&lt;br /&gt;
The beta function behaves asymptotically as B(x,&amp;amp;nbsp;y)&amp;amp;nbsp;~&amp;amp;nbsp;x&amp;lt;sup&amp;gt;−y&amp;lt;/sup&amp;gt; for large x and fixed y, which implies that for large values of k we have&lt;br /&gt;
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对于较大的x和固定的y，β函数表现为B(''x'',&amp;amp;nbsp;''y'')&amp;amp;nbsp;~&amp;amp;nbsp;''x''&amp;lt;sup&amp;gt;−''y''&amp;lt;/sup&amp;gt;，这意味着对于k的大值，我们有一个渐近的β函数&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k) \propto k^{-\gamma}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k) \propto k^{-\gamma}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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In other words, the preferential attachment process generates a &amp;quot;[[Long tail|long-tailed]]&amp;quot; distribution following a [[Pareto distribution]] or [[power law]] in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities,&amp;lt;ref name=SimonBiomet&amp;gt;{{cite journal | last=Simon | first=H. A. | title=On a class of skew distribution functions | journal=Biometrika | volume=42 | pages=425–440  | year=1955 | doi=10.1093/biomet/42.3-4.425  | issue=3–4 }}&amp;lt;/ref&amp;gt; the wealth of extremely wealthy individuals,&amp;lt;ref name=SimonBiomet /&amp;gt; the number of citations received by learned publications,&amp;lt;ref name=PriceJASIS&amp;gt;{{cite journal | last=Price | first=D. J. de S.  | title=A general theory of bibliometric and other cumulative advantage processes | journal=J. Amer. Soc. Inform. Sci. | volume=27 | pages=292–306 | year=1976 | url=http://garfield.library.upenn.edu/price/pricetheory1976.pdf | doi=10.1002/asi.4630270505 | issue=5}}&amp;lt;/ref&amp;gt; and the number of links to pages on the World Wide Web.&amp;lt;ref name=BAScience&amp;gt;{{cite journal | last=Barabási | first=A.-L. |author2=R. Albert | title=Emergence of scaling in random networks | journal=Science | volume=286 | pages=509–512 | year=1999 | arxiv=cond-mat/9910332 | doi=10.1126/science.286.5439.509 | issue=5439 | pmid=10521342| bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In other words, the preferential attachment process generates a &amp;quot;long-tailed&amp;quot; distribution following a Pareto distribution or power law in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities, the wealth of extremely wealthy individuals, and the number of links to pages on the World Wide Web.&lt;br /&gt;
&lt;br /&gt;
换句话说，优先链接过程在其尾部产生一个遵循'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;帕累托分布 Pareto Distribution&amp;lt;/font&amp;gt;'''或'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;幂定律 Power Law&amp;lt;/font&amp;gt;'''的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;“长尾”分布 Long-Tailed Distribution&amp;lt;/font&amp;gt;'''。这是历史上人们对优先链接感兴趣的主要原因: 经过经验的观察，物种分布和许多其他现象都遵循幂律分布，而优先链接过程是解释这种行为的主要候选机制。它还可能解释如城市规模的分布，极其富有的个人财富，以及万维网网页的链接数量等问题。&lt;br /&gt;
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The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;1) and gains new species in direct proportion to the number it already has (''a''&amp;amp;nbsp;=&amp;amp;nbsp;0), and hence ''P''(''k'')&amp;amp;nbsp;=&amp;amp;nbsp;B(''k'',&amp;amp;nbsp;''γ'')/B(''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;,&amp;amp;nbsp;''γ''&amp;amp;nbsp;−&amp;amp;nbsp;1) with ''γ''=2&amp;amp;nbsp;+&amp;amp;nbsp;1/''m''.  Similarly the Price model for scientific citations&amp;lt;ref name=PriceJASIS /&amp;gt; corresponds to the case ''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;0, ''a''&amp;amp;nbsp; =&amp;amp;nbsp;1 and the widely studied [[Barabási-Albert model]]&amp;lt;ref name=BAScience /&amp;gt; corresponds to ''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;''m'', ''a''&amp;amp;nbsp;=&amp;amp;nbsp;0.&lt;br /&gt;
&lt;br /&gt;
The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;1) and gains new species in direct proportion to the number it already has (a&amp;amp;nbsp;=&amp;amp;nbsp;0), and hence P(k)&amp;amp;nbsp;=&amp;amp;nbsp;B(k,&amp;amp;nbsp;γ)/B(k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;,&amp;amp;nbsp;γ&amp;amp;nbsp;−&amp;amp;nbsp;1) with γ=2&amp;amp;nbsp;+&amp;amp;nbsp;1/m.  Similarly the Price model for scientific citations corresponds to the case k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;0, a&amp;amp;nbsp; =&amp;amp;nbsp;1 and the widely studied Barabási-Albert model corresponds to k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;m, a&amp;amp;nbsp;=&amp;amp;nbsp;0.&lt;br /&gt;
&lt;br /&gt;
这里描述的一般模型包括作为特殊情况的许多其他特定模型。例如，在上面的种属例子中，每个属以一个单一的种(k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;1)开始，并且获得新的种(a&amp;amp;nbsp;=&amp;amp;nbsp;0) ，因此P(k)&amp;amp;nbsp;=&amp;amp;nbsp;B(k,&amp;amp;nbsp;γ)/B(k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;,&amp;amp;nbsp;γ&amp;amp;nbsp;−&amp;amp;nbsp;1)，γ=2&amp;amp;nbsp;+&amp;amp;nbsp;1/m. 。类似地，科学引文的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;价格模型 Price Model&amp;lt;/font&amp;gt;'''对应于k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;0，a&amp;amp;nbsp; =&amp;amp;nbsp;1 ，广泛研究的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Barabási-Albert 模型&amp;lt;/font&amp;gt;'''对应于 k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;m, a&amp;amp;nbsp;=&amp;amp;nbsp;0.。&lt;br /&gt;
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Preferential attachment is sometimes referred to as the [[Matthew effect]], but the two are not precisely equivalent.  The Matthew effect, first discussed by [[Robert K. Merton]],&amp;lt;ref&amp;gt;{{cite journal | last=Merton | first=Robert K. | authorlink=Robert K. Merton | title=The Matthew effect in science | journal=Science | volume=159 | pages=56–63 | year=1968 | doi=10.1126/science.159.3810.56 | pmid=17737466 | issue=3810| bibcode=1968Sci...159...56M }}&amp;lt;/ref&amp;gt; is named for a passage in the [[bible|biblical]] [[Gospel of Matthew]]: &amp;quot;For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him.&amp;quot; ([[Gospel of Matthew|Matthew]] [[s:Bible (New International Version)/Matthew#25:29|25:29]], [[New International Version]].)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.&lt;br /&gt;
&lt;br /&gt;
Preferential attachment is sometimes referred to as the Matthew effect, but the two are not precisely equivalent.  The Matthew effect, first discussed by Robert K. Merton, is named for a passage in the biblical Gospel of Matthew: &amp;quot;For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him.&amp;quot; (Matthew 25:29, New International Version.)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.&lt;br /&gt;
&lt;br /&gt;
优先链接有时被称为马太效应，但两者并不完全等同。马太效应，最早由罗伯特·金·莫顿讨论，是根据《圣经》中的一段马太福音命名的: 每个拥有的人都会得到更多。凡没有的就是他所有的，也必被夺去(马太福音25:29，新国际版)。优先链接过程不包括夺走部分。但是，由于马太效应背后的科学见解在任何情况下都是完全不同的，因此这一点可能没有意义。定性地说，它的目的不是描述一种机械的乘法效应，如优先链接，而是一种特定的人类行为。在这种行为中，人们更有可能给名人加分而不是给鲜为人知的人加分。马太效应的经典例子是两个不同的人同时做出科学发现时，其中一个是众所周知的，另一个则鲜为人知。据称，在这种情况下，人们往往更倾向于将这一发现归功于著名的科学家。因此，马太效应所要描述的现实世界现象与优先链接(虽然肯定与优先链接有关)是截然不同的。&lt;br /&gt;
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==History 历史==&lt;br /&gt;
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The first rigorous consideration of preferential attachment seems to be that of [[Udny Yule]] in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants.&amp;lt;ref name=YulePhilTrans /&amp;gt;  The process is sometimes called a &amp;quot;Yule process&amp;quot; in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.&lt;br /&gt;
&lt;br /&gt;
The first rigorous consideration of preferential attachment seems to be that of Udny Yule in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants.  The process is sometimes called a &amp;quot;Yule process&amp;quot; in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.&lt;br /&gt;
&lt;br /&gt;
1925年，那时的Udny Yule是第一个对优先链接进行严谨考量的人，他用它来解释被子植物属中物种数量的幂律分布。为了纪念他，这个过程有时被称为“Yule过程”。Yule能够证明这个过程产生了一个带有幂律尾巴的分布，但是他的证明的细节，按照今天的标准，是曲折和晦涩的，因为现代的随机过程理论工具当时还不存在，他被迫使用更加繁琐的证明方法。&lt;br /&gt;
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Most modern treatments of preferential attachment make use of the [[master equation]] method, whose use in this context was pioneered by [[Herbert A. Simon|Simon]] in 1955, in work on the distribution of sizes of cities and other phenomena.&amp;lt;ref name=SimonBiomet /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Most modern treatments of preferential attachment make use of the master equation method, whose use in this context was pioneered by Simon in 1955, in work on the distribution of sizes of cities and other phenomena.&lt;br /&gt;
&lt;br /&gt;
大多数现代优先链接的处理方法都使用了'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;主方程方法 Master Equation Method&amp;lt;/font&amp;gt;'''，这种方法在1955年由 Simon 首创，用于研究城市规模和其他现象的分布。&lt;br /&gt;
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The first application of preferential attachment to learned citations was given by [[Derek J. de Solla Price|Price]] in 1976.&amp;lt;ref name=PriceJASIS /&amp;gt;  (He referred to the process as a &amp;quot;cumulative advantage&amp;quot; process.)  His was also the first application of the process to the growth of a network, producing what would now be called a [[scale-free network]].  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including [[Lotka's law]] of scientific productivity and [[Bradford's law]] of journal use.&lt;br /&gt;
&lt;br /&gt;
The first application of preferential attachment to learned citations was given by Price in 1976.  (He referred to the process as a &amp;quot;cumulative advantage&amp;quot; process.)  His was also the first application of the process to the growth of a network, producing what would now be called a scale-free network.  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including Lotka's law of scientific productivity and Bradford's law of journal use.&lt;br /&gt;
&lt;br /&gt;
普赖斯于1976年首次将优先链接应用于学术引文。(他把这个过程称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;“优势累积” Cumulative Advantage&amp;lt;/font&amp;gt;'''过程。)他也是第一个将这一过程应用于网络发展的人，他创造了现在所谓的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Network&amp;lt;/font&amp;gt;'''。正是在网络增长的背景下，这一过程在今天得到了最频繁的研究。普莱斯还将优先链接作为许多其他现象中幂定律的可能解释，包括洛特卡的科学生产力定律和布拉德福德的期刊使用定律。&lt;br /&gt;
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The application of preferential attachment to the growth of the World Wide Web was proposed by [[BA model|Barabási and Albert]] in 1999.&amp;lt;ref name=BAScience /&amp;gt;  Barabási and Albert also coined the name &amp;quot;preferential attachment&amp;quot; by which the process is best known today and suggested that the process might apply to the growth of other networks as well. For growing networks, the precise functional form of preferential attachment can be estimated by [[maximum likelihood estimation]].&amp;lt;ref&amp;gt;{{cite journal |last1=Pham |first1=Thong |last2=Sheridan |first2=Paul |last3=Shimodaira |first3=Hidetoshi |title=PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks |journal=PLoS ONE |date=September 17, 2015 |volume=10 |issue=9 |doi=10.1371/journal.pone.0137796 |url=http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0137796 |pages=e0137796 |pmid=26378457 |pmc=4574777|bibcode=2015PLoSO..1037796P }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The application of preferential attachment to the growth of the World Wide Web was proposed by Barabási and Albert in 1999.&lt;br /&gt;
&lt;br /&gt;
1999年，Barabási 和 Albert 提出了优先链接在万维网发展中的应用。&lt;br /&gt;
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==See also==&lt;br /&gt;
&lt;br /&gt;
{{colbegin}}&lt;br /&gt;
&lt;br /&gt;
* [[Assortative mixing]]&lt;br /&gt;
关联矩阵&lt;br /&gt;
&lt;br /&gt;
* [[BA model]]&lt;br /&gt;
BA模型&lt;br /&gt;
&lt;br /&gt;
* [[Bose–Einstein condensation: a network theory approach]]&lt;br /&gt;
玻色-爱因斯坦凝聚：一种网络理论方法&lt;br /&gt;
&lt;br /&gt;
* [[Capital accumulation]]&lt;br /&gt;
资本积累&lt;br /&gt;
&lt;br /&gt;
* [[Chinese restaurant process]]&lt;br /&gt;
中餐厅过程&lt;br /&gt;
&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
&lt;br /&gt;
* [[Double jeopardy (marketing)]]&lt;br /&gt;
双重危险（营销）&lt;br /&gt;
&lt;br /&gt;
* [[Link-centric preferential attachment]]&lt;br /&gt;
以链接为中心的优先链接&lt;br /&gt;
&lt;br /&gt;
* [[Matthew effect (sociology)]]&lt;br /&gt;
马太效应（社会学）&lt;br /&gt;
&lt;br /&gt;
* [[Pitman–Yor process]]&lt;br /&gt;
Pitman-Yor过程&lt;br /&gt;
&lt;br /&gt;
* [[Power law]]&lt;br /&gt;
幂定律&lt;br /&gt;
&lt;br /&gt;
* [[Price's model]]&lt;br /&gt;
价格模型&lt;br /&gt;
&lt;br /&gt;
* [[Proof of stake]]&lt;br /&gt;
权益证明&lt;br /&gt;
&lt;br /&gt;
* [[Simon model]]&lt;br /&gt;
西蒙模型&lt;br /&gt;
&lt;br /&gt;
* [[Stochastic processes]]&lt;br /&gt;
随机过程&lt;br /&gt;
&lt;br /&gt;
* [[Wealth condensation]]&lt;br /&gt;
财富集聚&lt;br /&gt;
&lt;br /&gt;
* [[Yule–Simon distribution]]&lt;br /&gt;
Yule-Simon 分布&lt;br /&gt;
&lt;br /&gt;
* [[Bibliogram]]&lt;br /&gt;
书目图&lt;br /&gt;
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{{colend}}&lt;br /&gt;
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==References==&lt;br /&gt;
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&amp;lt;references/&amp;gt;&lt;br /&gt;
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{{DEFAULTSORT:Preferential Attachment}}&lt;br /&gt;
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[[Category:Stochastic processes]]&lt;br /&gt;
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Category:Stochastic processes&lt;br /&gt;
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类别: 随机过程&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Preferential attachment]]. Its edit history can be viewed at [[优先链接/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E4%BC%98%E5%85%88%E9%93%BE%E6%8E%A5&amp;diff=13596</id>
		<title>优先链接</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E4%BC%98%E5%85%88%E9%93%BE%E6%8E%A5&amp;diff=13596"/>
		<updated>2020-09-02T10:39:04Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
{{Redirect|Yule process|the type of birth process|Simple birth process}}&lt;br /&gt;
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A '''preferential attachment process''' is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not.  &amp;quot;Preferential attachment&amp;quot; is only the most recent of many names that have been given to such processes.  They are also referred to under the names &amp;quot;[[Udny Yule|Yule]] process&amp;quot;, &amp;quot;cumulative advantage&amp;quot;, &amp;quot;the rich get richer&amp;quot;, and, less correctly, the &amp;quot;[[Matthew effect (sociology)|Matthew effect]]&amp;quot;.  They are also related to [[Gibrat's law]]. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate [[power law]] distributions.&lt;br /&gt;
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A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not.  &amp;quot;Preferential attachment&amp;quot; is only the most recent of many names that have been given to such processes.  They are also referred to under the names &amp;quot;Yule process&amp;quot;, &amp;quot;cumulative advantage&amp;quot;, &amp;quot;the rich get richer&amp;quot;, and, less correctly, the &amp;quot;Matthew effect&amp;quot;.  They are also related to Gibrat's law. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions.&lt;br /&gt;
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优先链接过程是一类过程，其中一些数量，通常是某种形式的财富或信贷，是根据一些个人或物体已经拥有的数量来分布的，从而使那些已经富有的人比那些不富有的人得到更多。”优先链接”是该过程众多名称中最贴切其本质含义的名字。他们还被称为“Yule过程”、“优势累积”、“富人越来越富” ，以及不那么正确的“马太效应”。它们也与纪伯伦的法律有关。优先链接之所以受到科学家的关注，主要是因为它能在适当的条件下产生幂律分布。&lt;br /&gt;
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==Definition 定义==&lt;br /&gt;
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A preferential attachment process is a [[stochastic process|stochastic]] [[urn problem|urn process]], meaning a process in which discrete units of wealth, usually called &amp;quot;balls&amp;quot;, are added in a random or partly random fashion to a set of objects or containers, usually called &amp;quot;urns&amp;quot;.  A preferential attachment process is an urn process in which additional balls are added continuously to the system and are distributed among the urns as an increasing function of the number of balls the urns already have.  In the most commonly studied examples, the number of urns also increases continuously, although this is not a necessary condition for preferential attachment and examples have been studied with constant or even decreasing numbers of urns.&lt;br /&gt;
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A preferential attachment process is a stochastic urn process, meaning a process in which discrete units of wealth, usually called &amp;quot;balls&amp;quot;, are added in a random or partly random fashion to a set of objects or containers, usually called &amp;quot;urns&amp;quot;.  A preferential attachment process is an urn process in which additional balls are added continuously to the system and are distributed among the urns as an increasing function of the number of balls the urns already have.  In the most commonly studied examples, the number of urns also increases continuously, although this is not a necessary condition for preferential attachment and examples have been studied with constant or even decreasing numbers of urns.&lt;br /&gt;
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优先链接过程是一个随机的瓮过程，意味着在这个过程中，离散的财富单位，通常称为“球”，以随机或部分随机的方式添加到一组物体或容器，通常称为“骨灰瓮”。优先链接过程是一个瓮过程，在这个过程中，额外的球被不断地添加到系统中，并随着瓮中已有球数的增加而分布在瓮中。在最常研究的例子中，骨灰盒的数量也在不断增加，尽管这不是优先链接的必要条件，而且也有研究表示过骨灰盒数量不变甚至减少的例子。&lt;br /&gt;
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A classic example of a preferential attachment process is the growth in the number of [[species]] per [[genus]] in some higher [[taxon]] of biotic organisms.&amp;lt;ref name=YulePhilTrans&amp;gt;{{cite journal | last=Yule | first=G. U. | title=A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S | journal=[[Philosophical Transactions of the Royal Society B]] | volume=213 | pages=21–87 | year=1925 | doi=10.1098/rstb.1925.0002 | issue=402–410| doi-access=free }}&amp;lt;/ref&amp;gt;  New genera (&amp;quot;urns&amp;quot;) are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species (&amp;quot;balls&amp;quot;) are added as old ones [[speciation|speciate]] (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by [[Udny Yule|Yule]], is a ''[[linear]]'' preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.&lt;br /&gt;
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A classic example of a preferential attachment process is the growth in the number of species per genus in some higher taxon of biotic organisms.  New genera (&amp;quot;urns&amp;quot;) are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species (&amp;quot;balls&amp;quot;) are added as old ones speciate (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by Yule, is a linear preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.&lt;br /&gt;
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优先链接过程的一个典型例子是某些生物有机体的高级分类群中每个属的物种数量的增长。每当一个新出现的物种被认为与其前辈完全不同，不属于任何一个现有属时，新属(“ urns”)就被添加到分类单元中。新物种(”球”)与旧物种(即一分为二)一并加入，假设新物种与其亲本属于同一属(新属除外) ，新物种加入该属的概率将与该属已有的物种数量成正比。这个过程，首先被Yule研究，是一个线性优先链接的过程，因为新物种的繁殖率与其已经拥有的数量成线性关系。&lt;br /&gt;
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Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called [[Yule distribution]].  In the most general form of the process, balls are added to the system at an overall rate of ''m'' new balls for each new urn.  Each newly created urn starts out with ''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; balls and further balls are added to urns at a rate proportional to the number ''k'' that they already have plus a constant ''a''&amp;amp;nbsp;&amp;gt;&amp;amp;nbsp;−''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;.  With these definitions, the fraction ''P''(''k'') of urns having ''k'' balls in the limit of long time is given by&amp;lt;ref&amp;gt;{{cite journal | last=Newman | first=M. E. J. | title=Power laws, Pareto distributions and Zipf's law | journal=Contemporary Physics | volume=46 | pages=323–351 | year=2005 | arxiv=cond-mat/0412004 | doi=10.1080/00107510500052444 | issue=5 | bibcode=2005ConPh..46..323N}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called Yule distribution.  In the most general form of the process, balls are added to the system at an overall rate of m new balls for each new urn.  Each newly created urn starts out with k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; balls and further balls are added to urns at a rate proportional to the number k that they already have plus a constant a&amp;amp;nbsp;&amp;gt;&amp;amp;nbsp;−k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;.  With these definitions, the fraction P(k) of urns having k balls in the limit of long time is given by&lt;br /&gt;
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已知骨灰缸数量增加的线性优先链接过程，会按照所谓的尤尔分布在骨灰缸上产生球状分布。在最一般形式的过程中，球添加到系统中的整体速度为每m个新球到一个瓮中。每个新创建的瓮都以 k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; 球开始，球被不断地添加到骨灰瓮中，其速度与骨灰瓮的数量k，加上一个常数 ''a''&amp;amp;nbsp;&amp;gt;&amp;amp;nbsp;−''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;，成正比。利用这些定义，可以给出长时极限下具有k个球的骨灰瓮的分数p(k)&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_0+a,\gamma-1)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_0+a,\gamma-1)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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for ''k''&amp;amp;nbsp;≥&amp;amp;nbsp;''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; (and zero otherwise), where B(''x'',&amp;amp;nbsp;''y'') is the Euler [[beta function]]:&lt;br /&gt;
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for k&amp;amp;nbsp;≥&amp;amp;nbsp;k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; (and zero otherwise), where B(x,&amp;amp;nbsp;y) is the Euler beta function:&lt;br /&gt;
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对于k&amp;amp;nbsp;≥&amp;amp;nbsp;k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;(否则为0) ，其中 b (x，y)是 Euler beta 函数:&lt;br /&gt;
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&amp;lt;math&amp;gt; \mathrm{B}(x,y)={\Gamma(x)\Gamma(y)\over\Gamma(x+y)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt; \mathrm{B}(x,y)={\Gamma(x)\Gamma(y)\over\Gamma(x+y)},&amp;lt;/math&amp;gt;&lt;br /&gt;
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with Γ(''x'') being the standard [[gamma function]], and&lt;br /&gt;
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with Γ(x) being the standard gamma function, and&lt;br /&gt;
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(x)是标准伽马函数，并且&lt;br /&gt;
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&amp;lt;math&amp;gt;\gamma=2 + {k_0 + a\over m}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\gamma=2 + {k_0 + a\over m}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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The beta function behaves asymptotically as B(''x'',&amp;amp;nbsp;''y'')&amp;amp;nbsp;~&amp;amp;nbsp;''x''&amp;lt;sup&amp;gt;−''y''&amp;lt;/sup&amp;gt; for large ''x'' and fixed ''y'', which implies that for large values of ''k'' we have&lt;br /&gt;
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The beta function behaves asymptotically as B(x,&amp;amp;nbsp;y)&amp;amp;nbsp;~&amp;amp;nbsp;x&amp;lt;sup&amp;gt;−y&amp;lt;/sup&amp;gt; for large x and fixed y, which implies that for large values of k we have&lt;br /&gt;
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对于较大的x和固定的y，β函数表现为B(''x'',&amp;amp;nbsp;''y'')&amp;amp;nbsp;~&amp;amp;nbsp;''x''&amp;lt;sup&amp;gt;−''y''&amp;lt;/sup&amp;gt;，这意味着对于k的大值，我们有一个渐近的β函数&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k) \propto k^{-\gamma}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt; P(k) \propto k^{-\gamma}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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In other words, the preferential attachment process generates a &amp;quot;[[Long tail|long-tailed]]&amp;quot; distribution following a [[Pareto distribution]] or [[power law]] in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities,&amp;lt;ref name=SimonBiomet&amp;gt;{{cite journal | last=Simon | first=H. A. | title=On a class of skew distribution functions | journal=Biometrika | volume=42 | pages=425–440  | year=1955 | doi=10.1093/biomet/42.3-4.425  | issue=3–4 }}&amp;lt;/ref&amp;gt; the wealth of extremely wealthy individuals,&amp;lt;ref name=SimonBiomet /&amp;gt; the number of citations received by learned publications,&amp;lt;ref name=PriceJASIS&amp;gt;{{cite journal | last=Price | first=D. J. de S.  | title=A general theory of bibliometric and other cumulative advantage processes | journal=J. Amer. Soc. Inform. Sci. | volume=27 | pages=292–306 | year=1976 | url=http://garfield.library.upenn.edu/price/pricetheory1976.pdf | doi=10.1002/asi.4630270505 | issue=5}}&amp;lt;/ref&amp;gt; and the number of links to pages on the World Wide Web.&amp;lt;ref name=BAScience&amp;gt;{{cite journal | last=Barabási | first=A.-L. |author2=R. Albert | title=Emergence of scaling in random networks | journal=Science | volume=286 | pages=509–512 | year=1999 | arxiv=cond-mat/9910332 | doi=10.1126/science.286.5439.509 | issue=5439 | pmid=10521342| bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In other words, the preferential attachment process generates a &amp;quot;long-tailed&amp;quot; distribution following a Pareto distribution or power law in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities, the wealth of extremely wealthy individuals, and the number of links to pages on the World Wide Web.&lt;br /&gt;
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换句话说，优先链接过程在其尾部产生一个遵循帕累托分布或幂定律的“长尾”分布。这是历史上人们对优先链接行为感兴趣的主要原因: 经过经验的观察，物种分布和许多其他现象都遵循幂律分布，而优先链接过程是解释这种行为的主要候选机制。优先链接被认为是一个可能的候选解释，可能解释如城市规模的分布，极其富有的个人的财富，以及万维网网页的链接数量等问题。&lt;br /&gt;
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The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;1) and gains new species in direct proportion to the number it already has (''a''&amp;amp;nbsp;=&amp;amp;nbsp;0), and hence ''P''(''k'')&amp;amp;nbsp;=&amp;amp;nbsp;B(''k'',&amp;amp;nbsp;''γ'')/B(''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;,&amp;amp;nbsp;''γ''&amp;amp;nbsp;−&amp;amp;nbsp;1) with ''γ''=2&amp;amp;nbsp;+&amp;amp;nbsp;1/''m''.  Similarly the Price model for scientific citations&amp;lt;ref name=PriceJASIS /&amp;gt; corresponds to the case ''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;0, ''a''&amp;amp;nbsp; =&amp;amp;nbsp;1 and the widely studied [[Barabási-Albert model]]&amp;lt;ref name=BAScience /&amp;gt; corresponds to ''k''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;''m'', ''a''&amp;amp;nbsp;=&amp;amp;nbsp;0.&lt;br /&gt;
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The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;1) and gains new species in direct proportion to the number it already has (a&amp;amp;nbsp;=&amp;amp;nbsp;0), and hence P(k)&amp;amp;nbsp;=&amp;amp;nbsp;B(k,&amp;amp;nbsp;γ)/B(k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;,&amp;amp;nbsp;γ&amp;amp;nbsp;−&amp;amp;nbsp;1) with γ=2&amp;amp;nbsp;+&amp;amp;nbsp;1/m.  Similarly the Price model for scientific citations corresponds to the case k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;0, a&amp;amp;nbsp; =&amp;amp;nbsp;1 and the widely studied Barabási-Albert model corresponds to k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;m, a&amp;amp;nbsp;=&amp;amp;nbsp;0.&lt;br /&gt;
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这里描述的一般模型包括作为特殊情况的许多其他特定模型。例如，在上面的种属例子中，每个属以一个单一的种(k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;1)开始，并且获得新的种(a&amp;amp;nbsp;=&amp;amp;nbsp;0) ，因此P(k)&amp;amp;nbsp;=&amp;amp;nbsp;B(k,&amp;amp;nbsp;γ)/B(k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;,&amp;amp;nbsp;γ&amp;amp;nbsp;−&amp;amp;nbsp;1)，γ=2&amp;amp;nbsp;+&amp;amp;nbsp;1/m. 。类似地，科学引文的价格模型对应于k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;0，a&amp;amp;nbsp; =&amp;amp;nbsp;1 ，广泛研究的 Barabási-Albert 模型对应于 k&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;m, a&amp;amp;nbsp;=&amp;amp;nbsp;0.。&lt;br /&gt;
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Preferential attachment is sometimes referred to as the [[Matthew effect]], but the two are not precisely equivalent.  The Matthew effect, first discussed by [[Robert K. Merton]],&amp;lt;ref&amp;gt;{{cite journal | last=Merton | first=Robert K. | authorlink=Robert K. Merton | title=The Matthew effect in science | journal=Science | volume=159 | pages=56–63 | year=1968 | doi=10.1126/science.159.3810.56 | pmid=17737466 | issue=3810| bibcode=1968Sci...159...56M }}&amp;lt;/ref&amp;gt; is named for a passage in the [[bible|biblical]] [[Gospel of Matthew]]: &amp;quot;For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him.&amp;quot; ([[Gospel of Matthew|Matthew]] [[s:Bible (New International Version)/Matthew#25:29|25:29]], [[New International Version]].)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.&lt;br /&gt;
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Preferential attachment is sometimes referred to as the Matthew effect, but the two are not precisely equivalent.  The Matthew effect, first discussed by Robert K. Merton, is named for a passage in the biblical Gospel of Matthew: &amp;quot;For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him.&amp;quot; (Matthew 25:29, New International Version.)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.&lt;br /&gt;
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优先链接有时被称为马太效应，但两者并不完全等同。马太效应，最早由罗伯特·金·莫顿讨论，是根据《圣经》中的一段马太福音命名的: 每个拥有的人都会得到更多，他也将拥有更多。凡没有的就是他所有的，也必被夺去(马太福音25:29，新国际版)。优先链接过程不包括夺走部分。但是，由于马太效应背后的科学见解在任何情况下都是完全不同的，因此这一点可能没有意义。定性地说，它的目的不是描述一种机械的乘法效应，如优先链接，而是一种特定的人类行为。在这种行为中，人们更有可能给名人加分而不是给鲜为人知的人加分。马太效应的经典例子是两个不同的人同时做出科学发现时，其中一个是众所周知的，另一个则鲜为人知。据称，在这种情况下，人们往往更倾向于将这一发现归功于著名的科学家。因此，马太效应所要描述的现实世界现象与优先链接(虽然肯定与优先链接有关)是截然不同的。&lt;br /&gt;
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==History 历史==&lt;br /&gt;
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The first rigorous consideration of preferential attachment seems to be that of [[Udny Yule]] in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants.&amp;lt;ref name=YulePhilTrans /&amp;gt;  The process is sometimes called a &amp;quot;Yule process&amp;quot; in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.&lt;br /&gt;
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The first rigorous consideration of preferential attachment seems to be that of Udny Yule in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants.  The process is sometimes called a &amp;quot;Yule process&amp;quot; in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.&lt;br /&gt;
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1925年，那时的Udny Yule是第一个对优先链接进行严谨考量的人，他用它来解释被子植物属中物种数量的幂律分布。为了纪念他，这个过程有时被称为“Yule过程”。Yule 能够证明这个过程产生了一个带有幂律尾巴的分布，但是他的证明的细节，按照今天的标准，是扭曲和晦涩的，因为现代的随机过程理论工具当时还不存在，他被迫使用更加繁琐的证明方法。&lt;br /&gt;
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Most modern treatments of preferential attachment make use of the [[master equation]] method, whose use in this context was pioneered by [[Herbert A. Simon|Simon]] in 1955, in work on the distribution of sizes of cities and other phenomena.&amp;lt;ref name=SimonBiomet /&amp;gt;&lt;br /&gt;
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Most modern treatments of preferential attachment make use of the master equation method, whose use in this context was pioneered by Simon in 1955, in work on the distribution of sizes of cities and other phenomena.&lt;br /&gt;
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大多数现代优先链接的处理方法都使用了主方程方法，这种方法在1955年由 Simon 首创，用于研究城市规模和其他现象的分布。&lt;br /&gt;
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The first application of preferential attachment to learned citations was given by [[Derek J. de Solla Price|Price]] in 1976.&amp;lt;ref name=PriceJASIS /&amp;gt;  (He referred to the process as a &amp;quot;cumulative advantage&amp;quot; process.)  His was also the first application of the process to the growth of a network, producing what would now be called a [[scale-free network]].  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including [[Lotka's law]] of scientific productivity and [[Bradford's law]] of journal use.&lt;br /&gt;
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The first application of preferential attachment to learned citations was given by Price in 1976.  (He referred to the process as a &amp;quot;cumulative advantage&amp;quot; process.)  His was also the first application of the process to the growth of a network, producing what would now be called a scale-free network.  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including Lotka's law of scientific productivity and Bradford's law of journal use.&lt;br /&gt;
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普赖斯于1976年首次将优先链接应用于学术引文。(他把这个过程称为“优势累积”过程。)他也是第一个将这一过程应用于网络发展的人，他创造了现在所谓的无标度网络。正是在网络增长的背景下，这一过程在今天得到了最频繁的研究。普莱斯还将优先链接作为许多其他现象中幂定律的可能解释，包括洛特卡的科学生产力定律和布拉德福德的期刊使用定律。&lt;br /&gt;
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The application of preferential attachment to the growth of the World Wide Web was proposed by [[BA model|Barabási and Albert]] in 1999.&amp;lt;ref name=BAScience /&amp;gt;  Barabási and Albert also coined the name &amp;quot;preferential attachment&amp;quot; by which the process is best known today and suggested that the process might apply to the growth of other networks as well. For growing networks, the precise functional form of preferential attachment can be estimated by [[maximum likelihood estimation]].&amp;lt;ref&amp;gt;{{cite journal |last1=Pham |first1=Thong |last2=Sheridan |first2=Paul |last3=Shimodaira |first3=Hidetoshi |title=PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks |journal=PLoS ONE |date=September 17, 2015 |volume=10 |issue=9 |doi=10.1371/journal.pone.0137796 |url=http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0137796 |pages=e0137796 |pmid=26378457 |pmc=4574777|bibcode=2015PLoSO..1037796P }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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The application of preferential attachment to the growth of the World Wide Web was proposed by Barabási and Albert in 1999.&lt;br /&gt;
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1999年，Barabási 和 Albert 提出了优先链接在万维网发展中的应用。&lt;br /&gt;
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==See also==&lt;br /&gt;
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{{colbegin}}&lt;br /&gt;
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* [[Assortative mixing]]&lt;br /&gt;
关联矩阵&lt;br /&gt;
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* [[BA model]]&lt;br /&gt;
BA模型&lt;br /&gt;
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* [[Bose–Einstein condensation: a network theory approach]]&lt;br /&gt;
玻色-爱因斯坦凝聚：一种网络理论方法&lt;br /&gt;
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* [[Capital accumulation]]&lt;br /&gt;
资本积累&lt;br /&gt;
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* [[Chinese restaurant process]]&lt;br /&gt;
中餐厅过程&lt;br /&gt;
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* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
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* [[Double jeopardy (marketing)]]&lt;br /&gt;
双重危险（营销）&lt;br /&gt;
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* [[Link-centric preferential attachment]]&lt;br /&gt;
以链接为中心的优先链接&lt;br /&gt;
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* [[Matthew effect (sociology)]]&lt;br /&gt;
马太效应（社会学）&lt;br /&gt;
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* [[Pitman–Yor process]]&lt;br /&gt;
Pitman-Yor过程&lt;br /&gt;
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* [[Power law]]&lt;br /&gt;
幂定律&lt;br /&gt;
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* [[Price's model]]&lt;br /&gt;
价格模型&lt;br /&gt;
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* [[Proof of stake]]&lt;br /&gt;
权益证明&lt;br /&gt;
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* [[Simon model]]&lt;br /&gt;
西蒙模型&lt;br /&gt;
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* [[Stochastic processes]]&lt;br /&gt;
随机过程&lt;br /&gt;
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* [[Wealth condensation]]&lt;br /&gt;
财富集聚&lt;br /&gt;
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* [[Yule–Simon distribution]]&lt;br /&gt;
Yule-Simon 分布&lt;br /&gt;
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* [[Bibliogram]]&lt;br /&gt;
书目图&lt;br /&gt;
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{{colend}}&lt;br /&gt;
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==References==&lt;br /&gt;
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&amp;lt;references/&amp;gt;&lt;br /&gt;
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{{DEFAULTSORT:Preferential Attachment}}&lt;br /&gt;
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[[Category:Stochastic processes]]&lt;br /&gt;
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Category:Stochastic processes&lt;br /&gt;
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类别: 随机过程&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Preferential attachment]]. Its edit history can be viewed at [[优先链接/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%9B%BE%E7%9A%84%E7%BB%98%E5%88%B6_Graph_drawing&amp;diff=13527</id>
		<title>图的绘制 Graph drawing</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%9B%BE%E7%9A%84%E7%BB%98%E5%88%B6_Graph_drawing&amp;diff=13527"/>
		<updated>2020-09-01T05:59:21Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{about|the general subject of graph drawing|the annual research symposium|International Symposium on Graph Drawing}}&lt;br /&gt;
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[[File:WorldWideWebAroundWikipedia.png|thumb|&lt;br /&gt;
图1：Graphic representation of a minute fraction of the [[WWW]], demonstrating [[hyperlink]]s. 以图的形式展示万维网的一小部分]]&lt;br /&gt;
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'''Graph drawing''' is an area of [[mathematics]] and [[computer science]] combining methods from [[geometric graph theory]] and [[information visualization]] to derive two-dimensional depictions of [[graph (discrete mathematics)|graph]]s arising from applications such as [[social network analysis]], [[cartography]], [[linguistics]], and [[bioinformatics]].&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, pp. vii–viii; {{harvtxt|Herman|Melançon|Marshall|2000}}, Section 1.1, &amp;quot;Typical Application Areas&amp;quot;.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图的绘制 Graph Drawing&amp;lt;/font&amp;gt;''' is an area of '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;数学 Mathematics&amp;lt;/font&amp;gt;''' and '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;计算机科学 Computer Science&amp;lt;/font&amp;gt;''' combining methods from '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;几何图形理论 Geometric Graph Theory&amp;lt;/font&amp;gt;'''and '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;信息可视化 Information Visualization&amp;lt;/font&amp;gt;''' to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics.&lt;br /&gt;
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图形绘制是数学和计算机科学的一个交叉领域，它结合了几何图形理论和信息可视化的方法，对来自社会网络分析、制图学、语言学和生物信息学等各种应用场景进行二维描述。&lt;br /&gt;
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A drawing of a graph or '''network diagram''' is a pictorial representation of the [[vertex (graph theory)|vertices]] and [[edge (graph theory)|edges]] of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.&amp;lt;ref name=&amp;quot;dett6&amp;quot;&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p. 6.&amp;lt;/ref&amp;gt;  In the abstract, all that matters is which pairs of vertices are connected by edges.  In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and [[aesthetics]].&amp;lt;ref name=&amp;quot;dett-viii&amp;quot;/&amp;gt; The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's mental map.&amp;lt;ref&amp;gt;{{harvtxt|Misue|Eades|Lai|Sugiyama|1995}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.  In the abstract, all that matters is which pairs of vertices are connected by edges.  In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics.The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's mental map.&lt;br /&gt;
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图形或'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络图 Network Diagram&amp;lt;/font&amp;gt;'''的绘制是通过从图形抽象出的顶点和边表示的。这个图不应与图形本身混淆: 非常不同的图可以对应于同一个图形。抽象地说，重要的是哪一对顶点是由边连在一起。然而，在具体的绘图中，这些顶点和边的排列影响了它的可理解性、可用性、制造成本和美学。如果图形通过添加和删除边缘(动态绘制图形)而随时间变化，并且目标是保存用户的心理映射，那么问题会变得更糟。&lt;br /&gt;
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==Graphical conventions 画图公约==&lt;br /&gt;
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[[File:4node-digraph-natural.svg|thumb|upright=0.5|&lt;br /&gt;
图2：[[Directed graph]] with arrowheads showing edge directions 带箭头显示边方向的有向图]]&lt;br /&gt;
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Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as [[line segment]]s, [[Polygonal chain|polylines]], or curves in the [[Euclidean plane]].&amp;lt;ref name=&amp;quot;dett-viii&amp;quot;&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p. viii.&amp;lt;/ref&amp;gt; Node–link diagrams can be traced back to the 13th century work of [[Ramon Llull]], who drew diagrams of this type for [[complete graph]]s in order to analyze all pairwise combinations among sets of metaphysical concepts.&amp;lt;ref&amp;gt;{{citation|contribution=Two thousand years of combinatorics|first=Donald E.|last=Knuth|authorlink=Donald Knuth|pages=7–37|title=Combinatorics: Ancient and Modern|publisher=Oxford University Press|year=2013|editor1-first=Robin|editor1-last=Wilson|editor2-first=John J.|editor2-last=Watkins}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as line segments, polylines, or curves in the Euclidean plane. Node–link diagrams can be traced back to the 13th century work of Ramon Llull, who drew diagrams of this type for complete graphs in order to analyze all pairwise combinations among sets of metaphysical concepts.&lt;br /&gt;
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图经常绘制成节点链接图，其中顶点可表示为圆盘、框或文本标签，边表示为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;欧几里德平面 Euclidean Plane&amp;lt;/font&amp;gt;'''上的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;线段 Line Segments&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;折线 Polylines&amp;lt;/font&amp;gt;'''或曲线。节点链接图可以追溯到13世纪雷蒙 · 卢尔的作品，他为得到完全图，绘制了这种类型的图，以便分析形而上学概念集合之间的所有成对组合。&lt;br /&gt;
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In the case of [[directed graph]]s, [[Arrow (symbol)|arrowheads]] form a commonly used graphical convention to show their [[Orientability|orientation]];&amp;lt;ref name=&amp;quot;dett6&amp;quot;/&amp;gt; however, user studies have shown that other conventions such as tapering provide this information more effectively.&amp;lt;ref&amp;gt;{{harvtxt|Holten|van Wijk|2009}}; {{harvtxt|Holten|Isenberg|van Wijk|Fekete|2011}}.&amp;lt;/ref&amp;gt; [[Upward planar drawing]] uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary.&amp;lt;ref&amp;gt;{{harvtxt|Garg|Tamassia|1995}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In the case of directed graphs, arrowheads form a commonly used graphical convention to show their orientation; Upward planar drawing uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向图 Directed Graphs&amp;lt;/font&amp;gt;'''的情况下，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;箭头 Arrowheads&amp;lt;/font&amp;gt;'''约定俗成的来显示它们的方向; '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;向上平面绘图 Upward Planar Drawing&amp;lt;/font&amp;gt;'''使用以下约定：每个边都从较低的顶点定向到较高的顶点，因此不需要箭头。&lt;br /&gt;
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Alternative conventions to node–link diagrams include adjacency representations such as [[circle packing theorem|circle packings]], in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; [[intersection graph|intersection representations]] in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other; confluent drawings, in which edges are represented as smooth curves within mathematical [[train track (mathematics)|train tracks]]; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines;&amp;lt;ref name=&amp;quot;Longabaugh 2012&amp;quot;&amp;gt;{{harvtxt|Longabaugh|2012}}.&amp;lt;/ref&amp;gt; and visualizations of the [[adjacency matrix]] of the graph.&lt;br /&gt;
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Alternative conventions to node–link diagrams include adjacency representations such as circle packings, in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; intersection representations in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other; confluent drawings, in which edges are represented as smooth curves within mathematical train tracks; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines; and visualizations of the adjacency matrix of the graph.&lt;br /&gt;
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节点链接图可被其他画图方式替代，包括邻接表示法，如'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;圆填充 Circle Packings&amp;lt;/font&amp;gt;'''，其中顶点表示为平面上不相交的区域，边表示为区域之间的邻接; '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;交点表示 Intersection representations&amp;lt;/font&amp;gt;'''，其中顶点由非相交的几何对象表示，而边线由其交点表示；可见性表示，其中顶点由平面中的区域表示，边由彼此无障碍视线的区域表示; 汇流图，其中边表示为数学轨迹上的光滑曲线; 织物，其中顶点表示为水平线，边表示为垂直线; 以及图形'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;邻接矩阵 Adjiacency Matrix&amp;lt;/font&amp;gt;'''的可视化结点。&lt;br /&gt;
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==Quality measures 质量度量==&lt;br /&gt;
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Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Section 2.1.2, Aesthetics, pp. 14–16; {{harvtxt|Purchase|Cohen|James|1997}}.&amp;lt;/ref&amp;gt; In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.&lt;br /&gt;
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Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability. In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.&lt;br /&gt;
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为了找到客观评价图形美观性和可用性的方法，已经定义了许多不同的图形质量度量方法。除了指导针对同一图形的不同布局方法之间的选择之外，某些布局方法还尝试直接优化这些度量。&lt;br /&gt;
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[[File:4node-digraph-embed.svg|upright=0.5|thumb|&lt;br /&gt;
图3：[[Planar graph]] drawn without overlapping edges 没有重叠边的平面图]]&lt;br /&gt;
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*The [[Crossing number (graph theory)|crossing number]] of a drawing is the number of pairs of edges that cross each other.  If the graph is [[planar graph|planar]], then it is often convenient to draw it without any edge intersections; that is, in this case, a graph drawing represents a [[graph embedding]]. However, nonplanar graphs frequently arise in applications, so graph drawing algorithms must generally allow for edge crossings.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p 14.&amp;lt;/ref&amp;gt;&lt;br /&gt;
图的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;交叉数 Crossing Number&amp;lt;/font&amp;gt;'''是相互交叉的边对的数目。如果图形是平面的，那么通常可以方便地画出没有任何边缘相交的图形;也就是说，在本例中，图形绘制表示'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图形嵌入 Graph Embedding&amp;lt;/font&amp;gt;'''。然而，应用中经常出现非平面图，因此图形绘制算法通常必须考虑边缘交叉。&lt;br /&gt;
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*The [[Area (graph drawing)|area]] of a drawing is the size of its smallest [[bounding box]], relative to the closest distance between any two vertices. Drawings with smaller area are generally preferable to those with larger area, because they allow the features of the drawing to be shown at greater size and therefore more legibly. The [[aspect ratio]] of the bounding box may also be important.&lt;br /&gt;
一个图形的面积是它的最小'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;边界盒 Bounding Box&amp;lt;/font&amp;gt;'''的大小，相对于任意两个顶点之间的最近距离。面积小的画通常比面积大的画更可取，因为它们可以让图画的特征以更大的尺寸显示，因此更清晰。边界框的纵横比也很重要。&lt;br /&gt;
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*Symmetry display is the problem of finding [[Graph automorphism|symmetry group]]s within a given graph, and finding a drawing that displays as much of the symmetry as possible. Some layout methods automatically lead to symmetric drawings; alternatively, some drawing methods start by finding symmetries in the input graph and using them to construct a drawing.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p. 16.&amp;lt;/ref&amp;gt;&lt;br /&gt;
对称显示的问题是在给定的图形中找到'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;对称组 Symmetry Groups&amp;lt;/font&amp;gt;'''，并找到尽可能多地显示对称的绘图。一些布局方法自动导致对称图形;另外，一些绘图方法从查找输入图形中的对称性开始，并使用它们来构造绘图。&lt;br /&gt;
&lt;br /&gt;
*It is important that edges have shapes that are as simple as possible, to make it easier for the eye to follow them. In polyline drawings, the complexity of an edge may be measured by its [[Bend minimization|number of bends]], and many methods aim to provide drawings with few total bends or few bends per edge. Similarly for spline curves the complexity of an edge may be measured by the number of control points on the edge.&lt;br /&gt;
重要的是，边缘的形状要尽可能简单，以使眼睛更容易跟踪它们。在折线图中，一条边的复杂性可以通过它的弯曲数来衡量，许多方法的目的是提供很少的总弯曲或每条边很少弯曲的图。类似地，对于样条曲线，边的复杂性可以通过边上控制点的数量来度量。&lt;br /&gt;
&lt;br /&gt;
*Several commonly used quality measures concern lengths of edges: it is generally desirable to minimize the total length of the edges as well as the maximum length of any edge. Additionally, it may be preferable for the lengths of edges to be uniform rather than highly varied.&lt;br /&gt;
几种常用的质量度量方法与边的长度有关:通常需要最小化边的总长度以及任何边的最大长度。此外，最好使边缘的长度一致而不是变化很大。&lt;br /&gt;
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*[[Angular resolution (graph drawing)|Angular resolution]] is a measure of the sharpest angles in a graph drawing. If a graph has vertices with high [[degree (graph theory)|degree]] then it necessarily will have small angular resolution, but the angular resolution can be bounded below by a function of the degree.&amp;lt;ref name=&amp;quot;ps09&amp;quot;&amp;gt;{{harvtxt|Pach|Sharir|2009}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;角度分辨率 Angular Resolution&amp;lt;/font&amp;gt;'''是图形绘制中最锐角的度量。如果一个图的顶点高度高，那么它的角分辨率就会很小，但是角分辨率可以由角的函数来限定。&lt;br /&gt;
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*The [[slope number]] of a graph is the minimum number of distinct edge slopes needed in a drawing with straight line segment edges (allowing crossings). [[Cubic graph]]s have slope number at most four, but graphs of degree five may have unbounded slope number; it remains open whether the slope number of degree-4 graphs is bounded.&amp;lt;ref name=&amp;quot;ps09&amp;quot;/&amp;gt;&lt;br /&gt;
图的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;斜率数 Slope Number&amp;lt;/font&amp;gt;'''是在具有直线段边(允许交叉)的图中所需的明显边缘斜率的最小值。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;三次图 Cubic Graphs&amp;lt;/font&amp;gt;'''的斜率数最多为4，五次图的斜率数可能是无界的;4度图的斜率数是否有界仍然是开放的。&lt;br /&gt;
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==Layout methods 布局方法==&lt;br /&gt;
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[[File:Social Network Analysis Visualization.png|thumb|right|&lt;br /&gt;
图4：A force-based network visualization. 基于力的网络可视化 &amp;lt;ref&amp;gt;Published in {{Cite journal | volume = 10| issue = 3| last = Grandjean| first = Martin| title = La connaissance est un réseau| journal =Les Cahiers du Numérique| accessdate = 2014-10-15| date = 2014| pages = 37–54| url = http://www.cairn.info/resume.php?ID_ARTICLE=LCN_103_0037| doi=10.3166/lcn.10.3.37-54}}&amp;lt;/ref&amp;gt;]]&lt;br /&gt;
&lt;br /&gt;
There are many different graph layout strategies:&lt;br /&gt;
&lt;br /&gt;
There are many different graph layout strategies:&lt;br /&gt;
&lt;br /&gt;
有许多不同的图形布局策略:&lt;br /&gt;
&lt;br /&gt;
*In [[force-based layout]] systems, the graph drawing software modifies an initial vertex placement by continuously moving the vertices according to a system of forces based on physical metaphors related to systems of [[spring (device)|springs]] or [[molecular mechanics]]. Typically, these systems combine attractive forces between adjacent vertices with repulsive forces between all pairs of vertices, in order to seek a layout in which edge lengths are small while vertices are well-separated. These systems may perform [[gradient descent]] based minimization of an [[energy function]], or they may translate the forces directly into velocities or accelerations for the moving vertices.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Section 2.7, &amp;quot;The Force-Directed Approach&amp;quot;, pp. 29–30, and Chapter 10, &amp;quot;Force-Directed Methods&amp;quot;, pp. 303–326.&amp;lt;/ref&amp;gt;&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'基于力的布局系统 Force-Based Layout&amp;lt;/font&amp;gt;''中，图形绘制软件通过根据基于弹簧系统或分子力学的物理隐喻的力系统不断移动顶点来修改初始顶点的位置。通常，这些系统将相邻顶点之间的吸引力与所有对顶点之间的排斥力结合起来，以寻求一种边长较小而顶点分离良好的布局。这些系统可以执行'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;梯度下降 Gradient Descent&amp;lt;/font&amp;gt;'''最小化的一个能量函数，或者他们可以把力直接转化为节点的速度或加速度&lt;br /&gt;
&lt;br /&gt;
*[[Spectral layout]] methods use as coordinates the [[eigenvector]]s of a [[Matrix (mathematics)|matrix]] such as the [[Discrete Laplace operator|Laplacian]] derived from the [[adjacency matrix]] of the graph.&amp;lt;ref&amp;gt;{{harvtxt|Beckman|1994}}; {{harvtxt|Koren|2005}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;频谱布局方法 Spectral Layout&amp;lt;/font&amp;gt;'''使用从图的邻接矩阵派生的矩阵（例如Laplacian）的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;特征向量 Eigenvector&amp;lt;/font&amp;gt;'''作为坐标。&lt;br /&gt;
&lt;br /&gt;
*Orthogonal layout methods, which allow the edges of the graph to run horizontally or vertically, parallel to the coordinate axes of the layout. These methods were originally designed for [[VLSI]] and [[Printed circuit board|PCB]] layout problems but they have also been adapted for graph drawing. They typically involve a multiphase approach in which an input graph is planarized by replacing crossing points by vertices, a topological embedding of the planarized graph is found, edge orientations are chosen to minimize bends, vertices are placed consistently with these orientations, and finally a layout compaction stage reduces the area of the drawing.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Chapter 5, &amp;quot;Flow and Orthogonal Drawings&amp;quot;, pp. 137–170; {{harv|Eiglsperger|Fekete|Klau|2001}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
正交布局方法，允许图形的边缘水平或垂直运行，平行于布局的坐标轴。这些方法最初是为超大规模集成电路和PCB布局问题而设计的，但它们也适用于图形绘制。它们通常涉及到一种多阶段的方法，通过用顶点替换交叉点来平面化输入图，找到一个平面化图的拓扑嵌入，选择边的方向来最小化弯曲，顶点的放置始终与这些方向一致，最后一个布局压缩阶段减少&lt;br /&gt;
&lt;br /&gt;
*Tree layout algorithms these show a rooted [[tree structure|tree]]-like formation, suitable for [[tree (graph theory)|trees]]. Often, in a technique called &amp;quot;balloon layout&amp;quot;, the children of each node in the tree are drawn on a circle surrounding the node, with the radii of these circles diminishing at lower levels in the tree so that these circles do not overlap.&amp;lt;ref&amp;gt;{{harvtxt|Herman|Melançon|Marshall|2000}}, Section 2.2, &amp;quot;Traditional Layout – An Overview&amp;quot;.&amp;lt;/ref&amp;gt;&lt;br /&gt;
这些树布局算法显示了一种类似树的根状结构，适合树。通常，在一种称为气球布局的技术中，树中每个节点的子节点被画在围绕该节点的圆上，这些圆的半径在树的较低层次上递减，以便这些圆不会重叠。&lt;br /&gt;
&lt;br /&gt;
*[[Layered graph drawing]] methods (often called Sugiyama-style drawing) are best suited for [[directed acyclic graph]]s or graphs that are nearly acyclic, such as the graphs of dependencies between modules or functions in a software system. In these methods, the nodes of the graph are arranged into horizontal layers using methods such as the [[Coffman–Graham algorithm]], in such a way that most edges go downwards from one layer to the next; after this step, the nodes within each layer are arranged in order to minimize crossings.&amp;lt;ref&amp;gt;{{harvtxt|Sugiyama|Tagawa|Toda|1981}}; {{harvtxt|Bastert|Matuszewski|2001}}; {{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Chapter 9, &amp;quot;Layered Drawings of Digraphs&amp;quot;, pp. 265–302.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分层图绘制方法 Layered Graph Drawing&amp;lt;/font&amp;gt;'''(通常称为杉山式图)最适合于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向无环图 Directed Acyclic Graphs&amp;lt;/font&amp;gt;'''或接近无环的图，例如软件系统中模块或函数之间的依赖关系图。在这些方法中，图的节点使用Coffman Graham算法等方法被安排到水平层中，以这样的方式，大多数边从一层向下到下一层;在这一步之后，将每一层内的节点进行排列，以最小化交叉。&lt;br /&gt;
&lt;br /&gt;
[[File:Goldner-Harary-linear.svg|thumb|&lt;br /&gt;
图5：Arc diagram 弧线图]]&lt;br /&gt;
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&lt;br /&gt;
*[[Arc diagram]]s, a layout style dating back to the 1960s,&amp;lt;ref&amp;gt;{{harvtxt|Saaty|1964}}.&amp;lt;/ref&amp;gt; place vertices on a line; edges may be drawn as semicircles above or below the line, or as smooth curves linked together from multiple semicircles.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;弧图 Arc Diagram&amp;lt;/font&amp;gt;'''，一种布局风格可以追溯到20世纪60年代，将顶点放在一条直线上;边缘可以画成在直线上方或下方的半圆，也可以画成由多个半圆连接在一起的光滑曲线。&lt;br /&gt;
&lt;br /&gt;
*[[Circular layout]] methods place the vertices of the graph on a circle, choosing carefully the ordering of the vertices around the circle to reduce crossings and place adjacent vertices close to each other. Edges may be drawn either as chords of the circle or as arcs inside or outside of the circle. In some cases, multiple circles may be used.&amp;lt;ref&amp;gt;{{harvtxt|Doğrusöz|Madden|Madden|1997}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;圆形布局方法 Circular Layout&amp;lt;/font&amp;gt;'''将图形的顶点放置在一个圆上，仔细选择圆周上顶点的顺序，以减少交叉，并将相邻的顶点放置在彼此接近的位置。边可以画成圆的弦，也可以画成圆内外的弧线。在某些情况下，可以使用多个圆。&lt;br /&gt;
&lt;br /&gt;
*[[Dominance drawing]] places vertices in such a way that one vertex is upwards, rightwards, or both of another if and only if it is [[reachability|reachable]] from the other vertex. In this way, the layout style makes the reachability relation of the graph visually apparent.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Section 4.7, &amp;quot;Dominance Drawings&amp;quot;, pp. 112–127.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;优势图 Dominance Drawing&amp;lt;/font&amp;gt;'''当且仅当一个顶点可以从另一个顶点到达时，优势画顶点的方式是:一个顶点向上，或向右，或两个顶点同时向上。这样，布局风格使得图的可达性关系在视觉上变得明显。&lt;br /&gt;
&lt;br /&gt;
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==Application-specific graph drawings 专用图==&lt;br /&gt;
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Graphs and graph drawings arising in other areas of application include&lt;br /&gt;
&lt;br /&gt;
Graphs and graph drawings arising in other areas of application include&lt;br /&gt;
&lt;br /&gt;
在其他应用领域出现的图和图绘制包括&lt;br /&gt;
&lt;br /&gt;
* [[Sociogram]]s, drawings of a [[social network]], as often offered by [[social network analysis software]]&amp;lt;ref&amp;gt;{{Harvard citation text|Scott|2000}}; {{Harvard citation text|Brandes|Freeman|Wagner|2014}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会图 Sociogram&amp;lt;/font&amp;gt;'''，社会网络图，通常由社会网络分析软件提供&lt;br /&gt;
&lt;br /&gt;
* [[Hasse diagram]]s, a type of graph drawing specialized to [[partial order]]s&amp;lt;ref&amp;gt;{{Harvard citation text|Di Battista|Eades|Tamassia|Tollis|1994}}, pp. 15–16, and Chapter 6, &amp;quot;Flow and Upward Planarity&amp;quot;, pp. 171–214; {{Harvard citation text|Freese|2004}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;哈塞图 Hasse Diagram&amp;lt;/font&amp;gt;'''，一种专门用于部分顺序的图形绘制&lt;br /&gt;
&lt;br /&gt;
* [[Dessin d'enfant]]s, a type of graph drawing used in [[algebraic geometry]]{{sfnp|Zapponi|2003}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'Dessin d'enfants&amp;lt;/font&amp;gt;''，代数几何中使用的一种图形画法&lt;br /&gt;
&lt;br /&gt;
* [[State diagram]]s, graphical representations of [[finite-state machine]]s{{sfnp|Anderson|Head|2006}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;状态图 State Diagram&amp;lt;/font&amp;gt;'''，有限状态机的图形表示&lt;br /&gt;
&lt;br /&gt;
* [[Computer network diagram]]s, depictions of the nodes and connections in a [[computer network]]{{sfnp|Di Battista|Rimondini|2014}}&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;计算机网络图 Computer Network Diagram&amp;lt;/font&amp;gt;'''，描述计算机网络中的节点和连接&lt;br /&gt;
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* [[Flowchart]]s and [[DRAKON|drakon-charts]], drawings in which the nodes represent the steps of an [[algorithm]] and the edges represent [[control flow]] between steps.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;流程图 Flowchart&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;drakon图&amp;lt;/font&amp;gt;'''，其中节点表示算法的步骤，边表示步骤之间的控制流的绘图。&lt;br /&gt;
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* [[Data-flow diagram]]s, drawings in which the nodes represent the components of an [[information system]] and the edges represent the movement of information from one component to another.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;数据流程图 Data-Flow Diagram&amp;lt;/font&amp;gt;'''，其中节点表示信息系统的组件，边缘表示信息从一个组件到另一个组件的移动。&lt;br /&gt;
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* [[Bioinformatics]] including [[phylogenetic tree]]s, [[protein–protein interaction]] networks, and [[metabolic pathway]]s.{{sfnp|Bachmaier|Brandes|Schreiber|2014}}&lt;br /&gt;
生物信息学包括系统发育树，蛋白质，蛋白质相互作用网络，和代谢途径。&lt;br /&gt;
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In addition, the [[placement (electronic design automation)|placement]] and [[Routing (electronic design automation)|routing]] steps of [[electronic design automation]] (EDA) are similar in many ways to graph drawing, as is the problem of [[greedy embedding]] in [[distributed computing]], and the graph drawing literature includes several results borrowed from the EDA literature. However, these problems also differ in several important ways: for instance, in EDA, area minimization and signal length are more important than aesthetics, and the routing problem in EDA may have more than two terminals per net while the analogous problem in graph drawing generally only involves pairs of vertices for each edge.&lt;br /&gt;
&lt;br /&gt;
In addition, the placement and routing steps of electronic design automation (EDA) are similar in many ways to graph drawing, as is the problem of greedy embedding in distributed computing, and the graph drawing literature includes several results borrowed from the EDA literature. However, these problems also differ in several important ways: for instance, in EDA, area minimization and signal length are more important than aesthetics, and the routing problem in EDA may have more than two terminals per net while the analogous problem in graph drawing generally only involves pairs of vertices for each edge.&lt;br /&gt;
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此外，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;电子设计自动化（EDA）Electronic Design Automation&amp;lt;/font&amp;gt;'''的布局和布线步骤在很多方面与图形绘制相似，而在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分布式计算 Distributed Computing&amp;lt;/font&amp;gt;'''中'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;贪婪嵌入 Greedy Embedding&amp;lt;/font&amp;gt;'''的问题也是如此，并且图形绘制文献包括从EDA文献中借来的一些结果。然而，这些问题也在几个重要方面有所不同: 例如，在 EDA，面积最小化和信号长度比美观更重要，EDA中的路由问题可能每个网有两个以上的终端，而图绘制中的类似问题通常只涉及每条边的顶点对。&lt;br /&gt;
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==Software 软件==&lt;br /&gt;
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[[File:Gephi 0.9.1 Network Analysis and Visualization Software.png|thumb|&lt;br /&gt;
图6：A graph drawing interface 一个画图软件界面([[Gephi]] 0.9.1)]]Software, systems, and providers of systems for drawing graphs include:&lt;br /&gt;
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A graph drawing interface ([[Gephi 0.9.1)]]Software, systems, and providers of systems for drawing graphs include:&lt;br /&gt;
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软件、系统和绘制图形系统的提供者包括:&lt;br /&gt;
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&amp;lt;!-- DO ''not'' INCLUDE AN ENTRY HERE UNLESS IT ALREADY HAS A BLUE-LINKED ARTICLE --&amp;gt;&lt;br /&gt;
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&amp;lt;!-- DO not INCLUDE AN ENTRY HERE UNLESS IT ALREADY HAS A BLUE-LINKED ARTICLE --&amp;gt;&lt;br /&gt;
 &lt;br /&gt;
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* [[BioFabric]] open-source software for visualizing large networks by drawing nodes as horizontal lines.&lt;br /&gt;
BioFabric 开源软件，用于通过将节点绘制为水平线来可视化大型网络。&lt;br /&gt;
&lt;br /&gt;
* [[Cytoscape]], open-source software for visualizing molecular interaction networks&lt;br /&gt;
Cytoscape 可视化分子相互作用网络的开源软件&lt;br /&gt;
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* [[Gephi]], open-source network analysis and visualization software&lt;br /&gt;
Gephi 开源网络分析和可视化软件&lt;br /&gt;
&lt;br /&gt;
* [[graph-tool]], a [[Free Software|free/libre]] [[Python (programming language)|Python]] library for analysis of graphs.&lt;br /&gt;
Graph-Tool 一个免费/免费的Python库，用于分析图形&lt;br /&gt;
&lt;br /&gt;
* [[Graphviz]], an open-source graph drawing system from [[AT&amp;amp;T Corporation]]&amp;lt;ref&amp;gt;&amp;quot;Graphviz and Dynagraph – Static and Dynamic Graph Drawing Tools&amp;quot;, by John Ellson, Emden R. Gansner, Eleftherios Koutsofios, Stephen C. North, and Gordon Woodhull, in {{harvtxt| Jünger|Mutzel|2004}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
Graphviz 来自美国电话电报公司的开放源码绘图系统&lt;br /&gt;
&lt;br /&gt;
* [[Linkurious]], a commercial network analysis and visualization software for [[graph databases]]&lt;br /&gt;
Linkurious 图形数据库的商业网络分析和可视化软件&lt;br /&gt;
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* [[Mathematica]], a general purpose computation tool that includes 2D and 3D graph visualization and graph analysis tools.&amp;lt;ref&amp;gt;[http://reference.wolfram.com/mathematica/ref/GraphPlot.html GraphPlot] Mathematica documentation&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[http://reference.wolfram.com/mathematica/tutorial/GraphDrawingIntroduction.html Graph drawing tutorial]&amp;lt;/ref&amp;gt;&lt;br /&gt;
Mathematica 一个通用的计算工具，包括2D和3D图形可视化和图形分析工具。&lt;br /&gt;
&lt;br /&gt;
* [[Microsoft Automatic Graph Layout]], open-source .NET library (formerly called GLEE) for laying out graphs&amp;lt;ref&amp;gt;{{harvtxt|Nachmanson|Robertson|Lee|2008}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
Microsoft Automatic Graph Layout 用于图表布局的开放源码。net库(以前称为GLEE)&lt;br /&gt;
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* [[NetworkX]] is a Python library for studying graphs and networks.&lt;br /&gt;
NetworkX 用于研究图和网络的Python库&lt;br /&gt;
&lt;br /&gt;
* [[Tom Sawyer Software]]&amp;lt;ref&amp;gt;{{harvtxt|Madden|Madden|Powers|Himsolt|1996}}.&amp;lt;/ref&amp;gt; Tom Sawyer Perspectives is graphics-based software for building enterprise-class graph and data visualization and analysis applications. It is a Software Development Kit (SDK) with a graphics-based design and preview environment.&lt;br /&gt;
Tom Sawyer Perspectives是一款基于图形的软件，用于构建企业级图形和数据可视化和分析应用程序。它是一个软件开发工具包(SDK)，具有基于图形的设计和预览环境&lt;br /&gt;
&lt;br /&gt;
* [[Tulip (software)]],&amp;lt;ref&amp;gt;&amp;quot;Tulip – A Huge Graph Visualization Framework&amp;quot;, by David Auber, in {{harvtxt| Jünger|Mutzel|2004}}.&amp;lt;/ref&amp;gt; an open source data visualization tool&lt;br /&gt;
Tulip (software)开源数据可视化工具&lt;br /&gt;
yEd 具有图形布局功能的图形编辑器&lt;br /&gt;
&lt;br /&gt;
* [[yEd]], a graph editor with graph layout functionality&amp;lt;ref&amp;gt;&amp;quot;yFiles – Visualization and Automatic Layout of Graphs&amp;quot;, by Roland Wiese, Markus Eiglsperger, and Michael Kaufmann, in {{harvtxt| Jünger|Mutzel|2004}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
yEd 具有图形布局功能的图形编辑器&lt;br /&gt;
&lt;br /&gt;
* [[PGF/TikZ]] 3.0 with the &amp;lt;code&amp;gt;graphdrawing&amp;lt;/code&amp;gt; package (requires [[LuaTeX]]).&amp;lt;ref&amp;gt;{{harvtxt|Tantau|2013}}; see also the older [http://www.tcs.uni-luebeck.de/downloads/mitarbeiter/tantau/2012-gd-presentation.pdf GD 2012 presentation]&amp;lt;/ref&amp;gt;&lt;br /&gt;
PGF/TikZ 带graphdrawing软件包的3.0（需要LuaTeX）&lt;br /&gt;
&lt;br /&gt;
* [[LaNet-vi]], an open-source large network visualization software&lt;br /&gt;
LaNet-vi 开源大型网络可视化软件&lt;br /&gt;
&lt;br /&gt;
* [[Edraw Max]] 2D business technical diagramming software&lt;br /&gt;
Edraw Max 2D商业技术制图软件&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&lt;br /&gt;
;Footnotes&lt;br /&gt;
&lt;br /&gt;
Footnotes&lt;br /&gt;
&lt;br /&gt;
脚注&lt;br /&gt;
&lt;br /&gt;
{{reflist|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
;General references&lt;br /&gt;
&lt;br /&gt;
General references&lt;br /&gt;
&lt;br /&gt;
一般参考资料&lt;br /&gt;
&lt;br /&gt;
{{refbegin|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
最后1个 Di Battista 最初1个 Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
作者: 彼得 · 艾德斯&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
| last 3 Tamassia | first3 Roberto | author3-link Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
4 | last 4 Tollis | first4 Ioannis g.&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Computational Geometry (journal)|Computational Geometry: Theory and Applications]]&lt;br /&gt;
&lt;br /&gt;
 | journal = Computational Geometry: Theory and Applications&lt;br /&gt;
&lt;br /&gt;
计算几何: 理论与应用&lt;br /&gt;
&lt;br /&gt;
 | pages = 235–282&lt;br /&gt;
&lt;br /&gt;
 | pages = 235–282&lt;br /&gt;
&lt;br /&gt;
第235-282页&lt;br /&gt;
&lt;br /&gt;
 | title = Algorithms for Drawing Graphs: an Annotated Bibliography&lt;br /&gt;
&lt;br /&gt;
 | title = Algorithms for Drawing Graphs: an Annotated Bibliography&lt;br /&gt;
&lt;br /&gt;
图形绘制的算法: 附有注释的参考书目&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.cs.brown.edu/people/rt/gd.html&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.cs.brown.edu/people/rt/gd.html&lt;br /&gt;
&lt;br /&gt;
Http://www.cs.brown.edu/people/rt/gd.html&lt;br /&gt;
&lt;br /&gt;
 | volume = 4&lt;br /&gt;
&lt;br /&gt;
 | volume = 4&lt;br /&gt;
&lt;br /&gt;
第四卷&lt;br /&gt;
&lt;br /&gt;
 | issue = 5&lt;br /&gt;
&lt;br /&gt;
 | issue = 5&lt;br /&gt;
&lt;br /&gt;
第五期&lt;br /&gt;
&lt;br /&gt;
 | year = 1994 | doi=10.1016/0925-7721(94)00014-x}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 1994 | doi=10.1016/0925-7721(94)00014-x}}.&lt;br /&gt;
&lt;br /&gt;
1994年 | doi 10.1016 / 0925-7721(94)00014-x }。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
最后1个 Di Battista 最初1个 Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
作者: 彼得 · 艾德斯&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
| last 3 Tamassia | first3 Roberto | author3-link Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
4 | last 4 Tollis | first4 Ioannis g.&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-0-13-301615-4&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-0-13-301615-4&lt;br /&gt;
&lt;br /&gt;
| isbn 978-0-13-301615-4&lt;br /&gt;
&lt;br /&gt;
 | publisher = [[Prentice Hall]]&lt;br /&gt;
&lt;br /&gt;
 | publisher = Prentice Hall&lt;br /&gt;
&lt;br /&gt;
出版商 Prentice Hall&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Algorithms for the Visualization of Graphs&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Algorithms for the Visualization of Graphs&lt;br /&gt;
&lt;br /&gt;
图形绘制: 图形可视化的算法&lt;br /&gt;
&lt;br /&gt;
 | year = 1998}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 1998}}.&lt;br /&gt;
&lt;br /&gt;
1998年。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/2945.841119&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/2945.841119&lt;br /&gt;
&lt;br /&gt;
10.1109 / 2945.841119&lt;br /&gt;
&lt;br /&gt;
 | last1 = Herman | first1 = Ivan&lt;br /&gt;
&lt;br /&gt;
 | last1 = Herman | first1 = Ivan&lt;br /&gt;
&lt;br /&gt;
最后1个赫尔曼最初1个伊凡&lt;br /&gt;
&lt;br /&gt;
 | last2 = Melançon | first2 = Guy&lt;br /&gt;
&lt;br /&gt;
 | last2 = Melançon | first2 = Guy&lt;br /&gt;
&lt;br /&gt;
最后2个梅兰，最初2个盖伊&lt;br /&gt;
&lt;br /&gt;
 | last3 = Marshall | first3 = M. Scott&lt;br /&gt;
&lt;br /&gt;
 | last3 = Marshall | first3 = M. Scott&lt;br /&gt;
&lt;br /&gt;
3 m · 斯科特&lt;br /&gt;
&lt;br /&gt;
 | journal = IEEE Transactions on Visualization and Computer Graphics&lt;br /&gt;
&lt;br /&gt;
 | journal = IEEE Transactions on Visualization and Computer Graphics&lt;br /&gt;
&lt;br /&gt;
可视化与计算机图形学科学杂志&lt;br /&gt;
&lt;br /&gt;
 | pages = 24–43&lt;br /&gt;
&lt;br /&gt;
 | pages = 24–43&lt;br /&gt;
&lt;br /&gt;
第24-43页&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Visualization and Navigation in Information Visualization: A Survey&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Visualization and Navigation in Information Visualization: A Survey&lt;br /&gt;
&lt;br /&gt;
信息可视化的图形可视化和导航: 一个调查&lt;br /&gt;
&lt;br /&gt;
 | issue = 1&lt;br /&gt;
&lt;br /&gt;
 | issue = 1&lt;br /&gt;
&lt;br /&gt;
第一期&lt;br /&gt;
&lt;br /&gt;
 | volume = 6&lt;br /&gt;
&lt;br /&gt;
 | volume = 6&lt;br /&gt;
&lt;br /&gt;
第六卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2000 }}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2000 }}.&lt;br /&gt;
&lt;br /&gt;
2000年开始。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Jünger | first1 = Michael&lt;br /&gt;
&lt;br /&gt;
 | last1 = Jünger | first1 = Michael&lt;br /&gt;
&lt;br /&gt;
1 j nger | first1 Michael&lt;br /&gt;
&lt;br /&gt;
 | last2 = Mutzel | first2 = Petra | author2-link = Petra Mutzel&lt;br /&gt;
&lt;br /&gt;
 | last2 = Mutzel | first2 = Petra | author2-link = Petra Mutzel&lt;br /&gt;
&lt;br /&gt;
2 Petra | author2-link Petra Mutzel&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-00881-1&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-00881-1&lt;br /&gt;
&lt;br /&gt;
[国际标准图书馆编号978-3-540-00881-1]&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing Software&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing Software&lt;br /&gt;
&lt;br /&gt;
图形绘制软件&lt;br /&gt;
&lt;br /&gt;
 | year = 2004}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2004}}.&lt;br /&gt;
&lt;br /&gt;
2004年开始。&lt;br /&gt;
&lt;br /&gt;
*{{citation|title=Drawing Graphs: Methods and Models|editor1-first=Michael|editor1-last=Kaufmann|editor2-first=Dorothea|editor2-last=Wagner|editor2-link=Dorothea Wagner|publisher=Springer-Verlag|series=[[Lecture Notes in Computer Science]]|volume=2025|year=2001|doi=10.1007/3-540-44969-8|isbn=978-3-540-42062-0}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|url=http://cs.brown.edu/~rt/gdhandbook/|year=2014|access-date=2013-08-28|archive-url=https://web.archive.org/web/20130815181243/http://cs.brown.edu/~rt/gdhandbook/|archive-date=2013-08-15|url-status=dead}}.&lt;br /&gt;
&lt;br /&gt;
{{refend}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
;Specialized subtopics&lt;br /&gt;
&lt;br /&gt;
Specialized subtopics&lt;br /&gt;
&lt;br /&gt;
专门的副主题&lt;br /&gt;
&lt;br /&gt;
{{refbegin|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
*{{citation|title=Automata Theory with Modern Applications|first1=James Andrew|last1=Anderson|first2=Thomas J.|last2=Head|publisher=Cambridge University Press|year=2006|isbn=978-0-521-84887-9|pages=38–41|url=https://books.google.com/books?id=ikS8BLdLDxIC&amp;amp;pg=PA38}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Christian|last1=Bachmaier|first2=Ulrik|last2=Brandes|first3=Falk|last3=Schreiber|contribution=Biological Networks|pages=621–651|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|year=2014}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|contribution=Layered drawings of digraphs|first1=Oliver|last1=Bastert|first2=Christian|last2=Matuszewski|title=Drawing Graphs: Methods and Models|editor1-first=Michael|editor1-last=Kaufmann|editor2-first=Dorothea|editor2-last=Wagner|editor2-link=Dorothea Wagner|publisher=Springer-Verlag|series=Lecture Notes in Computer Science|volume=2025|year=2001|pages=87–120|doi=10.1007/3-540-44969-8_5|isbn=978-3-540-42062-0}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Beckman | first = Brian&lt;br /&gt;
&lt;br /&gt;
 | last = Beckman | first = Brian&lt;br /&gt;
&lt;br /&gt;
| 最后一个 Beckman | 第一个 Brian&lt;br /&gt;
&lt;br /&gt;
 | publisher = Microsoft Research&lt;br /&gt;
&lt;br /&gt;
 | publisher = Microsoft Research&lt;br /&gt;
&lt;br /&gt;
微软研究院&lt;br /&gt;
&lt;br /&gt;
 | series = Tech. Report MSR-TR-94-04&lt;br /&gt;
&lt;br /&gt;
 | series = Tech. Report MSR-TR-94-04&lt;br /&gt;
&lt;br /&gt;
系列技术。Msr-tr-94-04报告&lt;br /&gt;
&lt;br /&gt;
 | title = Theory of Spectral Graph Layout&lt;br /&gt;
&lt;br /&gt;
 | title = Theory of Spectral Graph Layout&lt;br /&gt;
&lt;br /&gt;
光谱图布局理论&lt;br /&gt;
&lt;br /&gt;
 | url = http://research.microsoft.com/apps/pubs/default.aspx?id=69611&lt;br /&gt;
&lt;br /&gt;
 | url = http://research.microsoft.com/apps/pubs/default.aspx?id=69611&lt;br /&gt;
&lt;br /&gt;
Http://research.microsoft.com/apps/pubs/default.aspx?id=69611&lt;br /&gt;
&lt;br /&gt;
 | year = 1994}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 1994}}.&lt;br /&gt;
&lt;br /&gt;
1994年。&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Ulrik|last1=Brandes|first2=Linton C.|last2=Freeman|first3=Dorothea|last3=Wagner|author3-link=Dorothea Wagner|contribution=Social Networks|pages=805–839|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|year=2014}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Giuseppe|last1=Di Battista|first2=Massimo|last2=Rimondini|contribution=Computer Networks|pages=763–803|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|year=2014}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Doğrusöz | first1 = Uğur&lt;br /&gt;
&lt;br /&gt;
 | last1 = Doğrusöz | first1 = Uğur&lt;br /&gt;
&lt;br /&gt;
 | last1 = Doğrusöz | first1 = Uğur&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Brendan&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Brendan&lt;br /&gt;
&lt;br /&gt;
2 Madden | first2 Brendan&lt;br /&gt;
&lt;br /&gt;
 | last3 = Madden | first3 = Patrick&lt;br /&gt;
&lt;br /&gt;
 | last3 = Madden | first3 = Patrick&lt;br /&gt;
&lt;br /&gt;
最后3个马登最初3个帕特里克&lt;br /&gt;
&lt;br /&gt;
 | editor-last = North | editor-first = Stephen&lt;br /&gt;
&lt;br /&gt;
 | editor-last = North | editor-first = Stephen&lt;br /&gt;
&lt;br /&gt;
| 编辑-最后的北 | 编辑-第一斯蒂芬&lt;br /&gt;
&lt;br /&gt;
 | contribution = Circular layout in the Graph Layout toolkit&lt;br /&gt;
&lt;br /&gt;
 | contribution = Circular layout in the Graph Layout toolkit&lt;br /&gt;
&lt;br /&gt;
| 贡献图形布局工具包中的循环布局&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-62495-3_40&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-62495-3_40&lt;br /&gt;
&lt;br /&gt;
10.1007 / 3-540-62495-340&lt;br /&gt;
&lt;br /&gt;
 | pages = 92–100&lt;br /&gt;
&lt;br /&gt;
 | pages = 92–100&lt;br /&gt;
&lt;br /&gt;
第92-100页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Symposium on Graph Drawing, GD '96 Berkeley, California, USA, September 18–20, 1996, Proceedings&lt;br /&gt;
&lt;br /&gt;
 | title = Symposium on Graph Drawing, GD '96 Berkeley, California, USA, September 18–20, 1996, Proceedings&lt;br /&gt;
&lt;br /&gt;
图形绘制研讨会，美国加州伯克利，1996年9月18-20日，美国国家科学院院刊&lt;br /&gt;
&lt;br /&gt;
 | volume = 1190&lt;br /&gt;
&lt;br /&gt;
 | volume = 1190&lt;br /&gt;
&lt;br /&gt;
第1190卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1997| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
 | year = 1997| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
1997年标题链接国际图形绘制专题研讨会&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-62495-0&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-62495-0&lt;br /&gt;
&lt;br /&gt;
| isbn 978-3-540-62495-0&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
免费访问&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Eiglsperger | first1 = Markus&lt;br /&gt;
&lt;br /&gt;
 | last1 = Eiglsperger | first1 = Markus&lt;br /&gt;
&lt;br /&gt;
1 | last 1 Eiglsperger | first1 Markus&lt;br /&gt;
&lt;br /&gt;
 | last2 = Fekete | first2 = Sándor&lt;br /&gt;
&lt;br /&gt;
 | last2 = Fekete | first2 = Sándor&lt;br /&gt;
&lt;br /&gt;
 | last2 = Fekete | first2 = Sándor&lt;br /&gt;
&lt;br /&gt;
 | last3 = Klau | first3 = Gunnar&lt;br /&gt;
&lt;br /&gt;
 | last3 = Klau | first3 = Gunnar&lt;br /&gt;
&lt;br /&gt;
3 Klau | first3 Gunnar&lt;br /&gt;
&lt;br /&gt;
 | contribution = Orthogonal graph drawing&lt;br /&gt;
&lt;br /&gt;
 | contribution = Orthogonal graph drawing&lt;br /&gt;
&lt;br /&gt;
| 贡献正交图绘制&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-44969-8_6&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-44969-8_6&lt;br /&gt;
&lt;br /&gt;
10.1007 / 3-540-44969-86&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Kaufmann | editor1-first = Michael&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Kaufmann | editor1-first = Michael&lt;br /&gt;
&lt;br /&gt;
1-last Kaufmann | editor1-first Michael&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Wagner | editor2-first = Dorothea | editor2-link = Dorothea Wagner&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Wagner | editor2-first = Dorothea | editor2-link = Dorothea Wagner&lt;br /&gt;
&lt;br /&gt;
2- 最后一位瓦格纳 | 2- 第一位多萝西娅 | 2- 链接多萝西娅 · 瓦格纳&lt;br /&gt;
&lt;br /&gt;
 | pages = 121–171&lt;br /&gt;
&lt;br /&gt;
 | pages = 121–171&lt;br /&gt;
&lt;br /&gt;
第121-171页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer Berlin / Heidelberg&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer Berlin / Heidelberg&lt;br /&gt;
&lt;br /&gt;
| 出版商斯普林格 · 柏林 / 海德堡&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing Graphs&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing Graphs&lt;br /&gt;
&lt;br /&gt;
| 题目绘制图形&lt;br /&gt;
&lt;br /&gt;
 | volume = 2025&lt;br /&gt;
&lt;br /&gt;
 | volume = 2025&lt;br /&gt;
&lt;br /&gt;
2025年&lt;br /&gt;
&lt;br /&gt;
 | year = 2001| isbn = 978-3-540-42062-0&lt;br /&gt;
&lt;br /&gt;
 | year = 2001| isbn = 978-3-540-42062-0&lt;br /&gt;
&lt;br /&gt;
2001 | isbn 978-3-540-42062-0&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Freese | first = Ralph&lt;br /&gt;
&lt;br /&gt;
 | last = Freese | first = Ralph&lt;br /&gt;
&lt;br /&gt;
最后一个自由 | 第一个拉尔夫&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Eklund | editor-first = Peter&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Eklund | editor-first = Peter&lt;br /&gt;
&lt;br /&gt;
| 编辑-最后埃克伦 | 编辑-第一彼得&lt;br /&gt;
&lt;br /&gt;
 | contribution = Automated lattice drawing&lt;br /&gt;
&lt;br /&gt;
 | contribution = Automated lattice drawing&lt;br /&gt;
&lt;br /&gt;
自动点阵绘图&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-24651-0_12&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-24651-0_12&lt;br /&gt;
&lt;br /&gt;
10.1007 / 978-3-540-24651-012&lt;br /&gt;
&lt;br /&gt;
 | pages = 589–590&lt;br /&gt;
&lt;br /&gt;
 | pages = 589–590&lt;br /&gt;
&lt;br /&gt;
第589-590页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Concept Lattices: Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, February 23-26, 2004, Proceedings&lt;br /&gt;
&lt;br /&gt;
 | title = Concept Lattices: Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, February 23-26, 2004, Proceedings&lt;br /&gt;
&lt;br /&gt;
| 题目概念格: 第二届形式概念分析国际会议，2004年，悉尼，澳大利亚，2004年2月23-26日，会议录&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.math.hawaii.edu/~ralph/Preprints/latdrawing.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.math.hawaii.edu/~ralph/Preprints/latdrawing.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.math.hawaii.edu/~ralph/preprints/latdrawing.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 2961&lt;br /&gt;
&lt;br /&gt;
 | volume = 2961&lt;br /&gt;
&lt;br /&gt;
第2961卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2004| isbn = 978-3-540-21043-6&lt;br /&gt;
&lt;br /&gt;
 | year = 2004| isbn = 978-3-540-21043-6&lt;br /&gt;
&lt;br /&gt;
2004 | isbn 978-3-540-21043-6&lt;br /&gt;
&lt;br /&gt;
 | citeseerx = 10.1.1.69.6245&lt;br /&gt;
&lt;br /&gt;
 | citeseerx = 10.1.1.69.6245&lt;br /&gt;
&lt;br /&gt;
10.1.1.69.6245&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Garg | first1 = Ashim&lt;br /&gt;
&lt;br /&gt;
 | last1 = Garg | first1 = Ashim&lt;br /&gt;
&lt;br /&gt;
1 | 最后1加格 | 首先1阿希姆&lt;br /&gt;
&lt;br /&gt;
 | last2 = Tamassia | first2 = Roberto&lt;br /&gt;
&lt;br /&gt;
 | last2 = Tamassia | first2 = Roberto&lt;br /&gt;
&lt;br /&gt;
2 Tamassia | first2 Roberto&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BF01108622&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BF01108622&lt;br /&gt;
&lt;br /&gt;
| doi 10.1007 / BF01108622&lt;br /&gt;
&lt;br /&gt;
 | issue = 2&lt;br /&gt;
&lt;br /&gt;
 | issue = 2&lt;br /&gt;
&lt;br /&gt;
第二期&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Order (journal)|Order]]&lt;br /&gt;
&lt;br /&gt;
 | journal = Order&lt;br /&gt;
&lt;br /&gt;
| 日志订单&lt;br /&gt;
&lt;br /&gt;
 | mr = 1354797&lt;br /&gt;
&lt;br /&gt;
 | mr = 1354797&lt;br /&gt;
&lt;br /&gt;
1354797先生&lt;br /&gt;
&lt;br /&gt;
 | pages = 109–133&lt;br /&gt;
&lt;br /&gt;
 | pages = 109–133&lt;br /&gt;
&lt;br /&gt;
第109-133页&lt;br /&gt;
&lt;br /&gt;
 | title = Upward planarity testing&lt;br /&gt;
&lt;br /&gt;
 | title = Upward planarity testing&lt;br /&gt;
&lt;br /&gt;
| 标题向上平面性测试&lt;br /&gt;
&lt;br /&gt;
 | volume = 12&lt;br /&gt;
&lt;br /&gt;
 | volume = 12&lt;br /&gt;
&lt;br /&gt;
第12卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1995| citeseerx = 10.1.1.10.2237&lt;br /&gt;
&lt;br /&gt;
 | year = 1995| citeseerx = 10.1.1.10.2237&lt;br /&gt;
&lt;br /&gt;
1995年10.1.1.10.2237&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Holten | first1 = Danny&lt;br /&gt;
&lt;br /&gt;
 | last1 = Holten | first1 = Danny&lt;br /&gt;
&lt;br /&gt;
 | last1 = Holten | first1 = Danny&lt;br /&gt;
&lt;br /&gt;
 | last2 = Isenberg | first2 = Petra&lt;br /&gt;
&lt;br /&gt;
 | last2 = Isenberg | first2 = Petra&lt;br /&gt;
&lt;br /&gt;
2 Isenberg | first2 Petra&lt;br /&gt;
&lt;br /&gt;
 | last3 = van Wijk | first3 = Jarke J. | author3-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 | last3 = van Wijk | first3 = Jarke J. | author3-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 | last3 = van Wijk | first3 = Jarke J.| author3-link Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 | last4 = Fekete | first4 = Jean-Daniel&lt;br /&gt;
&lt;br /&gt;
 | last4 = Fekete | first4 = Jean-Daniel&lt;br /&gt;
&lt;br /&gt;
最后4个 Fekete 最早4个 Jean-Daniel&lt;br /&gt;
&lt;br /&gt;
 | contribution = An extended evaluation of the readability of tapered, animated, and textured directed-edge representations in node-link graphs&lt;br /&gt;
&lt;br /&gt;
 | contribution = An extended evaluation of the readability of tapered, animated, and textured directed-edge representations in node-link graphs&lt;br /&gt;
&lt;br /&gt;
对节点链接图中渐变的、动画的和纹理的有向边表示的可读性的扩展评估&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/PACIFICVIS.2011.5742390&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/PACIFICVIS.2011.5742390&lt;br /&gt;
&lt;br /&gt;
10.1109 / PACIFICVIS. 2011.5742390&lt;br /&gt;
&lt;br /&gt;
 | pages = 195–202&lt;br /&gt;
&lt;br /&gt;
 | pages = 195–202&lt;br /&gt;
&lt;br /&gt;
第195-202页&lt;br /&gt;
&lt;br /&gt;
 | title = IEEE Pacific Visualization Symposium (PacificVis 2011)&lt;br /&gt;
&lt;br /&gt;
 | title = IEEE Pacific Visualization Symposium (PacificVis 2011)&lt;br /&gt;
&lt;br /&gt;
美国电气和电子工程师协会太平洋可视化研讨会(PacificVis 2011)&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.lri.fr/~isenberg/publications/papers/Holten_2011_AEP.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.lri.fr/~isenberg/publications/papers/Holten_2011_AEP.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.lri.fr/~isenberg/publications/papers/holten_2011_aep.pdf&lt;br /&gt;
&lt;br /&gt;
 | year = 2011| isbn = 978-1-61284-935-5&lt;br /&gt;
&lt;br /&gt;
 | year = 2011| isbn = 978-1-61284-935-5&lt;br /&gt;
&lt;br /&gt;
| 2011年 | isbn 978-1-61284-935-5&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 |last1        = Holten&lt;br /&gt;
&lt;br /&gt;
 |last1        = Holten&lt;br /&gt;
&lt;br /&gt;
1 Holten&lt;br /&gt;
&lt;br /&gt;
 |first1       = Danny&lt;br /&gt;
&lt;br /&gt;
 |first1       = Danny&lt;br /&gt;
&lt;br /&gt;
首先是丹尼&lt;br /&gt;
&lt;br /&gt;
 |last2        = van Wijk&lt;br /&gt;
&lt;br /&gt;
 |last2        = van Wijk&lt;br /&gt;
&lt;br /&gt;
2 van Wijk&lt;br /&gt;
&lt;br /&gt;
 |first2       = Jarke J.&lt;br /&gt;
&lt;br /&gt;
 |first2       = Jarke J.&lt;br /&gt;
&lt;br /&gt;
| first2 Jarke j.&lt;br /&gt;
&lt;br /&gt;
 |author2-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 |author2-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
| author2-link Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 |contribution = A user study on visualizing directed edges in graphs&lt;br /&gt;
&lt;br /&gt;
 |contribution = A user study on visualizing directed edges in graphs&lt;br /&gt;
&lt;br /&gt;
用户研究图形中的有向边可视化&lt;br /&gt;
&lt;br /&gt;
 |doi          = 10.1145/1518701.1519054&lt;br /&gt;
&lt;br /&gt;
 |doi          = 10.1145/1518701.1519054&lt;br /&gt;
&lt;br /&gt;
10.1145 / 1518701.1519054&lt;br /&gt;
&lt;br /&gt;
 |pages        = 2299–2308&lt;br /&gt;
&lt;br /&gt;
 |pages        = 2299–2308&lt;br /&gt;
&lt;br /&gt;
2299-2308页&lt;br /&gt;
&lt;br /&gt;
 |title        = Proceedings of the 27th International Conference on Human Factors in Computing Systems (CHI '09)&lt;br /&gt;
&lt;br /&gt;
 |title        = Proceedings of the 27th International Conference on Human Factors in Computing Systems (CHI '09)&lt;br /&gt;
&lt;br /&gt;
| 第27届计算机系统中的人为因素国际会议论文集(CHI’09)&lt;br /&gt;
&lt;br /&gt;
 |url          = http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |url          = http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |year         = 2009&lt;br /&gt;
&lt;br /&gt;
 |year         = 2009&lt;br /&gt;
&lt;br /&gt;
2009年&lt;br /&gt;
&lt;br /&gt;
 |url-status      = dead&lt;br /&gt;
&lt;br /&gt;
 |url-status      = dead&lt;br /&gt;
&lt;br /&gt;
状态死机&lt;br /&gt;
&lt;br /&gt;
 |archiveurl   = https://web.archive.org/web/20111106004500/http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |archiveurl   = https://web.archive.org/web/20111106004500/http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
Https://web.archive.org/web/20111106004500/http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |archivedate  = 2011-11-06&lt;br /&gt;
&lt;br /&gt;
 |archivedate  = 2011-11-06&lt;br /&gt;
&lt;br /&gt;
2011-11-06&lt;br /&gt;
&lt;br /&gt;
|isbn       = 9781605582467&lt;br /&gt;
&lt;br /&gt;
|isbn       = 9781605582467&lt;br /&gt;
&lt;br /&gt;
9781605582467&lt;br /&gt;
&lt;br /&gt;
 |citeseerx       = 10.1.1.212.5461&lt;br /&gt;
&lt;br /&gt;
 |citeseerx       = 10.1.1.212.5461&lt;br /&gt;
&lt;br /&gt;
10.1.1.212.5461&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Koren&lt;br /&gt;
&lt;br /&gt;
 | last = Koren&lt;br /&gt;
&lt;br /&gt;
最后一个韩国&lt;br /&gt;
&lt;br /&gt;
 | first = Yehuda&lt;br /&gt;
&lt;br /&gt;
 | first = Yehuda&lt;br /&gt;
&lt;br /&gt;
首先是耶胡达&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1016/j.camwa.2004.08.015&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1016/j.camwa.2004.08.015&lt;br /&gt;
&lt;br /&gt;
10.1016 / j.camwa. 2004.08&lt;br /&gt;
&lt;br /&gt;
 | issue = 11–12&lt;br /&gt;
&lt;br /&gt;
 | issue = 11–12&lt;br /&gt;
&lt;br /&gt;
第11-12期&lt;br /&gt;
&lt;br /&gt;
 | journal = Computers &amp;amp; Mathematics with Applications&lt;br /&gt;
&lt;br /&gt;
 | journal = Computers &amp;amp; Mathematics with Applications&lt;br /&gt;
&lt;br /&gt;
计算机与数学与应用》杂志&lt;br /&gt;
&lt;br /&gt;
 | mr = 2154691&lt;br /&gt;
&lt;br /&gt;
 | mr = 2154691&lt;br /&gt;
&lt;br /&gt;
2154691先生&lt;br /&gt;
&lt;br /&gt;
 | pages = 1867–1888&lt;br /&gt;
&lt;br /&gt;
 | pages = 1867–1888&lt;br /&gt;
&lt;br /&gt;
1867-1888页&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing graphs by eigenvectors: theory and practice&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing graphs by eigenvectors: theory and practice&lt;br /&gt;
&lt;br /&gt;
按特征向量绘图: 理论与实践&lt;br /&gt;
&lt;br /&gt;
 | url = https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
Https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/dblp-journals-camwa-koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 49&lt;br /&gt;
&lt;br /&gt;
 | volume = 49&lt;br /&gt;
&lt;br /&gt;
第49卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2005&lt;br /&gt;
&lt;br /&gt;
 | year = 2005&lt;br /&gt;
&lt;br /&gt;
2005年&lt;br /&gt;
&lt;br /&gt;
 | access-date = 2011-09-17&lt;br /&gt;
&lt;br /&gt;
 | access-date = 2011-09-17&lt;br /&gt;
&lt;br /&gt;
2011-09-17&lt;br /&gt;
&lt;br /&gt;
 | archive-url = https://web.archive.org/web/20120402131402/https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | archive-url = https://web.archive.org/web/20120402131402/https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
| 档案-url  https://web.archive.org/web/20120402131402/https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/dblp-journals-camwa-koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | archive-date = 2012-04-02&lt;br /&gt;
&lt;br /&gt;
 | archive-date = 2012-04-02&lt;br /&gt;
&lt;br /&gt;
2012-04-02&lt;br /&gt;
&lt;br /&gt;
 | url-status = dead&lt;br /&gt;
&lt;br /&gt;
 | url-status = dead&lt;br /&gt;
&lt;br /&gt;
状态死机&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Longabaugh | first = William&lt;br /&gt;
&lt;br /&gt;
 | last = Longabaugh | first = William&lt;br /&gt;
&lt;br /&gt;
最后一次朗格伯格 | 第一次威廉&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1186/1471-2105-13-275&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1186/1471-2105-13-275&lt;br /&gt;
&lt;br /&gt;
10.1186 / 1471-2105-13-275&lt;br /&gt;
&lt;br /&gt;
 | journal = BMC Bioinformatics&lt;br /&gt;
&lt;br /&gt;
 | journal = BMC Bioinformatics&lt;br /&gt;
&lt;br /&gt;
生物信息学杂志&lt;br /&gt;
&lt;br /&gt;
 | pages = 275&lt;br /&gt;
&lt;br /&gt;
 | pages = 275&lt;br /&gt;
&lt;br /&gt;
第275页&lt;br /&gt;
&lt;br /&gt;
 | title = Combing the hairball with BioFabric: a new approach for visualization of large networks&lt;br /&gt;
&lt;br /&gt;
 | title = Combing the hairball with BioFabric: a new approach for visualization of large networks&lt;br /&gt;
&lt;br /&gt;
用 BioFabric 梳理毛团: 一种大型网络可视化的新方法&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.biomedcentral.com/content/pdf/1471-2105-13-275.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.biomedcentral.com/content/pdf/1471-2105-13-275.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.biomedcentral.com/content/pdf/1471-2105-13-275.pdf&lt;br /&gt;
&lt;br /&gt;
 | pmid = 23102059&lt;br /&gt;
&lt;br /&gt;
 | pmid = 23102059&lt;br /&gt;
&lt;br /&gt;
23102059&lt;br /&gt;
&lt;br /&gt;
 | volume = 13&lt;br /&gt;
&lt;br /&gt;
 | volume = 13&lt;br /&gt;
&lt;br /&gt;
第13卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2012&lt;br /&gt;
&lt;br /&gt;
 | year = 2012&lt;br /&gt;
&lt;br /&gt;
2012年&lt;br /&gt;
&lt;br /&gt;
 | pmc=3574047}}.&lt;br /&gt;
&lt;br /&gt;
 | pmc=3574047}}.&lt;br /&gt;
&lt;br /&gt;
3574047}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Madden | first1 = Brendan&lt;br /&gt;
&lt;br /&gt;
 | last1 = Madden | first1 = Brendan&lt;br /&gt;
&lt;br /&gt;
最后一个疯狂橄榄球，最初一个布兰登&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Patrick&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Patrick&lt;br /&gt;
&lt;br /&gt;
2 Madden | first2 Patrick&lt;br /&gt;
&lt;br /&gt;
 | last3 = Powers | first3 = Steve&lt;br /&gt;
&lt;br /&gt;
 | last3 = Powers | first3 = Steve&lt;br /&gt;
&lt;br /&gt;
3 power | first3 Steve&lt;br /&gt;
&lt;br /&gt;
 | last4 = Himsolt | first4 = Michael&lt;br /&gt;
&lt;br /&gt;
 | last4 = Himsolt | first4 = Michael&lt;br /&gt;
&lt;br /&gt;
最后4希望 / 最初4迈克尔&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Brandenburg | editor-first = Franz J.&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Brandenburg | editor-first = Franz J.&lt;br /&gt;
&lt;br /&gt;
最后的勃兰登堡 | 最初的弗朗茨 j。&lt;br /&gt;
&lt;br /&gt;
 | contribution = Portable graph layout and editing&lt;br /&gt;
&lt;br /&gt;
 | contribution = Portable graph layout and editing&lt;br /&gt;
&lt;br /&gt;
便携式图形布局和编辑&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BFb0021822&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BFb0021822&lt;br /&gt;
&lt;br /&gt;
10.1007 / BFb0021822&lt;br /&gt;
&lt;br /&gt;
 | pages = 385–395&lt;br /&gt;
&lt;br /&gt;
 | pages = 385–395&lt;br /&gt;
&lt;br /&gt;
第385-395页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Symposium on Graph Drawing, GD '95, Passau, Germany, September 20–22, 1995, Proceedings&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Symposium on Graph Drawing, GD '95, Passau, Germany, September 20–22, 1995, Proceedings&lt;br /&gt;
&lt;br /&gt;
图形绘制: 图形绘制研讨会，GD’95，帕绍，1995年9月20-22日，美国国家科学院院刊&lt;br /&gt;
&lt;br /&gt;
 | volume = 1027&lt;br /&gt;
&lt;br /&gt;
 | volume = 1027&lt;br /&gt;
&lt;br /&gt;
第1027卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1996| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
 | year = 1996| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
1996年标题链接国际图形绘制专题研讨会&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-60723-6&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-60723-6&lt;br /&gt;
&lt;br /&gt;
[国际标准图书编号978-3-540-60723-6]&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
免费访问&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Misue | first1= K.&lt;br /&gt;
&lt;br /&gt;
 | last1 = Misue | first1= K.&lt;br /&gt;
&lt;br /&gt;
| 最后1米苏 | 最初1 k。&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades  | first2= P.&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades  | first2= P.&lt;br /&gt;
&lt;br /&gt;
2 Eades | first2 p.&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lai  | first3 = W.&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lai  | first3 = W.&lt;br /&gt;
&lt;br /&gt;
3 Lai | first3 w.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Sugiyama | first4 = K.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Sugiyama | first4 = K.&lt;br /&gt;
&lt;br /&gt;
最后4千。&lt;br /&gt;
&lt;br /&gt;
 | title = Layout Adjustment and the Mental Map&lt;br /&gt;
&lt;br /&gt;
 | title = Layout Adjustment and the Mental Map&lt;br /&gt;
&lt;br /&gt;
| 标题布局调整和心理地图&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Visual Languages and Computing&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Visual Languages and Computing&lt;br /&gt;
&lt;br /&gt;
视觉语言与计算杂志&lt;br /&gt;
&lt;br /&gt;
 | volume = 6 | number = 2 | pages = 183–210 | year = 1995 | doi=10.1006/jvlc.1995.1010}}.&lt;br /&gt;
&lt;br /&gt;
 | volume = 6 | number = 2 | pages = 183–210 | year = 1995 | doi=10.1006/jvlc.1995.1010}}.&lt;br /&gt;
&lt;br /&gt;
183-210 | year 1995 | doi 10.1006 / jvlc. 1995.1010}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Nachmanson | first1 = Lev&lt;br /&gt;
&lt;br /&gt;
 | last1 = Nachmanson | first1 = Lev&lt;br /&gt;
&lt;br /&gt;
最后1纳克曼森最初1列弗&lt;br /&gt;
&lt;br /&gt;
 | last2 = Robertson | first2 = George&lt;br /&gt;
&lt;br /&gt;
 | last2 = Robertson | first2 = George&lt;br /&gt;
&lt;br /&gt;
2 Robertson | first2 George&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lee | first3 = Bongshin&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lee | first3 = Bongshin&lt;br /&gt;
&lt;br /&gt;
3 | last 3 Lee | first3 bongsin&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Hong | editor1-first = Seok-Hee&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Hong | editor1-first = Seok-Hee&lt;br /&gt;
&lt;br /&gt;
| 编辑1-last Hong | 编辑1-first Seok-Hee&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki&lt;br /&gt;
&lt;br /&gt;
 | editor3-last = Quan | editor3-first = Wu&lt;br /&gt;
&lt;br /&gt;
 | editor3-last = Quan | editor3-first = Wu&lt;br /&gt;
&lt;br /&gt;
| 编辑3-last Quan | 编辑3-first Wu&lt;br /&gt;
&lt;br /&gt;
 | contribution = Drawing Graphs with GLEE&lt;br /&gt;
&lt;br /&gt;
 | contribution = Drawing Graphs with GLEE&lt;br /&gt;
&lt;br /&gt;
用 GLEE 绘制图表&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-77537-9_38&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-77537-9_38&lt;br /&gt;
&lt;br /&gt;
10.1007 / 978-3-540-77537-938&lt;br /&gt;
&lt;br /&gt;
 | pages = 389–394&lt;br /&gt;
&lt;br /&gt;
 | pages = 389–394&lt;br /&gt;
&lt;br /&gt;
第389-394页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing, 15th International Symposium, GD 2007, Sydney, Australia, September 24–26, 2007, Revised Papers&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing, 15th International Symposium, GD 2007, Sydney, Australia, September 24–26, 2007, Revised Papers&lt;br /&gt;
&lt;br /&gt;
15 th International Symposium，GD 2007，Sydney，Australia，September 24-26，Revised Papers&lt;br /&gt;
&lt;br /&gt;
 | contribution-url = ftp://ftp.research.microsoft.com/pub/TR/TR-2007-72.pdf&lt;br /&gt;
&lt;br /&gt;
 | contribution-url = ftp://ftp.research.microsoft.com/pub/TR/TR-2007-72.pdf&lt;br /&gt;
&lt;br /&gt;
| contribution-url ftp: / / ftp.research.microsoft. com / pub / tr / tr-2007-72.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 4875&lt;br /&gt;
&lt;br /&gt;
 | volume = 4875&lt;br /&gt;
&lt;br /&gt;
第4875卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2008| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
 | year = 2008| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
| 2008年 | 国际图形绘制专题研讨会链接&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-77536-2&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-77536-2&lt;br /&gt;
&lt;br /&gt;
[国际标准图书馆编号978-3-540-77536-2]&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
免费访问&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Pach | first1 = János | author1-link = János Pach&lt;br /&gt;
&lt;br /&gt;
 | last1 = Pach | first1 = János | author1-link = János Pach&lt;br /&gt;
&lt;br /&gt;
 | last1 = Pach | first1 = János | author1-link = János Pach&lt;br /&gt;
&lt;br /&gt;
 | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir&lt;br /&gt;
&lt;br /&gt;
 | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir&lt;br /&gt;
&lt;br /&gt;
2 | last 2 Sharir | first2 Micha | author2-link Micha Sharir&lt;br /&gt;
&lt;br /&gt;
 | contribution = 5.5 Angular resolution and slopes&lt;br /&gt;
&lt;br /&gt;
 | contribution = 5.5 Angular resolution and slopes&lt;br /&gt;
&lt;br /&gt;
| 贡献5.5角分辨率及斜坡&lt;br /&gt;
&lt;br /&gt;
 | pages = 126–127&lt;br /&gt;
&lt;br /&gt;
 | pages = 126–127&lt;br /&gt;
&lt;br /&gt;
第126-127页&lt;br /&gt;
&lt;br /&gt;
 | publisher = [[American Mathematical Society]]&lt;br /&gt;
&lt;br /&gt;
 | publisher = American Mathematical Society&lt;br /&gt;
&lt;br /&gt;
美国数学学会&lt;br /&gt;
&lt;br /&gt;
 | series = Mathematical Surveys and Monographs&lt;br /&gt;
&lt;br /&gt;
 | series = Mathematical Surveys and Monographs&lt;br /&gt;
&lt;br /&gt;
| 系列数学调查和专著&lt;br /&gt;
&lt;br /&gt;
 | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures&lt;br /&gt;
&lt;br /&gt;
 | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures&lt;br /&gt;
&lt;br /&gt;
离散几何及其算法应用: alcal 系列讲座&lt;br /&gt;
&lt;br /&gt;
 | volume = 152&lt;br /&gt;
&lt;br /&gt;
 | volume = 152&lt;br /&gt;
&lt;br /&gt;
第152卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2009}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2009}}.&lt;br /&gt;
&lt;br /&gt;
2009年开始。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Purchase&lt;br /&gt;
&lt;br /&gt;
 | last1 = Purchase&lt;br /&gt;
&lt;br /&gt;
| 最后一次购买&lt;br /&gt;
&lt;br /&gt;
 | first1 = H. C.&lt;br /&gt;
&lt;br /&gt;
 | first1 = H. C.&lt;br /&gt;
&lt;br /&gt;
| first1 H.c.&lt;br /&gt;
&lt;br /&gt;
 | author1-link = Helen Purchase&lt;br /&gt;
&lt;br /&gt;
 | author1-link = Helen Purchase&lt;br /&gt;
&lt;br /&gt;
| author1-link Helen Purchase&lt;br /&gt;
&lt;br /&gt;
 | last2 = Cohen&lt;br /&gt;
&lt;br /&gt;
 | last2 = Cohen&lt;br /&gt;
&lt;br /&gt;
2 Cohen&lt;br /&gt;
&lt;br /&gt;
 | first2 = R. F.&lt;br /&gt;
&lt;br /&gt;
 | first2 = R. F.&lt;br /&gt;
&lt;br /&gt;
| first2 R.f.&lt;br /&gt;
&lt;br /&gt;
 | last3 = James&lt;br /&gt;
&lt;br /&gt;
 | last3 = James&lt;br /&gt;
&lt;br /&gt;
最后3名詹姆斯&lt;br /&gt;
&lt;br /&gt;
 | first3 = M. I.&lt;br /&gt;
&lt;br /&gt;
 | first3 = M. I.&lt;br /&gt;
&lt;br /&gt;
第一季第三集。&lt;br /&gt;
&lt;br /&gt;
 | at = Article 4&lt;br /&gt;
&lt;br /&gt;
 | at = Article 4&lt;br /&gt;
&lt;br /&gt;
| 第四条&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1145/264216.264222&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1145/264216.264222&lt;br /&gt;
&lt;br /&gt;
10.1145 / 264216.264222&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Experimental Algorithmics&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Experimental Algorithmics&lt;br /&gt;
&lt;br /&gt;
实验算法杂志&lt;br /&gt;
&lt;br /&gt;
 | title = An experimental study of the basis for graph drawing algorithms&lt;br /&gt;
&lt;br /&gt;
 | title = An experimental study of the basis for graph drawing algorithms&lt;br /&gt;
&lt;br /&gt;
关于图形绘制算法基础的实验研究&lt;br /&gt;
&lt;br /&gt;
 | url = https://secure.cs.uvic.ca/twiki/pub/Research/Chisel/ComputationalAestheticsProject/Vol2Nbr4.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = https://secure.cs.uvic.ca/twiki/pub/Research/Chisel/ComputationalAestheticsProject/Vol2Nbr4.pdf&lt;br /&gt;
&lt;br /&gt;
Https://secure.cs.uvic.ca/twiki/pub/research/chisel/computationalaestheticsproject/vol2nbr4.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 2&lt;br /&gt;
&lt;br /&gt;
 | volume = 2&lt;br /&gt;
&lt;br /&gt;
第二卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1997&lt;br /&gt;
&lt;br /&gt;
 | year = 1997&lt;br /&gt;
&lt;br /&gt;
1997年&lt;br /&gt;
&lt;br /&gt;
 }}{{Dead link|date=January 2020 |bot=InternetArchiveBot |fix-attempted=yes }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Saaty | first = Thomas L. | authorlink = Thomas L. Saaty&lt;br /&gt;
&lt;br /&gt;
 | last = Saaty | first = Thomas L. | authorlink = Thomas L. Saaty&lt;br /&gt;
&lt;br /&gt;
作者链接: Thomas l. Saaty&lt;br /&gt;
&lt;br /&gt;
 | journal = Proc. Natl. Acad. Sci. U.S.A.&lt;br /&gt;
&lt;br /&gt;
 | journal = Proc. Natl. Acad. Sci. U.S.A.&lt;br /&gt;
&lt;br /&gt;
华尔街日报。自然。Acad.科学。美国。&lt;br /&gt;
&lt;br /&gt;
 | pages = 688–690&lt;br /&gt;
&lt;br /&gt;
 | pages = 688–690&lt;br /&gt;
&lt;br /&gt;
第688-690页&lt;br /&gt;
&lt;br /&gt;
 | title = The minimum number of intersections in complete graphs&lt;br /&gt;
&lt;br /&gt;
 | title = The minimum number of intersections in complete graphs&lt;br /&gt;
&lt;br /&gt;
| 标题完全图中的最小交叉点数&lt;br /&gt;
&lt;br /&gt;
 | volume = 52&lt;br /&gt;
&lt;br /&gt;
 | volume = 52&lt;br /&gt;
&lt;br /&gt;
第52卷&lt;br /&gt;
&lt;br /&gt;
 | issue = 3 | year = 1964&lt;br /&gt;
&lt;br /&gt;
 | issue = 3 | year = 1964&lt;br /&gt;
&lt;br /&gt;
第三期1964年&lt;br /&gt;
&lt;br /&gt;
 | doi=10.1073/pnas.52.3.688| pmid = 16591215 | pmc = 300329}}.&lt;br /&gt;
&lt;br /&gt;
 | doi=10.1073/pnas.52.3.688| pmid = 16591215 | pmc = 300329}}.&lt;br /&gt;
&lt;br /&gt;
10.1073 / pnas. 52.3.688 | pmid 16591215 | pmc 300329}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|title=Social network analysis: a handbook|first=John|last=Scott|edition=2nd|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|contribution=Sociograms and Graph Theory|url=https://books.google.com/books?id=Ww3_bKcz6kgC&amp;amp;pg=PA|pages=64–69}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Kozo|last1=Sugiyama|author1-link=Kozo Sugiyama|first2=Shôjirô|last2=Tagawa|first3=Mitsuhiko|last3=Toda|title=Methods for visual understanding of hierarchical system structures|journal=[[IEEE Systems, Man &amp;amp; Cybernetics Society|IEEE Transactions on Systems, Man, and Cybernetics]]|volume=SMC-11|issue=2|pages=109–125|year=1981|mr=0611436|doi=10.1109/TSMC.1981.4308636}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Tantau | first = Till&lt;br /&gt;
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 | last = Tantau | first = Till&lt;br /&gt;
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 | last = Tantau | first = Till&lt;br /&gt;
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 | doi = 10.7155/jgaa.00301&lt;br /&gt;
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 | doi = 10.7155/jgaa.00301&lt;br /&gt;
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10.7155 / jgaa. 00301&lt;br /&gt;
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 | issue = 4&lt;br /&gt;
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 | issue = 4&lt;br /&gt;
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第四期&lt;br /&gt;
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 | journal = [[Journal of Graph Algorithms and Applications]]&lt;br /&gt;
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 | journal = Journal of Graph Algorithms and Applications&lt;br /&gt;
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图形算法和应用杂志&lt;br /&gt;
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 | pages = 495–513&lt;br /&gt;
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 | pages = 495–513&lt;br /&gt;
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第495-513页&lt;br /&gt;
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 | title = Graph Drawing in TikZ&lt;br /&gt;
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 | title = Graph Drawing in TikZ&lt;br /&gt;
&lt;br /&gt;
| title Graph Drawing in TikZ&lt;br /&gt;
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 | volume = 17&lt;br /&gt;
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 | volume = 17&lt;br /&gt;
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第17卷&lt;br /&gt;
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 | year = 2013}}.&lt;br /&gt;
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 | year = 2013}}.&lt;br /&gt;
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2013年开始。&lt;br /&gt;
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* {{Citation | last1=Zapponi | first1=Leonardo | title=What is a Dessin d'Enfant | url=http://www.ams.org/notices/200307/what-is.pdf | date=August 2003 | journal=[[Notices of the American Mathematical Society]] | volume=50 | pages=788–789}}.&lt;br /&gt;
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{{refend}}&lt;br /&gt;
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==External links==&lt;br /&gt;
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* [http://graphx.codeplex.com GraphX library for .NET]: open-source WPF library for graph calculation and visualization. Supports many layout and edge routing algorithms.&lt;br /&gt;
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* [http://gdea.informatik.uni-koeln.de/ Graph drawing e-print archive]: including information on papers from all [[International Symposium on Graph Drawing|Graph Drawing symposia]].&lt;br /&gt;
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* {{dmoz|Science/Math/Combinatorics/Software/Graph_Drawing}} for many additional links related to graph drawing.&lt;br /&gt;
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{{Graph representations}}&lt;br /&gt;
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[[Category:Graph drawing| ]]&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Graph drawing]]. Its edit history can be viewed at [[图的绘制/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%9B%BE%E7%9A%84%E7%BB%98%E5%88%B6_Graph_drawing&amp;diff=13526</id>
		<title>图的绘制 Graph drawing</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%9B%BE%E7%9A%84%E7%BB%98%E5%88%B6_Graph_drawing&amp;diff=13526"/>
		<updated>2020-09-01T05:30:48Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{about|the general subject of graph drawing|the annual research symposium|International Symposium on Graph Drawing}}&lt;br /&gt;
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[[File:WorldWideWebAroundWikipedia.png|thumb|&lt;br /&gt;
图1：Graphic representation of a minute fraction of the [[WWW]], demonstrating [[hyperlink]]s. 以图的形式展示万维网的一小部分]]&lt;br /&gt;
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'''Graph drawing''' is an area of [[mathematics]] and [[computer science]] combining methods from [[geometric graph theory]] and [[information visualization]] to derive two-dimensional depictions of [[graph (discrete mathematics)|graph]]s arising from applications such as [[social network analysis]], [[cartography]], [[linguistics]], and [[bioinformatics]].&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, pp. vii–viii; {{harvtxt|Herman|Melançon|Marshall|2000}}, Section 1.1, &amp;quot;Typical Application Areas&amp;quot;.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics.&lt;br /&gt;
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图形绘制是数学和计算机科学的一个交叉领域，它结合了几何图形理论和信息可视化的方法，对来自社会网络分析、制图学、语言学和生物信息学等各种应用场景进行二维描述。&lt;br /&gt;
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A drawing of a graph or '''network diagram''' is a pictorial representation of the [[vertex (graph theory)|vertices]] and [[edge (graph theory)|edges]] of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.&amp;lt;ref name=&amp;quot;dett6&amp;quot;&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p. 6.&amp;lt;/ref&amp;gt;  In the abstract, all that matters is which pairs of vertices are connected by edges.  In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and [[aesthetics]].&amp;lt;ref name=&amp;quot;dett-viii&amp;quot;/&amp;gt; The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's mental map.&amp;lt;ref&amp;gt;{{harvtxt|Misue|Eades|Lai|Sugiyama|1995}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.  In the abstract, all that matters is which pairs of vertices are connected by edges.  In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics.&lt;br /&gt;
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图形或网络图的绘制是通过从图形抽象出的顶点和边表示的。这个图不应与图形本身混淆: 非常不同的图可以对应于同一个图形。抽象地说，重要的是哪一对顶点是由边连在一起。然而，在具体的绘图中，这些顶点和边的排列影响了它的可理解性、可用性、制造成本和美学。&lt;br /&gt;
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==Graphical conventions 画图公约==&lt;br /&gt;
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[[File:4node-digraph-natural.svg|thumb|upright=0.5|&lt;br /&gt;
图2：[[Directed graph]] with arrowheads showing edge directions 带箭头显示边方向的有向图]]&lt;br /&gt;
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Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as [[line segment]]s, [[Polygonal chain|polylines]], or curves in the [[Euclidean plane]].&amp;lt;ref name=&amp;quot;dett-viii&amp;quot;&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p. viii.&amp;lt;/ref&amp;gt; Node–link diagrams can be traced back to the 13th century work of [[Ramon Llull]], who drew diagrams of this type for [[complete graph]]s in order to analyze all pairwise combinations among sets of metaphysical concepts.&amp;lt;ref&amp;gt;{{citation|contribution=Two thousand years of combinatorics|first=Donald E.|last=Knuth|authorlink=Donald Knuth|pages=7–37|title=Combinatorics: Ancient and Modern|publisher=Oxford University Press|year=2013|editor1-first=Robin|editor1-last=Wilson|editor2-first=John J.|editor2-last=Watkins}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as line segments, polylines, or curves in the Euclidean plane. Node–link diagrams can be traced back to the 13th century work of Ramon Llull, who drew diagrams of this type for complete graphs in order to analyze all pairwise combinations among sets of metaphysical concepts.&lt;br /&gt;
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图经常绘制成节点链接图，其中顶点可表示为圆盘、框或文本标签，边表示为欧几里德平面上的线段、折线或曲线。节点链接图可以追溯到13世纪雷蒙 · 卢尔的作品，他为得到完全图，绘制了这种类型的图，以便分析形而上学概念集合之间的所有成对组合。&lt;br /&gt;
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In the case of [[directed graph]]s, [[Arrow (symbol)|arrowheads]] form a commonly used graphical convention to show their [[Orientability|orientation]];&amp;lt;ref name=&amp;quot;dett6&amp;quot;/&amp;gt; however, user studies have shown that other conventions such as tapering provide this information more effectively.&amp;lt;ref&amp;gt;{{harvtxt|Holten|van Wijk|2009}}; {{harvtxt|Holten|Isenberg|van Wijk|Fekete|2011}}.&amp;lt;/ref&amp;gt; [[Upward planar drawing]] uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary.&amp;lt;ref&amp;gt;{{harvtxt|Garg|Tamassia|1995}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In the case of directed graphs, arrowheads form a commonly used graphical convention to show their orientation; Upward planar drawing uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary.&lt;br /&gt;
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在有向图的情况下，箭头约定俗成的来显示它们的方向; 向上的平面绘图使用以下约定：每个边都从较低的顶点定向到较高的顶点，因此不需要箭头。&lt;br /&gt;
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Alternative conventions to node–link diagrams include adjacency representations such as [[circle packing theorem|circle packings]], in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; [[intersection graph|intersection representations]] in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other; confluent drawings, in which edges are represented as smooth curves within mathematical [[train track (mathematics)|train tracks]]; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines;&amp;lt;ref name=&amp;quot;Longabaugh 2012&amp;quot;&amp;gt;{{harvtxt|Longabaugh|2012}}.&amp;lt;/ref&amp;gt; and visualizations of the [[adjacency matrix]] of the graph.&lt;br /&gt;
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Alternative conventions to node–link diagrams include adjacency representations such as circle packings, in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; intersection representations in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other; confluent drawings, in which edges are represented as smooth curves within mathematical train tracks; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines; and visualizations of the adjacency matrix of the graph.&lt;br /&gt;
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节点链接图可被其他画图方式替代，包括邻接表示法，如圆填充，其中顶点表示为平面上不相交的区域，边表示为区域之间的邻接; 交点表示，其中顶点由非相交的几何对象表示，而边线由其交点表示；可见性表示，其中顶点由平面中的区域表示，边由彼此无障碍视线的区域表示; 汇流图，其中边表示为数学轨迹上的光滑曲线; 织物，其中顶点表示为水平线，边表示为垂直线; 以及图形邻接矩阵的可视化结点。&lt;br /&gt;
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==Quality measures 质量度量==&lt;br /&gt;
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Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Section 2.1.2, Aesthetics, pp. 14–16; {{harvtxt|Purchase|Cohen|James|1997}}.&amp;lt;/ref&amp;gt; In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.&lt;br /&gt;
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Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability. In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.&lt;br /&gt;
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为了找到客观评价图形美观性和可用性的方法，已经定义了许多不同的图形质量度量方法。除了指导针对同一图形的不同布局方法之间的选择之外，某些布局方法还尝试直接优化这些度量。&lt;br /&gt;
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[[File:4node-digraph-embed.svg|upright=0.5|thumb|&lt;br /&gt;
图3：[[Planar graph]] drawn without overlapping edges 没有重叠边的平面图]]&lt;br /&gt;
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*The [[Crossing number (graph theory)|crossing number]] of a drawing is the number of pairs of edges that cross each other.  If the graph is [[planar graph|planar]], then it is often convenient to draw it without any edge intersections; that is, in this case, a graph drawing represents a [[graph embedding]]. However, nonplanar graphs frequently arise in applications, so graph drawing algorithms must generally allow for edge crossings.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p 14.&amp;lt;/ref&amp;gt;&lt;br /&gt;
图的交叉数是相互交叉的边对的数目。如果图形是平面的，那么通常可以方便地画出没有任何边缘相交的图形;也就是说，在本例中，图形绘制表示嵌入图形。然而，应用中经常出现非平面图，因此图形绘制算法通常必须考虑边缘交叉。&lt;br /&gt;
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*The [[Area (graph drawing)|area]] of a drawing is the size of its smallest [[bounding box]], relative to the closest distance between any two vertices. Drawings with smaller area are generally preferable to those with larger area, because they allow the features of the drawing to be shown at greater size and therefore more legibly. The [[aspect ratio]] of the bounding box may also be important.&lt;br /&gt;
一个图形的面积是它的最小边界盒的大小，相对于任意两个顶点之间的最近距离。面积小的画通常比面积大的画更可取，因为它们可以让图画的特征以更大的尺寸显示，因此更清晰。边界框的纵横比也很重要。&lt;br /&gt;
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*Symmetry display is the problem of finding [[Graph automorphism|symmetry group]]s within a given graph, and finding a drawing that displays as much of the symmetry as possible. Some layout methods automatically lead to symmetric drawings; alternatively, some drawing methods start by finding symmetries in the input graph and using them to construct a drawing.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, p. 16.&amp;lt;/ref&amp;gt;&lt;br /&gt;
对称显示的问题是在给定的图形中找到对称组，并找到尽可能多地显示对称的绘图。一些布局方法自动导致对称图形;另外，一些绘图方法从查找输入图形中的对称性开始，并使用它们来构造绘图。&lt;br /&gt;
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*It is important that edges have shapes that are as simple as possible, to make it easier for the eye to follow them. In polyline drawings, the complexity of an edge may be measured by its [[Bend minimization|number of bends]], and many methods aim to provide drawings with few total bends or few bends per edge. Similarly for spline curves the complexity of an edge may be measured by the number of control points on the edge.&lt;br /&gt;
重要的是，边缘的形状要尽可能简单，以使眼睛更容易跟踪它们。在折线图中，一条边的复杂性可以通过它的弯曲数来衡量，许多方法的目的是提供很少的总弯曲或每条边很少弯曲的图。类似地，对于样条曲线，边的复杂性可以通过边上控制点的数量来度量。&lt;br /&gt;
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*Several commonly used quality measures concern lengths of edges: it is generally desirable to minimize the total length of the edges as well as the maximum length of any edge. Additionally, it may be preferable for the lengths of edges to be uniform rather than highly varied.&lt;br /&gt;
几种常用的质量度量方法与边的长度有关:通常需要最小化边的总长度以及任何边的最大长度。此外，最好使边缘的长度一致而不是变化很大。&lt;br /&gt;
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*[[Angular resolution (graph drawing)|Angular resolution]] is a measure of the sharpest angles in a graph drawing. If a graph has vertices with high [[degree (graph theory)|degree]] then it necessarily will have small angular resolution, but the angular resolution can be bounded below by a function of the degree.&amp;lt;ref name=&amp;quot;ps09&amp;quot;&amp;gt;{{harvtxt|Pach|Sharir|2009}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
角度分辨率是图形绘制中最锐角的度量。如果一个图的顶点高度高，那么它的角分辨率就会很小，但是角分辨率可以由角的函数来限定。&lt;br /&gt;
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*The [[slope number]] of a graph is the minimum number of distinct edge slopes needed in a drawing with straight line segment edges (allowing crossings). [[Cubic graph]]s have slope number at most four, but graphs of degree five may have unbounded slope number; it remains open whether the slope number of degree-4 graphs is bounded.&amp;lt;ref name=&amp;quot;ps09&amp;quot;/&amp;gt;&lt;br /&gt;
图的斜率数是在具有直线段边(允许交叉)的图中所需的明显边缘斜率的最小值。三次图的斜率数最多为4，五次图的斜率数可能是无界的;4度图的斜率数是否有界仍然是开放的。&lt;br /&gt;
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==Layout methods 布局方法==&lt;br /&gt;
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[[File:Social Network Analysis Visualization.png|thumb|right|&lt;br /&gt;
图4：A force-based network visualization. 基于力的网络可视化 &amp;lt;ref&amp;gt;Published in {{Cite journal | volume = 10| issue = 3| last = Grandjean| first = Martin| title = La connaissance est un réseau| journal =Les Cahiers du Numérique| accessdate = 2014-10-15| date = 2014| pages = 37–54| url = http://www.cairn.info/resume.php?ID_ARTICLE=LCN_103_0037| doi=10.3166/lcn.10.3.37-54}}&amp;lt;/ref&amp;gt;]]&lt;br /&gt;
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There are many different graph layout strategies:&lt;br /&gt;
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There are many different graph layout strategies:&lt;br /&gt;
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有许多不同的图形布局策略:&lt;br /&gt;
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*In [[force-based layout]] systems, the graph drawing software modifies an initial vertex placement by continuously moving the vertices according to a system of forces based on physical metaphors related to systems of [[spring (device)|springs]] or [[molecular mechanics]]. Typically, these systems combine attractive forces between adjacent vertices with repulsive forces between all pairs of vertices, in order to seek a layout in which edge lengths are small while vertices are well-separated. These systems may perform [[gradient descent]] based minimization of an [[energy function]], or they may translate the forces directly into velocities or accelerations for the moving vertices.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Section 2.7, &amp;quot;The Force-Directed Approach&amp;quot;, pp. 29–30, and Chapter 10, &amp;quot;Force-Directed Methods&amp;quot;, pp. 303–326.&amp;lt;/ref&amp;gt;&lt;br /&gt;
在基于力的布局系统中，图形绘制软件通过根据基于弹簧系统或分子力学的物理隐喻的力系统不断移动顶点来修改初始顶点的位置。通常，这些系统将相邻顶点之间的吸引力与所有对顶点之间的排斥力结合起来，以寻求一种边长较小而顶点分离良好的布局。这些系统可以执行梯度下降最小化的一个能量函数，或者他们可以把力直接转化为节点的速度或加速度&lt;br /&gt;
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*[[Spectral layout]] methods use as coordinates the [[eigenvector]]s of a [[Matrix (mathematics)|matrix]] such as the [[Discrete Laplace operator|Laplacian]] derived from the [[adjacency matrix]] of the graph.&amp;lt;ref&amp;gt;{{harvtxt|Beckman|1994}}; {{harvtxt|Koren|2005}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
频谱布局方法使用从图的邻接矩阵派生的矩阵（例如Laplacian）的特征向量作为坐标。&lt;br /&gt;
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*Orthogonal layout methods, which allow the edges of the graph to run horizontally or vertically, parallel to the coordinate axes of the layout. These methods were originally designed for [[VLSI]] and [[Printed circuit board|PCB]] layout problems but they have also been adapted for graph drawing. They typically involve a multiphase approach in which an input graph is planarized by replacing crossing points by vertices, a topological embedding of the planarized graph is found, edge orientations are chosen to minimize bends, vertices are placed consistently with these orientations, and finally a layout compaction stage reduces the area of the drawing.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Chapter 5, &amp;quot;Flow and Orthogonal Drawings&amp;quot;, pp. 137–170; {{harv|Eiglsperger|Fekete|Klau|2001}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
正交布局方法，允许图形的边缘水平或垂直运行，平行于布局的坐标轴。这些方法最初是为超大规模集成电路和PCB布局问题而设计的，但它们也适用于图形绘制。它们通常涉及到一种多阶段的方法，通过用顶点替换交叉点来平面化输入图，找到一个平面化图的拓扑嵌入，选择边的方向来最小化弯曲，顶点的放置始终与这些方向一致，最后一个布局压缩阶段减少&lt;br /&gt;
&lt;br /&gt;
*Tree layout algorithms these show a rooted [[tree structure|tree]]-like formation, suitable for [[tree (graph theory)|trees]]. Often, in a technique called &amp;quot;balloon layout&amp;quot;, the children of each node in the tree are drawn on a circle surrounding the node, with the radii of these circles diminishing at lower levels in the tree so that these circles do not overlap.&amp;lt;ref&amp;gt;{{harvtxt|Herman|Melançon|Marshall|2000}}, Section 2.2, &amp;quot;Traditional Layout – An Overview&amp;quot;.&amp;lt;/ref&amp;gt;&lt;br /&gt;
这些树布局算法显示了一种类似树的根状结构，适合树。通常，在一种称为气球布局的技术中，树中每个节点的子节点被画在围绕该节点的圆上，这些圆的半径在树的较低层次上递减，以便这些圆不会重叠。&lt;br /&gt;
&lt;br /&gt;
*[[Layered graph drawing]] methods (often called Sugiyama-style drawing) are best suited for [[directed acyclic graph]]s or graphs that are nearly acyclic, such as the graphs of dependencies between modules or functions in a software system. In these methods, the nodes of the graph are arranged into horizontal layers using methods such as the [[Coffman–Graham algorithm]], in such a way that most edges go downwards from one layer to the next; after this step, the nodes within each layer are arranged in order to minimize crossings.&amp;lt;ref&amp;gt;{{harvtxt|Sugiyama|Tagawa|Toda|1981}}; {{harvtxt|Bastert|Matuszewski|2001}}; {{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Chapter 9, &amp;quot;Layered Drawings of Digraphs&amp;quot;, pp. 265–302.&amp;lt;/ref&amp;gt;&lt;br /&gt;
分层图绘制方法(通常称为杉山式图)最适合于有向无环图或接近无环的图，例如软件系统中模块或函数之间的依赖关系图。在这些方法中，图的节点使用Coffman Graham算法等方法被安排到水平层中，以这样的方式，大多数边从一层向下到下一层;在这一步之后，将每一层内的节点进行排列，以最小化交叉。&lt;br /&gt;
&lt;br /&gt;
[[File:Goldner-Harary-linear.svg|thumb|&lt;br /&gt;
图5：Arc diagram 弧线图]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*[[Arc diagram]]s, a layout style dating back to the 1960s,&amp;lt;ref&amp;gt;{{harvtxt|Saaty|1964}}.&amp;lt;/ref&amp;gt; place vertices on a line; edges may be drawn as semicircles above or below the line, or as smooth curves linked together from multiple semicircles.&lt;br /&gt;
弧图，一种布局风格可以追溯到20世纪60年代，将顶点放在一条直线上;边缘可以画成在直线上方或下方的半圆，也可以画成由多个半圆连接在一起的光滑曲线。&lt;br /&gt;
&lt;br /&gt;
*[[Circular layout]] methods place the vertices of the graph on a circle, choosing carefully the ordering of the vertices around the circle to reduce crossings and place adjacent vertices close to each other. Edges may be drawn either as chords of the circle or as arcs inside or outside of the circle. In some cases, multiple circles may be used.&amp;lt;ref&amp;gt;{{harvtxt|Doğrusöz|Madden|Madden|1997}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
圆形布局方法将图形的顶点放置在一个圆上，仔细选择圆周上顶点的顺序，以减少交叉，并将相邻的顶点放置在彼此接近的位置。边可以画成圆的弦，也可以画成圆内外的弧线。在某些情况下，可以使用多个圆。&lt;br /&gt;
&lt;br /&gt;
*[[Dominance drawing]] places vertices in such a way that one vertex is upwards, rightwards, or both of another if and only if it is [[reachability|reachable]] from the other vertex. In this way, the layout style makes the reachability relation of the graph visually apparent.&amp;lt;ref&amp;gt;{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1994}}, Section 4.7, &amp;quot;Dominance Drawings&amp;quot;, pp. 112–127.&amp;lt;/ref&amp;gt;&lt;br /&gt;
当且仅当一个顶点可以从另一个顶点到达时，优势画顶点的方式是:一个顶点向上，或向右，或两个顶点同时向上。这样，布局风格使得图的可达性关系在视觉上变得明显。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Application-specific graph drawings 专用图==&lt;br /&gt;
&lt;br /&gt;
Graphs and graph drawings arising in other areas of application include&lt;br /&gt;
&lt;br /&gt;
Graphs and graph drawings arising in other areas of application include&lt;br /&gt;
&lt;br /&gt;
在其他应用领域出现的图和图绘制包括&lt;br /&gt;
&lt;br /&gt;
* [[Sociogram]]s, drawings of a [[social network]], as often offered by [[social network analysis software]]&amp;lt;ref&amp;gt;{{Harvard citation text|Scott|2000}}; {{Harvard citation text|Brandes|Freeman|Wagner|2014}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
社会图，社会网络图，通常由社会网络分析软件提供&lt;br /&gt;
&lt;br /&gt;
* [[Hasse diagram]]s, a type of graph drawing specialized to [[partial order]]s&amp;lt;ref&amp;gt;{{Harvard citation text|Di Battista|Eades|Tamassia|Tollis|1994}}, pp. 15–16, and Chapter 6, &amp;quot;Flow and Upward Planarity&amp;quot;, pp. 171–214; {{Harvard citation text|Freese|2004}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
哈塞图，一种专门用于部分顺序的图形绘制&lt;br /&gt;
&lt;br /&gt;
* [[Dessin d'enfant]]s, a type of graph drawing used in [[algebraic geometry]]{{sfnp|Zapponi|2003}}&lt;br /&gt;
Dessin d'enfants，代数几何中使用的一种图形画法&lt;br /&gt;
&lt;br /&gt;
* [[State diagram]]s, graphical representations of [[finite-state machine]]s{{sfnp|Anderson|Head|2006}}&lt;br /&gt;
状态图，有限状态机的图形表示&lt;br /&gt;
&lt;br /&gt;
* [[Computer network diagram]]s, depictions of the nodes and connections in a [[computer network]]{{sfnp|Di Battista|Rimondini|2014}}&lt;br /&gt;
计算机网络图，描述计算机网络中的节点和连接&lt;br /&gt;
&lt;br /&gt;
* [[Flowchart]]s and [[DRAKON|drakon-charts]], drawings in which the nodes represent the steps of an [[algorithm]] and the edges represent [[control flow]] between steps.&lt;br /&gt;
流程图和drakon图，其中节点表示算法的步骤，边表示步骤之间的控制流的绘图。&lt;br /&gt;
&lt;br /&gt;
* [[Data-flow diagram]]s, drawings in which the nodes represent the components of an [[information system]] and the edges represent the movement of information from one component to another.&lt;br /&gt;
数据流程图，其中节点表示信息系统的组件，边缘表示信息从一个组件到另一个组件的移动。&lt;br /&gt;
&lt;br /&gt;
* [[Bioinformatics]] including [[phylogenetic tree]]s, [[protein–protein interaction]] networks, and [[metabolic pathway]]s.{{sfnp|Bachmaier|Brandes|Schreiber|2014}}&lt;br /&gt;
生物信息学包括系统发育树，蛋白质，蛋白质相互作用网络，和代谢途径。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In addition, the [[placement (electronic design automation)|placement]] and [[Routing (electronic design automation)|routing]] steps of [[electronic design automation]] (EDA) are similar in many ways to graph drawing, as is the problem of [[greedy embedding]] in [[distributed computing]], and the graph drawing literature includes several results borrowed from the EDA literature. However, these problems also differ in several important ways: for instance, in EDA, area minimization and signal length are more important than aesthetics, and the routing problem in EDA may have more than two terminals per net while the analogous problem in graph drawing generally only involves pairs of vertices for each edge.&lt;br /&gt;
&lt;br /&gt;
In addition, the placement and routing steps of electronic design automation (EDA) are similar in many ways to graph drawing, as is the problem of greedy embedding in distributed computing, and the graph drawing literature includes several results borrowed from the EDA literature. However, these problems also differ in several important ways: for instance, in EDA, area minimization and signal length are more important than aesthetics, and the routing problem in EDA may have more than two terminals per net while the analogous problem in graph drawing generally only involves pairs of vertices for each edge.&lt;br /&gt;
&lt;br /&gt;
此外，电子设计自动化（EDA）的布局和布线步骤在很多方面与图形绘制相似，而在分布式计算中贪婪嵌入的问题也是如此，并且图形绘制文献包括从EDA文献中借来的一些结果。然而，这些问题也在几个重要方面有所不同: 例如，在 EDA，面积最小化和信号长度比美观更重要，EDA中的路由问题可能每个网有两个以上的终端，而图绘制中的类似问题通常只涉及每条边的顶点对。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Software 软件==&lt;br /&gt;
&lt;br /&gt;
[[File:Gephi 0.9.1 Network Analysis and Visualization Software.png|thumb|&lt;br /&gt;
图6：A graph drawing interface 一个画图软件界面([[Gephi]] 0.9.1)]]Software, systems, and providers of systems for drawing graphs include:&lt;br /&gt;
&lt;br /&gt;
A graph drawing interface ([[Gephi 0.9.1)]]Software, systems, and providers of systems for drawing graphs include:&lt;br /&gt;
&lt;br /&gt;
软件、系统和绘制图形系统的提供者包括:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!-- DO ''not'' INCLUDE AN ENTRY HERE UNLESS IT ALREADY HAS A BLUE-LINKED ARTICLE --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!-- DO not INCLUDE AN ENTRY HERE UNLESS IT ALREADY HAS A BLUE-LINKED ARTICLE --&amp;gt;&lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
* [[BioFabric]] open-source software for visualizing large networks by drawing nodes as horizontal lines.&lt;br /&gt;
BioFabric 开源软件，用于通过将节点绘制为水平线来可视化大型网络。&lt;br /&gt;
&lt;br /&gt;
* [[Cytoscape]], open-source software for visualizing molecular interaction networks&lt;br /&gt;
Cytoscape 可视化分子相互作用网络的开源软件&lt;br /&gt;
&lt;br /&gt;
* [[Gephi]], open-source network analysis and visualization software&lt;br /&gt;
Gephi 开源网络分析和可视化软件&lt;br /&gt;
&lt;br /&gt;
* [[graph-tool]], a [[Free Software|free/libre]] [[Python (programming language)|Python]] library for analysis of graphs.&lt;br /&gt;
Graph-Tool 一个免费/免费的Python库，用于分析图形&lt;br /&gt;
&lt;br /&gt;
* [[Graphviz]], an open-source graph drawing system from [[AT&amp;amp;T Corporation]]&amp;lt;ref&amp;gt;&amp;quot;Graphviz and Dynagraph – Static and Dynamic Graph Drawing Tools&amp;quot;, by John Ellson, Emden R. Gansner, Eleftherios Koutsofios, Stephen C. North, and Gordon Woodhull, in {{harvtxt| Jünger|Mutzel|2004}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
Graphviz 来自美国电话电报公司的开放源码绘图系统&lt;br /&gt;
&lt;br /&gt;
* [[Linkurious]], a commercial network analysis and visualization software for [[graph databases]]&lt;br /&gt;
Linkurious 图形数据库的商业网络分析和可视化软件&lt;br /&gt;
&lt;br /&gt;
* [[Mathematica]], a general purpose computation tool that includes 2D and 3D graph visualization and graph analysis tools.&amp;lt;ref&amp;gt;[http://reference.wolfram.com/mathematica/ref/GraphPlot.html GraphPlot] Mathematica documentation&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[http://reference.wolfram.com/mathematica/tutorial/GraphDrawingIntroduction.html Graph drawing tutorial]&amp;lt;/ref&amp;gt;&lt;br /&gt;
Mathematica 一个通用的计算工具，包括2D和3D图形可视化和图形分析工具。&lt;br /&gt;
&lt;br /&gt;
* [[Microsoft Automatic Graph Layout]], open-source .NET library (formerly called GLEE) for laying out graphs&amp;lt;ref&amp;gt;{{harvtxt|Nachmanson|Robertson|Lee|2008}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
Microsoft Automatic Graph Layout 用于图表布局的开放源码。net库(以前称为GLEE)&lt;br /&gt;
&lt;br /&gt;
* [[NetworkX]] is a Python library for studying graphs and networks.&lt;br /&gt;
NetworkX 用于研究图和网络的Python库&lt;br /&gt;
&lt;br /&gt;
* [[Tom Sawyer Software]]&amp;lt;ref&amp;gt;{{harvtxt|Madden|Madden|Powers|Himsolt|1996}}.&amp;lt;/ref&amp;gt; Tom Sawyer Perspectives is graphics-based software for building enterprise-class graph and data visualization and analysis applications. It is a Software Development Kit (SDK) with a graphics-based design and preview environment.&lt;br /&gt;
Tom Sawyer Perspectives是一款基于图形的软件，用于构建企业级图形和数据可视化和分析应用程序。它是一个软件开发工具包(SDK)，具有基于图形的设计和预览环境&lt;br /&gt;
&lt;br /&gt;
* [[Tulip (software)]],&amp;lt;ref&amp;gt;&amp;quot;Tulip – A Huge Graph Visualization Framework&amp;quot;, by David Auber, in {{harvtxt| Jünger|Mutzel|2004}}.&amp;lt;/ref&amp;gt; an open source data visualization tool&lt;br /&gt;
Tulip (software)开源数据可视化工具&lt;br /&gt;
yEd 具有图形布局功能的图形编辑器&lt;br /&gt;
&lt;br /&gt;
* [[yEd]], a graph editor with graph layout functionality&amp;lt;ref&amp;gt;&amp;quot;yFiles – Visualization and Automatic Layout of Graphs&amp;quot;, by Roland Wiese, Markus Eiglsperger, and Michael Kaufmann, in {{harvtxt| Jünger|Mutzel|2004}}.&amp;lt;/ref&amp;gt;&lt;br /&gt;
yEd 具有图形布局功能的图形编辑器&lt;br /&gt;
&lt;br /&gt;
* [[PGF/TikZ]] 3.0 with the &amp;lt;code&amp;gt;graphdrawing&amp;lt;/code&amp;gt; package (requires [[LuaTeX]]).&amp;lt;ref&amp;gt;{{harvtxt|Tantau|2013}}; see also the older [http://www.tcs.uni-luebeck.de/downloads/mitarbeiter/tantau/2012-gd-presentation.pdf GD 2012 presentation]&amp;lt;/ref&amp;gt;&lt;br /&gt;
PGF/TikZ 带graphdrawing软件包的3.0（需要LuaTeX）&lt;br /&gt;
&lt;br /&gt;
* [[LaNet-vi]], an open-source large network visualization software&lt;br /&gt;
LaNet-vi 开源大型网络可视化软件&lt;br /&gt;
&lt;br /&gt;
* [[Edraw Max]] 2D business technical diagramming software&lt;br /&gt;
Edraw Max 2D商业技术制图软件&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&lt;br /&gt;
;Footnotes&lt;br /&gt;
&lt;br /&gt;
Footnotes&lt;br /&gt;
&lt;br /&gt;
脚注&lt;br /&gt;
&lt;br /&gt;
{{reflist|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
;General references&lt;br /&gt;
&lt;br /&gt;
General references&lt;br /&gt;
&lt;br /&gt;
一般参考资料&lt;br /&gt;
&lt;br /&gt;
{{refbegin|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
最后1个 Di Battista 最初1个 Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
作者: 彼得 · 艾德斯&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
| last 3 Tamassia | first3 Roberto | author3-link Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
4 | last 4 Tollis | first4 Ioannis g.&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Computational Geometry (journal)|Computational Geometry: Theory and Applications]]&lt;br /&gt;
&lt;br /&gt;
 | journal = Computational Geometry: Theory and Applications&lt;br /&gt;
&lt;br /&gt;
计算几何: 理论与应用&lt;br /&gt;
&lt;br /&gt;
 | pages = 235–282&lt;br /&gt;
&lt;br /&gt;
 | pages = 235–282&lt;br /&gt;
&lt;br /&gt;
第235-282页&lt;br /&gt;
&lt;br /&gt;
 | title = Algorithms for Drawing Graphs: an Annotated Bibliography&lt;br /&gt;
&lt;br /&gt;
 | title = Algorithms for Drawing Graphs: an Annotated Bibliography&lt;br /&gt;
&lt;br /&gt;
图形绘制的算法: 附有注释的参考书目&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.cs.brown.edu/people/rt/gd.html&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.cs.brown.edu/people/rt/gd.html&lt;br /&gt;
&lt;br /&gt;
Http://www.cs.brown.edu/people/rt/gd.html&lt;br /&gt;
&lt;br /&gt;
 | volume = 4&lt;br /&gt;
&lt;br /&gt;
 | volume = 4&lt;br /&gt;
&lt;br /&gt;
第四卷&lt;br /&gt;
&lt;br /&gt;
 | issue = 5&lt;br /&gt;
&lt;br /&gt;
 | issue = 5&lt;br /&gt;
&lt;br /&gt;
第五期&lt;br /&gt;
&lt;br /&gt;
 | year = 1994 | doi=10.1016/0925-7721(94)00014-x}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 1994 | doi=10.1016/0925-7721(94)00014-x}}.&lt;br /&gt;
&lt;br /&gt;
1994年 | doi 10.1016 / 0925-7721(94)00014-x }。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last1 = Di Battista | first1 = Giuseppe&lt;br /&gt;
&lt;br /&gt;
最后1个 Di Battista 最初1个 Giuseppe&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades | first2 = Peter | author2-link = Peter Eades&lt;br /&gt;
&lt;br /&gt;
作者: 彼得 · 艾德斯&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last3 = Tamassia | first3 = Roberto | author3-link = Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
| last 3 Tamassia | first3 Roberto | author3-link Roberto Tamassia&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Tollis | first4 = Ioannis G.&lt;br /&gt;
&lt;br /&gt;
4 | last 4 Tollis | first4 Ioannis g.&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-0-13-301615-4&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-0-13-301615-4&lt;br /&gt;
&lt;br /&gt;
| isbn 978-0-13-301615-4&lt;br /&gt;
&lt;br /&gt;
 | publisher = [[Prentice Hall]]&lt;br /&gt;
&lt;br /&gt;
 | publisher = Prentice Hall&lt;br /&gt;
&lt;br /&gt;
出版商 Prentice Hall&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Algorithms for the Visualization of Graphs&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Algorithms for the Visualization of Graphs&lt;br /&gt;
&lt;br /&gt;
图形绘制: 图形可视化的算法&lt;br /&gt;
&lt;br /&gt;
 | year = 1998}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 1998}}.&lt;br /&gt;
&lt;br /&gt;
1998年。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/2945.841119&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/2945.841119&lt;br /&gt;
&lt;br /&gt;
10.1109 / 2945.841119&lt;br /&gt;
&lt;br /&gt;
 | last1 = Herman | first1 = Ivan&lt;br /&gt;
&lt;br /&gt;
 | last1 = Herman | first1 = Ivan&lt;br /&gt;
&lt;br /&gt;
最后1个赫尔曼最初1个伊凡&lt;br /&gt;
&lt;br /&gt;
 | last2 = Melançon | first2 = Guy&lt;br /&gt;
&lt;br /&gt;
 | last2 = Melançon | first2 = Guy&lt;br /&gt;
&lt;br /&gt;
最后2个梅兰，最初2个盖伊&lt;br /&gt;
&lt;br /&gt;
 | last3 = Marshall | first3 = M. Scott&lt;br /&gt;
&lt;br /&gt;
 | last3 = Marshall | first3 = M. Scott&lt;br /&gt;
&lt;br /&gt;
3 m · 斯科特&lt;br /&gt;
&lt;br /&gt;
 | journal = IEEE Transactions on Visualization and Computer Graphics&lt;br /&gt;
&lt;br /&gt;
 | journal = IEEE Transactions on Visualization and Computer Graphics&lt;br /&gt;
&lt;br /&gt;
可视化与计算机图形学科学杂志&lt;br /&gt;
&lt;br /&gt;
 | pages = 24–43&lt;br /&gt;
&lt;br /&gt;
 | pages = 24–43&lt;br /&gt;
&lt;br /&gt;
第24-43页&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Visualization and Navigation in Information Visualization: A Survey&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Visualization and Navigation in Information Visualization: A Survey&lt;br /&gt;
&lt;br /&gt;
信息可视化的图形可视化和导航: 一个调查&lt;br /&gt;
&lt;br /&gt;
 | issue = 1&lt;br /&gt;
&lt;br /&gt;
 | issue = 1&lt;br /&gt;
&lt;br /&gt;
第一期&lt;br /&gt;
&lt;br /&gt;
 | volume = 6&lt;br /&gt;
&lt;br /&gt;
 | volume = 6&lt;br /&gt;
&lt;br /&gt;
第六卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2000 }}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2000 }}.&lt;br /&gt;
&lt;br /&gt;
2000年开始。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Jünger | first1 = Michael&lt;br /&gt;
&lt;br /&gt;
 | last1 = Jünger | first1 = Michael&lt;br /&gt;
&lt;br /&gt;
1 j nger | first1 Michael&lt;br /&gt;
&lt;br /&gt;
 | last2 = Mutzel | first2 = Petra | author2-link = Petra Mutzel&lt;br /&gt;
&lt;br /&gt;
 | last2 = Mutzel | first2 = Petra | author2-link = Petra Mutzel&lt;br /&gt;
&lt;br /&gt;
2 Petra | author2-link Petra Mutzel&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-00881-1&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-00881-1&lt;br /&gt;
&lt;br /&gt;
[国际标准图书馆编号978-3-540-00881-1]&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing Software&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing Software&lt;br /&gt;
&lt;br /&gt;
图形绘制软件&lt;br /&gt;
&lt;br /&gt;
 | year = 2004}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2004}}.&lt;br /&gt;
&lt;br /&gt;
2004年开始。&lt;br /&gt;
&lt;br /&gt;
*{{citation|title=Drawing Graphs: Methods and Models|editor1-first=Michael|editor1-last=Kaufmann|editor2-first=Dorothea|editor2-last=Wagner|editor2-link=Dorothea Wagner|publisher=Springer-Verlag|series=[[Lecture Notes in Computer Science]]|volume=2025|year=2001|doi=10.1007/3-540-44969-8|isbn=978-3-540-42062-0}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|url=http://cs.brown.edu/~rt/gdhandbook/|year=2014|access-date=2013-08-28|archive-url=https://web.archive.org/web/20130815181243/http://cs.brown.edu/~rt/gdhandbook/|archive-date=2013-08-15|url-status=dead}}.&lt;br /&gt;
&lt;br /&gt;
{{refend}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
;Specialized subtopics&lt;br /&gt;
&lt;br /&gt;
Specialized subtopics&lt;br /&gt;
&lt;br /&gt;
专门的副主题&lt;br /&gt;
&lt;br /&gt;
{{refbegin|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
*{{citation|title=Automata Theory with Modern Applications|first1=James Andrew|last1=Anderson|first2=Thomas J.|last2=Head|publisher=Cambridge University Press|year=2006|isbn=978-0-521-84887-9|pages=38–41|url=https://books.google.com/books?id=ikS8BLdLDxIC&amp;amp;pg=PA38}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Christian|last1=Bachmaier|first2=Ulrik|last2=Brandes|first3=Falk|last3=Schreiber|contribution=Biological Networks|pages=621–651|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|year=2014}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|contribution=Layered drawings of digraphs|first1=Oliver|last1=Bastert|first2=Christian|last2=Matuszewski|title=Drawing Graphs: Methods and Models|editor1-first=Michael|editor1-last=Kaufmann|editor2-first=Dorothea|editor2-last=Wagner|editor2-link=Dorothea Wagner|publisher=Springer-Verlag|series=Lecture Notes in Computer Science|volume=2025|year=2001|pages=87–120|doi=10.1007/3-540-44969-8_5|isbn=978-3-540-42062-0}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Beckman | first = Brian&lt;br /&gt;
&lt;br /&gt;
 | last = Beckman | first = Brian&lt;br /&gt;
&lt;br /&gt;
| 最后一个 Beckman | 第一个 Brian&lt;br /&gt;
&lt;br /&gt;
 | publisher = Microsoft Research&lt;br /&gt;
&lt;br /&gt;
 | publisher = Microsoft Research&lt;br /&gt;
&lt;br /&gt;
微软研究院&lt;br /&gt;
&lt;br /&gt;
 | series = Tech. Report MSR-TR-94-04&lt;br /&gt;
&lt;br /&gt;
 | series = Tech. Report MSR-TR-94-04&lt;br /&gt;
&lt;br /&gt;
系列技术。Msr-tr-94-04报告&lt;br /&gt;
&lt;br /&gt;
 | title = Theory of Spectral Graph Layout&lt;br /&gt;
&lt;br /&gt;
 | title = Theory of Spectral Graph Layout&lt;br /&gt;
&lt;br /&gt;
光谱图布局理论&lt;br /&gt;
&lt;br /&gt;
 | url = http://research.microsoft.com/apps/pubs/default.aspx?id=69611&lt;br /&gt;
&lt;br /&gt;
 | url = http://research.microsoft.com/apps/pubs/default.aspx?id=69611&lt;br /&gt;
&lt;br /&gt;
Http://research.microsoft.com/apps/pubs/default.aspx?id=69611&lt;br /&gt;
&lt;br /&gt;
 | year = 1994}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 1994}}.&lt;br /&gt;
&lt;br /&gt;
1994年。&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Ulrik|last1=Brandes|first2=Linton C.|last2=Freeman|first3=Dorothea|last3=Wagner|author3-link=Dorothea Wagner|contribution=Social Networks|pages=805–839|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|year=2014}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Giuseppe|last1=Di Battista|first2=Massimo|last2=Rimondini|contribution=Computer Networks|pages=763–803|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|year=2014}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Doğrusöz | first1 = Uğur&lt;br /&gt;
&lt;br /&gt;
 | last1 = Doğrusöz | first1 = Uğur&lt;br /&gt;
&lt;br /&gt;
 | last1 = Doğrusöz | first1 = Uğur&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Brendan&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Brendan&lt;br /&gt;
&lt;br /&gt;
2 Madden | first2 Brendan&lt;br /&gt;
&lt;br /&gt;
 | last3 = Madden | first3 = Patrick&lt;br /&gt;
&lt;br /&gt;
 | last3 = Madden | first3 = Patrick&lt;br /&gt;
&lt;br /&gt;
最后3个马登最初3个帕特里克&lt;br /&gt;
&lt;br /&gt;
 | editor-last = North | editor-first = Stephen&lt;br /&gt;
&lt;br /&gt;
 | editor-last = North | editor-first = Stephen&lt;br /&gt;
&lt;br /&gt;
| 编辑-最后的北 | 编辑-第一斯蒂芬&lt;br /&gt;
&lt;br /&gt;
 | contribution = Circular layout in the Graph Layout toolkit&lt;br /&gt;
&lt;br /&gt;
 | contribution = Circular layout in the Graph Layout toolkit&lt;br /&gt;
&lt;br /&gt;
| 贡献图形布局工具包中的循环布局&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-62495-3_40&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-62495-3_40&lt;br /&gt;
&lt;br /&gt;
10.1007 / 3-540-62495-340&lt;br /&gt;
&lt;br /&gt;
 | pages = 92–100&lt;br /&gt;
&lt;br /&gt;
 | pages = 92–100&lt;br /&gt;
&lt;br /&gt;
第92-100页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Symposium on Graph Drawing, GD '96 Berkeley, California, USA, September 18–20, 1996, Proceedings&lt;br /&gt;
&lt;br /&gt;
 | title = Symposium on Graph Drawing, GD '96 Berkeley, California, USA, September 18–20, 1996, Proceedings&lt;br /&gt;
&lt;br /&gt;
图形绘制研讨会，美国加州伯克利，1996年9月18-20日，美国国家科学院院刊&lt;br /&gt;
&lt;br /&gt;
 | volume = 1190&lt;br /&gt;
&lt;br /&gt;
 | volume = 1190&lt;br /&gt;
&lt;br /&gt;
第1190卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1997| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
 | year = 1997| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
1997年标题链接国际图形绘制专题研讨会&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-62495-0&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-62495-0&lt;br /&gt;
&lt;br /&gt;
| isbn 978-3-540-62495-0&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
免费访问&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Eiglsperger | first1 = Markus&lt;br /&gt;
&lt;br /&gt;
 | last1 = Eiglsperger | first1 = Markus&lt;br /&gt;
&lt;br /&gt;
1 | last 1 Eiglsperger | first1 Markus&lt;br /&gt;
&lt;br /&gt;
 | last2 = Fekete | first2 = Sándor&lt;br /&gt;
&lt;br /&gt;
 | last2 = Fekete | first2 = Sándor&lt;br /&gt;
&lt;br /&gt;
 | last2 = Fekete | first2 = Sándor&lt;br /&gt;
&lt;br /&gt;
 | last3 = Klau | first3 = Gunnar&lt;br /&gt;
&lt;br /&gt;
 | last3 = Klau | first3 = Gunnar&lt;br /&gt;
&lt;br /&gt;
3 Klau | first3 Gunnar&lt;br /&gt;
&lt;br /&gt;
 | contribution = Orthogonal graph drawing&lt;br /&gt;
&lt;br /&gt;
 | contribution = Orthogonal graph drawing&lt;br /&gt;
&lt;br /&gt;
| 贡献正交图绘制&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-44969-8_6&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/3-540-44969-8_6&lt;br /&gt;
&lt;br /&gt;
10.1007 / 3-540-44969-86&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Kaufmann | editor1-first = Michael&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Kaufmann | editor1-first = Michael&lt;br /&gt;
&lt;br /&gt;
1-last Kaufmann | editor1-first Michael&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Wagner | editor2-first = Dorothea | editor2-link = Dorothea Wagner&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Wagner | editor2-first = Dorothea | editor2-link = Dorothea Wagner&lt;br /&gt;
&lt;br /&gt;
2- 最后一位瓦格纳 | 2- 第一位多萝西娅 | 2- 链接多萝西娅 · 瓦格纳&lt;br /&gt;
&lt;br /&gt;
 | pages = 121–171&lt;br /&gt;
&lt;br /&gt;
 | pages = 121–171&lt;br /&gt;
&lt;br /&gt;
第121-171页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer Berlin / Heidelberg&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer Berlin / Heidelberg&lt;br /&gt;
&lt;br /&gt;
| 出版商斯普林格 · 柏林 / 海德堡&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing Graphs&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing Graphs&lt;br /&gt;
&lt;br /&gt;
| 题目绘制图形&lt;br /&gt;
&lt;br /&gt;
 | volume = 2025&lt;br /&gt;
&lt;br /&gt;
 | volume = 2025&lt;br /&gt;
&lt;br /&gt;
2025年&lt;br /&gt;
&lt;br /&gt;
 | year = 2001| isbn = 978-3-540-42062-0&lt;br /&gt;
&lt;br /&gt;
 | year = 2001| isbn = 978-3-540-42062-0&lt;br /&gt;
&lt;br /&gt;
2001 | isbn 978-3-540-42062-0&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Freese | first = Ralph&lt;br /&gt;
&lt;br /&gt;
 | last = Freese | first = Ralph&lt;br /&gt;
&lt;br /&gt;
最后一个自由 | 第一个拉尔夫&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Eklund | editor-first = Peter&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Eklund | editor-first = Peter&lt;br /&gt;
&lt;br /&gt;
| 编辑-最后埃克伦 | 编辑-第一彼得&lt;br /&gt;
&lt;br /&gt;
 | contribution = Automated lattice drawing&lt;br /&gt;
&lt;br /&gt;
 | contribution = Automated lattice drawing&lt;br /&gt;
&lt;br /&gt;
自动点阵绘图&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-24651-0_12&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-24651-0_12&lt;br /&gt;
&lt;br /&gt;
10.1007 / 978-3-540-24651-012&lt;br /&gt;
&lt;br /&gt;
 | pages = 589–590&lt;br /&gt;
&lt;br /&gt;
 | pages = 589–590&lt;br /&gt;
&lt;br /&gt;
第589-590页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Concept Lattices: Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, February 23-26, 2004, Proceedings&lt;br /&gt;
&lt;br /&gt;
 | title = Concept Lattices: Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, February 23-26, 2004, Proceedings&lt;br /&gt;
&lt;br /&gt;
| 题目概念格: 第二届形式概念分析国际会议，2004年，悉尼，澳大利亚，2004年2月23-26日，会议录&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.math.hawaii.edu/~ralph/Preprints/latdrawing.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.math.hawaii.edu/~ralph/Preprints/latdrawing.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.math.hawaii.edu/~ralph/preprints/latdrawing.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 2961&lt;br /&gt;
&lt;br /&gt;
 | volume = 2961&lt;br /&gt;
&lt;br /&gt;
第2961卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2004| isbn = 978-3-540-21043-6&lt;br /&gt;
&lt;br /&gt;
 | year = 2004| isbn = 978-3-540-21043-6&lt;br /&gt;
&lt;br /&gt;
2004 | isbn 978-3-540-21043-6&lt;br /&gt;
&lt;br /&gt;
 | citeseerx = 10.1.1.69.6245&lt;br /&gt;
&lt;br /&gt;
 | citeseerx = 10.1.1.69.6245&lt;br /&gt;
&lt;br /&gt;
10.1.1.69.6245&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Garg | first1 = Ashim&lt;br /&gt;
&lt;br /&gt;
 | last1 = Garg | first1 = Ashim&lt;br /&gt;
&lt;br /&gt;
1 | 最后1加格 | 首先1阿希姆&lt;br /&gt;
&lt;br /&gt;
 | last2 = Tamassia | first2 = Roberto&lt;br /&gt;
&lt;br /&gt;
 | last2 = Tamassia | first2 = Roberto&lt;br /&gt;
&lt;br /&gt;
2 Tamassia | first2 Roberto&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BF01108622&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BF01108622&lt;br /&gt;
&lt;br /&gt;
| doi 10.1007 / BF01108622&lt;br /&gt;
&lt;br /&gt;
 | issue = 2&lt;br /&gt;
&lt;br /&gt;
 | issue = 2&lt;br /&gt;
&lt;br /&gt;
第二期&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Order (journal)|Order]]&lt;br /&gt;
&lt;br /&gt;
 | journal = Order&lt;br /&gt;
&lt;br /&gt;
| 日志订单&lt;br /&gt;
&lt;br /&gt;
 | mr = 1354797&lt;br /&gt;
&lt;br /&gt;
 | mr = 1354797&lt;br /&gt;
&lt;br /&gt;
1354797先生&lt;br /&gt;
&lt;br /&gt;
 | pages = 109–133&lt;br /&gt;
&lt;br /&gt;
 | pages = 109–133&lt;br /&gt;
&lt;br /&gt;
第109-133页&lt;br /&gt;
&lt;br /&gt;
 | title = Upward planarity testing&lt;br /&gt;
&lt;br /&gt;
 | title = Upward planarity testing&lt;br /&gt;
&lt;br /&gt;
| 标题向上平面性测试&lt;br /&gt;
&lt;br /&gt;
 | volume = 12&lt;br /&gt;
&lt;br /&gt;
 | volume = 12&lt;br /&gt;
&lt;br /&gt;
第12卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1995| citeseerx = 10.1.1.10.2237&lt;br /&gt;
&lt;br /&gt;
 | year = 1995| citeseerx = 10.1.1.10.2237&lt;br /&gt;
&lt;br /&gt;
1995年10.1.1.10.2237&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Holten | first1 = Danny&lt;br /&gt;
&lt;br /&gt;
 | last1 = Holten | first1 = Danny&lt;br /&gt;
&lt;br /&gt;
 | last1 = Holten | first1 = Danny&lt;br /&gt;
&lt;br /&gt;
 | last2 = Isenberg | first2 = Petra&lt;br /&gt;
&lt;br /&gt;
 | last2 = Isenberg | first2 = Petra&lt;br /&gt;
&lt;br /&gt;
2 Isenberg | first2 Petra&lt;br /&gt;
&lt;br /&gt;
 | last3 = van Wijk | first3 = Jarke J. | author3-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 | last3 = van Wijk | first3 = Jarke J. | author3-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 | last3 = van Wijk | first3 = Jarke J.| author3-link Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 | last4 = Fekete | first4 = Jean-Daniel&lt;br /&gt;
&lt;br /&gt;
 | last4 = Fekete | first4 = Jean-Daniel&lt;br /&gt;
&lt;br /&gt;
最后4个 Fekete 最早4个 Jean-Daniel&lt;br /&gt;
&lt;br /&gt;
 | contribution = An extended evaluation of the readability of tapered, animated, and textured directed-edge representations in node-link graphs&lt;br /&gt;
&lt;br /&gt;
 | contribution = An extended evaluation of the readability of tapered, animated, and textured directed-edge representations in node-link graphs&lt;br /&gt;
&lt;br /&gt;
对节点链接图中渐变的、动画的和纹理的有向边表示的可读性的扩展评估&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/PACIFICVIS.2011.5742390&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1109/PACIFICVIS.2011.5742390&lt;br /&gt;
&lt;br /&gt;
10.1109 / PACIFICVIS. 2011.5742390&lt;br /&gt;
&lt;br /&gt;
 | pages = 195–202&lt;br /&gt;
&lt;br /&gt;
 | pages = 195–202&lt;br /&gt;
&lt;br /&gt;
第195-202页&lt;br /&gt;
&lt;br /&gt;
 | title = IEEE Pacific Visualization Symposium (PacificVis 2011)&lt;br /&gt;
&lt;br /&gt;
 | title = IEEE Pacific Visualization Symposium (PacificVis 2011)&lt;br /&gt;
&lt;br /&gt;
美国电气和电子工程师协会太平洋可视化研讨会(PacificVis 2011)&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.lri.fr/~isenberg/publications/papers/Holten_2011_AEP.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.lri.fr/~isenberg/publications/papers/Holten_2011_AEP.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.lri.fr/~isenberg/publications/papers/holten_2011_aep.pdf&lt;br /&gt;
&lt;br /&gt;
 | year = 2011| isbn = 978-1-61284-935-5&lt;br /&gt;
&lt;br /&gt;
 | year = 2011| isbn = 978-1-61284-935-5&lt;br /&gt;
&lt;br /&gt;
| 2011年 | isbn 978-1-61284-935-5&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 |last1        = Holten&lt;br /&gt;
&lt;br /&gt;
 |last1        = Holten&lt;br /&gt;
&lt;br /&gt;
1 Holten&lt;br /&gt;
&lt;br /&gt;
 |first1       = Danny&lt;br /&gt;
&lt;br /&gt;
 |first1       = Danny&lt;br /&gt;
&lt;br /&gt;
首先是丹尼&lt;br /&gt;
&lt;br /&gt;
 |last2        = van Wijk&lt;br /&gt;
&lt;br /&gt;
 |last2        = van Wijk&lt;br /&gt;
&lt;br /&gt;
2 van Wijk&lt;br /&gt;
&lt;br /&gt;
 |first2       = Jarke J.&lt;br /&gt;
&lt;br /&gt;
 |first2       = Jarke J.&lt;br /&gt;
&lt;br /&gt;
| first2 Jarke j.&lt;br /&gt;
&lt;br /&gt;
 |author2-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 |author2-link = Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
| author2-link Jack van Wijk&lt;br /&gt;
&lt;br /&gt;
 |contribution = A user study on visualizing directed edges in graphs&lt;br /&gt;
&lt;br /&gt;
 |contribution = A user study on visualizing directed edges in graphs&lt;br /&gt;
&lt;br /&gt;
用户研究图形中的有向边可视化&lt;br /&gt;
&lt;br /&gt;
 |doi          = 10.1145/1518701.1519054&lt;br /&gt;
&lt;br /&gt;
 |doi          = 10.1145/1518701.1519054&lt;br /&gt;
&lt;br /&gt;
10.1145 / 1518701.1519054&lt;br /&gt;
&lt;br /&gt;
 |pages        = 2299–2308&lt;br /&gt;
&lt;br /&gt;
 |pages        = 2299–2308&lt;br /&gt;
&lt;br /&gt;
2299-2308页&lt;br /&gt;
&lt;br /&gt;
 |title        = Proceedings of the 27th International Conference on Human Factors in Computing Systems (CHI '09)&lt;br /&gt;
&lt;br /&gt;
 |title        = Proceedings of the 27th International Conference on Human Factors in Computing Systems (CHI '09)&lt;br /&gt;
&lt;br /&gt;
| 第27届计算机系统中的人为因素国际会议论文集(CHI’09)&lt;br /&gt;
&lt;br /&gt;
 |url          = http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |url          = http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |year         = 2009&lt;br /&gt;
&lt;br /&gt;
 |year         = 2009&lt;br /&gt;
&lt;br /&gt;
2009年&lt;br /&gt;
&lt;br /&gt;
 |url-status      = dead&lt;br /&gt;
&lt;br /&gt;
 |url-status      = dead&lt;br /&gt;
&lt;br /&gt;
状态死机&lt;br /&gt;
&lt;br /&gt;
 |archiveurl   = https://web.archive.org/web/20111106004500/http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |archiveurl   = https://web.archive.org/web/20111106004500/http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
Https://web.archive.org/web/20111106004500/http://www.win.tue.nl/~dholten/papers/directed_edges_chi.pdf&lt;br /&gt;
&lt;br /&gt;
 |archivedate  = 2011-11-06&lt;br /&gt;
&lt;br /&gt;
 |archivedate  = 2011-11-06&lt;br /&gt;
&lt;br /&gt;
2011-11-06&lt;br /&gt;
&lt;br /&gt;
|isbn       = 9781605582467&lt;br /&gt;
&lt;br /&gt;
|isbn       = 9781605582467&lt;br /&gt;
&lt;br /&gt;
9781605582467&lt;br /&gt;
&lt;br /&gt;
 |citeseerx       = 10.1.1.212.5461&lt;br /&gt;
&lt;br /&gt;
 |citeseerx       = 10.1.1.212.5461&lt;br /&gt;
&lt;br /&gt;
10.1.1.212.5461&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Koren&lt;br /&gt;
&lt;br /&gt;
 | last = Koren&lt;br /&gt;
&lt;br /&gt;
最后一个韩国&lt;br /&gt;
&lt;br /&gt;
 | first = Yehuda&lt;br /&gt;
&lt;br /&gt;
 | first = Yehuda&lt;br /&gt;
&lt;br /&gt;
首先是耶胡达&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1016/j.camwa.2004.08.015&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1016/j.camwa.2004.08.015&lt;br /&gt;
&lt;br /&gt;
10.1016 / j.camwa. 2004.08&lt;br /&gt;
&lt;br /&gt;
 | issue = 11–12&lt;br /&gt;
&lt;br /&gt;
 | issue = 11–12&lt;br /&gt;
&lt;br /&gt;
第11-12期&lt;br /&gt;
&lt;br /&gt;
 | journal = Computers &amp;amp; Mathematics with Applications&lt;br /&gt;
&lt;br /&gt;
 | journal = Computers &amp;amp; Mathematics with Applications&lt;br /&gt;
&lt;br /&gt;
计算机与数学与应用》杂志&lt;br /&gt;
&lt;br /&gt;
 | mr = 2154691&lt;br /&gt;
&lt;br /&gt;
 | mr = 2154691&lt;br /&gt;
&lt;br /&gt;
2154691先生&lt;br /&gt;
&lt;br /&gt;
 | pages = 1867–1888&lt;br /&gt;
&lt;br /&gt;
 | pages = 1867–1888&lt;br /&gt;
&lt;br /&gt;
1867-1888页&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing graphs by eigenvectors: theory and practice&lt;br /&gt;
&lt;br /&gt;
 | title = Drawing graphs by eigenvectors: theory and practice&lt;br /&gt;
&lt;br /&gt;
按特征向量绘图: 理论与实践&lt;br /&gt;
&lt;br /&gt;
 | url = https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
Https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/dblp-journals-camwa-koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 49&lt;br /&gt;
&lt;br /&gt;
 | volume = 49&lt;br /&gt;
&lt;br /&gt;
第49卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2005&lt;br /&gt;
&lt;br /&gt;
 | year = 2005&lt;br /&gt;
&lt;br /&gt;
2005年&lt;br /&gt;
&lt;br /&gt;
 | access-date = 2011-09-17&lt;br /&gt;
&lt;br /&gt;
 | access-date = 2011-09-17&lt;br /&gt;
&lt;br /&gt;
2011-09-17&lt;br /&gt;
&lt;br /&gt;
 | archive-url = https://web.archive.org/web/20120402131402/https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | archive-url = https://web.archive.org/web/20120402131402/https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/DBLP-journals-camwa-Koren05.pdf&lt;br /&gt;
&lt;br /&gt;
| 档案-url  https://web.archive.org/web/20120402131402/https://akpublic.research.att.com/areas/visualization/papers_videos/pdf/dblp-journals-camwa-koren05.pdf&lt;br /&gt;
&lt;br /&gt;
 | archive-date = 2012-04-02&lt;br /&gt;
&lt;br /&gt;
 | archive-date = 2012-04-02&lt;br /&gt;
&lt;br /&gt;
2012-04-02&lt;br /&gt;
&lt;br /&gt;
 | url-status = dead&lt;br /&gt;
&lt;br /&gt;
 | url-status = dead&lt;br /&gt;
&lt;br /&gt;
状态死机&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Longabaugh | first = William&lt;br /&gt;
&lt;br /&gt;
 | last = Longabaugh | first = William&lt;br /&gt;
&lt;br /&gt;
最后一次朗格伯格 | 第一次威廉&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1186/1471-2105-13-275&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1186/1471-2105-13-275&lt;br /&gt;
&lt;br /&gt;
10.1186 / 1471-2105-13-275&lt;br /&gt;
&lt;br /&gt;
 | journal = BMC Bioinformatics&lt;br /&gt;
&lt;br /&gt;
 | journal = BMC Bioinformatics&lt;br /&gt;
&lt;br /&gt;
生物信息学杂志&lt;br /&gt;
&lt;br /&gt;
 | pages = 275&lt;br /&gt;
&lt;br /&gt;
 | pages = 275&lt;br /&gt;
&lt;br /&gt;
第275页&lt;br /&gt;
&lt;br /&gt;
 | title = Combing the hairball with BioFabric: a new approach for visualization of large networks&lt;br /&gt;
&lt;br /&gt;
 | title = Combing the hairball with BioFabric: a new approach for visualization of large networks&lt;br /&gt;
&lt;br /&gt;
用 BioFabric 梳理毛团: 一种大型网络可视化的新方法&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.biomedcentral.com/content/pdf/1471-2105-13-275.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = http://www.biomedcentral.com/content/pdf/1471-2105-13-275.pdf&lt;br /&gt;
&lt;br /&gt;
Http://www.biomedcentral.com/content/pdf/1471-2105-13-275.pdf&lt;br /&gt;
&lt;br /&gt;
 | pmid = 23102059&lt;br /&gt;
&lt;br /&gt;
 | pmid = 23102059&lt;br /&gt;
&lt;br /&gt;
23102059&lt;br /&gt;
&lt;br /&gt;
 | volume = 13&lt;br /&gt;
&lt;br /&gt;
 | volume = 13&lt;br /&gt;
&lt;br /&gt;
第13卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2012&lt;br /&gt;
&lt;br /&gt;
 | year = 2012&lt;br /&gt;
&lt;br /&gt;
2012年&lt;br /&gt;
&lt;br /&gt;
 | pmc=3574047}}.&lt;br /&gt;
&lt;br /&gt;
 | pmc=3574047}}.&lt;br /&gt;
&lt;br /&gt;
3574047}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Madden | first1 = Brendan&lt;br /&gt;
&lt;br /&gt;
 | last1 = Madden | first1 = Brendan&lt;br /&gt;
&lt;br /&gt;
最后一个疯狂橄榄球，最初一个布兰登&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Patrick&lt;br /&gt;
&lt;br /&gt;
 | last2 = Madden | first2 = Patrick&lt;br /&gt;
&lt;br /&gt;
2 Madden | first2 Patrick&lt;br /&gt;
&lt;br /&gt;
 | last3 = Powers | first3 = Steve&lt;br /&gt;
&lt;br /&gt;
 | last3 = Powers | first3 = Steve&lt;br /&gt;
&lt;br /&gt;
3 power | first3 Steve&lt;br /&gt;
&lt;br /&gt;
 | last4 = Himsolt | first4 = Michael&lt;br /&gt;
&lt;br /&gt;
 | last4 = Himsolt | first4 = Michael&lt;br /&gt;
&lt;br /&gt;
最后4希望 / 最初4迈克尔&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Brandenburg | editor-first = Franz J.&lt;br /&gt;
&lt;br /&gt;
 | editor-last = Brandenburg | editor-first = Franz J.&lt;br /&gt;
&lt;br /&gt;
最后的勃兰登堡 | 最初的弗朗茨 j。&lt;br /&gt;
&lt;br /&gt;
 | contribution = Portable graph layout and editing&lt;br /&gt;
&lt;br /&gt;
 | contribution = Portable graph layout and editing&lt;br /&gt;
&lt;br /&gt;
便携式图形布局和编辑&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BFb0021822&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/BFb0021822&lt;br /&gt;
&lt;br /&gt;
10.1007 / BFb0021822&lt;br /&gt;
&lt;br /&gt;
 | pages = 385–395&lt;br /&gt;
&lt;br /&gt;
 | pages = 385–395&lt;br /&gt;
&lt;br /&gt;
第385-395页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Symposium on Graph Drawing, GD '95, Passau, Germany, September 20–22, 1995, Proceedings&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing: Symposium on Graph Drawing, GD '95, Passau, Germany, September 20–22, 1995, Proceedings&lt;br /&gt;
&lt;br /&gt;
图形绘制: 图形绘制研讨会，GD’95，帕绍，1995年9月20-22日，美国国家科学院院刊&lt;br /&gt;
&lt;br /&gt;
 | volume = 1027&lt;br /&gt;
&lt;br /&gt;
 | volume = 1027&lt;br /&gt;
&lt;br /&gt;
第1027卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1996| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
 | year = 1996| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
1996年标题链接国际图形绘制专题研讨会&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-60723-6&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-60723-6&lt;br /&gt;
&lt;br /&gt;
[国际标准图书编号978-3-540-60723-6]&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
免费访问&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Misue | first1= K.&lt;br /&gt;
&lt;br /&gt;
 | last1 = Misue | first1= K.&lt;br /&gt;
&lt;br /&gt;
| 最后1米苏 | 最初1 k。&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades  | first2= P.&lt;br /&gt;
&lt;br /&gt;
 | last2 = Eades  | first2= P.&lt;br /&gt;
&lt;br /&gt;
2 Eades | first2 p.&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lai  | first3 = W.&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lai  | first3 = W.&lt;br /&gt;
&lt;br /&gt;
3 Lai | first3 w.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Sugiyama | first4 = K.&lt;br /&gt;
&lt;br /&gt;
 | last4 = Sugiyama | first4 = K.&lt;br /&gt;
&lt;br /&gt;
最后4千。&lt;br /&gt;
&lt;br /&gt;
 | title = Layout Adjustment and the Mental Map&lt;br /&gt;
&lt;br /&gt;
 | title = Layout Adjustment and the Mental Map&lt;br /&gt;
&lt;br /&gt;
| 标题布局调整和心理地图&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Visual Languages and Computing&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Visual Languages and Computing&lt;br /&gt;
&lt;br /&gt;
视觉语言与计算杂志&lt;br /&gt;
&lt;br /&gt;
 | volume = 6 | number = 2 | pages = 183–210 | year = 1995 | doi=10.1006/jvlc.1995.1010}}.&lt;br /&gt;
&lt;br /&gt;
 | volume = 6 | number = 2 | pages = 183–210 | year = 1995 | doi=10.1006/jvlc.1995.1010}}.&lt;br /&gt;
&lt;br /&gt;
183-210 | year 1995 | doi 10.1006 / jvlc. 1995.1010}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Nachmanson | first1 = Lev&lt;br /&gt;
&lt;br /&gt;
 | last1 = Nachmanson | first1 = Lev&lt;br /&gt;
&lt;br /&gt;
最后1纳克曼森最初1列弗&lt;br /&gt;
&lt;br /&gt;
 | last2 = Robertson | first2 = George&lt;br /&gt;
&lt;br /&gt;
 | last2 = Robertson | first2 = George&lt;br /&gt;
&lt;br /&gt;
2 Robertson | first2 George&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lee | first3 = Bongshin&lt;br /&gt;
&lt;br /&gt;
 | last3 = Lee | first3 = Bongshin&lt;br /&gt;
&lt;br /&gt;
3 | last 3 Lee | first3 bongsin&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Hong | editor1-first = Seok-Hee&lt;br /&gt;
&lt;br /&gt;
 | editor1-last = Hong | editor1-first = Seok-Hee&lt;br /&gt;
&lt;br /&gt;
| 编辑1-last Hong | 编辑1-first Seok-Hee&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki&lt;br /&gt;
&lt;br /&gt;
 | editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki&lt;br /&gt;
&lt;br /&gt;
 | editor3-last = Quan | editor3-first = Wu&lt;br /&gt;
&lt;br /&gt;
 | editor3-last = Quan | editor3-first = Wu&lt;br /&gt;
&lt;br /&gt;
| 编辑3-last Quan | 编辑3-first Wu&lt;br /&gt;
&lt;br /&gt;
 | contribution = Drawing Graphs with GLEE&lt;br /&gt;
&lt;br /&gt;
 | contribution = Drawing Graphs with GLEE&lt;br /&gt;
&lt;br /&gt;
用 GLEE 绘制图表&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-77537-9_38&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1007/978-3-540-77537-9_38&lt;br /&gt;
&lt;br /&gt;
10.1007 / 978-3-540-77537-938&lt;br /&gt;
&lt;br /&gt;
 | pages = 389–394&lt;br /&gt;
&lt;br /&gt;
 | pages = 389–394&lt;br /&gt;
&lt;br /&gt;
第389-394页&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | publisher = Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
| 出版商 Springer-Verlag&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
 | series = Lecture Notes in Computer Science&lt;br /&gt;
&lt;br /&gt;
| 计算机科学系列讲稿&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing, 15th International Symposium, GD 2007, Sydney, Australia, September 24–26, 2007, Revised Papers&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing, 15th International Symposium, GD 2007, Sydney, Australia, September 24–26, 2007, Revised Papers&lt;br /&gt;
&lt;br /&gt;
15 th International Symposium，GD 2007，Sydney，Australia，September 24-26，Revised Papers&lt;br /&gt;
&lt;br /&gt;
 | contribution-url = ftp://ftp.research.microsoft.com/pub/TR/TR-2007-72.pdf&lt;br /&gt;
&lt;br /&gt;
 | contribution-url = ftp://ftp.research.microsoft.com/pub/TR/TR-2007-72.pdf&lt;br /&gt;
&lt;br /&gt;
| contribution-url ftp: / / ftp.research.microsoft. com / pub / tr / tr-2007-72.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 4875&lt;br /&gt;
&lt;br /&gt;
 | volume = 4875&lt;br /&gt;
&lt;br /&gt;
第4875卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2008| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
 | year = 2008| title-link = International Symposium on Graph Drawing&lt;br /&gt;
&lt;br /&gt;
| 2008年 | 国际图形绘制专题研讨会链接&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-77536-2&lt;br /&gt;
&lt;br /&gt;
 | isbn = 978-3-540-77536-2&lt;br /&gt;
&lt;br /&gt;
[国际标准图书馆编号978-3-540-77536-2]&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
 | doi-access = free&lt;br /&gt;
&lt;br /&gt;
免费访问&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
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 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Pach | first1 = János | author1-link = János Pach&lt;br /&gt;
&lt;br /&gt;
 | last1 = Pach | first1 = János | author1-link = János Pach&lt;br /&gt;
&lt;br /&gt;
 | last1 = Pach | first1 = János | author1-link = János Pach&lt;br /&gt;
&lt;br /&gt;
 | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir&lt;br /&gt;
&lt;br /&gt;
 | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir&lt;br /&gt;
&lt;br /&gt;
2 | last 2 Sharir | first2 Micha | author2-link Micha Sharir&lt;br /&gt;
&lt;br /&gt;
 | contribution = 5.5 Angular resolution and slopes&lt;br /&gt;
&lt;br /&gt;
 | contribution = 5.5 Angular resolution and slopes&lt;br /&gt;
&lt;br /&gt;
| 贡献5.5角分辨率及斜坡&lt;br /&gt;
&lt;br /&gt;
 | pages = 126–127&lt;br /&gt;
&lt;br /&gt;
 | pages = 126–127&lt;br /&gt;
&lt;br /&gt;
第126-127页&lt;br /&gt;
&lt;br /&gt;
 | publisher = [[American Mathematical Society]]&lt;br /&gt;
&lt;br /&gt;
 | publisher = American Mathematical Society&lt;br /&gt;
&lt;br /&gt;
美国数学学会&lt;br /&gt;
&lt;br /&gt;
 | series = Mathematical Surveys and Monographs&lt;br /&gt;
&lt;br /&gt;
 | series = Mathematical Surveys and Monographs&lt;br /&gt;
&lt;br /&gt;
| 系列数学调查和专著&lt;br /&gt;
&lt;br /&gt;
 | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures&lt;br /&gt;
&lt;br /&gt;
 | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures&lt;br /&gt;
&lt;br /&gt;
离散几何及其算法应用: alcal 系列讲座&lt;br /&gt;
&lt;br /&gt;
 | volume = 152&lt;br /&gt;
&lt;br /&gt;
 | volume = 152&lt;br /&gt;
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第152卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2009}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2009}}.&lt;br /&gt;
&lt;br /&gt;
2009年开始。&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Purchase&lt;br /&gt;
&lt;br /&gt;
 | last1 = Purchase&lt;br /&gt;
&lt;br /&gt;
| 最后一次购买&lt;br /&gt;
&lt;br /&gt;
 | first1 = H. C.&lt;br /&gt;
&lt;br /&gt;
 | first1 = H. C.&lt;br /&gt;
&lt;br /&gt;
| first1 H.c.&lt;br /&gt;
&lt;br /&gt;
 | author1-link = Helen Purchase&lt;br /&gt;
&lt;br /&gt;
 | author1-link = Helen Purchase&lt;br /&gt;
&lt;br /&gt;
| author1-link Helen Purchase&lt;br /&gt;
&lt;br /&gt;
 | last2 = Cohen&lt;br /&gt;
&lt;br /&gt;
 | last2 = Cohen&lt;br /&gt;
&lt;br /&gt;
2 Cohen&lt;br /&gt;
&lt;br /&gt;
 | first2 = R. F.&lt;br /&gt;
&lt;br /&gt;
 | first2 = R. F.&lt;br /&gt;
&lt;br /&gt;
| first2 R.f.&lt;br /&gt;
&lt;br /&gt;
 | last3 = James&lt;br /&gt;
&lt;br /&gt;
 | last3 = James&lt;br /&gt;
&lt;br /&gt;
最后3名詹姆斯&lt;br /&gt;
&lt;br /&gt;
 | first3 = M. I.&lt;br /&gt;
&lt;br /&gt;
 | first3 = M. I.&lt;br /&gt;
&lt;br /&gt;
第一季第三集。&lt;br /&gt;
&lt;br /&gt;
 | at = Article 4&lt;br /&gt;
&lt;br /&gt;
 | at = Article 4&lt;br /&gt;
&lt;br /&gt;
| 第四条&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1145/264216.264222&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1145/264216.264222&lt;br /&gt;
&lt;br /&gt;
10.1145 / 264216.264222&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Experimental Algorithmics&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Experimental Algorithmics&lt;br /&gt;
&lt;br /&gt;
实验算法杂志&lt;br /&gt;
&lt;br /&gt;
 | title = An experimental study of the basis for graph drawing algorithms&lt;br /&gt;
&lt;br /&gt;
 | title = An experimental study of the basis for graph drawing algorithms&lt;br /&gt;
&lt;br /&gt;
关于图形绘制算法基础的实验研究&lt;br /&gt;
&lt;br /&gt;
 | url = https://secure.cs.uvic.ca/twiki/pub/Research/Chisel/ComputationalAestheticsProject/Vol2Nbr4.pdf&lt;br /&gt;
&lt;br /&gt;
 | url = https://secure.cs.uvic.ca/twiki/pub/Research/Chisel/ComputationalAestheticsProject/Vol2Nbr4.pdf&lt;br /&gt;
&lt;br /&gt;
Https://secure.cs.uvic.ca/twiki/pub/research/chisel/computationalaestheticsproject/vol2nbr4.pdf&lt;br /&gt;
&lt;br /&gt;
 | volume = 2&lt;br /&gt;
&lt;br /&gt;
 | volume = 2&lt;br /&gt;
&lt;br /&gt;
第二卷&lt;br /&gt;
&lt;br /&gt;
 | year = 1997&lt;br /&gt;
&lt;br /&gt;
 | year = 1997&lt;br /&gt;
&lt;br /&gt;
1997年&lt;br /&gt;
&lt;br /&gt;
 }}{{Dead link|date=January 2020 |bot=InternetArchiveBot |fix-attempted=yes }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Saaty | first = Thomas L. | authorlink = Thomas L. Saaty&lt;br /&gt;
&lt;br /&gt;
 | last = Saaty | first = Thomas L. | authorlink = Thomas L. Saaty&lt;br /&gt;
&lt;br /&gt;
作者链接: Thomas l. Saaty&lt;br /&gt;
&lt;br /&gt;
 | journal = Proc. Natl. Acad. Sci. U.S.A.&lt;br /&gt;
&lt;br /&gt;
 | journal = Proc. Natl. Acad. Sci. U.S.A.&lt;br /&gt;
&lt;br /&gt;
华尔街日报。自然。Acad.科学。美国。&lt;br /&gt;
&lt;br /&gt;
 | pages = 688–690&lt;br /&gt;
&lt;br /&gt;
 | pages = 688–690&lt;br /&gt;
&lt;br /&gt;
第688-690页&lt;br /&gt;
&lt;br /&gt;
 | title = The minimum number of intersections in complete graphs&lt;br /&gt;
&lt;br /&gt;
 | title = The minimum number of intersections in complete graphs&lt;br /&gt;
&lt;br /&gt;
| 标题完全图中的最小交叉点数&lt;br /&gt;
&lt;br /&gt;
 | volume = 52&lt;br /&gt;
&lt;br /&gt;
 | volume = 52&lt;br /&gt;
&lt;br /&gt;
第52卷&lt;br /&gt;
&lt;br /&gt;
 | issue = 3 | year = 1964&lt;br /&gt;
&lt;br /&gt;
 | issue = 3 | year = 1964&lt;br /&gt;
&lt;br /&gt;
第三期1964年&lt;br /&gt;
&lt;br /&gt;
 | doi=10.1073/pnas.52.3.688| pmid = 16591215 | pmc = 300329}}.&lt;br /&gt;
&lt;br /&gt;
 | doi=10.1073/pnas.52.3.688| pmid = 16591215 | pmc = 300329}}.&lt;br /&gt;
&lt;br /&gt;
10.1073 / pnas. 52.3.688 | pmid 16591215 | pmc 300329}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|title=Social network analysis: a handbook|first=John|last=Scott|edition=2nd|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|contribution=Sociograms and Graph Theory|url=https://books.google.com/books?id=Ww3_bKcz6kgC&amp;amp;pg=PA|pages=64–69}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=Kozo|last1=Sugiyama|author1-link=Kozo Sugiyama|first2=Shôjirô|last2=Tagawa|first3=Mitsuhiko|last3=Toda|title=Methods for visual understanding of hierarchical system structures|journal=[[IEEE Systems, Man &amp;amp; Cybernetics Society|IEEE Transactions on Systems, Man, and Cybernetics]]|volume=SMC-11|issue=2|pages=109–125|year=1981|mr=0611436|doi=10.1109/TSMC.1981.4308636}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last = Tantau | first = Till&lt;br /&gt;
&lt;br /&gt;
 | last = Tantau | first = Till&lt;br /&gt;
&lt;br /&gt;
 | last = Tantau | first = Till&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.7155/jgaa.00301&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.7155/jgaa.00301&lt;br /&gt;
&lt;br /&gt;
10.7155 / jgaa. 00301&lt;br /&gt;
&lt;br /&gt;
 | issue = 4&lt;br /&gt;
&lt;br /&gt;
 | issue = 4&lt;br /&gt;
&lt;br /&gt;
第四期&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Journal of Graph Algorithms and Applications]]&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of Graph Algorithms and Applications&lt;br /&gt;
&lt;br /&gt;
图形算法和应用杂志&lt;br /&gt;
&lt;br /&gt;
 | pages = 495–513&lt;br /&gt;
&lt;br /&gt;
 | pages = 495–513&lt;br /&gt;
&lt;br /&gt;
第495-513页&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing in TikZ&lt;br /&gt;
&lt;br /&gt;
 | title = Graph Drawing in TikZ&lt;br /&gt;
&lt;br /&gt;
| title Graph Drawing in TikZ&lt;br /&gt;
&lt;br /&gt;
 | volume = 17&lt;br /&gt;
&lt;br /&gt;
 | volume = 17&lt;br /&gt;
&lt;br /&gt;
第17卷&lt;br /&gt;
&lt;br /&gt;
 | year = 2013}}.&lt;br /&gt;
&lt;br /&gt;
 | year = 2013}}.&lt;br /&gt;
&lt;br /&gt;
2013年开始。&lt;br /&gt;
&lt;br /&gt;
* {{Citation | last1=Zapponi | first1=Leonardo | title=What is a Dessin d'Enfant | url=http://www.ams.org/notices/200307/what-is.pdf | date=August 2003 | journal=[[Notices of the American Mathematical Society]] | volume=50 | pages=788–789}}.&lt;br /&gt;
&lt;br /&gt;
{{refend}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==External links==&lt;br /&gt;
&lt;br /&gt;
* [http://graphx.codeplex.com GraphX library for .NET]: open-source WPF library for graph calculation and visualization. Supports many layout and edge routing algorithms.&lt;br /&gt;
&lt;br /&gt;
* [http://gdea.informatik.uni-koeln.de/ Graph drawing e-print archive]: including information on papers from all [[International Symposium on Graph Drawing|Graph Drawing symposia]].&lt;br /&gt;
&lt;br /&gt;
* {{dmoz|Science/Math/Combinatorics/Software/Graph_Drawing}} for many additional links related to graph drawing.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
{{Graph representations}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph drawing| ]]&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Graph drawing]]. Its edit history can be viewed at [[图的绘制/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%8A%A8%E6%80%81%E7%BD%91%E7%BB%9C%E5%88%86%E6%9E%90&amp;diff=13460</id>
		<title>动态网络分析</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%8A%A8%E6%80%81%E7%BD%91%E7%BB%9C%E5%88%86%E6%9E%90&amp;diff=13460"/>
		<updated>2020-08-29T05:39:44Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{More citations needed|date=April 2009}}&lt;br /&gt;
&lt;br /&gt;
'''Dynamic network analysis''' ('''DNA''') is an emergent scientific field that brings together traditional [[social network analysis]] (SNA), [[link analysis]] (LA), [[social simulation]] and [[multi-agent systems]] (MAS) within [[network science]] and [[network theory]]. There are two aspects of this field. The first is the [[statistical analysis]] of DNA data. The second is the utilization of simulation to address issues of network dynamics.  DNA networks vary from traditional social networks in that they are larger, dynamic, multi-mode, multi-plex networks, and may contain varying levels of [[uncertainty]]. The main difference of DNA to SNA is that DNA takes interactions of social features conditioning structure and behavior of networks into account. DNA is tied to temporal analysis but temporal analysis is not necessarily tied to DNA, as changes in networks sometimes result from external factors which are independent of social features found in networks. One of the most notable and earliest of cases in the use of DNA is in Sampson's monastery study, where he took snapshots of the same network from different intervals and observed and analyzed the evolution of the network.&amp;lt;ref&amp;gt;Harrison C. White, 1992, Identity and control: A structural theory of social action. Princeton University Press.&amp;lt;/ref&amp;gt; An early study of the dynamics of link utilization in very large-scale complex networks provides evidence of dynamic centrality, dynamic motifs, and cycles of social interactions.&amp;lt;ref&amp;gt;Dan Braha, Yaneer Bar‐Yam, 2006, [https://static1.squarespace.com/static/5b68a4e4a2772c2a206180a1/t/5c5de3faf4e1fc43e7b3d21e/1549657083988/Complexity_Braha_Original_w_Cover.pdf “From centrality to temporary fame: Dynamic centrality in complex networks,”] Complexity, 12(2), 59-63.&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;Dan Braha, Yaneer Bar-Yam 2009, [https://static1.squarespace.com/static/5b68a4e4a2772c2a206180a1/t/5c5de35feb393115883fdb54/1549656928806/DN_June_12_2008_necsi_Web.pdf Time-dependent complex networks: Dynamic centrality, dynamic motifs, and cycles of social interactions.] In Adaptive Networks (pp. 39-50). Springer, Berlin, Heidelberg.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Dynamic network analysis (DNA) is an emergent scientific field that brings together traditional social network analysis (SNA), link analysis (LA), social simulation and multi-agent systems (MAS) within network science and network theory. There are two aspects of this field. The first is the statistical analysis of DNA data. The second is the utilization of simulation to address issues of network dynamics.  DNA networks vary from traditional social networks in that they are larger, dynamic, multi-mode, multi-plex networks, and may contain varying levels of uncertainty. The main difference of DNA to SNA is that DNA takes interactions of social features conditioning structure and behavior of networks into account. DNA is tied to temporal analysis but temporal analysis is not necessarily tied to DNA, as changes in networks sometimes result from external factors which are independent of social features found in networks. One of the most notable and earliest of cases in the use of DNA is in Sampson's monastery study, where he took snapshots of the same network from different intervals and observed and analyzed the evolution of the network. An early study of the dynamics of link utilization in very large-scale complex networks provides evidence of dynamic centrality, dynamic motifs, and cycles of social interactions.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;动态网络分析 Dynamic network analysis&amp;lt;/font&amp;gt;'''是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络科学 Network Science&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络理论 Network Theory&amp;lt;/font&amp;gt;'''中将传统的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会网络分析 Social Network Analysis SNA&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;链路分析 Link Analysis LA&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会模拟 Social Simulation&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;多主体系统 Multi-Agent Systems MAS&amp;lt;/font&amp;gt;'''相结合的新兴科学领域。这个领域有两个方向。首先是动态网络数据的统计分析。第二是利用仿真来解决网络动态问题。动态网络不同于传统的社会网络，因为它们更大、更动态、多模式的多重网络，并且可能包含不同程度的不确定性。DNA 与 SNA 的主要区别在于，动态网络分析考虑了社会特征的相互作用，从而调节了网络的结构和行为。动态网络分析与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;时间分析 Temporal Analysis&amp;lt;/font&amp;gt;'''有关，但时间分析并不一定与动态网络分析有关，因为网络的变化有时是由外部因素造成的，这些外部因素与网络中的社会特征无关。Sampson的修道院研究是DNA使用中最著名和最早的案例之一，他在该研究中从不同间隔拍摄了同一网络的快照，并观察并分析了网络的演变。对超大型复杂网络中动态的链接利用的早期研究提供了'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;动态中心性 Dynamic Centrality&amp;lt;/font&amp;gt;'''，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;动态主题 Dynamic Motifs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社交互动周期 Cycles of Social Interactions&amp;lt;/font&amp;gt;'''的证据。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
DNA statistical tools are generally optimized for large-scale networks and admit the analysis of multiple networks simultaneously in which, there are multiple types of [[Node (networking)|nodes]] (multi-node) and multiple types of links (multi-plex). Multi-node multi-plex networks are generally referred to as&lt;br /&gt;
&lt;br /&gt;
DNA statistical tools are generally optimized for large-scale networks and admit the analysis of multiple networks simultaneously in which, there are multiple types of nodes (multi-node) and multiple types of links (multi-plex). Multi-node multi-plex networks are generally referred to as&lt;br /&gt;
&lt;br /&gt;
DNA统计工具通常可针对大规模网络进行优化，同时允许对多个具有多种类型的节点(多节点)和多种类型的链接(多边)的网络进行分析。多节点多重网络通常被称为&lt;br /&gt;
&lt;br /&gt;
meta-networks or high-dimensional networks. In contrast, SNA statistical tools focus on single or at most two mode data and facilitate the analysis of only one type of link at a time.&lt;br /&gt;
&lt;br /&gt;
meta-networks or high-dimensional networks. In contrast, SNA statistical tools focus on single or at most two mode data and facilitate the analysis of only one type of link at a time.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;元网络 Meta-Network&amp;lt;/font&amp;gt;'''或'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;高维网络 High-Dimensional Networks&amp;lt;/font&amp;gt;'''。相比之下，SNA统计工具侧重于单一模式或至多两种模式的数据，一次只便于分析一种类型的链接。&lt;br /&gt;
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DNA statistical tools tend to provide more measures to the user, because they have measures that use data drawn from multiple networks simultaneously. Latent space models (Sarkar and Moore, 2005)&amp;lt;ref name=&amp;quot;social&amp;quot;/&amp;gt; and agent-based simulation are often used to examine dynamic social networks (Carley et al., 2009).&amp;lt;ref name=&amp;quot;Etiology&amp;quot;/&amp;gt; From a computer simulation perspective, nodes in DNA are like atoms in quantum theory, nodes can be, though need not be, treated as probabilistic.  Whereas nodes in a traditional SNA model are [[wikt:static|static]], nodes in a DNA model have the ability to learn.  Properties change over time; nodes can adapt: A company's employees can learn new skills and increase their value to the network; or, capture one terrorist and three more are forced to improvise. Change propagates from one node to the next and so on.  DNA adds the element of a network's evolution and considers the circumstances under which change is likely to occur.&lt;br /&gt;
&lt;br /&gt;
DNA statistical tools tend to provide more measures to the user, because they have measures that use data drawn from multiple networks simultaneously. Latent space models (Sarkar and Moore, 2005) and agent-based simulation are often used to examine dynamic social networks (Carley et al., 2009). From a computer simulation perspective, nodes in DNA are like atoms in quantum theory, nodes can be, though need not be, treated as probabilistic.  Whereas nodes in a traditional SNA model are static, nodes in a DNA model have the ability to learn.  Properties change over time; nodes can adapt: A company's employees can learn new skills and increase their value to the network; or, capture one terrorist and three more are forced to improvise. Change propagates from one node to the next and so on.  DNA adds the element of a network's evolution and considers the circumstances under which change is likely to occur.&lt;br /&gt;
&lt;br /&gt;
DNA统计工具倾向于为用户提供更多的测量方法，因为它们可以同时使用来自多个网络的数据。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;潜在空间模型 Latent Space Models&amp;lt;/font&amp;gt;'''(Sarkar 和 Moore，2005年)和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基于代理的模拟 Agent-Based Simulation&amp;lt;/font&amp;gt;'''经常被用来检查动态的社会网络(Carley 等人，2009年)。从计算机仿真的角度来看，DNA中的节点就像量子理论中的原子一样，尽管不一定需要将节点视为概率的。传统SNA模型中的节点是静态的，而DNA模型中的节点具有学习能力。特性随时间变化; 节点可以随之适应: 一个公司的员工可以学习新的技能，增加他们对网络的价值; 或者，抓捕了一名恐怖分子，另外三人被迫临时合作。变化从一个节点传播到下一个节点，依此类推。DNA增加了网络进化的元素，并考虑了可能发生变化的环境。&lt;br /&gt;
&lt;br /&gt;
[[Image:DynamicNetworkAnalysisExample.jpg|right|340px|thumb|&lt;br /&gt;
图1：An example of a multi-entity, multi-network, dynamic network diagram 多实体、多网络、动态网络图示]]&lt;br /&gt;
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There are three main features to dynamic network analysis that distinguish it from standard social network analysis. First, rather than just using social networks, DNA looks at meta-networks. Second, agent-based modeling and other forms of simulations are often used to explore how networks evolve and adapt as well as the impact of interventions on those networks. Third, the links in the network are not binary; in fact, in many cases they represent the probability that there is a link.&lt;br /&gt;
&lt;br /&gt;
There are three main features to dynamic network analysis that distinguish it from standard social network analysis. First, rather than just using social networks, DNA looks at meta-networks. Second, agent-based modeling and other forms of simulations are often used to explore how networks evolve and adapt as well as the impact of interventions on those networks. Third, the links in the network are not binary; in fact, in many cases they represent the probability that there is a link.&lt;br /&gt;
&lt;br /&gt;
动态网络分析与标准社会网络分析相比，有三个主要特征。首先，DNA研究的不仅仅是社交网络，而是元网络。其次，基于代理的建模和其他形式的模拟常用于探索网络如何演变和适应以及干预对这些网络的影响。第三，网络中的链接不是二进制的; 事实上，在许多情况下，它们代表了链接存在的概率。&lt;br /&gt;
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==Meta-network 元网络==&lt;br /&gt;
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A meta-network is a multi-mode, multi-link, multi-level network. Multi-mode means that there are many types of nodes; e.g., nodes people and locations. Multi-link means that there are many types of links; e.g., friendship and advice. Multi-level means that some nodes may be members of other nodes, such as a network composed of people and organizations and one of the links is who is a member of which organization.&lt;br /&gt;
&lt;br /&gt;
A meta-network is a multi-mode, multi-link, multi-level network. Multi-mode means that there are many types of nodes; e.g., nodes people and locations. Multi-link means that there are many types of links; e.g., friendship and advice. Multi-level means that some nodes may be members of other nodes, such as a network composed of people and organizations and one of the links is who is a member of which organization.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;元网络 Meta-Network&amp;lt;/font&amp;gt;'''是一个多模式、多链路、多层次的网络。多模式意味着有许多类型的节点; 例如，人和位置。多链接意味着有许多类型的链接，例如，友谊和建议。多层级意味着某些节点可能是其他节点的成员，例如由人员和组织组成的网络，而链接之一就是谁是哪个组织的成员。&lt;br /&gt;
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While different researchers use different modes, common modes reflect who, what, when, where, why and how. A simple example of a meta-network is the PCANS formulation with people, tasks, and resources.&amp;lt;ref name=&amp;quot;pcans&amp;quot;/&amp;gt; A more detailed formulation considers people, tasks, resources, knowledge, and organizations.&amp;lt;ref name=&amp;quot;smartAgents&amp;quot;/&amp;gt; The ORA tool was developed to support meta-network analysis.&amp;lt;ref name=&amp;quot;toolkit&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
While different researchers use different modes, common modes reflect who, what, when, where, why and how. A simple example of a meta-network is the PCANS formulation with people, tasks, and resources. A more detailed formulation considers people, tasks, resources, knowledge, and organizations. The ORA tool was developed to support meta-network analysis.&lt;br /&gt;
&lt;br /&gt;
虽然不同的研究人员使用不同的模式，共同的模式却都反映了谁，什么，什么时候，在哪里，为什么和如何。元网络的一个简单示例是包含人、任务和资源的 PCANS 公式。更详细的公式考虑人员、任务、资源、知识和组织。ORA工具是为支持元网络分析而开发的。&lt;br /&gt;
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==Illustrative problems that people in the DNA area work on   DNA领域的人们正在研究的说明性问题==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* Developing metrics and statistics to assess and identify change within and across networks.&lt;br /&gt;
开发度量和统计数据，以评估和识别网络内部和网络之间的变化。&lt;br /&gt;
&lt;br /&gt;
* Developing and validating simulations to study network change, evolution, adaptation, decay. See [[Computer simulation and organizational studies]]&lt;br /&gt;
开发和验证仿真以研究网络的变化，演变，适应性，衰减。&lt;br /&gt;
&lt;br /&gt;
* Developing and testing theory of network change, evolution, adaptation, decay&amp;lt;ref&amp;gt;{{cite journal|author=Majdandzic, A.|title=Spontaneous recovery in dynamical networks|journal=Nature Physics|volume=10|pages=34–38|year=2013|doi=10.1038/nphys2819|display-authors=etal}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
网络变化，演化，适应，衰减的开发和测试理论&lt;br /&gt;
&lt;br /&gt;
* Developing and validating formal models of network generation and evolution&lt;br /&gt;
开发和验证网络生成和演化的正式模型&lt;br /&gt;
&lt;br /&gt;
* Developing techniques to visualize network change overall or at the node or group level&lt;br /&gt;
开发技术以可视化整体或节点或组级别的网络变化&lt;br /&gt;
&lt;br /&gt;
* Developing statistical techniques to see whether differences observed over time in networks are due to simply different samples from a distribution of links and nodes or changes over time in the underlying distribution of links and nodes&lt;br /&gt;
开发统计技术，以了解随着时间的推移，在网络中观察到的差异是否仅仅是因为链接和节点分布的样本不同，还是因为链接和节点的底层分布随时间的变化&lt;br /&gt;
&lt;br /&gt;
* Developing control processes for networks over time&lt;br /&gt;
在时间维度上，为网络开发控制过程&lt;br /&gt;
&lt;br /&gt;
* Developing algorithms to change distributions of links in networks over time&lt;br /&gt;
在时间维度上，开发算法来改变网络中的链接分布&lt;br /&gt;
&lt;br /&gt;
* Developing algorithms to track groups in networks over time&lt;br /&gt;
在时间维度上，开发算法来跟踪网络中的群体&lt;br /&gt;
&lt;br /&gt;
* Developing tools to extract or locate networks from various data sources such as texts&lt;br /&gt;
开发从各种数据源(如文本)中提取或定位网络的工具&lt;br /&gt;
&lt;br /&gt;
* Developing statistically valid measurements on networks over time&lt;br /&gt;
在时间维度上，在网络上开发有效的统计度量&lt;br /&gt;
&lt;br /&gt;
* Examining the robustness of network metrics under various types of missing data&lt;br /&gt;
检查网络指标在不同类型缺失数据下的鲁棒性&lt;br /&gt;
&lt;br /&gt;
* Empirical studies of multi-mode multi-link multi-time period networks&lt;br /&gt;
多模式、多链路、多时段网络的实证研究&lt;br /&gt;
&lt;br /&gt;
* Examining networks as probabilistic time-variant phenomena&lt;br /&gt;
检验网络的概率时变现象&lt;br /&gt;
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* Forecasting change in existing networks&lt;br /&gt;
预测现有网络的变化&lt;br /&gt;
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* Identifying trails through time given a sequence of networks&lt;br /&gt;
在给定网络序列的时间内识别路径&lt;br /&gt;
&lt;br /&gt;
* Identifying changes in node criticality given a sequence of networks anything else related to multi-mode multi-link multi-time period networks&lt;br /&gt;
在给定的网络序列中识别节点临界性的变化，任何与多模式、多链路、多时段网络相关的东西&lt;br /&gt;
&lt;br /&gt;
* Studying random walks on temporal networks&amp;lt;ref name=&amp;quot;rw&amp;quot; /&amp;gt;&lt;br /&gt;
研究时间网络上的随机游动&lt;br /&gt;
&lt;br /&gt;
* Quantifying structural properties of contact sequences in dynamic networks, which influence dynamical processes&amp;lt;ref name=&amp;quot;betweenness&amp;quot;/&amp;gt;&lt;br /&gt;
动态网络中影响动态过程的接触序列结构特性的量化&lt;br /&gt;
&lt;br /&gt;
*Assessment of covert activity&amp;lt;ref name=&amp;quot;covert&amp;quot; /&amp;gt; and dark networks&amp;lt;ref name=&amp;quot;dark&amp;quot; /&amp;gt;&lt;br /&gt;
秘密活动的评估&lt;br /&gt;
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*Citational analysis&amp;lt;ref name=&amp;quot;citation&amp;quot; /&amp;gt;&lt;br /&gt;
引文分析&lt;br /&gt;
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*Social media analysis&amp;lt;ref name=&amp;quot;socialmedia&amp;quot; /&amp;gt;&lt;br /&gt;
社交媒体分析&lt;br /&gt;
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*Assessment of public health systems&amp;lt;ref name=&amp;quot;health&amp;quot; /&amp;gt;&lt;br /&gt;
公共卫生系统的评估&lt;br /&gt;
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*Analysis of hospital safety outcomes&amp;lt;ref name=&amp;quot;hospital&amp;quot; /&amp;gt;&lt;br /&gt;
医院安全结果分析&lt;br /&gt;
&lt;br /&gt;
*Assessment of the structure of ethnic violence from news data&amp;lt;ref name=&amp;quot;ethnic&amp;quot; /&amp;gt;&lt;br /&gt;
从新闻数据中评估种族暴力的结构&lt;br /&gt;
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*Assessment of terror groups&amp;lt;ref name=&amp;quot;terror&amp;quot; /&amp;gt;&lt;br /&gt;
对恐怖组织的评估&lt;br /&gt;
&lt;br /&gt;
*Online social decay of social interactions&amp;lt;ref name=&amp;quot;socialdecay&amp;quot; /&amp;gt;&lt;br /&gt;
社交互动的在线社交衰退&lt;br /&gt;
&lt;br /&gt;
*Visualization of large financial networks over time&amp;lt;ref name=&amp;quot;dnb&amp;quot; /&amp;gt;&lt;br /&gt;
大型金融网络随时间的可视化&lt;br /&gt;
&lt;br /&gt;
*Modelling of classroom interactions in schools&amp;lt;ref name=&amp;quot;edu&amp;quot; /&amp;gt;&lt;br /&gt;
学校课堂互动的建模&lt;br /&gt;
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== See also ==&lt;br /&gt;
&lt;br /&gt;
* [[Graph dynamical system]]&lt;br /&gt;
图动态系统&lt;br /&gt;
&lt;br /&gt;
* [[International Network for Social Network Analysis]]&lt;br /&gt;
国际社交网络的分析&lt;br /&gt;
&lt;br /&gt;
* [[Kathleen M. Carley]]&lt;br /&gt;
凯瑟琳·卡利&lt;br /&gt;
&lt;br /&gt;
* [[Network dynamics]]&lt;br /&gt;
网络动力学&lt;br /&gt;
&lt;br /&gt;
* [[Network science]]&lt;br /&gt;
网络科学&lt;br /&gt;
&lt;br /&gt;
* [[Sequential dynamical system]]ios13.3 deca mield(8)&lt;br /&gt;
顺序动力学系统&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
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{{Reflist|refs=&lt;br /&gt;
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{{Reflist|refs=&lt;br /&gt;
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{通货再膨胀 | 参考文献&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;pcans&amp;quot;&amp;gt;David Krackhardt and Kathleen M. Carley, 1998, &amp;quot;A PCANS Model of Structure in Organization,&amp;quot; In proceedings of the 1998 International Symposium on Command and Control Research and Technology, Monterey, CA, June 1998, Evidence Based Research, Vienna, VA, Pp. 113-119.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;smartAgents&amp;quot;&amp;gt;Kathleen M. Carley, 2002, &amp;quot;Smart Agents and Organizations of the Future,&amp;quot; The Handbook of New Media. Edited by Leah Lievrouw and Sonia Livingstone (Eds.), Thousand Oaks, CA, Sage, Ch. 12: 206-220.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;rw&amp;quot;&amp;gt;Michele Starnini, Andrea Baronchelli, Alain Barrat, 2012, Random walks on temporal networks. Phys. Rev. E 85, 056115, http://link.aps.org/doi/10.1103/PhysRevE.85.056115&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;betweenness&amp;quot;&amp;gt;René Pfitzner, Ingo Scholtes, Antonios Garas, Claudio Juan Tessone, Frank Schweitzer, 2012, &amp;quot;Betweenness Preference: Quantifying Correlations in the Topological Dynamics of Temporal Networks&amp;quot;, Physical Review Letters, Vol. 110, May 10, 2013.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;social&amp;quot;&amp;gt;Purnamrita Sarkar and Andrew W. Moore. 2005. Dynamic social network analysis using latent space models. SIGKDD Explor. Newsl. 7, 2 (December 2005), 31-40.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;covert&amp;quot;&amp;gt;Carley, Kathleen M., Michael K., Martin and John P. Hancock, 2009, &amp;quot;Dynamic Network Analysis Applied to Experiments from the Decision Architectures Research Environment,&amp;quot;  Advanced Decision Architectures for the Warfigher: Foundation and Technology, Ch. 4.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;Etiology&amp;quot;&amp;gt;Kathleen M. Carley, Michael K. Martin and Brian Hirshman, 2009, &amp;quot;The Etiology of Social Change,&amp;quot; Topics in Cognitive Science, 1.4:621-650&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;dark&amp;quot;&amp;gt;Everton, Sean, 2012, Disrupting Dark Networks, Cambridge University Press, New York, NY&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;citation&amp;quot;&amp;gt;Kas, Miray, Kathleen M. Carley and L. Richard Carley, 2012, &amp;quot;Who was Where, When? Spatiotemporal Analysis of Researcher Mobility in Nuclear Science,&amp;quot; In proceedings of the International Workshop on Spatio Temporal data Integration and Retrieval (STIR 2012), held in conjunction&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;citation&amp;quot;&amp;gt;Kas, Miray, Kathleen M. Carley and L. Richard Carley, 2012, &amp;quot;Who was Where, When? Spatiotemporal Analysis of Researcher Mobility in Nuclear Science,&amp;quot; In proceedings of the International Workshop on Spatio Temporal data Integration and Retrieval (STIR 2012), held in conjunction&lt;br /&gt;
&lt;br /&gt;
Miray，Kathleen m. Carley and l. Richard Carley，2012，“ Who was Where，When？核科学中研究人员流动性的时空分析” ，在国际时空数据整合与检索研讨会(STIR，2012)的会议记录中，同时举行&lt;br /&gt;
&lt;br /&gt;
with ICDE 2012, April 1, 2012, Washington D.C.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
with ICDE 2012, April 1, 2012, Washington D.C.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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2012，April 1,2012，Washington d.c. / ref&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;terror&amp;quot;&amp;gt;Kenney, Michael J., John Horgan, Cale Horne, Peter Vining, Kathleen M. Carley, Michael Bigrigg, [[Mia Bloom]], Kurt Braddock, 2012, Organizational adaptation in an activist network: Social networks, leadership, and change in al-Muhajiroun, Applied Ergonomics, 44(5):739-747.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&lt;br /&gt;
&amp;lt;ref name=&amp;quot;health&amp;quot;&amp;gt;Merrill, Jacqueline, Mark G. Orr, Christie Y. Jeon, Rosalind V. Wilson, Jonathan Storrick and Kathleen M. Carley, 2012, &amp;quot;Topology of Local Health Officials’ Advice Networks: Mind the Gaps,&amp;quot; Journal of Public Health Management Practice, 18(6): 602–608&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;ethnic&amp;quot;&amp;gt;Van Holt, Tracy, Jeffrey C. Johnson, Jamie Brinkley, Kathleen M. Carley and Janna Caspersen, 2012, &amp;quot;Structure of ethnic violence in Sudan: an automated content, meta-network and geospatial analytical approach,&amp;quot; Computational and Mathematical Organization Theory, 18:340-355.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;hospital&amp;quot;&amp;gt;Effken, Judith A.,Sheila Gephart and Kathleen M. Carley, 2013, &amp;quot;Using ORA to Assess the Relationship of Handoffs to Quality and Safety Outcomes,&amp;quot; Computers, Informatics, Nursing. 31(1): 36-44.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;socialmedia&amp;quot;&amp;gt;Carley, Kathleen. M., Jürgen Pfeffer, [[Huan Liu]], Fred Morstatter, Rebecca Goolsby, 2013, Near Real Time Assessment of Social Media Using Geo-Temporal Network Analytics, In Proceedings of 2013 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM), August 25–28, 2013, Niagara Falls, Canada.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&lt;br /&gt;
&amp;lt;ref name=&amp;quot;toolkit&amp;quot;&amp;gt;Kathleen M. Carley. 2014. &amp;quot;ORA: A Toolkit for Dynamic Network Analysis and Visualization,&amp;quot; In Reda Alhajj and Jon Rokne (Eds.) Encyclopedia of Social Network Analysis and Mining, Springer.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;socialdecay&amp;quot;&amp;gt;M. Abufouda, K. A. Zweig .&amp;quot;A Theoretical Model for Understanding the Dynamics of Online Social Networks Decay&amp;quot;. arXiv preprint arXiv:1610.01538.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;dnb&amp;quot;&amp;gt;{{cite journal|last1=Heijmans|first1=Ronald|last2=Heuver|first2=Richard|last3=Levallois|first3=Clement|last4=van Lelyveld|first4=Iman|title=Dynamic visualization of large financial networks|journal=The Journal of Network Theory in Finance|volume=2|issue=2|year=2016|pages=57–79|issn=2055-7795|doi=10.21314/JNTF.2016.017}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;edu&amp;quot;&amp;gt;Christian Bokhove, 2016, &amp;quot;Exploring classroom interaction with dynamic social network analysis&amp;quot;, International Journal of Research &amp;amp; Method in Education, doi:10.1080/1743727X.2016.1192116.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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== Further reading ==&lt;br /&gt;
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*Kathleen M. Carley, 2003, &amp;quot;Dynamic Network Analysis&amp;quot; in Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers, Ronald Breiger, Kathleen Carley, and Philippa Pattison, (Eds.) Committee on Human Factors, National Research Council, National Research Council. Pp.&amp;amp;nbsp;133–145, Washington, DC.&lt;br /&gt;
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*Kathleen M. Carley, 2002, &amp;quot;Smart Agents and Organizations of the Future&amp;quot;  The Handbook of New Media. Edited by Leah Lievrouw and Sonia Livingstone, Ch. 12, pp.&amp;amp;nbsp;206–220, Thousand Oaks, CA, Sage.&lt;br /&gt;
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*Kathleen M. Carley, Jana Diesner, Jeffrey Reminga, Maksim Tsvetovat, 2008, Toward an Interoperable Dynamic Network Analysis Toolkit, DSS Special Issue on Cyberinfrastructure for Homeland Security: Advances in Information Sharing, Data Mining, and Collaboration Systems. [http://www.sciencedirect.com/science/journal/01679236 Decision Support Systems] 43(4):1324-1347 ([http://www.sciencedirect.com/science?_ob=ArticleURL&amp;amp;_udi=B6V8S-4KGG5P7-1&amp;amp;_user=4422&amp;amp;_coverDate=08%2F31%2F2007&amp;amp;_rdoc=20&amp;amp;_fmt=high&amp;amp;_orig=browse&amp;amp;_srch=doc-info(%23toc%235878%232007%23999569995%23665759%23FLA%23display%23Volume)&amp;amp;_cdi=5878&amp;amp;_sort=d&amp;amp;_docanchor=&amp;amp;_ct=52&amp;amp;_acct=C000059600&amp;amp;_version=1&amp;amp;_urlVersion=0&amp;amp;_userid=4422&amp;amp;md5=9459e84d7a8863039c7abd5065266250 article 20]{{dead link|date=March 2019|bot=medic}}{{cbignore|bot=medic}})&lt;br /&gt;
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* Dan Braha and Yaneer Bar-Yam, 2006, [https://static1.squarespace.com/static/5b68a4e4a2772c2a206180a1/t/5c5def5124a694e75e6109bc/1549659985911/Theory_Time_Dependent.pdf &amp;quot;Time-Dependent Complex Networks.&amp;quot;]&lt;br /&gt;
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*Terrill L. Frantz, Kathleen M. Carley. 2009, Toward A Confidence Estimate For The Most-Central-Actor Finding. Academy of Management Annual Conference, Chicago, IL, USA, 7–11 August. (Awarded the Sage Publications/RM Division Best Student Paper Award)&lt;br /&gt;
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*Petter Holme, Jari Saramäki, 2011, &amp;quot;Temporal networks&amp;quot;. https://arxiv.org/abs/1108.1780&lt;br /&gt;
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*C. Aggarwal, K. Subbian, 2014, &amp;quot;Evolutionary Network Analysis: A Survey&amp;quot;. ACM Computing Surveys, 47(1). ([http://charuaggarwal.net/CSUR-2013-0157.pdf pdf])&lt;br /&gt;
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== External links ==&lt;br /&gt;
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* [http://www.eecs.harvard.edu/%7Eparkes/RadcliffeSeminar.htm Radcliffe Exploratory Seminar on Dynamic Networks]&lt;br /&gt;
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* [http://www.casos.cs.cmu.edu/ Center for Computational Analysis of Social and Organizational Systems (CASOS)]&lt;br /&gt;
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[[Category:Computer network analysis]]&lt;br /&gt;
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Category:Computer network analysis&lt;br /&gt;
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类别: 计算机网络分析&lt;br /&gt;
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[[Category:Social statistics]]&lt;br /&gt;
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Category:Social statistics&lt;br /&gt;
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类别: 社会统计&lt;br /&gt;
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[[Category:Methods in sociology]]&lt;br /&gt;
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Category:Methods in sociology&lt;br /&gt;
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范畴: 社会学方法&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Dynamic network analysis]]. Its edit history can be viewed at [[动态网络分析/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%8A%A8%E6%80%81%E7%BD%91%E7%BB%9C%E5%88%86%E6%9E%90&amp;diff=13459</id>
		<title>动态网络分析</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%8A%A8%E6%80%81%E7%BD%91%E7%BB%9C%E5%88%86%E6%9E%90&amp;diff=13459"/>
		<updated>2020-08-29T05:37:47Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{More citations needed|date=April 2009}}&lt;br /&gt;
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'''Dynamic network analysis''' ('''DNA''') is an emergent scientific field that brings together traditional [[social network analysis]] (SNA), [[link analysis]] (LA), [[social simulation]] and [[multi-agent systems]] (MAS) within [[network science]] and [[network theory]]. There are two aspects of this field. The first is the [[statistical analysis]] of DNA data. The second is the utilization of simulation to address issues of network dynamics.  DNA networks vary from traditional social networks in that they are larger, dynamic, multi-mode, multi-plex networks, and may contain varying levels of [[uncertainty]]. The main difference of DNA to SNA is that DNA takes interactions of social features conditioning structure and behavior of networks into account. DNA is tied to temporal analysis but temporal analysis is not necessarily tied to DNA, as changes in networks sometimes result from external factors which are independent of social features found in networks. One of the most notable and earliest of cases in the use of DNA is in Sampson's monastery study, where he took snapshots of the same network from different intervals and observed and analyzed the evolution of the network.&amp;lt;ref&amp;gt;Harrison C. White, 1992, Identity and control: A structural theory of social action. Princeton University Press.&amp;lt;/ref&amp;gt; An early study of the dynamics of link utilization in very large-scale complex networks provides evidence of dynamic centrality, dynamic motifs, and cycles of social interactions.&amp;lt;ref&amp;gt;Dan Braha, Yaneer Bar‐Yam, 2006, [https://static1.squarespace.com/static/5b68a4e4a2772c2a206180a1/t/5c5de3faf4e1fc43e7b3d21e/1549657083988/Complexity_Braha_Original_w_Cover.pdf “From centrality to temporary fame: Dynamic centrality in complex networks,”] Complexity, 12(2), 59-63.&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;Dan Braha, Yaneer Bar-Yam 2009, [https://static1.squarespace.com/static/5b68a4e4a2772c2a206180a1/t/5c5de35feb393115883fdb54/1549656928806/DN_June_12_2008_necsi_Web.pdf Time-dependent complex networks: Dynamic centrality, dynamic motifs, and cycles of social interactions.] In Adaptive Networks (pp. 39-50). Springer, Berlin, Heidelberg.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Dynamic network analysis (DNA) is an emergent scientific field that brings together traditional social network analysis (SNA), link analysis (LA), social simulation and multi-agent systems (MAS) within network science and network theory. There are two aspects of this field. The first is the statistical analysis of DNA data. The second is the utilization of simulation to address issues of network dynamics.  DNA networks vary from traditional social networks in that they are larger, dynamic, multi-mode, multi-plex networks, and may contain varying levels of uncertainty. The main difference of DNA to SNA is that DNA takes interactions of social features conditioning structure and behavior of networks into account. DNA is tied to temporal analysis but temporal analysis is not necessarily tied to DNA, as changes in networks sometimes result from external factors which are independent of social features found in networks. One of the most notable and earliest of cases in the use of DNA is in Sampson's monastery study, where he took snapshots of the same network from different intervals and observed and analyzed the evolution of the network. An early study of the dynamics of link utilization in very large-scale complex networks provides evidence of dynamic centrality, dynamic motifs, and cycles of social interactions.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;动态网络分析 Dynamic network analysis&amp;lt;/font&amp;gt;'''是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络科学 Network Science&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络理论 Network Theory&amp;lt;/font&amp;gt;'''中将传统的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会网络分析 Social Network Analysis SNA&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;链路分析 Link Analysis LA&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会模拟 Social Simulation&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;多主体系统 Multi-Agent Systems MAS&amp;lt;/font&amp;gt;'''相结合的新兴科学领域。这个领域有两个方向。首先是动态网络数据的统计分析。第二是利用仿真来解决网络动态问题。动态网络不同于传统的社会网络，因为它们更大、更动态、多模式的多重网络，并且可能包含不同程度的不确定性。DNA 与 SNA 的主要区别在于，动态网络分析考虑了社会特征的相互作用，从而调节了网络的结构和行为。动态网络分析与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;时间分析 Temporal Analysis&amp;lt;/font&amp;gt;'''有关，但时间分析并不一定与动态网络分析有关，因为网络的变化有时是由外部因素造成的，这些外部因素与网络中的社会特征无关。Sampson的修道院研究是DNA使用中最著名和最早的案例之一，他在该研究中从不同间隔拍摄了同一网络的快照，并观察并分析了网络的演变。对超大型复杂网络中动态的链接利用的早期研究提供了'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;动态中心性 Dynamic Centrality&amp;lt;/font&amp;gt;'''，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;动态主题 Dynamic Motifs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社交互动周期 Cycles of Social Interactions&amp;lt;/font&amp;gt;'''的证据。&lt;br /&gt;
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DNA statistical tools are generally optimized for large-scale networks and admit the analysis of multiple networks simultaneously in which, there are multiple types of [[Node (networking)|nodes]] (multi-node) and multiple types of links (multi-plex). Multi-node multi-plex networks are generally referred to as&lt;br /&gt;
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DNA statistical tools are generally optimized for large-scale networks and admit the analysis of multiple networks simultaneously in which, there are multiple types of nodes (multi-node) and multiple types of links (multi-plex). Multi-node multi-plex networks are generally referred to as&lt;br /&gt;
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DNA统计工具通常可针对大规模网络进行优化，同时允许对多个具有多种类型的节点(多节点)和多种类型的链接(多边)的网络进行分析。多节点多重网络通常被称为&lt;br /&gt;
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meta-networks or high-dimensional networks. In contrast, SNA statistical tools focus on single or at most two mode data and facilitate the analysis of only one type of link at a time.&lt;br /&gt;
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meta-networks or high-dimensional networks. In contrast, SNA statistical tools focus on single or at most two mode data and facilitate the analysis of only one type of link at a time.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;元网络 Meta-Network&amp;lt;/font&amp;gt;'''或'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;高维网络 High-Dimensional Networks&amp;lt;/font&amp;gt;'''。相比之下，SNA统计工具侧重于单一模式或至多两种模式的数据，一次只便于分析一种类型的链接。&lt;br /&gt;
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DNA statistical tools tend to provide more measures to the user, because they have measures that use data drawn from multiple networks simultaneously. Latent space models (Sarkar and Moore, 2005)&amp;lt;ref name=&amp;quot;social&amp;quot;/&amp;gt; and agent-based simulation are often used to examine dynamic social networks (Carley et al., 2009).&amp;lt;ref name=&amp;quot;Etiology&amp;quot;/&amp;gt; From a computer simulation perspective, nodes in DNA are like atoms in quantum theory, nodes can be, though need not be, treated as probabilistic.  Whereas nodes in a traditional SNA model are [[wikt:static|static]], nodes in a DNA model have the ability to learn.  Properties change over time; nodes can adapt: A company's employees can learn new skills and increase their value to the network; or, capture one terrorist and three more are forced to improvise. Change propagates from one node to the next and so on.  DNA adds the element of a network's evolution and considers the circumstances under which change is likely to occur.&lt;br /&gt;
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DNA statistical tools tend to provide more measures to the user, because they have measures that use data drawn from multiple networks simultaneously. Latent space models (Sarkar and Moore, 2005) and agent-based simulation are often used to examine dynamic social networks (Carley et al., 2009). From a computer simulation perspective, nodes in DNA are like atoms in quantum theory, nodes can be, though need not be, treated as probabilistic.  Whereas nodes in a traditional SNA model are static, nodes in a DNA model have the ability to learn.  Properties change over time; nodes can adapt: A company's employees can learn new skills and increase their value to the network; or, capture one terrorist and three more are forced to improvise. Change propagates from one node to the next and so on.  DNA adds the element of a network's evolution and considers the circumstances under which change is likely to occur.&lt;br /&gt;
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DNA统计工具倾向于为用户提供更多的测量方法，因为它们可以同时使用来自多个网络的数据。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;潜在空间模型 Latent Space Models&amp;lt;/font&amp;gt;'''(Sarkar 和 Moore，2005年)和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基于代理的模拟 Agent-Based Simulation&amp;lt;/font&amp;gt;'''经常被用来检查动态的社会网络(Carley 等人，2009年)。从计算机仿真的角度来看，DNA中的节点就像量子理论中的原子一样，尽管不一定需要将节点视为概率的。传统SNA模型中的节点是静态的，而DNA模型中的节点具有学习能力。特性随时间变化; 节点可以随之适应: 一个公司的员工可以学习新的技能，增加他们对网络的价值; 或者，抓捕了一名恐怖分子，另外三人被迫临时合作。变化从一个节点传播到下一个节点，依此类推。DNA增加了网络进化的元素，并考虑了可能发生变化的环境。&lt;br /&gt;
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[[Image:DynamicNetworkAnalysisExample.jpg|right|340px|thumb|&lt;br /&gt;
图1：An example of a multi-entity, multi-network, dynamic network diagram 多实体、多网络、动态网络图示]]&lt;br /&gt;
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There are three main features to dynamic network analysis that distinguish it from standard social network analysis. First, rather than just using social networks, DNA looks at meta-networks. Second, agent-based modeling and other forms of simulations are often used to explore how networks evolve and adapt as well as the impact of interventions on those networks. Third, the links in the network are not binary; in fact, in many cases they represent the probability that there is a link.&lt;br /&gt;
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There are three main features to dynamic network analysis that distinguish it from standard social network analysis. First, rather than just using social networks, DNA looks at meta-networks. Second, agent-based modeling and other forms of simulations are often used to explore how networks evolve and adapt as well as the impact of interventions on those networks. Third, the links in the network are not binary; in fact, in many cases they represent the probability that there is a link.&lt;br /&gt;
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动态网络分析与标准社会网络分析相比，有三个主要特征。首先，DNA研究的不仅仅是社交网络，而是元网络。其次，基于代理的建模和其他形式的模拟常用于探索网络如何演变和适应以及干预对这些网络的影响。第三，网络中的链接不是二进制的; 事实上，在许多情况下，它们代表了链接存在的概率。&lt;br /&gt;
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==Meta-network 元网络==&lt;br /&gt;
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A meta-network is a multi-mode, multi-link, multi-level network. Multi-mode means that there are many types of nodes; e.g., nodes people and locations. Multi-link means that there are many types of links; e.g., friendship and advice. Multi-level means that some nodes may be members of other nodes, such as a network composed of people and organizations and one of the links is who is a member of which organization.&lt;br /&gt;
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A meta-network is a multi-mode, multi-link, multi-level network. Multi-mode means that there are many types of nodes; e.g., nodes people and locations. Multi-link means that there are many types of links; e.g., friendship and advice. Multi-level means that some nodes may be members of other nodes, such as a network composed of people and organizations and one of the links is who is a member of which organization.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;元网络 Meta-Network&amp;lt;/font&amp;gt;'''是一个多模式、多链路、多层次的网络。多模式意味着有许多类型的节点; 例如，人和位置。多链接意味着有许多类型的链接，例如，友谊和建议。多层级意味着某些节点可能是其他节点的成员，例如由人员和组织组成的网络，而链接之一就是谁是哪个组织的成员。&lt;br /&gt;
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While different researchers use different modes, common modes reflect who, what, when, where, why and how. A simple example of a meta-network is the PCANS formulation with people, tasks, and resources.&amp;lt;ref name=&amp;quot;pcans&amp;quot;/&amp;gt; A more detailed formulation considers people, tasks, resources, knowledge, and organizations.&amp;lt;ref name=&amp;quot;smartAgents&amp;quot;/&amp;gt; The ORA tool was developed to support meta-network analysis.&amp;lt;ref name=&amp;quot;toolkit&amp;quot; /&amp;gt;&lt;br /&gt;
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While different researchers use different modes, common modes reflect who, what, when, where, why and how. A simple example of a meta-network is the PCANS formulation with people, tasks, and resources. A more detailed formulation considers people, tasks, resources, knowledge, and organizations. The ORA tool was developed to support meta-network analysis.&lt;br /&gt;
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虽然不同的研究人员使用不同的模式，共同的模式却都反映了谁，什么，什么时候，在哪里，为什么和如何。元网络的一个简单示例是包含人、任务和资源的 PCANS 公式。更详细的公式考虑人员、任务、资源、知识和组织。ORA工具是为支持元网络分析而开发的。&lt;br /&gt;
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==Illustrative problems that people in the DNA area work on DNA领域的人们正在研究的说明性问题==&lt;br /&gt;
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* Developing metrics and statistics to assess and identify change within and across networks.&lt;br /&gt;
开发度量和统计数据，以评估和识别网络内部和网络之间的变化。&lt;br /&gt;
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* Developing and validating simulations to study network change, evolution, adaptation, decay. See [[Computer simulation and organizational studies]]&lt;br /&gt;
开发和验证仿真以研究网络的变化，演变，适应性，衰减。&lt;br /&gt;
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* Developing and testing theory of network change, evolution, adaptation, decay&amp;lt;ref&amp;gt;{{cite journal|author=Majdandzic, A.|title=Spontaneous recovery in dynamical networks|journal=Nature Physics|volume=10|pages=34–38|year=2013|doi=10.1038/nphys2819|display-authors=etal}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
网络变化，演化，适应，衰减的开发和测试理论&lt;br /&gt;
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* Developing and validating formal models of network generation and evolution&lt;br /&gt;
开发和验证网络生成和演化的正式模型&lt;br /&gt;
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* Developing techniques to visualize network change overall or at the node or group level&lt;br /&gt;
开发技术以可视化整体或节点或组级别的网络变化&lt;br /&gt;
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* Developing statistical techniques to see whether differences observed over time in networks are due to simply different samples from a distribution of links and nodes or changes over time in the underlying distribution of links and nodes&lt;br /&gt;
开发统计技术，以了解随着时间的推移，在网络中观察到的差异是否仅仅是因为链接和节点分布的样本不同，还是因为链接和节点的底层分布随时间的变化&lt;br /&gt;
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* Developing control processes for networks over time&lt;br /&gt;
在时间维度上，为网络开发控制过程&lt;br /&gt;
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* Developing algorithms to change distributions of links in networks over time&lt;br /&gt;
在时间维度上，开发算法来改变网络中的链接分布&lt;br /&gt;
&lt;br /&gt;
* Developing algorithms to track groups in networks over time&lt;br /&gt;
在时间维度上，开发算法来跟踪网络中的群体&lt;br /&gt;
&lt;br /&gt;
* Developing tools to extract or locate networks from various data sources such as texts&lt;br /&gt;
开发从各种数据源(如文本)中提取或定位网络的工具&lt;br /&gt;
&lt;br /&gt;
* Developing statistically valid measurements on networks over time&lt;br /&gt;
在时间维度上，在网络上开发有效的统计度量&lt;br /&gt;
&lt;br /&gt;
* Examining the robustness of network metrics under various types of missing data&lt;br /&gt;
检查网络指标在不同类型缺失数据下的鲁棒性&lt;br /&gt;
&lt;br /&gt;
* Empirical studies of multi-mode multi-link multi-time period networks&lt;br /&gt;
多模式、多链路、多时段网络的实证研究&lt;br /&gt;
&lt;br /&gt;
* Examining networks as probabilistic time-variant phenomena&lt;br /&gt;
检验网络的概率时变现象&lt;br /&gt;
&lt;br /&gt;
* Forecasting change in existing networks&lt;br /&gt;
预测现有网络的变化&lt;br /&gt;
&lt;br /&gt;
* Identifying trails through time given a sequence of networks&lt;br /&gt;
在给定网络序列的时间内识别路径&lt;br /&gt;
&lt;br /&gt;
* Identifying changes in node criticality given a sequence of networks anything else related to multi-mode multi-link multi-time period networks&lt;br /&gt;
在给定的网络序列中识别节点临界性的变化，任何与多模式、多链路、多时段网络相关的东西&lt;br /&gt;
&lt;br /&gt;
* Studying random walks on temporal networks&amp;lt;ref name=&amp;quot;rw&amp;quot; /&amp;gt;&lt;br /&gt;
研究时间网络上的随机游动&lt;br /&gt;
&lt;br /&gt;
* Quantifying structural properties of contact sequences in dynamic networks, which influence dynamical processes&amp;lt;ref name=&amp;quot;betweenness&amp;quot;/&amp;gt;&lt;br /&gt;
动态网络中影响动态过程的接触序列结构特性的量化&lt;br /&gt;
&lt;br /&gt;
*Assessment of covert activity&amp;lt;ref name=&amp;quot;covert&amp;quot; /&amp;gt; and dark networks&amp;lt;ref name=&amp;quot;dark&amp;quot; /&amp;gt;&lt;br /&gt;
秘密活动的评估&lt;br /&gt;
&lt;br /&gt;
*Citational analysis&amp;lt;ref name=&amp;quot;citation&amp;quot; /&amp;gt;&lt;br /&gt;
引文分析&lt;br /&gt;
&lt;br /&gt;
*Social media analysis&amp;lt;ref name=&amp;quot;socialmedia&amp;quot; /&amp;gt;&lt;br /&gt;
社交媒体分析&lt;br /&gt;
&lt;br /&gt;
*Assessment of public health systems&amp;lt;ref name=&amp;quot;health&amp;quot; /&amp;gt;&lt;br /&gt;
公共卫生系统的评估&lt;br /&gt;
&lt;br /&gt;
*Analysis of hospital safety outcomes&amp;lt;ref name=&amp;quot;hospital&amp;quot; /&amp;gt;&lt;br /&gt;
医院安全结果分析&lt;br /&gt;
&lt;br /&gt;
*Assessment of the structure of ethnic violence from news data&amp;lt;ref name=&amp;quot;ethnic&amp;quot; /&amp;gt;&lt;br /&gt;
从新闻数据中评估种族暴力的结构&lt;br /&gt;
&lt;br /&gt;
*Assessment of terror groups&amp;lt;ref name=&amp;quot;terror&amp;quot; /&amp;gt;&lt;br /&gt;
对恐怖组织的评估&lt;br /&gt;
&lt;br /&gt;
*Online social decay of social interactions&amp;lt;ref name=&amp;quot;socialdecay&amp;quot; /&amp;gt;&lt;br /&gt;
社交互动的在线社交衰退&lt;br /&gt;
&lt;br /&gt;
*Visualization of large financial networks over time&amp;lt;ref name=&amp;quot;dnb&amp;quot; /&amp;gt;&lt;br /&gt;
大型金融网络随时间的可视化&lt;br /&gt;
&lt;br /&gt;
*Modelling of classroom interactions in schools&amp;lt;ref name=&amp;quot;edu&amp;quot; /&amp;gt;&lt;br /&gt;
学校课堂互动的建模&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
* [[Graph dynamical system]]&lt;br /&gt;
图动态系统&lt;br /&gt;
&lt;br /&gt;
* [[International Network for Social Network Analysis]]&lt;br /&gt;
国际社交网络的分析&lt;br /&gt;
&lt;br /&gt;
* [[Kathleen M. Carley]]&lt;br /&gt;
凯瑟琳·卡利&lt;br /&gt;
&lt;br /&gt;
* [[Network dynamics]]&lt;br /&gt;
网络动力学&lt;br /&gt;
&lt;br /&gt;
* [[Network science]]&lt;br /&gt;
网络科学&lt;br /&gt;
&lt;br /&gt;
* [[Sequential dynamical system]]ios13.3 deca mield(8)&lt;br /&gt;
顺序动力学系统&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&lt;br /&gt;
{{Reflist|refs=&lt;br /&gt;
&lt;br /&gt;
{{Reflist|refs=&lt;br /&gt;
&lt;br /&gt;
{通货再膨胀 | 参考文献&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;pcans&amp;quot;&amp;gt;David Krackhardt and Kathleen M. Carley, 1998, &amp;quot;A PCANS Model of Structure in Organization,&amp;quot; In proceedings of the 1998 International Symposium on Command and Control Research and Technology, Monterey, CA, June 1998, Evidence Based Research, Vienna, VA, Pp. 113-119.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;smartAgents&amp;quot;&amp;gt;Kathleen M. Carley, 2002, &amp;quot;Smart Agents and Organizations of the Future,&amp;quot; The Handbook of New Media. Edited by Leah Lievrouw and Sonia Livingstone (Eds.), Thousand Oaks, CA, Sage, Ch. 12: 206-220.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;rw&amp;quot;&amp;gt;Michele Starnini, Andrea Baronchelli, Alain Barrat, 2012, Random walks on temporal networks. Phys. Rev. E 85, 056115, http://link.aps.org/doi/10.1103/PhysRevE.85.056115&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;betweenness&amp;quot;&amp;gt;René Pfitzner, Ingo Scholtes, Antonios Garas, Claudio Juan Tessone, Frank Schweitzer, 2012, &amp;quot;Betweenness Preference: Quantifying Correlations in the Topological Dynamics of Temporal Networks&amp;quot;, Physical Review Letters, Vol. 110, May 10, 2013.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;social&amp;quot;&amp;gt;Purnamrita Sarkar and Andrew W. Moore. 2005. Dynamic social network analysis using latent space models. SIGKDD Explor. Newsl. 7, 2 (December 2005), 31-40.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;covert&amp;quot;&amp;gt;Carley, Kathleen M., Michael K., Martin and John P. Hancock, 2009, &amp;quot;Dynamic Network Analysis Applied to Experiments from the Decision Architectures Research Environment,&amp;quot;  Advanced Decision Architectures for the Warfigher: Foundation and Technology, Ch. 4.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;Etiology&amp;quot;&amp;gt;Kathleen M. Carley, Michael K. Martin and Brian Hirshman, 2009, &amp;quot;The Etiology of Social Change,&amp;quot; Topics in Cognitive Science, 1.4:621-650&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;dark&amp;quot;&amp;gt;Everton, Sean, 2012, Disrupting Dark Networks, Cambridge University Press, New York, NY&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;citation&amp;quot;&amp;gt;Kas, Miray, Kathleen M. Carley and L. Richard Carley, 2012, &amp;quot;Who was Where, When? Spatiotemporal Analysis of Researcher Mobility in Nuclear Science,&amp;quot; In proceedings of the International Workshop on Spatio Temporal data Integration and Retrieval (STIR 2012), held in conjunction&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;citation&amp;quot;&amp;gt;Kas, Miray, Kathleen M. Carley and L. Richard Carley, 2012, &amp;quot;Who was Where, When? Spatiotemporal Analysis of Researcher Mobility in Nuclear Science,&amp;quot; In proceedings of the International Workshop on Spatio Temporal data Integration and Retrieval (STIR 2012), held in conjunction&lt;br /&gt;
&lt;br /&gt;
Miray，Kathleen m. Carley and l. Richard Carley，2012，“ Who was Where，When？核科学中研究人员流动性的时空分析” ，在国际时空数据整合与检索研讨会(STIR，2012)的会议记录中，同时举行&lt;br /&gt;
&lt;br /&gt;
with ICDE 2012, April 1, 2012, Washington D.C.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
with ICDE 2012, April 1, 2012, Washington D.C.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
2012，April 1,2012，Washington d.c. / ref&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;terror&amp;quot;&amp;gt;Kenney, Michael J., John Horgan, Cale Horne, Peter Vining, Kathleen M. Carley, Michael Bigrigg, [[Mia Bloom]], Kurt Braddock, 2012, Organizational adaptation in an activist network: Social networks, leadership, and change in al-Muhajiroun, Applied Ergonomics, 44(5):739-747.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;health&amp;quot;&amp;gt;Merrill, Jacqueline, Mark G. Orr, Christie Y. Jeon, Rosalind V. Wilson, Jonathan Storrick and Kathleen M. Carley, 2012, &amp;quot;Topology of Local Health Officials’ Advice Networks: Mind the Gaps,&amp;quot; Journal of Public Health Management Practice, 18(6): 602–608&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;ethnic&amp;quot;&amp;gt;Van Holt, Tracy, Jeffrey C. Johnson, Jamie Brinkley, Kathleen M. Carley and Janna Caspersen, 2012, &amp;quot;Structure of ethnic violence in Sudan: an automated content, meta-network and geospatial analytical approach,&amp;quot; Computational and Mathematical Organization Theory, 18:340-355.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;hospital&amp;quot;&amp;gt;Effken, Judith A.,Sheila Gephart and Kathleen M. Carley, 2013, &amp;quot;Using ORA to Assess the Relationship of Handoffs to Quality and Safety Outcomes,&amp;quot; Computers, Informatics, Nursing. 31(1): 36-44.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;socialmedia&amp;quot;&amp;gt;Carley, Kathleen. M., Jürgen Pfeffer, [[Huan Liu]], Fred Morstatter, Rebecca Goolsby, 2013, Near Real Time Assessment of Social Media Using Geo-Temporal Network Analytics, In Proceedings of 2013 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM), August 25–28, 2013, Niagara Falls, Canada.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;toolkit&amp;quot;&amp;gt;Kathleen M. Carley. 2014. &amp;quot;ORA: A Toolkit for Dynamic Network Analysis and Visualization,&amp;quot; In Reda Alhajj and Jon Rokne (Eds.) Encyclopedia of Social Network Analysis and Mining, Springer.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;socialdecay&amp;quot;&amp;gt;M. Abufouda, K. A. Zweig .&amp;quot;A Theoretical Model for Understanding the Dynamics of Online Social Networks Decay&amp;quot;. arXiv preprint arXiv:1610.01538.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;dnb&amp;quot;&amp;gt;{{cite journal|last1=Heijmans|first1=Ronald|last2=Heuver|first2=Richard|last3=Levallois|first3=Clement|last4=van Lelyveld|first4=Iman|title=Dynamic visualization of large financial networks|journal=The Journal of Network Theory in Finance|volume=2|issue=2|year=2016|pages=57–79|issn=2055-7795|doi=10.21314/JNTF.2016.017}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;edu&amp;quot;&amp;gt;Christian Bokhove, 2016, &amp;quot;Exploring classroom interaction with dynamic social network analysis&amp;quot;, International Journal of Research &amp;amp; Method in Education, doi:10.1080/1743727X.2016.1192116.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Further reading ==&lt;br /&gt;
&lt;br /&gt;
*Kathleen M. Carley, 2003, &amp;quot;Dynamic Network Analysis&amp;quot; in Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers, Ronald Breiger, Kathleen Carley, and Philippa Pattison, (Eds.) Committee on Human Factors, National Research Council, National Research Council. Pp.&amp;amp;nbsp;133–145, Washington, DC.&lt;br /&gt;
&lt;br /&gt;
*Kathleen M. Carley, 2002, &amp;quot;Smart Agents and Organizations of the Future&amp;quot;  The Handbook of New Media. Edited by Leah Lievrouw and Sonia Livingstone, Ch. 12, pp.&amp;amp;nbsp;206–220, Thousand Oaks, CA, Sage.&lt;br /&gt;
&lt;br /&gt;
*Kathleen M. Carley, Jana Diesner, Jeffrey Reminga, Maksim Tsvetovat, 2008, Toward an Interoperable Dynamic Network Analysis Toolkit, DSS Special Issue on Cyberinfrastructure for Homeland Security: Advances in Information Sharing, Data Mining, and Collaboration Systems. [http://www.sciencedirect.com/science/journal/01679236 Decision Support Systems] 43(4):1324-1347 ([http://www.sciencedirect.com/science?_ob=ArticleURL&amp;amp;_udi=B6V8S-4KGG5P7-1&amp;amp;_user=4422&amp;amp;_coverDate=08%2F31%2F2007&amp;amp;_rdoc=20&amp;amp;_fmt=high&amp;amp;_orig=browse&amp;amp;_srch=doc-info(%23toc%235878%232007%23999569995%23665759%23FLA%23display%23Volume)&amp;amp;_cdi=5878&amp;amp;_sort=d&amp;amp;_docanchor=&amp;amp;_ct=52&amp;amp;_acct=C000059600&amp;amp;_version=1&amp;amp;_urlVersion=0&amp;amp;_userid=4422&amp;amp;md5=9459e84d7a8863039c7abd5065266250 article 20]{{dead link|date=March 2019|bot=medic}}{{cbignore|bot=medic}})&lt;br /&gt;
&lt;br /&gt;
* Dan Braha and Yaneer Bar-Yam, 2006, [https://static1.squarespace.com/static/5b68a4e4a2772c2a206180a1/t/5c5def5124a694e75e6109bc/1549659985911/Theory_Time_Dependent.pdf &amp;quot;Time-Dependent Complex Networks.&amp;quot;]&lt;br /&gt;
&lt;br /&gt;
*Terrill L. Frantz, Kathleen M. Carley. 2009, Toward A Confidence Estimate For The Most-Central-Actor Finding. Academy of Management Annual Conference, Chicago, IL, USA, 7–11 August. (Awarded the Sage Publications/RM Division Best Student Paper Award)&lt;br /&gt;
&lt;br /&gt;
*Petter Holme, Jari Saramäki, 2011, &amp;quot;Temporal networks&amp;quot;. https://arxiv.org/abs/1108.1780&lt;br /&gt;
&lt;br /&gt;
*C. Aggarwal, K. Subbian, 2014, &amp;quot;Evolutionary Network Analysis: A Survey&amp;quot;. ACM Computing Surveys, 47(1). ([http://charuaggarwal.net/CSUR-2013-0157.pdf pdf])&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== External links ==&lt;br /&gt;
&lt;br /&gt;
* [http://www.eecs.harvard.edu/%7Eparkes/RadcliffeSeminar.htm Radcliffe Exploratory Seminar on Dynamic Networks]&lt;br /&gt;
&lt;br /&gt;
* [http://www.casos.cs.cmu.edu/ Center for Computational Analysis of Social and Organizational Systems (CASOS)]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
{{Social networking}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
{{DEFAULTSORT:Dynamic Network Analysis}}&lt;br /&gt;
&lt;br /&gt;
[[Category:Computer network analysis]]&lt;br /&gt;
&lt;br /&gt;
Category:Computer network analysis&lt;br /&gt;
&lt;br /&gt;
类别: 计算机网络分析&lt;br /&gt;
&lt;br /&gt;
[[Category:Social statistics]]&lt;br /&gt;
&lt;br /&gt;
Category:Social statistics&lt;br /&gt;
&lt;br /&gt;
类别: 社会统计&lt;br /&gt;
&lt;br /&gt;
[[Category:Methods in sociology]]&lt;br /&gt;
&lt;br /&gt;
Category:Methods in sociology&lt;br /&gt;
&lt;br /&gt;
范畴: 社会学方法&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Dynamic network analysis]]. Its edit history can be viewed at [[动态网络分析/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13391</id>
		<title>枢纽</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13391"/>
		<updated>2020-08-28T06:09:01Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{redirect|Network hub|the Ethernet technology|Ethernet hub}}&lt;br /&gt;
&lt;br /&gt;
In [[network science]], a '''hub''' is a [[Node (networking)|node]] with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf Barabási, Albert-László. ''Network Science: Graph Theory''., p. 27]&amp;lt;/ref&amp;gt; While hubs cannot be observed in a random network, they are expected to emerge in [[scale-free network]]s. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the [[network topology]]. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
&lt;br /&gt;
In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the network topology. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络科学 Network Science&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点 Hub&amp;lt;/font&amp;gt;'''是链接数大大超过平均值的节点。枢纽节点是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Property of Networks&amp;lt;/font&amp;gt;'''的重要特性。虽然在随机网络中不存在枢纽节点，但是它们却存在于无标度网络中。无标度网络中枢纽节点的涌现与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;幂律分布 Power-Law Distribution&amp;lt;/font&amp;gt;'''有关，其对网络的拓扑结构有着重要的影响。枢纽节点可以在很多真实的网络中找到，比如大脑网络或者互联网。&lt;br /&gt;
&lt;br /&gt;
[[File:Network representation of brain connectivity.JPG|thumb|right|&lt;br /&gt;
&lt;br /&gt;
图1：Network representation of brain connectivity. Hubs are highlighted &lt;br /&gt;
表示大脑连接性的网络，其中枢纽节点被突出显示]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[File:Internet map 4096.png|thumb|150px|right|&lt;br /&gt;
&lt;br /&gt;
图2:Partial map of the Internet based on the January 15, 2005. Hubs are highlighted &lt;br /&gt;
2005年1月15日的因特网局部图，枢纽节点被突出显示]]&lt;br /&gt;
&lt;br /&gt;
A hub is a component of a network with a high-degree [[Vertex (graph theory)|node]]. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links ([[Degree (graph theory)|degrees]]) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size ''N'' of the network and average degree ''&amp;lt;k&amp;gt;'' constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree ''k'' is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree ''k'' while the other nodes have a small number of links.&lt;br /&gt;
&lt;br /&gt;
A hub is a component of a network with a high-degree node. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links (degrees) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size N of the network and average degree &amp;lt;k&amp;gt; constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree k is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree k while the other nodes have a small number of links.&lt;br /&gt;
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一个枢纽节点是网络中具有高度值的组件。与网络中的其他节点相比，枢纽节点的链接数要大得多。无尺度网络中枢纽的链接数(度)远远高于随机网络中链接数最大的节点，在保持网络的大小''N''和平均度 ''&amp;lt;k&amp;gt;''不变的情况下。枢纽的存在是随机网络和无标度网络的最大区别。在随机网络中，每个节点的度''k''是可比的，因此不可能出现枢纽节点。在无标度网络中，少数节点(即枢纽节点)具有高度值 ''k''，而其他节点只有少量的链接。&lt;br /&gt;
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== Emergence 涌现==&lt;br /&gt;
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[[Image:Scale-free network sample.png|thumb|Example of a random network and a scale-free network|400px|right|&lt;br /&gt;
图3：Random network (a) and scale-free network (b). In the scale-free network, the larger hubs are highlighted. &lt;br /&gt;
随机网络（a）和无标度网络（b），在无标度网络中，大型枢纽被突出显示]]&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks ([[Barabási–Albert model]]) are different from random networks ([[Erdős–Rényi model]]) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks (Barabási–Albert model) are different from random networks (Erdős–Rényi model) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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枢纽节点的出现可以用无标度网络和随机网络的区别来解释。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks Barabási-Albert model&amp;lt;/font&amp;gt;'''与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机网络 Random Networks Erdős–Rényi model&amp;lt;/font&amp;gt;'''在两个方面有所不同: (a)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;增长 Growth&amp;lt;/font&amp;gt;'''，(b)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;优先连接 Preferential Attachment&amp;lt;/font&amp;gt;'''。{ Cite journal&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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Http://www.nd.edu/~networks/publication%20categories/03%20journal%20articles/physics/statisticalmechanics_rev%20of%20modern%20physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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1 = r é ka Albert | first1 = r é ka | last1 = Albert&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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2 = albert-lászló Barabási | first2 = albert-lászló | last2 = Barabási&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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| title = 复杂网络的统计力学&lt;br /&gt;
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 | journal = [[Reviews of Modern Physics]]&lt;br /&gt;
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 | journal = Reviews of Modern Physics&lt;br /&gt;
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| 杂志 = 现代物理学评论&lt;br /&gt;
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 | volume = 74&lt;br /&gt;
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 | volume = 74&lt;br /&gt;
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74&lt;br /&gt;
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 | pages = 47–97&lt;br /&gt;
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 | pages = 47–97&lt;br /&gt;
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| 页数 = 47-97&lt;br /&gt;
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 | year = 2002&lt;br /&gt;
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 | year = 2002&lt;br /&gt;
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2002年&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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10.1103/RevModPhys. 74.47&lt;br /&gt;
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 | bibcode=2002RvMP...74...47A&lt;br /&gt;
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 | bibcode=2002RvMP...74...47A&lt;br /&gt;
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2002RvMP... 74... 47A&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* (a) Scale-free networks assume a continuous growth of the number of nodes ''N'', compared to random networks which assume a fixed number of nodes. In scale-free networks the degree of the largest hub rises polynomially with the size of the network. Therefore, the degree of a hub can be high in a scale-free network. In random networks the degree of the largest node rises logaritmically (or slower) with N, thus the hub number will be small even in a very large network.&lt;br /&gt;
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无标度网络假设节点数“N”持续增长，而随机网络假设节点数为固定数目。在无标度网络中，最大枢纽节点的度数随着网络的大小而多项式地上升。因此，在无标度网络中，枢纽节点的度数可以很高。在随机网络中，最大节点的度数随着N的增大而呈线性增大(或变慢)，因此即使在一个非常大的随机网络中，枢纽节点的数量也会很小。&lt;br /&gt;
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* (b) A new node in a scale-free network has a tendency to link to a node with a higher degree, compared to a new node in a random network which links itself to a random node. This process is called [[preferential attachment]]. The tendency of a new node to link to a node with a high degree ''k'' is characterized by [[Power law|power-law distribution]] (also known as rich-gets-richer process). This idea was introduced by [[Vilfredo Pareto]] and it explained why a small percentage of the population earns most of the money. This process is present in networks as well, for example 80 percent of web links point to 15 percent of webpages. The emergence of scale-free networks is not typical only of networks created by human action, but also of such networks as metabolic networks or illness networks.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 8.[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf]&amp;lt;/ref&amp;gt; This phenomenon may be explained by the example of hubs on the World Wide Web such as Facebook or Google. These webpages are very well known and therefore the tendency of other webpages pointing to them is much higher than linking to random small webpages.&lt;br /&gt;
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无标度网络中的新节点，与随机网络中的新节点相比，更倾向于连接到具有更高度的节点。这个过程称为优先依附。新节点连接到高度值''k''节点的趋势可被看做幂律分布(也被称作，越富有就越富有的过程)。这个想法是由Vilfredo Pareto提出的，它解释了为什么一小部分人赚了大部分的钱。这个过程也存在于网络中，例如80%的网络链接指向15%的网页。无标度网络的不仅是符合的人类活动网络，也可以拟合包括代谢网络或疾病网络。这种现象可以用像Facebook或谷歌这样的万维网中心的例子来解释。这些网页是非常有名的，因此其他网页指向他们的倾向比指向到随机的小网页要高得多。&lt;br /&gt;
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The mathematical explanation for [[Barabási–Albert model]]:&lt;br /&gt;
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The mathematical explanation for Barabási–Albert model:&lt;br /&gt;
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Barabási-Albert 模型的数学解释:&lt;br /&gt;
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[[File:Barabasi Albert model.gif|thumb|300px|&lt;br /&gt;
图4：The steps of the growth of the network according to the Barabasi–Albert model (&amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt;) &lt;br /&gt;
根据 Barabasi-Albert 模型 &amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt; 分析了网络生长的步骤]]&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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这个网络从一个初始的由&amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt;个节点构成的连接网络开始。&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&amp;lt;ref name=&amp;quot;RMP&amp;quot;/&amp;gt;&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&lt;br /&gt;
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每次向网络添加一个新节点。每个新的节点都被连接到&amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt;现有的节点，其概率与现有节点已经拥有的链接数量成正比。形式上，新节点与节点&amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt;相连的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
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where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
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where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
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其中&amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt;是节点&amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt;的度数，这个和取自所有预先存在的节点&amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt;。（分母的结果是当前网络中边的两倍）。&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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网络中枢纽节点的涌现也与时间有关。在无标度网络中，较早出现的节点比后来者更有可能成为枢纽。这种现象被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;先发优势 First-Mover Advantage&amp;lt;/font&amp;gt;'''，它解释了为什么一些节点成为枢纽，而一些没有。然而，在一个真实的网络中，出现的时间并不是影响枢纽规模的唯一因素。例如，在谷歌成为全球最大的互联网中心8年后，Facebook 出现了，然而在2011年，Facebook 成为了全球最大的互联网中心。因此，在实际网络中，枢纽的增长和规模也取决于各种属性，如节点的流行程度、质量或老化程度。&lt;br /&gt;
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== Attributes 属性==&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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无标度网络中的枢纽节点有几个属性&lt;br /&gt;
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=== Shortening the path lengths in a network 缩短网络中的路径距离===&lt;br /&gt;
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The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 23.[http://barabasi.com/networksciencebook/content/book_chapter_4.pdf]&amp;lt;/ref&amp;gt; Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
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The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes. Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
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网络中可观察到的枢纽节点越多，节点之间的距离就越小。在无标度网络中，枢纽是连接度小的节点的桥梁。由于无标度网络中随机节点间的距离很小，我们称无标度网络为“小”或“超小”网络。虽然在一个非常小的网络中，路径距离的差异可能并不明显，但是在大型随机网络和无标度网络中，路径距离的差异是显著的。&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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无标度网络中的平均路径长度:&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点的老化 Aging of hubs (nodes)&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_6.pdf Barabási, Albert-László. ''Network Science: Evolving Networks''., p. 3]&amp;lt;/ref&amp;gt; The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).{{Citation needed|date=May 2016}}&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks. The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).&lt;br /&gt;
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这种现象在真实网络中有所出现，当旧的枢纽被隐藏在网络中时。这种现象导致了网络演化和拓扑结构的变化。老化现象的例子可参照 Facebook 超越了网络中最大的中心 Google (自2000年以来最大的节点)。&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;鲁棒性和攻击容忍度 Robustness and Attack Tolerance&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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随机故障的节点或定向攻击网络关键组成部分的枢纽。在网络枢纽节点的随机失效过程中，网络的鲁棒性是由节点的随机失效决定的。&amp;lt; ref &amp;gt; { cite journal | title = Resilience of the Internet to Random breaks | &lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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日记 = 体育。牧师。李特 | &lt;br /&gt;
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year=2000|&lt;br /&gt;
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2000年&lt;br /&gt;
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1 = Reoven | &lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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最后 = Cohen | &lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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首先2 = k。2 = Erez | first3 = d.3 = ben-Avraham | first4 = s | last4 = Havlin | authorlink4 = Shlomo Havlin | &lt;br /&gt;
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volume=85|&lt;br /&gt;
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85 | &lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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页数 = 4626-8&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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Doi = 10.1103/physrvlett. 85.4626 | &lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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Http://link.aps.org/doi/10.1103/physrevlett.85.4626&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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| bibcode = 2000PhRvL. . 85.4626 c | arxiv = cond-mat/0007048 | pmid = 11082612}&lt;br /&gt;
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&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
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由于枢纽与大量小度节点共存，随机故障删除它的几率很小。而去除小节点对网络的完整性影响不大。即使随机删除会作用到上枢纽，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络碎片化 Fragmantation of Network&amp;lt;/font&amp;gt;'''的可能性也非常小，因为剩余的枢纽仍能将网络连接在一起。在这种情况下，枢纽就是无标度网络的中坚力量。&lt;br /&gt;
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During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system.&amp;lt;ref&amp;gt;{{Cite journal|author1=S. V. Buldyrev |author2=R. Parshani |author3=G. Paul |author4=H. E. Stanley |author5=S. Havlin |title = Catastrophic cascade of failures in interdependent networks|journal = Nature |volume = 464 |pages = 1025–28 |year = 2010 | doi=10.1038/nature08932 |issue=7291|url=http://havlin.biu.ac.il/Publications.php?keyword=Catastrophic+cascade+of+failures+in+interdependent+networks&amp;amp;year=*&amp;amp;match=all |pmid=20393559|arxiv=1012.0206 |bibcode=2010Natur.464.1025B }}&amp;lt;/ref&amp;gt; On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
&lt;br /&gt;
During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system. On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
&lt;br /&gt;
在对枢纽进行有针对性的攻击时，网络的完整性会相对较快地崩溃。由于小节点主要连接到枢纽节点，对最大枢纽的攻击会在短时间内破坏网络。2008年的金融市场崩溃就是这种网络失灵的一个例子，当时最大的参与者(枢纽)的破产导致了整个系统的持续崩溃。另一方面，去除恐怖主义网络中的枢纽节点可能会产生积极的影响，目标节点的消失可能会摧毁整个恐怖组织。网络的攻击容忍度可以通过连接其外围节点来提高，但是它需要加倍的链接数量。&lt;br /&gt;
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=== Degree correlation 度相关===&lt;br /&gt;
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The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
&lt;br /&gt;
The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全度相关 Perfect Degree Correlation&amp;lt;/font&amp;gt;'''意味着每个度值为k的节点只连接到相同度值的节点。节点之间的这种连通性决定了网络的拓扑结构，从而影响了网络的鲁棒性。如果枢纽节点之间的链接数量与偶然预期的相同，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中性网络 Neutral Network&amp;lt;/font&amp;gt;'''。如果枢纽节点倾向于相互连接，同时避免链接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;选型网络 Assortative Network&amp;lt;/font&amp;gt;'''。这个网络抵抗攻击的能力相对较强，因为枢纽节点形成了一个核心组，这对枢纽节点的移除来说是更加冗余的。如果枢纽节点避免相互连接，同时连接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;非选型网络 Disassortative Network&amp;lt;/font&amp;gt;'''。这个网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中心辐射特征 Hub-and-Spoke Character&amp;lt;/font&amp;gt;'''。因此，如果我们删除这种类型网络中的枢纽节点，可能会破坏或摧毁整个网络。&lt;br /&gt;
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==='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;传播效应 Spreading phenomenon&amp;lt;/font&amp;gt;'''  ===&lt;br /&gt;
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The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction &amp;lt;ref&amp;gt;{{Cite journal|first1=Duygu |last1=Balcan |first2=Hao |last2=Hu |first3=Bruno |last3=Goncalves |first4=Paolo |last4=Bajardi |first5=Chiara |last5=Poletto |first6=Jose J.|last6=Ramasco|first7=Daniela|last7=Paolotti|first8=Nicola|last8=Perra |first9=Michele |last9=Tizzoni |first10=Wouter |last10=Van den Broeck|first11=Vittoria |last11=Colizza|first12=Alessandro |last12=Vespignani|date=14 September 2009|title=Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility|journal=BMC Medicine |volume=7 |issue=45 |page=29 |doi=10.1186/1741-7015-7-45 |id= |pmid=19744314 |pmc=2755471 |arxiv=0909.2417}}&amp;lt;/ref&amp;gt; may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
&lt;br /&gt;
The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction  may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
&lt;br /&gt;
这些枢纽节点还负责有效地在网络上传播材料。在疾病传播或信息流的分析中，枢纽被称为超级传播者。超级传播者可能会产生积极的影响，如有效的信息流动，但在 H1N1或艾滋病等流行病传播的情况下也会产生毁灭性的影响。数学模型，如 H1H1流行病预测模型，基于人类流动网络，传染性，或人与人之间的社会互动，可以让我们预测疾病的传播。在根除疾病方面，枢纽也很重要。在无标度网络中，枢纽是最有可能被感染的，因为他们有大量的连接。在枢纽被感染后，它将疾病广播到它所链接的节点。因此，选择性免疫那些枢纽节点可能是消灭传播性疾病的成本效益策略。&lt;br /&gt;
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== References ==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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[[Category:Network theory]]&lt;br /&gt;
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Category:Network theory&lt;br /&gt;
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范畴: 网络理论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Hub (network science)]]. Its edit history can be viewed at [[枢纽节点/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13390</id>
		<title>枢纽</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13390"/>
		<updated>2020-08-28T06:08:05Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{redirect|Network hub|the Ethernet technology|Ethernet hub}}&lt;br /&gt;
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In [[network science]], a '''hub''' is a [[Node (networking)|node]] with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf Barabási, Albert-László. ''Network Science: Graph Theory''., p. 27]&amp;lt;/ref&amp;gt; While hubs cannot be observed in a random network, they are expected to emerge in [[scale-free network]]s. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the [[network topology]]. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the network topology. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络科学 Network Science&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点 Hub&amp;lt;/font&amp;gt;'''是链接数大大超过平均值的节点。枢纽节点是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Property of Networks&amp;lt;/font&amp;gt;'''的重要特性。虽然在随机网络中不存在枢纽节点，但是它们却存在于无标度网络中。无标度网络中枢纽节点的涌现与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;幂律分布 Power-Law Distribution&amp;lt;/font&amp;gt;'''有关，其对网络的拓扑结构有着重要的影响。枢纽节点可以在很多真实的网络中找到，比如大脑网络或者互联网。&lt;br /&gt;
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[[File:Network representation of brain connectivity.JPG|thumb|right|&lt;br /&gt;
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图1：Network representation of brain connectivity. Hubs are highlighted 表示大脑连接性的网络，其中枢纽节点被突出显示]]&lt;br /&gt;
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[[File:Internet map 4096.png|thumb|150px|right|&lt;br /&gt;
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图2:Partial map of the Internet based on the January 15, 2005. Hubs are highlighted 2005年1月15日的因特网局部图，枢纽节点被突出显示]]&lt;br /&gt;
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A hub is a component of a network with a high-degree [[Vertex (graph theory)|node]]. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links ([[Degree (graph theory)|degrees]]) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size ''N'' of the network and average degree ''&amp;lt;k&amp;gt;'' constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree ''k'' is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree ''k'' while the other nodes have a small number of links.&lt;br /&gt;
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A hub is a component of a network with a high-degree node. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links (degrees) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size N of the network and average degree &amp;lt;k&amp;gt; constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree k is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree k while the other nodes have a small number of links.&lt;br /&gt;
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一个枢纽节点是网络中具有高度值的组件。与网络中的其他节点相比，枢纽节点的链接数要大得多。无尺度网络中枢纽的链接数(度)远远高于随机网络中链接数最大的节点，在保持网络的大小''N''和平均度 ''&amp;lt;k&amp;gt;''不变的情况下。枢纽的存在是随机网络和无标度网络的最大区别。在随机网络中，每个节点的度''k''是可比的，因此不可能出现枢纽节点。在无标度网络中，少数节点(即枢纽节点)具有高度值 ''k''，而其他节点只有少量的链接。&lt;br /&gt;
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== Emergence 涌现==&lt;br /&gt;
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[[Image:Scale-free network sample.png|thumb|Example of a random network and a scale-free network|400px|right|&lt;br /&gt;
图3：Random network (a) and scale-free network (b). In the scale-free network, the larger hubs are highlighted. 随机网络（a）和无标度网络（b），在无标度网络中，大型枢纽被突出显示]]&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks ([[Barabási–Albert model]]) are different from random networks ([[Erdős–Rényi model]]) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks (Barabási–Albert model) are different from random networks (Erdős–Rényi model) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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枢纽节点的出现可以用无标度网络和随机网络的区别来解释。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks Barabási-Albert model&amp;lt;/font&amp;gt;'''与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机网络 Random Networks Erdős–Rényi model&amp;lt;/font&amp;gt;'''在两个方面有所不同: (a)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;增长 Growth&amp;lt;/font&amp;gt;'''，(b)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;优先连接 Preferential Attachment&amp;lt;/font&amp;gt;'''。{ Cite journal&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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Http://www.nd.edu/~networks/publication%20categories/03%20journal%20articles/physics/statisticalmechanics_rev%20of%20modern%20physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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1 = r é ka Albert | first1 = r é ka | last1 = Albert&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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2 = albert-lászló Barabási | first2 = albert-lászló | last2 = Barabási&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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| title = 复杂网络的统计力学&lt;br /&gt;
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 | journal = [[Reviews of Modern Physics]]&lt;br /&gt;
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 | journal = Reviews of Modern Physics&lt;br /&gt;
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| 杂志 = 现代物理学评论&lt;br /&gt;
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 | volume = 74&lt;br /&gt;
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 | pages = 47–97&lt;br /&gt;
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| 页数 = 47-97&lt;br /&gt;
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 | year = 2002&lt;br /&gt;
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2002年&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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10.1103/RevModPhys. 74.47&lt;br /&gt;
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 | bibcode=2002RvMP...74...47A&lt;br /&gt;
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2002RvMP... 74... 47A&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* (a) Scale-free networks assume a continuous growth of the number of nodes ''N'', compared to random networks which assume a fixed number of nodes. In scale-free networks the degree of the largest hub rises polynomially with the size of the network. Therefore, the degree of a hub can be high in a scale-free network. In random networks the degree of the largest node rises logaritmically (or slower) with N, thus the hub number will be small even in a very large network.&lt;br /&gt;
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无标度网络假设节点数“N”持续增长，而随机网络假设节点数为固定数目。在无标度网络中，最大枢纽节点的度数随着网络的大小而多项式地上升。因此，在无标度网络中，枢纽节点的度数可以很高。在随机网络中，最大节点的度数随着N的增大而呈线性增大(或变慢)，因此即使在一个非常大的随机网络中，枢纽节点的数量也会很小。&lt;br /&gt;
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* (b) A new node in a scale-free network has a tendency to link to a node with a higher degree, compared to a new node in a random network which links itself to a random node. This process is called [[preferential attachment]]. The tendency of a new node to link to a node with a high degree ''k'' is characterized by [[Power law|power-law distribution]] (also known as rich-gets-richer process). This idea was introduced by [[Vilfredo Pareto]] and it explained why a small percentage of the population earns most of the money. This process is present in networks as well, for example 80 percent of web links point to 15 percent of webpages. The emergence of scale-free networks is not typical only of networks created by human action, but also of such networks as metabolic networks or illness networks.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 8.[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf]&amp;lt;/ref&amp;gt; This phenomenon may be explained by the example of hubs on the World Wide Web such as Facebook or Google. These webpages are very well known and therefore the tendency of other webpages pointing to them is much higher than linking to random small webpages.&lt;br /&gt;
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无标度网络中的新节点，与随机网络中的新节点相比，更倾向于连接到具有更高度的节点。这个过程称为优先依附。新节点连接到高度值''k''节点的趋势可被看做幂律分布(也被称作，越富有就越富有的过程)。这个想法是由Vilfredo Pareto提出的，它解释了为什么一小部分人赚了大部分的钱。这个过程也存在于网络中，例如80%的网络链接指向15%的网页。无标度网络的不仅是符合的人类活动网络，也可以拟合包括代谢网络或疾病网络。这种现象可以用像Facebook或谷歌这样的万维网中心的例子来解释。这些网页是非常有名的，因此其他网页指向他们的倾向比指向到随机的小网页要高得多。&lt;br /&gt;
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The mathematical explanation for [[Barabási–Albert model]]:&lt;br /&gt;
&lt;br /&gt;
The mathematical explanation for Barabási–Albert model:&lt;br /&gt;
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Barabási-Albert 模型的数学解释:&lt;br /&gt;
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[[File:Barabasi Albert model.gif|thumb|300px|&lt;br /&gt;
图4：The steps of the growth of the network according to the Barabasi–Albert model (&amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt;) 根据 Barabasi-Albert 模型 &amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt; 分析了网络生长的步骤]]&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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这个网络从一个初始的由&amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt;个节点构成的连接网络开始。&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&amp;lt;ref name=&amp;quot;RMP&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&lt;br /&gt;
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每次向网络添加一个新节点。每个新的节点都被连接到&amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt;现有的节点，其概率与现有节点已经拥有的链接数量成正比。形式上，新节点与节点&amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt;相连的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&lt;br /&gt;
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where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
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其中&amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt;是节点&amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt;的度数，这个和取自所有预先存在的节点&amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt;。（分母的结果是当前网络中边的两倍）。&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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网络中枢纽节点的涌现也与时间有关。在无标度网络中，较早出现的节点比后来者更有可能成为枢纽。这种现象被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;先发优势 First-Mover Advantage&amp;lt;/font&amp;gt;'''，它解释了为什么一些节点成为枢纽，而一些没有。然而，在一个真实的网络中，出现的时间并不是影响枢纽规模的唯一因素。例如，在谷歌成为全球最大的互联网中心8年后，Facebook 出现了，然而在2011年，Facebook 成为了全球最大的互联网中心。因此，在实际网络中，枢纽的增长和规模也取决于各种属性，如节点的流行程度、质量或老化程度。&lt;br /&gt;
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== Attributes 属性==&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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无标度网络中的枢纽节点有几个属性&lt;br /&gt;
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=== Shortening the path lengths in a network 缩短网络中的路径距离===&lt;br /&gt;
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The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 23.[http://barabasi.com/networksciencebook/content/book_chapter_4.pdf]&amp;lt;/ref&amp;gt; Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
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The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes. Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
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网络中可观察到的枢纽节点越多，节点之间的距离就越小。在无标度网络中，枢纽是连接度小的节点的桥梁。由于无标度网络中随机节点间的距离很小，我们称无标度网络为“小”或“超小”网络。虽然在一个非常小的网络中，路径距离的差异可能并不明显，但是在大型随机网络和无标度网络中，路径距离的差异是显著的。&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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无标度网络中的平均路径长度:&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点的老化 Aging of hubs (nodes)&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_6.pdf Barabási, Albert-László. ''Network Science: Evolving Networks''., p. 3]&amp;lt;/ref&amp;gt; The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).{{Citation needed|date=May 2016}}&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks. The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).&lt;br /&gt;
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这种现象在真实网络中有所出现，当旧的枢纽被隐藏在网络中时。这种现象导致了网络演化和拓扑结构的变化。老化现象的例子可参照 Facebook 超越了网络中最大的中心 Google (自2000年以来最大的节点)。&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;鲁棒性和攻击容忍度 Robustness and Attack Tolerance&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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随机故障的节点或定向攻击网络关键组成部分的枢纽。在网络枢纽节点的随机失效过程中，网络的鲁棒性是由节点的随机失效决定的。&amp;lt; ref &amp;gt; { cite journal | title = Resilience of the Internet to Random breaks | &lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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日记 = 体育。牧师。李特 | &lt;br /&gt;
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year=2000|&lt;br /&gt;
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year=2000|&lt;br /&gt;
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2000年&lt;br /&gt;
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first=Reoven|&lt;br /&gt;
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first=Reoven|&lt;br /&gt;
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1 = Reoven | &lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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最后 = Cohen | &lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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首先2 = k。2 = Erez | first3 = d.3 = ben-Avraham | first4 = s | last4 = Havlin | authorlink4 = Shlomo Havlin | &lt;br /&gt;
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volume=85|&lt;br /&gt;
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volume=85|&lt;br /&gt;
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85 | &lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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页数 = 4626-8&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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Doi = 10.1103/physrvlett. 85.4626 | &lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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Http://link.aps.org/doi/10.1103/physrevlett.85.4626&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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| bibcode = 2000PhRvL. . 85.4626 c | arxiv = cond-mat/0007048 | pmid = 11082612}&lt;br /&gt;
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&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
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&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
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由于枢纽与大量小度节点共存，随机故障删除它的几率很小。而去除小节点对网络的完整性影响不大。即使随机删除会作用到上枢纽，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络碎片化 Fragmantation of Network&amp;lt;/font&amp;gt;'''的可能性也非常小，因为剩余的枢纽仍能将网络连接在一起。在这种情况下，枢纽就是无标度网络的中坚力量。&lt;br /&gt;
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During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system.&amp;lt;ref&amp;gt;{{Cite journal|author1=S. V. Buldyrev |author2=R. Parshani |author3=G. Paul |author4=H. E. Stanley |author5=S. Havlin |title = Catastrophic cascade of failures in interdependent networks|journal = Nature |volume = 464 |pages = 1025–28 |year = 2010 | doi=10.1038/nature08932 |issue=7291|url=http://havlin.biu.ac.il/Publications.php?keyword=Catastrophic+cascade+of+failures+in+interdependent+networks&amp;amp;year=*&amp;amp;match=all |pmid=20393559|arxiv=1012.0206 |bibcode=2010Natur.464.1025B }}&amp;lt;/ref&amp;gt; On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
&lt;br /&gt;
During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system. On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
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在对枢纽进行有针对性的攻击时，网络的完整性会相对较快地崩溃。由于小节点主要连接到枢纽节点，对最大枢纽的攻击会在短时间内破坏网络。2008年的金融市场崩溃就是这种网络失灵的一个例子，当时最大的参与者(枢纽)的破产导致了整个系统的持续崩溃。另一方面，去除恐怖主义网络中的枢纽节点可能会产生积极的影响，目标节点的消失可能会摧毁整个恐怖组织。网络的攻击容忍度可以通过连接其外围节点来提高，但是它需要加倍的链接数量。&lt;br /&gt;
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=== Degree correlation 度相关===&lt;br /&gt;
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The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
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The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全度相关 Perfect Degree Correlation&amp;lt;/font&amp;gt;'''意味着每个度值为k的节点只连接到相同度值的节点。节点之间的这种连通性决定了网络的拓扑结构，从而影响了网络的鲁棒性。如果枢纽节点之间的链接数量与偶然预期的相同，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中性网络 Neutral Network&amp;lt;/font&amp;gt;'''。如果枢纽节点倾向于相互连接，同时避免链接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;选型网络 Assortative Network&amp;lt;/font&amp;gt;'''。这个网络抵抗攻击的能力相对较强，因为枢纽节点形成了一个核心组，这对枢纽节点的移除来说是更加冗余的。如果枢纽节点避免相互连接，同时连接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;非选型网络 Disassortative Network&amp;lt;/font&amp;gt;'''。这个网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中心辐射特征 Hub-and-Spoke Character&amp;lt;/font&amp;gt;'''。因此，如果我们删除这种类型网络中的枢纽节点，可能会破坏或摧毁整个网络。&lt;br /&gt;
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==='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;传播效应 Spreading phenomenon&amp;lt;/font&amp;gt;'''  ===&lt;br /&gt;
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The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction &amp;lt;ref&amp;gt;{{Cite journal|first1=Duygu |last1=Balcan |first2=Hao |last2=Hu |first3=Bruno |last3=Goncalves |first4=Paolo |last4=Bajardi |first5=Chiara |last5=Poletto |first6=Jose J.|last6=Ramasco|first7=Daniela|last7=Paolotti|first8=Nicola|last8=Perra |first9=Michele |last9=Tizzoni |first10=Wouter |last10=Van den Broeck|first11=Vittoria |last11=Colizza|first12=Alessandro |last12=Vespignani|date=14 September 2009|title=Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility|journal=BMC Medicine |volume=7 |issue=45 |page=29 |doi=10.1186/1741-7015-7-45 |id= |pmid=19744314 |pmc=2755471 |arxiv=0909.2417}}&amp;lt;/ref&amp;gt; may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
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The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction  may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
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这些枢纽节点还负责有效地在网络上传播材料。在疾病传播或信息流的分析中，枢纽被称为超级传播者。超级传播者可能会产生积极的影响，如有效的信息流动，但在 H1N1或艾滋病等流行病传播的情况下也会产生毁灭性的影响。数学模型，如 H1H1流行病预测模型，基于人类流动网络，传染性，或人与人之间的社会互动，可以让我们预测疾病的传播。在根除疾病方面，枢纽也很重要。在无标度网络中，枢纽是最有可能被感染的，因为他们有大量的连接。在枢纽被感染后，它将疾病广播到它所链接的节点。因此，选择性免疫那些枢纽节点可能是消灭传播性疾病的成本效益策略。&lt;br /&gt;
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== References ==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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[[Category:Network theory]]&lt;br /&gt;
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Category:Network theory&lt;br /&gt;
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范畴: 网络理论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Hub (network science)]]. Its edit history can be viewed at [[枢纽节点/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13389</id>
		<title>枢纽</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13389"/>
		<updated>2020-08-28T06:05:41Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{redirect|Network hub|the Ethernet technology|Ethernet hub}}&lt;br /&gt;
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In [[network science]], a '''hub''' is a [[Node (networking)|node]] with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf Barabási, Albert-László. ''Network Science: Graph Theory''., p. 27]&amp;lt;/ref&amp;gt; While hubs cannot be observed in a random network, they are expected to emerge in [[scale-free network]]s. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the [[network topology]]. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the network topology. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络科学 Network Science&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点 Hub&amp;lt;/font&amp;gt;'''是链接数大大超过平均值的节点。枢纽节点是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Property of Networks&amp;lt;/font&amp;gt;'''的重要特性。虽然在随机网络中不存在枢纽节点，但是它们却存在于无标度网络中。无标度网络中枢纽节点的涌现与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;幂律分布 Power-Law Distribution&amp;lt;/font&amp;gt;'''有关，其对网络的拓扑结构有着重要的影响。枢纽节点可以在很多真实的网络中找到，比如大脑网络或者互联网。&lt;br /&gt;
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[[File:Network representation of brain connectivity.JPG|thumb|&lt;br /&gt;
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图1：Network representation of brain connectivity. Hubs are highlighted|150px|right] 表示大脑连接性的网络，其中枢纽节点被突出显示]]&lt;br /&gt;
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[[File:Internet map 4096.png|thumb|Partial map of the Internet based on the January 15, 2005. Hubs are highlighted|150px|right]&lt;br /&gt;
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图2:2005年1月15日的因特网局部图，枢纽节点被突出显示]]&lt;br /&gt;
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A hub is a component of a network with a high-degree [[Vertex (graph theory)|node]]. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links ([[Degree (graph theory)|degrees]]) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size ''N'' of the network and average degree ''&amp;lt;k&amp;gt;'' constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree ''k'' is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree ''k'' while the other nodes have a small number of links.&lt;br /&gt;
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A hub is a component of a network with a high-degree node. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links (degrees) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size N of the network and average degree &amp;lt;k&amp;gt; constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree k is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree k while the other nodes have a small number of links.&lt;br /&gt;
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一个枢纽节点是网络中具有高度值的组件。与网络中的其他节点相比，枢纽节点的链接数要大得多。无尺度网络中枢纽的链接数(度)远远高于随机网络中链接数最大的节点，在保持网络的大小''N''和平均度 ''&amp;lt;k&amp;gt;''不变的情况下。枢纽的存在是随机网络和无标度网络的最大区别。在随机网络中，每个节点的度''k''是可比的，因此不可能出现枢纽节点。在无标度网络中，少数节点(即枢纽节点)具有高度值 ''k''，而其他节点只有少量的链接。&lt;br /&gt;
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== Emergence 涌现==&lt;br /&gt;
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[[Image:Scale-free network sample.png|thumb|Example of a random network and a scale-free network|400px|right|&lt;br /&gt;
图3：Random network (a) and scale-free network (b). In the scale-free network, the larger hubs are highlighted. 随机网络（a）和无标度网络（b），在无标度网络中，大型枢纽被突出显示]]&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks ([[Barabási–Albert model]]) are different from random networks ([[Erdős–Rényi model]]) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks (Barabási–Albert model) are different from random networks (Erdős–Rényi model) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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枢纽节点的出现可以用无标度网络和随机网络的区别来解释。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks Barabási-Albert model&amp;lt;/font&amp;gt;'''与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机网络 Random Networks Erdős–Rényi model&amp;lt;/font&amp;gt;'''在两个方面有所不同: (a)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;增长 Growth&amp;lt;/font&amp;gt;'''，(b)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;优先连接 Preferential Attachment&amp;lt;/font&amp;gt;'''。{ Cite journal&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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Http://www.nd.edu/~networks/publication%20categories/03%20journal%20articles/physics/statisticalmechanics_rev%20of%20modern%20physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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1 = r é ka Albert | first1 = r é ka | last1 = Albert&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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2 = albert-lászló Barabási | first2 = albert-lászló | last2 = Barabási&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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| title = 复杂网络的统计力学&lt;br /&gt;
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 | journal = [[Reviews of Modern Physics]]&lt;br /&gt;
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 | journal = Reviews of Modern Physics&lt;br /&gt;
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| 杂志 = 现代物理学评论&lt;br /&gt;
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 | volume = 74&lt;br /&gt;
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 | pages = 47–97&lt;br /&gt;
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 | pages = 47–97&lt;br /&gt;
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| 页数 = 47-97&lt;br /&gt;
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 | year = 2002&lt;br /&gt;
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2002年&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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10.1103/RevModPhys. 74.47&lt;br /&gt;
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 | bibcode=2002RvMP...74...47A&lt;br /&gt;
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2002RvMP... 74... 47A&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* (a) Scale-free networks assume a continuous growth of the number of nodes ''N'', compared to random networks which assume a fixed number of nodes. In scale-free networks the degree of the largest hub rises polynomially with the size of the network. Therefore, the degree of a hub can be high in a scale-free network. In random networks the degree of the largest node rises logaritmically (or slower) with N, thus the hub number will be small even in a very large network.&lt;br /&gt;
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无标度网络假设节点数“N”持续增长，而随机网络假设节点数为固定数目。在无标度网络中，最大枢纽节点的度数随着网络的大小而多项式地上升。因此，在无标度网络中，枢纽节点的度数可以很高。在随机网络中，最大节点的度数随着N的增大而呈线性增大(或变慢)，因此即使在一个非常大的随机网络中，枢纽节点的数量也会很小。&lt;br /&gt;
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* (b) A new node in a scale-free network has a tendency to link to a node with a higher degree, compared to a new node in a random network which links itself to a random node. This process is called [[preferential attachment]]. The tendency of a new node to link to a node with a high degree ''k'' is characterized by [[Power law|power-law distribution]] (also known as rich-gets-richer process). This idea was introduced by [[Vilfredo Pareto]] and it explained why a small percentage of the population earns most of the money. This process is present in networks as well, for example 80 percent of web links point to 15 percent of webpages. The emergence of scale-free networks is not typical only of networks created by human action, but also of such networks as metabolic networks or illness networks.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 8.[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf]&amp;lt;/ref&amp;gt; This phenomenon may be explained by the example of hubs on the World Wide Web such as Facebook or Google. These webpages are very well known and therefore the tendency of other webpages pointing to them is much higher than linking to random small webpages.&lt;br /&gt;
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无标度网络中的新节点，与随机网络中的新节点相比，更倾向于连接到具有更高度的节点。这个过程称为优先依附。新节点连接到高度值''k''节点的趋势可被看做幂律分布(也被称作，越富有就越富有的过程)。这个想法是由Vilfredo Pareto提出的，它解释了为什么一小部分人赚了大部分的钱。这个过程也存在于网络中，例如80%的网络链接指向15%的网页。无标度网络的不仅是符合的人类活动网络，也可以拟合包括代谢网络或疾病网络。这种现象可以用像Facebook或谷歌这样的万维网中心的例子来解释。这些网页是非常有名的，因此其他网页指向他们的倾向比指向到随机的小网页要高得多。&lt;br /&gt;
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The mathematical explanation for [[Barabási–Albert model]]:&lt;br /&gt;
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The mathematical explanation for Barabási–Albert model:&lt;br /&gt;
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Barabási-Albert 模型的数学解释:&lt;br /&gt;
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[[File:Barabasi Albert model.gif|thumb|300px|&lt;br /&gt;
图4：The steps of the growth of the network according to the Barabasi–Albert model (&amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt;) 根据 Barabasi-Albert 模型 &amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt; 分析了网络生长的步骤]]&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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这个网络从一个初始的由&amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt;个节点构成的连接网络开始。&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&amp;lt;ref name=&amp;quot;RMP&amp;quot;/&amp;gt;&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&lt;br /&gt;
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每次向网络添加一个新节点。每个新的节点都被连接到&amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt;现有的节点，其概率与现有节点已经拥有的链接数量成正比。形式上，新节点与节点&amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt;相连的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
&lt;br /&gt;
其中&amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt;是节点&amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt;的度数，这个和取自所有预先存在的节点&amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt;。（分母的结果是当前网络中边的两倍）。&lt;br /&gt;
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&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
&lt;br /&gt;
Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
&lt;br /&gt;
网络中枢纽节点的涌现也与时间有关。在无标度网络中，较早出现的节点比后来者更有可能成为枢纽。这种现象被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;先发优势 First-Mover Advantage&amp;lt;/font&amp;gt;'''，它解释了为什么一些节点成为枢纽，而一些没有。然而，在一个真实的网络中，出现的时间并不是影响枢纽规模的唯一因素。例如，在谷歌成为全球最大的互联网中心8年后，Facebook 出现了，然而在2011年，Facebook 成为了全球最大的互联网中心。因此，在实际网络中，枢纽的增长和规模也取决于各种属性，如节点的流行程度、质量或老化程度。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Attributes 属性==&lt;br /&gt;
&lt;br /&gt;
There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
&lt;br /&gt;
There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
&lt;br /&gt;
无标度网络中的枢纽节点有几个属性&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
=== Shortening the path lengths in a network 缩短网络中的路径距离===&lt;br /&gt;
&lt;br /&gt;
The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 23.[http://barabasi.com/networksciencebook/content/book_chapter_4.pdf]&amp;lt;/ref&amp;gt; Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
&lt;br /&gt;
The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes. Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
&lt;br /&gt;
网络中可观察到的枢纽节点越多，节点之间的距离就越小。在无标度网络中，枢纽是连接度小的节点的桥梁。由于无标度网络中随机节点间的距离很小，我们称无标度网络为“小”或“超小”网络。虽然在一个非常小的网络中，路径距离的差异可能并不明显，但是在大型随机网络和无标度网络中，路径距离的差异是显著的。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Average path length in scale-free networks:&lt;br /&gt;
&lt;br /&gt;
Average path length in scale-free networks:&lt;br /&gt;
&lt;br /&gt;
无标度网络中的平均路径长度:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点的老化 Aging of hubs (nodes)&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
&lt;br /&gt;
The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_6.pdf Barabási, Albert-László. ''Network Science: Evolving Networks''., p. 3]&amp;lt;/ref&amp;gt; The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).{{Citation needed|date=May 2016}}&lt;br /&gt;
&lt;br /&gt;
The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks. The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).&lt;br /&gt;
&lt;br /&gt;
这种现象在真实网络中有所出现，当旧的枢纽被隐藏在网络中时。这种现象导致了网络演化和拓扑结构的变化。老化现象的例子可参照 Facebook 超越了网络中最大的中心 Google (自2000年以来最大的节点)。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;鲁棒性和攻击容忍度 Robustness and Attack Tolerance&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
&lt;br /&gt;
During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
&lt;br /&gt;
During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
&lt;br /&gt;
随机故障的节点或定向攻击网络关键组成部分的枢纽。在网络枢纽节点的随机失效过程中，网络的鲁棒性是由节点的随机失效决定的。&amp;lt; ref &amp;gt; { cite journal | title = Resilience of the Internet to Random breaks | &lt;br /&gt;
&lt;br /&gt;
journal=Phys. Rev. Lett.|&lt;br /&gt;
&lt;br /&gt;
journal=Phys. Rev. Lett.|&lt;br /&gt;
&lt;br /&gt;
日记 = 体育。牧师。李特 | &lt;br /&gt;
&lt;br /&gt;
year=2000|&lt;br /&gt;
&lt;br /&gt;
year=2000|&lt;br /&gt;
&lt;br /&gt;
2000年&lt;br /&gt;
&lt;br /&gt;
first=Reoven|&lt;br /&gt;
&lt;br /&gt;
first=Reoven|&lt;br /&gt;
&lt;br /&gt;
1 = Reoven | &lt;br /&gt;
&lt;br /&gt;
last=Cohen|&lt;br /&gt;
&lt;br /&gt;
last=Cohen|&lt;br /&gt;
&lt;br /&gt;
最后 = Cohen | &lt;br /&gt;
&lt;br /&gt;
first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
&lt;br /&gt;
first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
&lt;br /&gt;
首先2 = k。2 = Erez | first3 = d.3 = ben-Avraham | first4 = s | last4 = Havlin | authorlink4 = Shlomo Havlin | &lt;br /&gt;
&lt;br /&gt;
volume=85|&lt;br /&gt;
&lt;br /&gt;
volume=85|&lt;br /&gt;
&lt;br /&gt;
85 | &lt;br /&gt;
&lt;br /&gt;
pages=4626–8|&lt;br /&gt;
&lt;br /&gt;
pages=4626–8|&lt;br /&gt;
&lt;br /&gt;
页数 = 4626-8&lt;br /&gt;
&lt;br /&gt;
doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
&lt;br /&gt;
doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
&lt;br /&gt;
Doi = 10.1103/physrvlett. 85.4626 | &lt;br /&gt;
&lt;br /&gt;
url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
&lt;br /&gt;
url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
&lt;br /&gt;
Http://link.aps.org/doi/10.1103/physrevlett.85.4626&lt;br /&gt;
&lt;br /&gt;
|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
&lt;br /&gt;
|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
&lt;br /&gt;
| bibcode = 2000PhRvL. . 85.4626 c | arxiv = cond-mat/0007048 | pmid = 11082612}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
&lt;br /&gt;
由于枢纽与大量小度节点共存，随机故障删除它的几率很小。而去除小节点对网络的完整性影响不大。即使随机删除会作用到上枢纽，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络碎片化 Fragmantation of Network&amp;lt;/font&amp;gt;'''的可能性也非常小，因为剩余的枢纽仍能将网络连接在一起。在这种情况下，枢纽就是无标度网络的中坚力量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system.&amp;lt;ref&amp;gt;{{Cite journal|author1=S. V. Buldyrev |author2=R. Parshani |author3=G. Paul |author4=H. E. Stanley |author5=S. Havlin |title = Catastrophic cascade of failures in interdependent networks|journal = Nature |volume = 464 |pages = 1025–28 |year = 2010 | doi=10.1038/nature08932 |issue=7291|url=http://havlin.biu.ac.il/Publications.php?keyword=Catastrophic+cascade+of+failures+in+interdependent+networks&amp;amp;year=*&amp;amp;match=all |pmid=20393559|arxiv=1012.0206 |bibcode=2010Natur.464.1025B }}&amp;lt;/ref&amp;gt; On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
&lt;br /&gt;
During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system. On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
&lt;br /&gt;
在对枢纽进行有针对性的攻击时，网络的完整性会相对较快地崩溃。由于小节点主要连接到枢纽节点，对最大枢纽的攻击会在短时间内破坏网络。2008年的金融市场崩溃就是这种网络失灵的一个例子，当时最大的参与者(枢纽)的破产导致了整个系统的持续崩溃。另一方面，去除恐怖主义网络中的枢纽节点可能会产生积极的影响，目标节点的消失可能会摧毁整个恐怖组织。网络的攻击容忍度可以通过连接其外围节点来提高，但是它需要加倍的链接数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
=== Degree correlation 度相关===&lt;br /&gt;
&lt;br /&gt;
The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
&lt;br /&gt;
The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全度相关 Perfect Degree Correlation&amp;lt;/font&amp;gt;'''意味着每个度值为k的节点只连接到相同度值的节点。节点之间的这种连通性决定了网络的拓扑结构，从而影响了网络的鲁棒性。如果枢纽节点之间的链接数量与偶然预期的相同，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中性网络 Neutral Network&amp;lt;/font&amp;gt;'''。如果枢纽节点倾向于相互连接，同时避免链接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;选型网络 Assortative Network&amp;lt;/font&amp;gt;'''。这个网络抵抗攻击的能力相对较强，因为枢纽节点形成了一个核心组，这对枢纽节点的移除来说是更加冗余的。如果枢纽节点避免相互连接，同时连接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;非选型网络 Disassortative Network&amp;lt;/font&amp;gt;'''。这个网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中心辐射特征 Hub-and-Spoke Character&amp;lt;/font&amp;gt;'''。因此，如果我们删除这种类型网络中的枢纽节点，可能会破坏或摧毁整个网络。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;传播效应 Spreading phenomenon&amp;lt;/font&amp;gt;'''  ===&lt;br /&gt;
&lt;br /&gt;
The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction &amp;lt;ref&amp;gt;{{Cite journal|first1=Duygu |last1=Balcan |first2=Hao |last2=Hu |first3=Bruno |last3=Goncalves |first4=Paolo |last4=Bajardi |first5=Chiara |last5=Poletto |first6=Jose J.|last6=Ramasco|first7=Daniela|last7=Paolotti|first8=Nicola|last8=Perra |first9=Michele |last9=Tizzoni |first10=Wouter |last10=Van den Broeck|first11=Vittoria |last11=Colizza|first12=Alessandro |last12=Vespignani|date=14 September 2009|title=Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility|journal=BMC Medicine |volume=7 |issue=45 |page=29 |doi=10.1186/1741-7015-7-45 |id= |pmid=19744314 |pmc=2755471 |arxiv=0909.2417}}&amp;lt;/ref&amp;gt; may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
&lt;br /&gt;
The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction  may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
&lt;br /&gt;
这些枢纽节点还负责有效地在网络上传播材料。在疾病传播或信息流的分析中，枢纽被称为超级传播者。超级传播者可能会产生积极的影响，如有效的信息流动，但在 H1N1或艾滋病等流行病传播的情况下也会产生毁灭性的影响。数学模型，如 H1H1流行病预测模型，基于人类流动网络，传染性，或人与人之间的社会互动，可以让我们预测疾病的传播。在根除疾病方面，枢纽也很重要。在无标度网络中，枢纽是最有可能被感染的，因为他们有大量的连接。在枢纽被感染后，它将疾病广播到它所链接的节点。因此，选择性免疫那些枢纽节点可能是消灭传播性疾病的成本效益策略。&lt;br /&gt;
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== References ==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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[[Category:Network theory]]&lt;br /&gt;
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Category:Network theory&lt;br /&gt;
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范畴: 网络理论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Hub (network science)]]. Its edit history can be viewed at [[枢纽节点/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13388</id>
		<title>枢纽</title>
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		<updated>2020-08-28T06:04:54Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{redirect|Network hub|the Ethernet technology|Ethernet hub}}&lt;br /&gt;
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In [[network science]], a '''hub''' is a [[Node (networking)|node]] with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf Barabási, Albert-László. ''Network Science: Graph Theory''., p. 27]&amp;lt;/ref&amp;gt; While hubs cannot be observed in a random network, they are expected to emerge in [[scale-free network]]s. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the [[network topology]]. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the network topology. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络科学 Network Science&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点 Hub&amp;lt;/font&amp;gt;'''是链接数大大超过平均值的节点。枢纽节点是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Property of Networks&amp;lt;/font&amp;gt;'''的重要特性。虽然在随机网络中不存在枢纽节点，但是它们却存在于无标度网络中。无标度网络中枢纽节点的涌现与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;幂律分布 Power-Law Distribution&amp;lt;/font&amp;gt;'''有关，其对网络的拓扑结构有着重要的影响。枢纽节点可以在很多真实的网络中找到，比如大脑网络或者互联网。&lt;br /&gt;
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[[File:Network representation of brain connectivity.JPG|thumb|&lt;br /&gt;
图1：Network representation of brain connectivity. Hubs are highlighted|150px|right] 表示大脑连接性的网络，其中枢纽节点被突出显示]]&lt;br /&gt;
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[[File:Internet map 4096.png|thumb|Partial map of the Internet based on the January 15, 2005. Hubs are highlighted|150px|right]&lt;br /&gt;
图2:2005年1月15日的因特网局部图，枢纽节点被突出显示]]&lt;br /&gt;
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A hub is a component of a network with a high-degree [[Vertex (graph theory)|node]]. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links ([[Degree (graph theory)|degrees]]) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size ''N'' of the network and average degree ''&amp;lt;k&amp;gt;'' constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree ''k'' is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree ''k'' while the other nodes have a small number of links.&lt;br /&gt;
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A hub is a component of a network with a high-degree node. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links (degrees) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size N of the network and average degree &amp;lt;k&amp;gt; constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree k is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree k while the other nodes have a small number of links.&lt;br /&gt;
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一个枢纽节点是网络中具有高度值的组件。与网络中的其他节点相比，枢纽节点的链接数要大得多。无尺度网络中枢纽的链接数(度)远远高于随机网络中链接数最大的节点，在保持网络的大小''N''和平均度 ''&amp;lt;k&amp;gt;''不变的情况下。枢纽的存在是随机网络和无标度网络的最大区别。在随机网络中，每个节点的度''k''是可比的，因此不可能出现枢纽节点。在无标度网络中，少数节点(即枢纽节点)具有高度值 ''k''，而其他节点只有少量的链接。&lt;br /&gt;
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== Emergence 涌现==&lt;br /&gt;
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[[Image:Scale-free network sample.png|thumb|Example of a random network and a scale-free network|400px|right|&lt;br /&gt;
图3：Random network (a) and scale-free network (b). In the scale-free network, the larger hubs are highlighted. 随机网络（a）和无标度网络（b），在无标度网络中，大型枢纽被突出显示]]&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks ([[Barabási–Albert model]]) are different from random networks ([[Erdős–Rényi model]]) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks (Barabási–Albert model) are different from random networks (Erdős–Rényi model) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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枢纽节点的出现可以用无标度网络和随机网络的区别来解释。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks Barabási-Albert model&amp;lt;/font&amp;gt;'''与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机网络 Random Networks Erdős–Rényi model&amp;lt;/font&amp;gt;'''在两个方面有所不同: (a)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;增长 Growth&amp;lt;/font&amp;gt;'''，(b)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;优先连接 Preferential Attachment&amp;lt;/font&amp;gt;'''。{ Cite journal&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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Http://www.nd.edu/~networks/publication%20categories/03%20journal%20articles/physics/statisticalmechanics_rev%20of%20modern%20physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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1 = r é ka Albert | first1 = r é ka | last1 = Albert&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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2 = albert-lászló Barabási | first2 = albert-lászló | last2 = Barabási&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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| title = 复杂网络的统计力学&lt;br /&gt;
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 | journal = [[Reviews of Modern Physics]]&lt;br /&gt;
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 | journal = Reviews of Modern Physics&lt;br /&gt;
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| 杂志 = 现代物理学评论&lt;br /&gt;
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 | volume = 74&lt;br /&gt;
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 | volume = 74&lt;br /&gt;
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74&lt;br /&gt;
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 | pages = 47–97&lt;br /&gt;
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 | pages = 47–97&lt;br /&gt;
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| 页数 = 47-97&lt;br /&gt;
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 | year = 2002&lt;br /&gt;
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 | year = 2002&lt;br /&gt;
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2002年&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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10.1103/RevModPhys. 74.47&lt;br /&gt;
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 | bibcode=2002RvMP...74...47A&lt;br /&gt;
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 | bibcode=2002RvMP...74...47A&lt;br /&gt;
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2002RvMP... 74... 47A&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* (a) Scale-free networks assume a continuous growth of the number of nodes ''N'', compared to random networks which assume a fixed number of nodes. In scale-free networks the degree of the largest hub rises polynomially with the size of the network. Therefore, the degree of a hub can be high in a scale-free network. In random networks the degree of the largest node rises logaritmically (or slower) with N, thus the hub number will be small even in a very large network.&lt;br /&gt;
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无标度网络假设节点数“N”持续增长，而随机网络假设节点数为固定数目。在无标度网络中，最大枢纽节点的度数随着网络的大小而多项式地上升。因此，在无标度网络中，枢纽节点的度数可以很高。在随机网络中，最大节点的度数随着N的增大而呈线性增大(或变慢)，因此即使在一个非常大的随机网络中，枢纽节点的数量也会很小。&lt;br /&gt;
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* (b) A new node in a scale-free network has a tendency to link to a node with a higher degree, compared to a new node in a random network which links itself to a random node. This process is called [[preferential attachment]]. The tendency of a new node to link to a node with a high degree ''k'' is characterized by [[Power law|power-law distribution]] (also known as rich-gets-richer process). This idea was introduced by [[Vilfredo Pareto]] and it explained why a small percentage of the population earns most of the money. This process is present in networks as well, for example 80 percent of web links point to 15 percent of webpages. The emergence of scale-free networks is not typical only of networks created by human action, but also of such networks as metabolic networks or illness networks.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 8.[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf]&amp;lt;/ref&amp;gt; This phenomenon may be explained by the example of hubs on the World Wide Web such as Facebook or Google. These webpages are very well known and therefore the tendency of other webpages pointing to them is much higher than linking to random small webpages.&lt;br /&gt;
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无标度网络中的新节点，与随机网络中的新节点相比，更倾向于连接到具有更高度的节点。这个过程称为优先依附。新节点连接到高度值''k''节点的趋势可被看做幂律分布(也被称作，越富有就越富有的过程)。这个想法是由Vilfredo Pareto提出的，它解释了为什么一小部分人赚了大部分的钱。这个过程也存在于网络中，例如80%的网络链接指向15%的网页。无标度网络的不仅是符合的人类活动网络，也可以拟合包括代谢网络或疾病网络。这种现象可以用像Facebook或谷歌这样的万维网中心的例子来解释。这些网页是非常有名的，因此其他网页指向他们的倾向比指向到随机的小网页要高得多。&lt;br /&gt;
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The mathematical explanation for [[Barabási–Albert model]]:&lt;br /&gt;
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The mathematical explanation for Barabási–Albert model:&lt;br /&gt;
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Barabási-Albert 模型的数学解释:&lt;br /&gt;
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[[File:Barabasi Albert model.gif|thumb|300px|&lt;br /&gt;
图4：The steps of the growth of the network according to the Barabasi–Albert model (&amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt;) 根据 Barabasi-Albert 模型 &amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt; 分析了网络生长的步骤]]&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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这个网络从一个初始的由&amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt;个节点构成的连接网络开始。&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&amp;lt;ref name=&amp;quot;RMP&amp;quot;/&amp;gt;&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&lt;br /&gt;
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每次向网络添加一个新节点。每个新的节点都被连接到&amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt;现有的节点，其概率与现有节点已经拥有的链接数量成正比。形式上，新节点与节点&amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt;相连的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
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where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
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where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
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其中&amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt;是节点&amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt;的度数，这个和取自所有预先存在的节点&amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt;。（分母的结果是当前网络中边的两倍）。&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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网络中枢纽节点的涌现也与时间有关。在无标度网络中，较早出现的节点比后来者更有可能成为枢纽。这种现象被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;先发优势 First-Mover Advantage&amp;lt;/font&amp;gt;'''，它解释了为什么一些节点成为枢纽，而一些没有。然而，在一个真实的网络中，出现的时间并不是影响枢纽规模的唯一因素。例如，在谷歌成为全球最大的互联网中心8年后，Facebook 出现了，然而在2011年，Facebook 成为了全球最大的互联网中心。因此，在实际网络中，枢纽的增长和规模也取决于各种属性，如节点的流行程度、质量或老化程度。&lt;br /&gt;
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== Attributes 属性==&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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无标度网络中的枢纽节点有几个属性&lt;br /&gt;
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=== Shortening the path lengths in a network 缩短网络中的路径距离===&lt;br /&gt;
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The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 23.[http://barabasi.com/networksciencebook/content/book_chapter_4.pdf]&amp;lt;/ref&amp;gt; Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
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The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes. Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
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网络中可观察到的枢纽节点越多，节点之间的距离就越小。在无标度网络中，枢纽是连接度小的节点的桥梁。由于无标度网络中随机节点间的距离很小，我们称无标度网络为“小”或“超小”网络。虽然在一个非常小的网络中，路径距离的差异可能并不明显，但是在大型随机网络和无标度网络中，路径距离的差异是显著的。&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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无标度网络中的平均路径长度:&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点的老化 Aging of hubs (nodes)&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_6.pdf Barabási, Albert-László. ''Network Science: Evolving Networks''., p. 3]&amp;lt;/ref&amp;gt; The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).{{Citation needed|date=May 2016}}&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks. The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).&lt;br /&gt;
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这种现象在真实网络中有所出现，当旧的枢纽被隐藏在网络中时。这种现象导致了网络演化和拓扑结构的变化。老化现象的例子可参照 Facebook 超越了网络中最大的中心 Google (自2000年以来最大的节点)。&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;鲁棒性和攻击容忍度 Robustness and Attack Tolerance&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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随机故障的节点或定向攻击网络关键组成部分的枢纽。在网络枢纽节点的随机失效过程中，网络的鲁棒性是由节点的随机失效决定的。&amp;lt; ref &amp;gt; { cite journal | title = Resilience of the Internet to Random breaks | &lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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日记 = 体育。牧师。李特 | &lt;br /&gt;
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year=2000|&lt;br /&gt;
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year=2000|&lt;br /&gt;
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2000年&lt;br /&gt;
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first=Reoven|&lt;br /&gt;
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first=Reoven|&lt;br /&gt;
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1 = Reoven | &lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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最后 = Cohen | &lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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首先2 = k。2 = Erez | first3 = d.3 = ben-Avraham | first4 = s | last4 = Havlin | authorlink4 = Shlomo Havlin | &lt;br /&gt;
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volume=85|&lt;br /&gt;
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volume=85|&lt;br /&gt;
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85 | &lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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页数 = 4626-8&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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Doi = 10.1103/physrvlett. 85.4626 | &lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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Http://link.aps.org/doi/10.1103/physrevlett.85.4626&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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| bibcode = 2000PhRvL. . 85.4626 c | arxiv = cond-mat/0007048 | pmid = 11082612}&lt;br /&gt;
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&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
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&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
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由于枢纽与大量小度节点共存，随机故障删除它的几率很小。而去除小节点对网络的完整性影响不大。即使随机删除会作用到上枢纽，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络碎片化 Fragmantation of Network&amp;lt;/font&amp;gt;'''的可能性也非常小，因为剩余的枢纽仍能将网络连接在一起。在这种情况下，枢纽就是无标度网络的中坚力量。&lt;br /&gt;
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During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system.&amp;lt;ref&amp;gt;{{Cite journal|author1=S. V. Buldyrev |author2=R. Parshani |author3=G. Paul |author4=H. E. Stanley |author5=S. Havlin |title = Catastrophic cascade of failures in interdependent networks|journal = Nature |volume = 464 |pages = 1025–28 |year = 2010 | doi=10.1038/nature08932 |issue=7291|url=http://havlin.biu.ac.il/Publications.php?keyword=Catastrophic+cascade+of+failures+in+interdependent+networks&amp;amp;year=*&amp;amp;match=all |pmid=20393559|arxiv=1012.0206 |bibcode=2010Natur.464.1025B }}&amp;lt;/ref&amp;gt; On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
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During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system. On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
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在对枢纽进行有针对性的攻击时，网络的完整性会相对较快地崩溃。由于小节点主要连接到枢纽节点，对最大枢纽的攻击会在短时间内破坏网络。2008年的金融市场崩溃就是这种网络失灵的一个例子，当时最大的参与者(枢纽)的破产导致了整个系统的持续崩溃。另一方面，去除恐怖主义网络中的枢纽节点可能会产生积极的影响，目标节点的消失可能会摧毁整个恐怖组织。网络的攻击容忍度可以通过连接其外围节点来提高，但是它需要加倍的链接数量。&lt;br /&gt;
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=== Degree correlation 度相关===&lt;br /&gt;
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The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
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The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全度相关 Perfect Degree Correlation&amp;lt;/font&amp;gt;'''意味着每个度值为k的节点只连接到相同度值的节点。节点之间的这种连通性决定了网络的拓扑结构，从而影响了网络的鲁棒性。如果枢纽节点之间的链接数量与偶然预期的相同，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中性网络 Neutral Network&amp;lt;/font&amp;gt;'''。如果枢纽节点倾向于相互连接，同时避免链接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;选型网络 Assortative Network&amp;lt;/font&amp;gt;'''。这个网络抵抗攻击的能力相对较强，因为枢纽节点形成了一个核心组，这对枢纽节点的移除来说是更加冗余的。如果枢纽节点避免相互连接，同时连接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;非选型网络 Disassortative Network&amp;lt;/font&amp;gt;'''。这个网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中心辐射特征 Hub-and-Spoke Character&amp;lt;/font&amp;gt;'''。因此，如果我们删除这种类型网络中的枢纽节点，可能会破坏或摧毁整个网络。&lt;br /&gt;
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==='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;传播效应 Spreading phenomenon&amp;lt;/font&amp;gt;'''  ===&lt;br /&gt;
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The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction &amp;lt;ref&amp;gt;{{Cite journal|first1=Duygu |last1=Balcan |first2=Hao |last2=Hu |first3=Bruno |last3=Goncalves |first4=Paolo |last4=Bajardi |first5=Chiara |last5=Poletto |first6=Jose J.|last6=Ramasco|first7=Daniela|last7=Paolotti|first8=Nicola|last8=Perra |first9=Michele |last9=Tizzoni |first10=Wouter |last10=Van den Broeck|first11=Vittoria |last11=Colizza|first12=Alessandro |last12=Vespignani|date=14 September 2009|title=Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility|journal=BMC Medicine |volume=7 |issue=45 |page=29 |doi=10.1186/1741-7015-7-45 |id= |pmid=19744314 |pmc=2755471 |arxiv=0909.2417}}&amp;lt;/ref&amp;gt; may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
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The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction  may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
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这些枢纽节点还负责有效地在网络上传播材料。在疾病传播或信息流的分析中，枢纽被称为超级传播者。超级传播者可能会产生积极的影响，如有效的信息流动，但在 H1N1或艾滋病等流行病传播的情况下也会产生毁灭性的影响。数学模型，如 H1H1流行病预测模型，基于人类流动网络，传染性，或人与人之间的社会互动，可以让我们预测疾病的传播。在根除疾病方面，枢纽也很重要。在无标度网络中，枢纽是最有可能被感染的，因为他们有大量的连接。在枢纽被感染后，它将疾病广播到它所链接的节点。因此，选择性免疫那些枢纽节点可能是消灭传播性疾病的成本效益策略。&lt;br /&gt;
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== References ==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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[[Category:Network theory]]&lt;br /&gt;
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Category:Network theory&lt;br /&gt;
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范畴: 网络理论&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Hub (network science)]]. Its edit history can be viewed at [[枢纽节点/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13387</id>
		<title>枢纽</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13387"/>
		<updated>2020-08-28T05:57:17Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{redirect|Network hub|the Ethernet technology|Ethernet hub}}&lt;br /&gt;
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In [[network science]], a '''hub''' is a [[Node (networking)|node]] with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf Barabási, Albert-László. ''Network Science: Graph Theory''., p. 27]&amp;lt;/ref&amp;gt; While hubs cannot be observed in a random network, they are expected to emerge in [[scale-free network]]s. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the [[network topology]]. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the network topology. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络科学 Network Science&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点 Hub&amp;lt;/font&amp;gt;'''是链接数大大超过平均值的节点。枢纽节点是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Property of Networks&amp;lt;/font&amp;gt;'''的重要特性。虽然在随机网络中不存在枢纽节点，但是它们却存在于无标度网络中。无标度网络中枢纽节点的涌现与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;幂律分布 Power-Law Distribution&amp;lt;/font&amp;gt;'''有关，其对网络的拓扑结构有着重要的影响。枢纽节点可以在很多真实的网络中找到，比如大脑网络或者互联网。&lt;br /&gt;
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[[File:Network representation of brain connectivity.JPG|thumb|&lt;br /&gt;
图1：Network representation of brain connectivity. Hubs are highlighted|150px|right] 表示大脑连接性的网络，其中枢纽节点被突出显示]&lt;br /&gt;
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[[File:Internet map 4096.png|thumb|Partial map of the Internet based on the January 15, 2005. Hubs are highlighted|150px|right]&lt;br /&gt;
图2:2005年1月15日的因特网局部图，枢纽节点被突出显示]&lt;br /&gt;
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A hub is a component of a network with a high-degree [[Vertex (graph theory)|node]]. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links ([[Degree (graph theory)|degrees]]) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size ''N'' of the network and average degree ''&amp;lt;k&amp;gt;'' constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree ''k'' is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree ''k'' while the other nodes have a small number of links.&lt;br /&gt;
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A hub is a component of a network with a high-degree node. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links (degrees) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size N of the network and average degree &amp;lt;k&amp;gt; constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree k is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree k while the other nodes have a small number of links.&lt;br /&gt;
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一个枢纽节点是网络中具有高度值的组件。与网络中的其他节点相比，枢纽节点的链接数要大得多。无尺度网络中枢纽的链接数(度)远远高于随机网络中链接数最大的节点，在保持网络的大小''N''和平均度 ''&amp;lt;k&amp;gt;''不变的情况下。枢纽的存在是随机网络和无标度网络的最大区别。在随机网络中，每个节点的度''k''是可比的，因此不可能出现枢纽节点。在无标度网络中，少数节点(即枢纽节点)具有高度值 ''k''，而其他节点只有少量的链接。&lt;br /&gt;
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== Emergence 涌现==&lt;br /&gt;
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[[Image:Scale-free network sample.png|thumb|Example of a random network and a scale-free network|400px|right|&lt;br /&gt;
图3：Random network (a) and scale-free network (b). In the scale-free network, the larger hubs are highlighted. 随机网络（a）和无标度网络（b），在无标度网络中，大型枢纽被突出显示]]&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks ([[Barabási–Albert model]]) are different from random networks ([[Erdős–Rényi model]]) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks (Barabási–Albert model) are different from random networks (Erdős–Rényi model) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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枢纽节点的出现可以用无标度网络和随机网络的区别来解释。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks Barabási-Albert model&amp;lt;/font&amp;gt;'''与'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机网络 Random Networks Erdős–Rényi model&amp;lt;/font&amp;gt;'''在两个方面有所不同: (a)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;增长 Growth&amp;lt;/font&amp;gt;'''，(b)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;优先连接 Preferential Attachment&amp;lt;/font&amp;gt;'''。{ Cite journal&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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Http://www.nd.edu/~networks/publication%20categories/03%20journal%20articles/physics/statisticalmechanics_rev%20of%20modern%20physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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1 = r é ka Albert | first1 = r é ka | last1 = Albert&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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2 = albert-lászló Barabási | first2 = albert-lászló | last2 = Barabási&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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| title = 复杂网络的统计力学&lt;br /&gt;
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 | journal = [[Reviews of Modern Physics]]&lt;br /&gt;
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 | journal = Reviews of Modern Physics&lt;br /&gt;
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| 杂志 = 现代物理学评论&lt;br /&gt;
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 | volume = 74&lt;br /&gt;
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 | volume = 74&lt;br /&gt;
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74&lt;br /&gt;
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 | pages = 47–97&lt;br /&gt;
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 | pages = 47–97&lt;br /&gt;
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| 页数 = 47-97&lt;br /&gt;
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 | year = 2002&lt;br /&gt;
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 | year = 2002&lt;br /&gt;
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2002年&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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10.1103/RevModPhys. 74.47&lt;br /&gt;
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 | bibcode=2002RvMP...74...47A&lt;br /&gt;
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 | bibcode=2002RvMP...74...47A&lt;br /&gt;
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2002RvMP... 74... 47A&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* (a) Scale-free networks assume a continuous growth of the number of nodes ''N'', compared to random networks which assume a fixed number of nodes. In scale-free networks the degree of the largest hub rises polynomially with the size of the network. Therefore, the degree of a hub can be high in a scale-free network. In random networks the degree of the largest node rises logaritmically (or slower) with N, thus the hub number will be small even in a very large network.&lt;br /&gt;
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无标度网络假设节点数“N”持续增长，而随机网络假设节点数为固定数目。在无标度网络中，最大枢纽节点的度数随着网络的大小而多项式地上升。因此，在无标度网络中，枢纽节点的度数可以很高。在随机网络中，最大节点的度数随着N的增大而呈线性增大(或变慢)，因此即使在一个非常大的随机网络中，枢纽节点的数量也会很小。&lt;br /&gt;
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* (b) A new node in a scale-free network has a tendency to link to a node with a higher degree, compared to a new node in a random network which links itself to a random node. This process is called [[preferential attachment]]. The tendency of a new node to link to a node with a high degree ''k'' is characterized by [[Power law|power-law distribution]] (also known as rich-gets-richer process). This idea was introduced by [[Vilfredo Pareto]] and it explained why a small percentage of the population earns most of the money. This process is present in networks as well, for example 80 percent of web links point to 15 percent of webpages. The emergence of scale-free networks is not typical only of networks created by human action, but also of such networks as metabolic networks or illness networks.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 8.[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf]&amp;lt;/ref&amp;gt; This phenomenon may be explained by the example of hubs on the World Wide Web such as Facebook or Google. These webpages are very well known and therefore the tendency of other webpages pointing to them is much higher than linking to random small webpages.&lt;br /&gt;
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无标度网络中的新节点，与随机网络中的新节点相比，更倾向于连接到具有更高度的节点。这个过程称为优先依附。新节点连接到高度值''k''节点的趋势可被看做幂律分布(也被称作，越富有就越富有的过程)。这个想法是由Vilfredo Pareto提出的，它解释了为什么一小部分人赚了大部分的钱。这个过程也存在于网络中，例如80%的网络链接指向15%的网页。无标度网络的不仅是符合的人类活动网络，也可以拟合包括代谢网络或疾病网络。这种现象可以用像Facebook或谷歌这样的万维网中心的例子来解释。这些网页是非常有名的，因此其他网页指向他们的倾向比指向到随机的小网页要高得多。&lt;br /&gt;
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The mathematical explanation for [[Barabási–Albert model]]:&lt;br /&gt;
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The mathematical explanation for Barabási–Albert model:&lt;br /&gt;
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Barabási-Albert 模型的数学解释:&lt;br /&gt;
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[[File:Barabasi Albert model.gif|thumb|300px|&lt;br /&gt;
图4：The steps of the growth of the network according to the Barabasi–Albert model (&amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt;) 根据 Barabasi-Albert 模型 &amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt; 分析了网络生长的步骤]]&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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这个网络从一个初始的由&amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt;个节点构成的连接网络开始。&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&amp;lt;ref name=&amp;quot;RMP&amp;quot;/&amp;gt;&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&lt;br /&gt;
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每次向网络添加一个新节点。每个新的节点都被连接到&amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt;现有的节点，其概率与现有节点已经拥有的链接数量成正比。形式上，新节点与节点&amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt;相连的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
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where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
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where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
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其中&amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt;是节点&amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt;的度数，这个和取自所有预先存在的节点&amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt;。（分母的结果是当前网络中边的两倍）。&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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网络中枢纽节点的涌现也与时间有关。在无标度网络中，较早出现的节点比后来者更有可能成为枢纽。这种现象被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;先发优势 First-Mover Advantage&amp;lt;/font&amp;gt;'''，它解释了为什么一些节点成为枢纽，而一些没有。然而，在一个真实的网络中，出现的时间并不是影响枢纽规模的唯一因素。例如，在谷歌成为全球最大的互联网中心8年后，Facebook 出现了，然而在2011年，Facebook 成为了全球最大的互联网中心。因此，在实际网络中，枢纽的增长和规模也取决于各种属性，如节点的流行程度、质量或老化程度。&lt;br /&gt;
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== Attributes 属性==&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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无标度网络中的枢纽节点有几个属性&lt;br /&gt;
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=== Shortening the path lengths in a network 缩短网络中的路径距离===&lt;br /&gt;
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The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 23.[http://barabasi.com/networksciencebook/content/book_chapter_4.pdf]&amp;lt;/ref&amp;gt; Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
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The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes. Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
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网络中可观察到的枢纽节点越多，节点之间的距离就越小。在无标度网络中，枢纽是连接度小的节点的桥梁。由于无标度网络中随机节点间的距离很小，我们称无标度网络为“小”或“超小”网络。虽然在一个非常小的网络中，路径距离的差异可能并不明显，但是在大型随机网络和无标度网络中，路径距离的差异是显著的。&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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无标度网络中的平均路径长度:&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;枢纽节点的老化 Aging of hubs (nodes)&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_6.pdf Barabási, Albert-László. ''Network Science: Evolving Networks''., p. 3]&amp;lt;/ref&amp;gt; The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).{{Citation needed|date=May 2016}}&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks. The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).&lt;br /&gt;
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这种现象在真实网络中有所出现，当旧的枢纽被隐藏在网络中时。这种现象导致了网络演化和拓扑结构的变化。老化现象的例子可参照 Facebook 超越了网络中最大的中心 Google (自2000年以来最大的节点)。&lt;br /&gt;
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=== '''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;鲁棒性和攻击容忍度 Robustness and Attack Tolerance&amp;lt;/font&amp;gt;''' ===&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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随机故障的节点或定向攻击网络关键组成部分的枢纽。在网络枢纽节点的随机失效过程中，网络的鲁棒性是由节点的随机失效决定的。&amp;lt; ref &amp;gt; { cite journal | title = Resilience of the Internet to Random breaks | &lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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日记 = 体育。牧师。李特 | &lt;br /&gt;
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year=2000|&lt;br /&gt;
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year=2000|&lt;br /&gt;
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2000年&lt;br /&gt;
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first=Reoven|&lt;br /&gt;
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first=Reoven|&lt;br /&gt;
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1 = Reoven | &lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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最后 = Cohen | &lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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首先2 = k。2 = Erez | first3 = d.3 = ben-Avraham | first4 = s | last4 = Havlin | authorlink4 = Shlomo Havlin | &lt;br /&gt;
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volume=85|&lt;br /&gt;
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volume=85|&lt;br /&gt;
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85 | &lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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页数 = 4626-8&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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Doi = 10.1103/physrvlett. 85.4626 | &lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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Http://link.aps.org/doi/10.1103/physrevlett.85.4626&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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| bibcode = 2000PhRvL. . 85.4626 c | arxiv = cond-mat/0007048 | pmid = 11082612}&lt;br /&gt;
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&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
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&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
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由于枢纽与大量小度节点共存，随机故障删除它的几率很小。而去除小节点对网络的完整性影响不大。即使随机删除会作用到上枢纽，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络碎片化 Fragmantation of Network&amp;lt;/font&amp;gt;'''的可能性也非常小，因为剩余的枢纽仍能将网络连接在一起。在这种情况下，枢纽就是无标度网络的中坚力量。&lt;br /&gt;
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During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system.&amp;lt;ref&amp;gt;{{Cite journal|author1=S. V. Buldyrev |author2=R. Parshani |author3=G. Paul |author4=H. E. Stanley |author5=S. Havlin |title = Catastrophic cascade of failures in interdependent networks|journal = Nature |volume = 464 |pages = 1025–28 |year = 2010 | doi=10.1038/nature08932 |issue=7291|url=http://havlin.biu.ac.il/Publications.php?keyword=Catastrophic+cascade+of+failures+in+interdependent+networks&amp;amp;year=*&amp;amp;match=all |pmid=20393559|arxiv=1012.0206 |bibcode=2010Natur.464.1025B }}&amp;lt;/ref&amp;gt; On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
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During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system. On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
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在对枢纽进行有针对性的攻击时，网络的完整性会相对较快地崩溃。由于小节点主要连接到枢纽节点，对最大枢纽的攻击会在短时间内破坏网络。2008年的金融市场崩溃就是这种网络失灵的一个例子，当时最大的参与者(枢纽)的破产导致了整个系统的持续崩溃。另一方面，去除恐怖主义网络中的枢纽节点可能会产生积极的影响，目标节点的消失可能会摧毁整个恐怖组织。网络的攻击容忍度可以通过连接其外围节点来提高，但是它需要加倍的链接数量。&lt;br /&gt;
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=== Degree correlation 度相关===&lt;br /&gt;
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The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
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The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全度相关 Perfect Degree Correlation&amp;lt;/font&amp;gt;'''意味着每个度值为k的节点只连接到相同度值的节点。节点之间的这种连通性决定了网络的拓扑结构，从而影响了网络的鲁棒性。如果枢纽节点之间的链接数量与偶然预期的相同，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中性网络 Neutral Network&amp;lt;/font&amp;gt;'''。如果枢纽节点倾向于相互连接，同时避免链接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;选型网络 Assortative Network&amp;lt;/font&amp;gt;'''。这个网络抵抗攻击的能力相对较强，因为枢纽节点形成了一个核心组，这对枢纽节点的移除来说是更加冗余的。如果枢纽节点避免相互连接，同时连接到小度的节点，我们将这个网络称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;非选型网络 Disassortative Network&amp;lt;/font&amp;gt;'''。这个网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;中心辐射特征 Hub-and-Spoke Character&amp;lt;/font&amp;gt;'''。因此，如果我们删除这种类型网络中的枢纽节点，可能会破坏或摧毁整个网络。&lt;br /&gt;
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==='''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;传播效应 Spreading phenomenon&amp;lt;/font&amp;gt;'''  ===&lt;br /&gt;
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The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction &amp;lt;ref&amp;gt;{{Cite journal|first1=Duygu |last1=Balcan |first2=Hao |last2=Hu |first3=Bruno |last3=Goncalves |first4=Paolo |last4=Bajardi |first5=Chiara |last5=Poletto |first6=Jose J.|last6=Ramasco|first7=Daniela|last7=Paolotti|first8=Nicola|last8=Perra |first9=Michele |last9=Tizzoni |first10=Wouter |last10=Van den Broeck|first11=Vittoria |last11=Colizza|first12=Alessandro |last12=Vespignani|date=14 September 2009|title=Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility|journal=BMC Medicine |volume=7 |issue=45 |page=29 |doi=10.1186/1741-7015-7-45 |id= |pmid=19744314 |pmc=2755471 |arxiv=0909.2417}}&amp;lt;/ref&amp;gt; may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
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The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction  may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
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这些枢纽节点还负责有效地在网络上传播材料。在疾病传播或信息流的分析中，枢纽被称为超级传播者。超级传播者可能会产生积极的影响，如有效的信息流动，但在 H1N1或艾滋病等流行病传播的情况下也会产生毁灭性的影响。数学模型，如 H1H1流行病预测模型，基于人类流动网络，传染性，或人与人之间的社会互动，可以让我们预测疾病的传播。在根除疾病方面，枢纽也很重要。在无标度网络中，枢纽是最有可能被感染的，因为他们有大量的连接。在枢纽被感染后，它将疾病广播到它所链接的节点。因此，选择性免疫那些枢纽节点可能是消灭传播性疾病的成本效益策略。&lt;br /&gt;
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== References ==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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[[Category:Network theory]]&lt;br /&gt;
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Category:Network theory&lt;br /&gt;
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范畴: 网络理论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Hub (network science)]]. Its edit history can be viewed at [[枢纽节点/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13375</id>
		<title>枢纽</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%9E%A2%E7%BA%BD&amp;diff=13375"/>
		<updated>2020-08-26T08:57:24Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
{{redirect|Network hub|the Ethernet technology|Ethernet hub}}&lt;br /&gt;
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In [[network science]], a '''hub''' is a [[Node (networking)|node]] with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf Barabási, Albert-László. ''Network Science: Graph Theory''., p. 27]&amp;lt;/ref&amp;gt; While hubs cannot be observed in a random network, they are expected to emerge in [[scale-free network]]s. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the [[network topology]]. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the network topology. Hubs can be found in many real networks, such as Brain Network or Internet.&lt;br /&gt;
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在网络科学中，枢纽节点是链接数大大超过平均值的节点。枢纽节点是无标度网络的重要特性。虽然在随机网络中不存在枢纽节点，但是它们却存在于无标度网络中。无标度网络中枢纽节点的涌现与幂律分布有关，其对网络的拓扑结构有着重要的影响。枢纽节点可以在很多真实的网络中找到，比如大脑网络或者互联网。&lt;br /&gt;
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[[File:Network representation of brain connectivity.JPG|thumb|&lt;br /&gt;
图1：Network representation of brain connectivity. Hubs are highlighted|150px|right] 表示大脑连接性的网络，其中枢纽节点被突出显示]&lt;br /&gt;
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[[File:Internet map 4096.png|thumb|Partial map of the Internet based on the January 15, 2005. Hubs are highlighted|150px|right]&lt;br /&gt;
图2:2005年1月15日的因特网局部图，枢纽节点被突出显示]&lt;br /&gt;
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A hub is a component of a network with a high-degree [[Vertex (graph theory)|node]]. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links ([[Degree (graph theory)|degrees]]) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size ''N'' of the network and average degree ''&amp;lt;k&amp;gt;'' constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree ''k'' is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree ''k'' while the other nodes have a small number of links.&lt;br /&gt;
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A hub is a component of a network with a high-degree node. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links (degrees) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size N of the network and average degree &amp;lt;k&amp;gt; constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree k is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree k while the other nodes have a small number of links.&lt;br /&gt;
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一个枢纽节点是网络中具有高度值的一个组件。与网络中的其他节点相比，枢纽节点的链接数要大得多。无尺度网络中枢纽的链接数(度)远远高于随机网络中链接数最大的节点，在保持网络的大小''N''和平均度 ''&amp;lt;k&amp;gt;''不变的情况下。枢纽的存在是随机网络和无标度网络的最大区别。在随机网络中，每个节点的度''k''是可比的，因此不可能出现枢纽节点。在无标度网络中，少数节点(即枢纽节点)具有高度值 ''k''，而其他节点只有少量的链接。&lt;br /&gt;
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== Emergence 涌现==&lt;br /&gt;
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[[Image:Scale-free network sample.png|thumb|Example of a random network and a scale-free network|400px|right|&lt;br /&gt;
图3：Random network (a) and scale-free network (b). In the scale-free network, the larger hubs are highlighted. 随机网络（a）和无标度网络（b），在无标度网络中，大型枢纽被突出显示]]&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks ([[Barabási–Albert model]]) are different from random networks ([[Erdős–Rényi model]]) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks (Barabási–Albert model) are different from random networks (Erdős–Rényi model) in two aspects: (a) growth, (b) preferential attachment.&amp;lt;ref name=RMP&amp;gt;{{Cite journal&lt;br /&gt;
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枢纽节点的出现可以用无标度网络和随机网络的区别来解释。无标度网络(Barabási-Albert 模型)与随机网络(Erdős–Rényi 模型)在两个方面有所不同: (a)增长，(b)优先连接。{ Cite journal&lt;br /&gt;
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 | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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Http://www.nd.edu/~networks/publication%20categories/03%20journal%20articles/physics/statisticalmechanics_rev%20of%20modern%20physics%2074,%2047%20(2002).pdf&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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 | authorlink1 = Réka Albert | first1 = Réka | last1 = Albert&lt;br /&gt;
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1 = r é ka Albert | first1 = r é ka | last1 = Albert&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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 | authorlink2 = Albert-László Barabási | first2 = Albert-László | last2 = Barabási&lt;br /&gt;
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2 = albert-lászló Barabási | first2 = albert-lászló | last2 = Barabási&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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 | title = Statistical mechanics of complex networks&lt;br /&gt;
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| title = 复杂网络的统计力学&lt;br /&gt;
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 | journal = [[Reviews of Modern Physics]]&lt;br /&gt;
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| 杂志 = 现代物理学评论&lt;br /&gt;
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| 页数 = 47-97&lt;br /&gt;
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2002年&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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 | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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10.1103/RevModPhys. 74.47&lt;br /&gt;
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2002RvMP... 74... 47A&lt;br /&gt;
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|arxiv = cond-mat/0106096 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* (a) Scale-free networks assume a continuous growth of the number of nodes ''N'', compared to random networks which assume a fixed number of nodes. In scale-free networks the degree of the largest hub rises polynomially with the size of the network. Therefore, the degree of a hub can be high in a scale-free network. In random networks the degree of the largest node rises logaritmically (or slower) with N, thus the hub number will be small even in a very large network.&lt;br /&gt;
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无标度网络假设节点数“N”持续增长，而随机网络假设节点数为固定数目。在无标度网络中，最大枢纽节点的度数随着网络的大小而多项式地上升。因此，在无标度网络中，枢纽节点的度数可以很高。在随机网络中，最大节点的度数随着N的增大而呈线性增大(或变慢)，因此即使在一个非常大的随机网络中，枢纽节点的数量也会很小。&lt;br /&gt;
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* (b) A new node in a scale-free network has a tendency to link to a node with a higher degree, compared to a new node in a random network which links itself to a random node. This process is called [[preferential attachment]]. The tendency of a new node to link to a node with a high degree ''k'' is characterized by [[Power law|power-law distribution]] (also known as rich-gets-richer process). This idea was introduced by [[Vilfredo Pareto]] and it explained why a small percentage of the population earns most of the money. This process is present in networks as well, for example 80 percent of web links point to 15 percent of webpages. The emergence of scale-free networks is not typical only of networks created by human action, but also of such networks as metabolic networks or illness networks.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 8.[http://barabasi.com/networksciencebook/content/book_chapter_2.pdf]&amp;lt;/ref&amp;gt; This phenomenon may be explained by the example of hubs on the World Wide Web such as Facebook or Google. These webpages are very well known and therefore the tendency of other webpages pointing to them is much higher than linking to random small webpages.&lt;br /&gt;
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无标度网络中的新节点，与随机网络中的新节点相比，更倾向于连接到具有更高度的节点。这个过程称为优先依附。新节点连接到高度值''k''节点的趋势可被看做幂律分布(也被称作，越富有就越富有的过程)。这个想法是由Vilfredo Pareto提出的，它解释了为什么一小部分人赚了大部分的钱。这个过程也存在于网络中，例如80%的网络链接指向15%的网页。无标度网络的不仅是符合的人类活动网络，也可以拟合包括代谢网络或疾病网络。这种现象可以用像Facebook或谷歌这样的万维网中心的例子来解释。这些网页是非常有名的，因此其他网页指向他们的倾向比指向到随机的小网页要高得多。&lt;br /&gt;
&lt;br /&gt;
The mathematical explanation for [[Barabási–Albert model]]:&lt;br /&gt;
&lt;br /&gt;
The mathematical explanation for Barabási–Albert model:&lt;br /&gt;
&lt;br /&gt;
Barabási-Albert 模型的数学解释:&lt;br /&gt;
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[[File:Barabasi Albert model.gif|thumb|300px|&lt;br /&gt;
图4：The steps of the growth of the network according to the Barabasi–Albert model (&amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt;) 根据 Barabasi-Albert 模型 &amp;lt;math&amp;gt;m_0=m=2&amp;lt;/math&amp;gt; 分析了网络生长的步骤]]&lt;br /&gt;
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The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
&lt;br /&gt;
The network begins with an initial connected network of &amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt; nodes.&lt;br /&gt;
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这个网络从一个初始的由&amp;lt;math&amp;gt;m_0&amp;lt;/math&amp;gt;个节点构成的连接网络开始。&lt;br /&gt;
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New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&amp;lt;ref name=&amp;quot;RMP&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
New nodes are added to the network one at a time. Each new node is connected to &amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt; existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability &amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt; that the new node is connected to node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; is&lt;br /&gt;
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每次向网络添加一个新节点。每个新的节点都被连接到&amp;lt;math&amp;gt;m \le m_0&amp;lt;/math&amp;gt;现有的节点，其概率与现有节点已经拥有的链接数量成正比。形式上，新节点与节点&amp;lt;math&amp;gt;p_i&amp;lt;/math&amp;gt;相连的概率是&lt;br /&gt;
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&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;p_i = \frac{k_i}{\sum_j k_j},&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt; is the degree of the node &amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt; and the sum is taken over all pre-existing nodes &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt; (i.e. the denominator results in twice the current number of edges in the network).&lt;br /&gt;
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其中&amp;lt;math&amp;gt;k_i&amp;lt;/math&amp;gt;是节点&amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt;的度数，这个和取自所有预先存在的节点&amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt;。（分母的结果是当前网络中边的两倍）。&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.&lt;br /&gt;
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网络中枢纽节点的涌现也与时间有关。在无标度网络中，较早出现的节点比后来者更有可能成为枢纽。这种现象被称为先发优势，它解释了为什么一些节点成为枢纽，而一些没有。然而，在一个真实的网络中，出现的时间并不是影响枢纽规模的唯一因素。例如，在谷歌成为全球最大的互联网中心8年后，Facebook 出现了，然而在2011年，Facebook 成为了全球最大的互联网中心。因此，在实际网络中，枢纽的增长和规模也取决于各种属性，如节点的流行程度、质量或老化程度。&lt;br /&gt;
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== Attributes 属性==&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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There are several attributes of Hubs in a Scale-Free Network&lt;br /&gt;
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无标度网络中的枢纽节点有几个属性&lt;br /&gt;
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=== Shortening the path lengths in a network 缩短网络中的路径距离===&lt;br /&gt;
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The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes.&amp;lt;ref&amp;gt;Barabási, Albert-László. ''Network Science: The Scale-Free Property''., p. 23.[http://barabasi.com/networksciencebook/content/book_chapter_4.pdf]&amp;lt;/ref&amp;gt; Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
&lt;br /&gt;
The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes. Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as &amp;quot;small&amp;quot; or &amp;quot;ultra small&amp;quot;. While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.&lt;br /&gt;
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网络中可观察到的枢纽节点越多，节点之间的距离就越小。在无标度网络中，枢纽是连接度小的节点的桥梁。由于无标度网络中随机节点间的距离很小，我们称无标度网络为“小”或“超小”网络。虽然在一个非常小的网络中，路径距离的差异可能并不明显，但是在大型随机网络和无标度网络中，路径距离的差异是显著的。&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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Average path length in scale-free networks:&lt;br /&gt;
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无标度网络中的平均路径长度:&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\ell\sim\frac{\ln N}{\ln \ln N}.&amp;lt;/math&amp;gt;&lt;br /&gt;
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=== Aging of hubs (nodes) 枢纽节点的老化===&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks.&amp;lt;ref&amp;gt;[http://barabasi.com/networksciencebook/content/book_chapter_6.pdf Barabási, Albert-László. ''Network Science: Evolving Networks''., p. 3]&amp;lt;/ref&amp;gt; The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).{{Citation needed|date=May 2016}}&lt;br /&gt;
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The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks. The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).&lt;br /&gt;
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这种现象在真实网络中出现，当旧的枢纽被隐藏在网络中时。这种现象导致了网络演化和拓扑结构的变化。老化现象的例子可参照 Facebook 超越了网络中最大的中心 Google (自2000年以来最大的节点)。&lt;br /&gt;
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=== Robustness and Attack Tolerance 鲁棒性和攻击容忍度===&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.&amp;lt;ref&amp;gt;{{cite journal|title=Resilience of the Internet to Random Breakdowns|&lt;br /&gt;
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随机故障的节点或定向攻击的枢纽是网络的关键组成部分。在网络中枢节点的随机失效过程中，网络的鲁棒性是由节点的随机失效决定的。&amp;lt; ref &amp;gt; { cite journal | title = Resilience of the Internet to Random breaks | &lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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journal=Phys. Rev. Lett.|&lt;br /&gt;
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日记 = 体育。牧师。李特 | &lt;br /&gt;
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year=2000|&lt;br /&gt;
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year=2000|&lt;br /&gt;
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2000年&lt;br /&gt;
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first=Reoven|&lt;br /&gt;
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first=Reoven|&lt;br /&gt;
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1 = Reoven | &lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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last=Cohen|&lt;br /&gt;
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最后 = Cohen | &lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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first2=K. |last2=Erez |first3=D. |last3=ben-Avraham |first4=S.|last4=Havlin|authorlink4=Shlomo Havlin|&lt;br /&gt;
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首先2 = k。2 = Erez | first3 = d.3 = ben-Avraham | first4 = s | last4 = Havlin | authorlink4 = Shlomo Havlin | &lt;br /&gt;
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volume=85|&lt;br /&gt;
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volume=85|&lt;br /&gt;
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85 | &lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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pages=4626–8|&lt;br /&gt;
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页数 = 4626-8&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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doi= 10.1103/PhysRevLett.85.4626|&lt;br /&gt;
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Doi = 10.1103/physrvlett. 85.4626 | &lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626&lt;br /&gt;
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Http://link.aps.org/doi/10.1103/physrevlett.85.4626&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 |pmid=11082612}}&lt;br /&gt;
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| bibcode = 2000PhRvL. . 85.4626 c | arxiv = cond-mat/0007048 | pmid = 11082612}&lt;br /&gt;
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&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
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&amp;lt;/ref&amp;gt; The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.&lt;br /&gt;
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由于枢纽与大量小度节点共存，随机故障删除它的几率很小。去除小节点对网络的完整性影响不大。即使随机删除会作用到上枢纽，网络碎片化的可能性也非常小，因为剩余的枢纽仍能将网络连接在一起。在这种情况下，枢纽就是无标度网络的中坚力量。&lt;br /&gt;
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During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system.&amp;lt;ref&amp;gt;{{Cite journal|author1=S. V. Buldyrev |author2=R. Parshani |author3=G. Paul |author4=H. E. Stanley |author5=S. Havlin |title = Catastrophic cascade of failures in interdependent networks|journal = Nature |volume = 464 |pages = 1025–28 |year = 2010 | doi=10.1038/nature08932 |issue=7291|url=http://havlin.biu.ac.il/Publications.php?keyword=Catastrophic+cascade+of+failures+in+interdependent+networks&amp;amp;year=*&amp;amp;match=all |pmid=20393559|arxiv=1012.0206 |bibcode=2010Natur.464.1025B }}&amp;lt;/ref&amp;gt; On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
&lt;br /&gt;
During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system. On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.&lt;br /&gt;
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在对枢纽进行有针对性的攻击时，网络的完整性会相对较快地崩溃。由于小节点主要连接到枢纽节点，对最大枢纽的攻击会在短时间内破坏网络。2008年的金融市场崩溃就是这种网络失灵的一个例子，当时最大的参与者(枢纽)的破产导致了整个系统的持续崩溃。另一方面，去除恐怖主义网络中的枢纽节点可能会产生积极的影响，目标节点的消失可能会摧毁整个恐怖组织。网络的攻击容忍度可以通过连接其外围节点来提高，但是它需要加倍的链接数量。&lt;br /&gt;
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=== Degree correlation 度相关===&lt;br /&gt;
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The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
&lt;br /&gt;
The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.&lt;br /&gt;
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完全度相关意味着每个度值为k的节点只连接到相同度值的节点。节点之间的这种连通性决定了网络的拓扑结构，从而影响了网络的鲁棒性。如果枢纽节点之间的链接数量与偶然预期的相同，我们将这个网络称为中性网络。如果枢纽节点倾向于相互连接，同时避免链接到小度的节点，我们将这个网络称为选型网络。这个网络抵抗攻击的能力相对较强，因为枢纽节点形成了一个核心组，这对枢纽节点的移除来说是更加冗余的。如果枢纽节点避免相互连接，同时连接到小度的节点，我们将这个网络称为非选型网络。这个网络具有中心辐射特征。因此，如果我们删除这种类型网络中的枢纽节点，可能会破坏或摧毁整个网络。&lt;br /&gt;
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=== Spreading phenomenon ===&lt;br /&gt;
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The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction &amp;lt;ref&amp;gt;{{Cite journal|first1=Duygu |last1=Balcan |first2=Hao |last2=Hu |first3=Bruno |last3=Goncalves |first4=Paolo |last4=Bajardi |first5=Chiara |last5=Poletto |first6=Jose J.|last6=Ramasco|first7=Daniela|last7=Paolotti|first8=Nicola|last8=Perra |first9=Michele |last9=Tizzoni |first10=Wouter |last10=Van den Broeck|first11=Vittoria |last11=Colizza|first12=Alessandro |last12=Vespignani|date=14 September 2009|title=Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility|journal=BMC Medicine |volume=7 |issue=45 |page=29 |doi=10.1186/1741-7015-7-45 |id= |pmid=19744314 |pmc=2755471 |arxiv=0909.2417}}&amp;lt;/ref&amp;gt; may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
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The hubs are also responsible for effective spreading of material on network. In an analysis of disease spreading or information flow, hubs are referred to as super-spreaders. Super-spreaders may have a positive impact, such as effective information flow, but also devastating in a case of epidemic spreading such as H1N1 or AIDS. The mathematical models such as model of H1H1 Epidemic prediction  may allow us to predict the spread of diseases based on human mobility networks, infectiousness, or social interactions among humans. Hubs are also important in the eradication of disease. In a scale-free network hubs are most likely to be infected, because of the large number of connections they have. After the hub is infected, it broadcasts the disease to the nodes it is linked to. Therefore, the selective immunization of hubs may be the cost-effective strategy in eradication of spreading disease.&lt;br /&gt;
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这些枢纽节点还负责有效地在网络上传播材料。在疾病传播或信息流的分析中，枢纽被称为超级传播者。超级传播者可能会产生积极的影响，如有效的信息流动，但在 H1N1或艾滋病等流行病传播的情况下也会产生毁灭性的影响。数学模型，如 H1H1流行病预测模型，基于人类流动网络，传染性，或人与人之间的社会互动，可以让我们预测疾病的传播。在根除疾病方面，枢纽也很重要。在无标度网络中，枢纽是最有可能被感染的，因为他们有大量的连接。在枢纽被感染后，它将疾病广播到它所链接的节点。因此，选择性免疫那些枢纽节点可能是消灭传播性疾病的成本效益策略。&lt;br /&gt;
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== References ==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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[[Category:Network theory]]&lt;br /&gt;
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Category:Network theory&lt;br /&gt;
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范畴: 网络理论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Hub (network science)]]. Its edit history can be viewed at [[枢纽节点/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E7%BB%9F%E8%AE%A1%E5%8A%9B%E5%AD%A6&amp;diff=13049</id>
		<title>统计力学</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E7%BB%9F%E8%AE%A1%E5%8A%9B%E5%AD%A6&amp;diff=13049"/>
		<updated>2020-08-20T02:13:48Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;已由佳欣初步翻译&lt;br /&gt;
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{{Short description|Physics of large number of particles' statistical behavior}}&lt;br /&gt;
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{{Statistical mechanics}}&lt;br /&gt;
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'''Statistical mechanics''' is one of the pillars of modern [[physics]]. It is necessary for the fundamental study of any physical system that has many [[degrees of freedom]]. The approach is based on [[Statistics|statistical]] methods, [[probability theory]] and the [[Microscopic scale|microscopic]] physical laws.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;{{NoteTag|The term ''statistical mechanics'' is sometimes used to refer to only ''statistical thermodynamics''. This article takes the broader view. By some definitions, ''[[statistical physics]]'' is an even broader term which statistically studies any type of physical system, but is often taken to be synonymous with statistical mechanics.}}&lt;br /&gt;
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Statistical mechanics is one of the pillars of modern physics. It is necessary for the fundamental study of any physical system that has many degrees of freedom. The approach is based on statistical methods, probability theory and the microscopic physical laws.&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计力学 Statistical Mechanics&amp;lt;/font&amp;gt;是现代物理学的支柱之一。对于具有多个&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;自由度 Degrees of Freedom&amp;lt;/font&amp;gt;的物理系统的基础研究，统计力学是不可或缺的。统计力学的方法是基于&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计学方法 Statistical Methods&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;概率论 Probability Theory&amp;lt;/font&amp;gt;和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微观物理定律 Microscopic Physical Laws&amp;lt;/font&amp;gt;。&lt;br /&gt;
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It can be used to explain the [[thermodynamics|thermodynamic]] behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as '''statistical thermodynamics''' or '''equilibrium statistical mechanics'''.&lt;br /&gt;
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It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.&lt;br /&gt;
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统计力学一个分支可以用来解释大系统的热力学行为。该分支完善和扩展了经典热力学，被称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计热力学 Statistical Thermodynamics&amp;lt;/font&amp;gt;或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;平衡态统计力学 Equilibrium Statistical Mechanics&amp;lt;/font&amp;gt;。&lt;br /&gt;
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Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as [[heat capacity]]) to microscopic behavior, whereas, in [[classical thermodynamics]], the only available option would be to measure and tabulate such quantities for various materials.&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt;&lt;br /&gt;
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Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.&lt;br /&gt;
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统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热容 Heat Capacity&amp;lt;/font&amp;gt;)与微观行为联系起来。而在&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典热力学 Classical Thermodynamics&amp;lt;/font&amp;gt;中，唯一可行的选择就是测量和列出各种材料的热力学量。&lt;br /&gt;
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Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as '''non-equilibrium statistical mechanics''' (sometimes called '''statistical dynamics''') deals with the issue of microscopically modelling the speed of [[irreversible process]]es that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The [[fluctuation–dissipation theorem]] is the basic knowledge obtained from applying [[non-equilibrium statistical mechanics]] to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.&lt;br /&gt;
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Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics (sometimes called statistical dynamics) deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.&lt;br /&gt;
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统计力学也可以用来研究非平衡的系统。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;非平衡统计力学 Non-equilibrium Statistical Mechanics&amp;lt;/font&amp;gt;(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;涨落-耗散定理 Fluctuation–Dissipation Theorem&amp;lt;/font&amp;gt;是人们从非平衡态统计力学中获得的基本知识，它是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。&lt;br /&gt;
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== Principles: mechanics and ensembles 原理：力学和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;系综 Ensembles&amp;lt;/font&amp;gt;==&lt;br /&gt;
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{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}&lt;br /&gt;
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In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts:&lt;br /&gt;
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In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:&lt;br /&gt;
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在物理学中，有两种力学被广泛研究: &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典力学 Classical Mechanics&amp;lt;/font&amp;gt;和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;量子力学 Quantum Mechanics&amp;lt;/font&amp;gt;。对于这两种力学，标准的数学方法与两个概念有关&lt;br /&gt;
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# The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics).&lt;br /&gt;
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The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).&lt;br /&gt;
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力学系统在给定时间内的完整状态，用数学表示为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;相空间 Phase Space&amp;lt;/font&amp;gt;中的点(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;纯量子态矢量 Pure Quantum State Vector&amp;lt;/font&amp;gt;(量子力学)。&lt;br /&gt;
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# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)&lt;br /&gt;
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An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)&lt;br /&gt;
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一个运动方程描述状态在时间上的演化: &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;哈密尔顿方程 Hamilton's Equations&amp;lt;/font&amp;gt;(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;含时薛定谔方程 Time-dependent Schrödinger Equation&amp;lt;/font&amp;gt;(量子力学)&lt;br /&gt;
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Using these two concepts, the state at any other time, past or future, can in principle be calculated.&lt;br /&gt;
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Using these two concepts, the state at any other time, past or future, can in principle be calculated.&lt;br /&gt;
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使用这两个概念，系统在任何时间的状态，无论过去或未来，原则上都可以计算出来。&lt;br /&gt;
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There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.&lt;br /&gt;
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There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.&lt;br /&gt;
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然而，这些定律与日常生活经验之间存在着脱节。因为对于在人类尺度上进行的过程(例如化学反应)，我们没有必要(甚至在理论上也不可能)在微观层面上准确地知道每个分子所在的位置及其速度。统计力学通过增加一些对于系统状态的不确定性，填补了力学定律和人类不完全知识的实践经验之间的脱节。&lt;br /&gt;
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Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the [[Statistical ensemble (mathematical physics)|statistical ensemble]], which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a [[probability distribution]] over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a [[phase space]] with [[canonical coordinates]]. In quantum statistical mechanics, the ensemble is a probability distribution over pure states,{{NoteTag|The probabilities in quantum statistical mechanics should not be confused with [[quantum superposition]]. While a quantum ensemble can contain states with quantum superpositions, a single quantum state cannot be used to represent an ensemble.}} and can be compactly summarized as a [[density matrix]].&lt;br /&gt;
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Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix.&lt;br /&gt;
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普通力学只考虑单一状态的行为，而统计力学引入了&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计系综 Statistical Ensemble&amp;lt;/font&amp;gt;，它是系统在各种状态下的大量虚拟、独立的拷贝的集合。系综是一个覆盖系统所有可能状态的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;概率分布 Probability Distribution&amp;lt;/font&amp;gt;。在经典的统计力学中，系综是相点上的概率分布(与普通力学中的单相点相反) ，通常表现为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;正则坐标 Canonical Coordinates&amp;lt;/font&amp;gt;下相空间中的分布。在量子统计力学中，系综是纯态上的概率分布，可以简单地概括为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;密度矩阵 Density Matrix&amp;lt;/font&amp;gt;。 &lt;br /&gt;
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As is usual for probabilities, the ensemble can be interpreted in different ways:&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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As is usual for probabilities, the ensemble can be interpreted in different ways:&lt;br /&gt;
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与通常的概率一样，系综可以用不同的方式来解释:&lt;br /&gt;
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* an ensemble can be taken to represent the various possible states that a ''single system'' could be in ([[epistemic probability]], a form of knowledge), or&lt;br /&gt;
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* the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner ([[empirical probability]]), in the limit of an infinite number of trials.&lt;br /&gt;
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* 系综可以表示&amp;quot;单个系统&amp;quot;的所有可能状态（&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;认识概率 Epistemic Probability&amp;lt;/font&amp;gt;，知识的一种形式），或者&lt;br /&gt;
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* 系综的元素可以理解为在无限次试验的极限下，在类似但不完全受控的独立系统中，重复进行实验得到的系统的状态（&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经验概率 Empirical Probability&amp;lt;/font&amp;gt;）。&lt;br /&gt;
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These two meanings are equivalent for many purposes, and will be used interchangeably in this article.&lt;br /&gt;
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These two meanings are equivalent for many purposes, and will be used interchangeably in this article.&lt;br /&gt;
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这两种意义在很多情况下是等价的，在本文中可以互换使用。&lt;br /&gt;
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However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the [[Liouville's theorem (Hamiltonian)|Liouville equation]] (classical mechanics) or the [[von Neumann equation]] (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.&lt;br /&gt;
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However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.&lt;br /&gt;
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然而，这种概率是可被解释的，系综中的每个随时间演化的状态都可以由运动方程给出。因此，系综本身(状态的概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;刘维尔方程 Liouville Equation&amp;lt;/font&amp;gt;(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;冯·诺依曼方程 Von Neumann Equation&amp;lt;/font&amp;gt;(量子力学)给出。这些方程是简单地通过应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统的概率随时间演化过程中是守恒的。&lt;br /&gt;
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One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as ''equilibrium ensembles'' and their condition is known as ''statistical equilibrium''. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state.{{NoteTag|Statistical equilibrium should not be confused with ''[[mechanical equilibrium]]''. The latter occurs when a mechanical system has completely ceased to evolve even on a microscopic scale, due to being in a state with a perfect balancing of forces. Statistical equilibrium generally involves states that are very far from mechanical equilibrium.}} The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.&lt;br /&gt;
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One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.&lt;br /&gt;
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有一种特殊的系综是不随时间演化的。这样的系综称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;平衡系综 Equilibrium Ensembles&amp;lt;/font&amp;gt;，它们的状态称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计平衡 Statistical Equilibrium&amp;lt;/font&amp;gt;。如果对于每个状态，无论是未来还是过去，该系综都包含在内，并且其概率等于处于该状态的概率，则出现统计平衡的情况。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学研究更一般情况下的可以随时间演化的系综，以及（或）非孤立系统的系综。&lt;br /&gt;
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== Statistical thermodynamics 统计热力学==&lt;br /&gt;
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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the [[classical thermodynamics]] of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in [[thermodynamic equilibrium]], and the microscopic behaviours and motions occurring inside the material.&lt;br /&gt;
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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.&lt;br /&gt;
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统计热力学(也称为平衡态统计力学)的主要目标是根据组成某材料的粒子的性质和它们之间的相互作用，推导出材料的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典热力学 Classical Thermodynamics&amp;lt;/font&amp;gt;。换句话说，统计热力学提供了热力学平衡态中物质的宏观性质与物质内部微观行为和运动之间的联系。&lt;br /&gt;
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Whereas statistical mechanics proper involves dynamics, here the attention is focussed on ''statistical equilibrium'' (steady state). Statistical equilibrium does not mean that the particles have stopped moving ([[mechanical equilibrium]]), rather, only that the ensemble is not evolving.&lt;br /&gt;
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Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.&lt;br /&gt;
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然而统计力学本身就涉及到动态变化，此时的关注点集中在统计平衡(稳态)上。统计平衡并不意味着粒子已经停止运动(力学平衡) ，相反，只是系综没有进化。&lt;br /&gt;
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=== Fundamental postulate 基本假设===&lt;br /&gt;
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A [[sufficient condition|sufficient]] (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).&lt;br /&gt;
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孤立系统统计平衡的一个充分(但不是必要)条件是，其概率分布只是某些守恒量(总能量、总粒子数等)的函数。&lt;br /&gt;
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There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt; Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.&lt;br /&gt;
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There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.&lt;br /&gt;
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有许多不同的平衡系综可以考虑，但只有一些适用于热力学。为了说明为什么给定系统的系综具有这样或那样的形式，还需要一些额外的假设。&lt;br /&gt;
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A common approach found in many textbooks is to take the ''equal a priori probability postulate''.&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt; This postulate states that&lt;br /&gt;
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A common approach found in many textbooks is to take the equal a priori probability postulate. This postulate states that&lt;br /&gt;
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在许多教科书中常见的一种方法是采用&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;先验概率相等假设 Equal A Priori Probability Postulate&amp;lt;/font&amp;gt;。这个假设表明&lt;br /&gt;
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''For an isolated system with an exactly known energy and exactly known composition, the system can be found with ''equal probability'' in any [[microstate (statistical mechanics)|microstate]] consistent with that knowledge.''&lt;br /&gt;
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For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge.&lt;br /&gt;
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对于一个已知精确能量和组成的孤立系统，可以在任何符合条件的微观状态下等概率的找到该系统。&lt;br /&gt;
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The equal a priori probability postulate therefore provides a motivation for the [[microcanonical ensemble]] described below. There are various arguments in favour of the equal a priori probability postulate:&lt;br /&gt;
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The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:&lt;br /&gt;
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因此，先验概率相等假设为下面描述的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微正则系综 Microcanonical Ensemble&amp;lt;/font&amp;gt;提供了一个动力。有各种各样的论据支持先验概率相等假设:&lt;br /&gt;
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* [[Ergodic hypothesis]]: An ergodic system is one that evolves over time to explore &amp;quot;all accessible&amp;quot; states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.&lt;br /&gt;
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* [[Principle of indifference]]: In the absence of any further information, we can only assign equal probabilities to each compatible situation.&lt;br /&gt;
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* [[Maximum entropy thermodynamics|Maximum information entropy]]: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest [[Gibbs entropy]] ([[information entropy]]).&amp;lt;ref&amp;gt;{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;各态历经假设 Ergodic hypothesis&amp;lt;/font&amp;gt;]]:各态历经系统是一种随着时间的演化而探索“所有可到达”状态的系统:所有具有相同能量和组成的状态。在各态历经系统中，微正则系综是唯一可能的具有固定能量的平衡系综。这种方法的适用性有限，因为大多数系统不是各态历经的。&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;无差别原则 Principle of Indifference&amp;lt;/font&amp;gt;]]: 在没有更多信息的情况下，我们只能对每一个相容的情况分配相等的概率。&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;最大热力学熵|最大信息熵 Maximum Entropy Thermodynamics|Maximum Information Entropy&amp;lt;/font&amp;gt;]]: 无差异原则的一个更详细的版本表明，正确的系综是与已知信息兼容且具有最大[[吉布斯熵]] ([[信息熵]])的系综。&amp;lt;ref&amp;gt;{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Other fundamental postulates for statistical mechanics have also been proposed.&amp;lt;ref name=&amp;quot;uffink&amp;quot;/&amp;gt;&lt;br /&gt;
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Other fundamental postulates for statistical mechanics have also been proposed.&lt;br /&gt;
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其他关于统计力学的基本假设也有被提出。&lt;br /&gt;
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===Three thermodynamic ensembles 三种热力学系综===&lt;br /&gt;
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{{main|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}&lt;br /&gt;
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There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt; These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.&lt;br /&gt;
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There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.&lt;br /&gt;
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对于任何有限体积的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;孤立系统 Isolated System&amp;lt;/font&amp;gt;，可以定义三种简单形式的平衡系综。这些是统计热力学中最经常讨论的系综。在宏观极限(定义如下) ，它们都与经典热力学有对应。&lt;br /&gt;
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; [[Microcanonical ensemble]]&lt;br /&gt;
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Microcanonical Ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微正则系综 Microcanonical Ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.&lt;br /&gt;
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describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.&lt;br /&gt;
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描述了一个具有精确给定能量和固定成分(精确数量的粒子)的系统。微正则系综中，与能量和组成相一致的每个可能状态的概率是相等的。&lt;br /&gt;
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; [[Canonical ensemble]]&lt;br /&gt;
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Canonical ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;正则系综 Canonical Ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system of fixed composition that is in [[thermal equilibrium]]{{NoteTag|The transitive thermal equilibrium (as in, &amp;quot;X is thermal equilibrium with Y&amp;quot;) used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.}} with a [[heat bath]] of a precise [[thermodynamic temperature|temperature]]. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.&lt;br /&gt;
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describes a system of fixed composition that is in thermal equilibrium with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.&lt;br /&gt;
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描述了一个固定成分的系统，这个系统与一个精确温度的热浴形成热平衡。正则系综包含能量不同但组成完全相同的状态; 根据总能量的不同，系综中不同的状态被赋予不同的概率。&lt;br /&gt;
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; [[Grand canonical ensemble]]&lt;br /&gt;
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Grand canonical ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;巨正则系综 Grand Canonical Ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise [[chemical potential]]s for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.&lt;br /&gt;
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describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.&lt;br /&gt;
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描述了一个具有非固定成分(不确定粒子数)在热库中处于热力学和化学平衡的系统。热库具有精确的温度，各种类型的粒子具有精确的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;化学势 Chemical Potential&amp;lt;/font&amp;gt;。巨正则系综包含不同能量和状态的大量粒子; 根据总能量和粒子数的不同，系综中不同状态的概率也不同。&lt;br /&gt;
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For systems containing many particles (the [[thermodynamic limit]]), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used.&amp;lt;ref name=&amp;quot;Reif&amp;quot;&amp;gt;{{cite book | last = Reif | first = F. | title = Fundamentals of Statistical and Thermal Physics | publisher = McGraw–Hill | year = 1965 | isbn = 9780070518001 | page = [https://archive.org/details/fundamentalsofst00fred/page/227 227] | url-access = registration | url = https://archive.org/details/fundamentalsofst00fred/page/227 }}&amp;lt;/ref&amp;gt; The Gibbs theorem about equivalence of ensembles&amp;lt;ref&amp;gt;{{cite journal |doi=10.1007/s10955-015-1212-2|title=Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels|journal=Journal of Statistical Physics|volume=159|issue=5|pages=987–1016|year=2015|last1=Touchette|first1=Hugo|arxiv=1403.6608}}&amp;lt;/ref&amp;gt; was developed into the theory of [[concentration of measure]] phenomenon,&amp;lt;ref&amp;gt;{{cite book |doi=10.1090/surv/089|title=The Concentration of Measure Phenomenon|volume=89|series=Mathematical Surveys and Monographs|year=2005|isbn=9780821837924|last1=Ledoux|first1=Michel}}.&amp;lt;/ref&amp;gt; which has applications in many areas of science, from functional analysis to methods of [[artificial intelligence]] and [[big data]] technology.&amp;lt;ref&amp;gt;{{cite journal |doi=10.1098/rsta.2017.0237|pmc=5869543|title=Blessing of dimensionality: Mathematical foundations of the statistical physics of data|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=376|issue=2118|pages=20170237|year=2018|last1=Gorban|first1=A. N.|last2=Tyukin|first2=I. Y.}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.&lt;br /&gt;
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对于包含大量粒子的系统(&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热力学极限 Thermodynamic Limit&amp;lt;/font&amp;gt;) ，上面列出的三种系综都倾向于体现出相同的行为。因此，使用哪种系综只是一个简单的数学问题。发展成为(&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;测度现象集中理论 Concentration of Measure Phenomenon&amp;lt;/font&amp;gt;系综等价的吉布斯定理，在从函数分析到人工智能和大数据技术等许多科学领域都有广泛的应用。&lt;br /&gt;
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Important cases where the thermodynamic ensembles ''do not'' give identical results include:&lt;br /&gt;
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Important cases where the thermodynamic ensembles do not give identical results include:&lt;br /&gt;
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热力学系综不能给出相同结果的重要例子包括:&lt;br /&gt;
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* Microscopic systems.&lt;br /&gt;
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* Large systems at a phase transition.&lt;br /&gt;
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* Large systems with long-range interactions.&lt;br /&gt;
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* 微观系统&lt;br /&gt;
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* 处于相变的宏观系统&lt;br /&gt;
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* 长程关联的宏观系统&lt;br /&gt;
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In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&lt;br /&gt;
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In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.&lt;br /&gt;
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在这些情况下必须选择正确的热力学系综，因为这些系综之间不仅在涨落的大小方面有可观测的差异，而且在平均量方面都有可观察的差异，如粒子数的分布。正确的系综是对应于该系统的准备和表征的方式ーー换句话说，系综反映我们对系统的认知。&lt;br /&gt;
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:{| class=&amp;quot;wikitable sortable mw-collapsible mw-collapsed&amp;quot;&lt;br /&gt;
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{| class=&amp;quot;wikitable sortable mw-collapsible mw-collapsed&amp;quot;&lt;br /&gt;
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{ | class“ wikable sortable mw-collable mw-collapse”&lt;br /&gt;
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! rowspan=&amp;quot;2&amp;quot;|&lt;br /&gt;
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! rowspan=&amp;quot;2&amp;quot;|&lt;br /&gt;
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!Rowspan“2” | &lt;br /&gt;
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! colspan=&amp;quot;3&amp;quot;| Thermodynamic ensembles&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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! colspan=&amp;quot;3&amp;quot;| Thermodynamic ensembles&lt;br /&gt;
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!Colspan“3” | 热力学系数&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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! [[Microcanonical ensemble|Microcanonical]]&lt;br /&gt;
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! Microcanonical&lt;br /&gt;
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!微正则化&lt;br /&gt;
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! [[Canonical ensemble|Canonical]]&lt;br /&gt;
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! Canonical&lt;br /&gt;
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!典范&lt;br /&gt;
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! [[Grand canonical ensemble|Grand canonical]]&lt;br /&gt;
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! Grand canonical&lt;br /&gt;
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!巨正典&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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! Fixed variables&lt;br /&gt;
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! Fixed variables&lt;br /&gt;
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!固定变量&lt;br /&gt;
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| &amp;lt;center&amp;gt;{{math|''N, E, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
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| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
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中心，中心&lt;br /&gt;
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| &amp;lt;center&amp;gt;{{math|''N, T, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
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| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
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中心，中心&lt;br /&gt;
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| &amp;lt;center&amp;gt;{{math|''μ, T, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
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| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
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中心，中心&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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! Microscopic features&lt;br /&gt;
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! Microscopic features&lt;br /&gt;
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!微观特征&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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“ div class”“ plainlist”&lt;br /&gt;
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*&amp;lt;center&amp;gt;Number of [[Microstate (statistical mechanics)|microstates]]&amp;lt;/center&amp;gt;&lt;br /&gt;
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*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt; W &amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
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&amp;lt;/div&amp;gt;&lt;br /&gt;
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&amp;lt;/div&amp;gt;&lt;br /&gt;
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/ div&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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“ div class”“ plainlist”&lt;br /&gt;
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*&amp;lt;center&amp;gt;[[Canonical partition function]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;Z = \sum_k e^{- E_k / k_B T}&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
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&amp;lt;/div&amp;gt;&lt;br /&gt;
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/ div&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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“ div class”“ plainlist”&lt;br /&gt;
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*&amp;lt;center&amp;gt;[[Grand partition function]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\mathcal Z = \sum_k e^{ -(E_k - \mu N_k) /k_B T}&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
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/ div&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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! Macroscopic function&lt;br /&gt;
&lt;br /&gt;
! Macroscopic function&lt;br /&gt;
&lt;br /&gt;
!宏观功能&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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“ div class”“ plainlist”&lt;br /&gt;
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*&amp;lt;center&amp;gt;[[Boltzmann entropy]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;S =  k_B \log W&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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“ div class”“ plainlist”&lt;br /&gt;
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*&amp;lt;center&amp;gt;[[Helmholtz free energy]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;F = - k_B T \log Z&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
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&amp;lt;/div&amp;gt;&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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“ div class”“ plainlist”&lt;br /&gt;
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*&amp;lt;center&amp;gt;[[Grand potential]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\Omega =- k_B T \log \mathcal Z &amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
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&amp;lt;/div&amp;gt;&lt;br /&gt;
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=== Calculation Methods 计算方法===&lt;br /&gt;
&lt;br /&gt;
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.&lt;br /&gt;
&lt;br /&gt;
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.&lt;br /&gt;
&lt;br /&gt;
一旦计算出一个系统的特征状态函数，该系统就被“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是对最一般的(和现实的)情况进行精确的求解实在是太复杂了。存在各种方法来近似真实的系综，并且计算平均量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
====Exact 精确解====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are some cases which allow exact solutions.&lt;br /&gt;
&lt;br /&gt;
There are some cases which allow exact solutions.&lt;br /&gt;
&lt;br /&gt;
有些情况可以得到精确解。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics).&lt;br /&gt;
&lt;br /&gt;
* Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably, idealized gases of non-interacting particles have this property, allowing exact derivations of [[Maxwell–Boltzmann statistics]], [[Fermi–Dirac statistics]], and [[Bose–Einstein statistics]].&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few [[toy model]]s.&amp;lt;ref&amp;gt;{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}&amp;lt;/ref&amp;gt; Some examples include the [[Bethe ansatz]], [[square-lattice Ising model]] in zero field, [[hard hexagon model]].&lt;br /&gt;
&lt;br /&gt;
* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。&lt;br /&gt;
&lt;br /&gt;
* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;麦克斯韦–玻尔兹曼统计 Maxwell–Boltzmann Statistics&amp;lt;/font&amp;gt;]]， [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;费米-狄拉克统计 Fermi–Dirac Statistics&amp;lt;/font&amp;gt;]]，和[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;波色-爱因斯坦统计 Bose–Einstein Statistics&amp;lt;/font&amp;gt;]]。&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。&amp;lt;ref&amp;gt;{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}&amp;lt;/ref&amp;gt; 一些例子包括[[Bethe ansatz]]，零场下的[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt; 二维格点伊辛模型 Square-Lattice Ising Model&amp;lt;/font&amp;gt;]]，[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;硬六边形模型 Hard Hexagon Model&amp;lt;/font&amp;gt;]]。&lt;br /&gt;
&lt;br /&gt;
====Monte Carlo 蒙特卡罗方法====&lt;br /&gt;
&lt;br /&gt;
{{main|Monte Carlo method}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
One approximate approach that is particularly well suited to computers is the [[Monte Carlo method]], which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.&lt;br /&gt;
&lt;br /&gt;
One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.&lt;br /&gt;
&lt;br /&gt;
一个特别适合于计算机的近似方法是&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;蒙特卡罗方法 Monte Carlo Method&amp;lt;/font&amp;gt;，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态可构成系统全部状态集的代表样本，就可以得到近似的特征函数。随着随机样本数量的增加，误差可以降低到任意低的水平。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* The [[Metropolis–Hastings algorithm]] is a classic Monte Carlo method which was initially used to sample the canonical ensemble.&lt;br /&gt;
&lt;br /&gt;
* [[Path integral Monte Carlo]], also used to sample the canonical ensemble.&lt;br /&gt;
&lt;br /&gt;
* [[Metropolis–Hastings 算法]] 是一种经典的蒙特卡罗算法，最初用于正则系综的采样。&lt;br /&gt;
&lt;br /&gt;
* [[路径积分蒙特卡罗方法]] 也可以用于正则系综的采样。&lt;br /&gt;
&lt;br /&gt;
==== Other 其他====&lt;br /&gt;
&lt;br /&gt;
* For rarefied non-ideal gases, approaches such as the [[cluster expansion]] use [[perturbation theory]] to include the effect of weak interactions, leading to a [[virial expansion]].&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* For dense fluids, another approximate approach is based on reduced distribution functions, in particular the [[radial distribution function]].&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* [[Molecular dynamics]] computer simulations can be used to calculate [[microcanonical ensemble]] averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions.&lt;br /&gt;
&lt;br /&gt;
* Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.&lt;br /&gt;
&lt;br /&gt;
* 对于稀薄的非理想气体，[[团簇膨胀 cluster expansion]]等方法使用[[微扰理论 perturbation theory]]来涵盖弱相互作用的影响，得到[[维里展开 virial expansion]]。&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是[[径向分布函数 radial distribution function]]。&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* [[分子动力学 Molecular dynamics]] 计算机模拟可以用来计算各态历经系统中的[[微正则系综]]平均。对于包含与随机热浴有连接的系统，他们还可以在正则和巨正则条件下建模。&lt;br /&gt;
&lt;br /&gt;
* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。&lt;br /&gt;
&lt;br /&gt;
== Non-equilibrium statistical mechanics 非平衡态统计力学==&lt;br /&gt;
&lt;br /&gt;
{{see also|Non-equilibrium thermodynamics}}&lt;br /&gt;
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There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:&lt;br /&gt;
&lt;br /&gt;
There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:&lt;br /&gt;
&lt;br /&gt;
有许多令人感兴趣的物理现象都涉及到失去平衡的准热力学过程，例如:&lt;br /&gt;
&lt;br /&gt;
* [[Thermal conduction|heat transport by the internal motions in a material]], driven by a temperature imbalance,&lt;br /&gt;
&lt;br /&gt;
* [[Electrical conduction|electric currents carried by the motion of charges in a conductor]], driven by a voltage imbalance,&lt;br /&gt;
&lt;br /&gt;
* spontaneous [[chemical reaction]]s driven by a decrease in free energy,&lt;br /&gt;
&lt;br /&gt;
* [[friction]], [[dissipation]], [[quantum decoherence]],&lt;br /&gt;
&lt;br /&gt;
* systems being pumped by external forces ([[optical pumping]], etc.),&lt;br /&gt;
&lt;br /&gt;
* and irreversible processes in general.&lt;br /&gt;
&lt;br /&gt;
* [[热传导|材料内部的物质运动来传导热量]]，由热度不平衡来驱动，&lt;br /&gt;
&lt;br /&gt;
* [[导电|导体内电荷的运动产生电流]]，由电压不平衡来驱动，&lt;br /&gt;
&lt;br /&gt;
* 自发的 [[化学反应]]，由自由能的下降驱动，&lt;br /&gt;
&lt;br /&gt;
* [[摩擦]]，[[耗散]]，[[量子退相干]，&lt;br /&gt;
&lt;br /&gt;
* 由外力泵送的系统([光泵]]等)&lt;br /&gt;
&lt;br /&gt;
* 以及一般意义的可逆过程&lt;br /&gt;
&lt;br /&gt;
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)&lt;br /&gt;
&lt;br /&gt;
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)&lt;br /&gt;
&lt;br /&gt;
所有这些过程都是以特征速率随时间发生，这些速率对于工程来说非常重要。非平衡态统计力学研究领域关注的是在微观水平上理解这些非平衡过程。(统计热力学只能用来计算在外部不平衡被消除，整体回归到平衡状态之后的最终结果。)&lt;br /&gt;
&lt;br /&gt;
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In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as [[Liouville's theorem (Hamiltonian)|Liouville's equation]] or its quantum equivalent, the [[von Neumann equation]]. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's [[Gibbs entropy]] is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.&lt;br /&gt;
&lt;br /&gt;
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.&lt;br /&gt;
&lt;br /&gt;
原则上，非平衡态统计力学在数学上可以是精确的: 孤立系统的系综根据确定性方程随时间演化，如刘维尔方程或其量子等价、冯·诺依曼方程。这些方程是将运动力学方程独立应用于系综中每个状态的结果。不幸的是，这些系综演化方程继承了底层动力学运动的大部分复杂性，因此很难得到精确解。此外，系综演化方程是完全可逆的，不会破坏信息(系综的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;吉布斯熵 Gibbs entropy&amp;lt;/font&amp;gt;被保留)。为了在模拟不可逆过程中取得进展，除了概率和可逆力学外，还必须考虑其他因素。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.&lt;br /&gt;
&lt;br /&gt;
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.&lt;br /&gt;
&lt;br /&gt;
因此，非平衡力学是一个活跃的理论研究领域，因为这些额外假设的有效范围仍将继续探索。在下面的小节中描述了一些方法。&lt;br /&gt;
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=== Stochastic methods 随机方法===&lt;br /&gt;
&lt;br /&gt;
One approach to non-equilibrium statistical mechanics is to incorporate [[stochastic]] (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from [[Black hole information paradox|hypothetical situations involving black holes]], a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as [[Chaos theory|chaotic]] or [[pseudorandom]] influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.&lt;br /&gt;
&lt;br /&gt;
One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.&lt;br /&gt;
&lt;br /&gt;
处理非平衡态统计力学的一个方法是将随机行为引入系统。随机行为可以破坏系综中包含的信息。虽然这在技术上是不准确的(除了涉及黑洞的假设情况外，黑洞系统本身不会导致信息丢失) ，但这种随机性是为了反映出，随着时间的推移，感兴趣的信息会在系统内部转化为微妙的相关性，或者系统与环境之间的相关性。这些关联表现为对感兴趣的变量的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;混沌 Chaos&amp;lt;/font&amp;gt;或伪随机的影响。用适当的随机性取代这些相关性，计算可以变得容易得多。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
|1 = ''[[Boltzmann transport equation]]'': An early form of stochastic mechanics appeared even before the term &amp;quot;statistical mechanics&amp;quot; had been coined, in studies of [[kinetic theory of gases|kinetic theory]]. [[James Clerk Maxwell]] had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. [[Ludwig Boltzmann]] subsequently showed that, by taking this [[molecular chaos]] for granted as a complete randomization, the motions of particles in a gas would follow a simple [[Boltzmann transport equation]] that would rapidly restore a gas to an equilibrium state (see [[H-theorem]]).&lt;br /&gt;
&lt;br /&gt;
|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term &amp;quot;statistical mechanics&amp;quot; had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).&lt;br /&gt;
&lt;br /&gt;
&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;玻尔兹曼输运方程 Boltzmann Transport Equation&amp;lt;/font&amp;gt;: 在动力学理论研究中，早期的随机力学甚至在“统计力学”一词被创造之前就已经出现了。詹姆斯·克拉克·麦克斯韦已经证明分子碰撞会导致气体内部明显的混沌运动。路德维希·玻尔兹曼随后证明，如果把这种分子混沌理所当然地看作是一种完全的随机化，那么气体中粒子的运动将遵循一个简单的玻尔兹曼输运方程，这个方程将使气体迅速恢复到平衡状态(见H-定理)。&lt;br /&gt;
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The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the &amp;quot;interesting&amp;quot; information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped [[semiconductor]]s (in [[transistor]]s), where the electrons are indeed analogous to a rarefied gas.&lt;br /&gt;
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The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the &amp;quot;interesting&amp;quot; information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas.&lt;br /&gt;
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玻耳兹曼输运方程及其相关方法是非平衡态统计力学的重要工具，因为它们极其简单。这些近似方法在“感兴趣的”信息立即(在一次碰撞之后)变成微妙关联的系统中非常有效，这种关联本质上限制它们为稀薄气体。玻耳兹曼输运方程被发现在模拟轻掺杂半导体(晶体管)的电子输运中非常有用，其中的电子确实类似于稀薄气体。&lt;br /&gt;
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A quantum technique related in theme is the [[random phase approximation]].&lt;br /&gt;
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A quantum technique related in theme is the random phase approximation.&lt;br /&gt;
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一个与主题相关的量子技术是&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;随机相位近似 Random Phase Approximation&amp;lt;/font&amp;gt;。&lt;br /&gt;
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|2 = ''[[BBGKY hierarchy]]'':&lt;br /&gt;
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|2 = BBGKY hierarchy:&lt;br /&gt;
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2 BBGKY 层级:&lt;br /&gt;
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In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The [[BBGKY hierarchy]] (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.&lt;br /&gt;
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In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.&lt;br /&gt;
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在液体和稠密气体中，不能在一次碰撞后立即丢掉粒子之间的关联。BBGKY 层级结构(&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;层级结构 Bogoliubov-Born-Green-Kirkwood-Yvon&amp;lt;/font&amp;gt;)提供了一种推导玻尔兹曼型方程的方法，但也可以将它们扩展到稀薄气体情况之外，包括在几次碰撞之后的相关性。&lt;br /&gt;
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|3 = ''[[Keldysh formalism]]'' (a.k.a. NEGF—non-equilibrium Green functions):&lt;br /&gt;
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|3 = Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions):&lt;br /&gt;
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| 3 Keldysh 公式。NEGF—非平衡态格林函数):&lt;br /&gt;
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A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic [[quantum transport]] calculations.&lt;br /&gt;
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A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic quantum transport calculations.&lt;br /&gt;
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人们在 Keldysh 公式中发明了包含随机动力学的量子方法，这种方法常用于电子量子输运计算。&lt;br /&gt;
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|4 = Stochastic [[Liouville's theorem (Hamiltonian)|Liouville equation]]&lt;br /&gt;
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|4 = Stochastic Liouville equation&lt;br /&gt;
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| 4随机 刘维尔方程&lt;br /&gt;
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=== Near-equilibrium methods 近平衡态方法 ===&lt;br /&gt;
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Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in [[linear response theory]]. A remarkable result, as formalized by the [[fluctuation-dissipation theorem]], is that the response of a system when near equilibrium is precisely related to the [[Statistical fluctuations|fluctuations]] that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or &amp;quot;know&amp;quot; how it came to be away from equilibrium.&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;{{rp|664}}&lt;br /&gt;
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Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or &amp;quot;know&amp;quot; how it came to be away from equilibrium.&lt;br /&gt;
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非平衡态统计力学模型处理的另一类重要的系统，是对平衡态仅有非常轻微扰动的系统。在很小的扰动下，响应可以用&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;线性响应理论 Linear Response Theory&amp;lt;/font&amp;gt;进行分析。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;涨落-耗散定理 Fluctuation-Dissipation Theorem&amp;lt;/font&amp;gt;是其中一个重要的结果，近平衡态系统的响应与系统总体平衡时的涨落准确相关。从本质上讲，一个系统如果稍微偏离平衡，无论是由于外力还是由于涨落，都会以同样的方式向平衡方向弛豫。因为这个系统无法区分偏离和回归，也“不知道”它是如何偏离平衡的。&lt;br /&gt;
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This provides an indirect avenue for obtaining numbers such as [[Ohm's law|ohmic conductivity]] and [[thermal conductivity]] by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.&lt;br /&gt;
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This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.&lt;br /&gt;
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这提供了一个间接的方法，通过从平衡态统计力学中提取结果来获得诸如欧姆电导率和热导率之类的物理量。由于平衡态统计力学在数学上有很好的定义，而且(在某些情况下)更易于计算，因此在近平衡态统计力学中，涨落-耗散关联可以成为一种方便的计算捷径。&lt;br /&gt;
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A few of the theoretical tools used to make this connection include:&lt;br /&gt;
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A few of the theoretical tools used to make this connection include:&lt;br /&gt;
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用于建立这种关联的一些理论工具包括:&lt;br /&gt;
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* [[Fluctuation–dissipation theorem]]&lt;br /&gt;
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* [[Onsager reciprocal relations]]&lt;br /&gt;
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* [[Green–Kubo relations]]&lt;br /&gt;
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* [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker formalism]]&lt;br /&gt;
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* [[Mori–Zwanzig formalism]]&lt;br /&gt;
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* [[涨落-耗散定理]]&lt;br /&gt;
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* [[昂萨格倒易关系]]&lt;br /&gt;
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* [[格林-库伯关系]]&lt;br /&gt;
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* [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker 公式]]&lt;br /&gt;
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* [[Mori–Zwanzig 公式]]&lt;br /&gt;
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=== Hybrid methods 组合方法===&lt;br /&gt;
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An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects ([[weak localization]], [[conductance fluctuations]]) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic [[dephasing]] by interactions between various electrons by use of the Keldysh method.&amp;lt;ref&amp;gt;{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid =  | pmc = |bibcode = 1982JPhC...15.7367A }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid =  | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.&lt;br /&gt;
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一种先进的方法使用了随机方法和线性响应理论的结合。例如，计算电子系统电导中的量子相干效应(弱局域化，电导涨落)的一种方法是使用 Green-Kubo 关系，包括随机退相的各种电子之间的相互作用，使用Keldysh 方法。&lt;br /&gt;
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==Applications outside thermodynamics==&lt;br /&gt;
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The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:&lt;br /&gt;
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The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:&lt;br /&gt;
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系综也可以用来分析系统状态有不确定性的一般机械系统。在下列情况中也会使用系综:&lt;br /&gt;
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* [[propagation of uncertainty]] over time,&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt;&lt;br /&gt;
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* [[regression analysis]] of gravitational [[orbit]]s,&lt;br /&gt;
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* [[ensemble forecasting]] of weather,&lt;br /&gt;
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* dynamics of [[neural networks]],&lt;br /&gt;
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* bounded-rational [[potential game]]s in game theory and economics.&lt;br /&gt;
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* 不确定性随时间传播，&lt;br /&gt;
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* 引力[[轨道]]的[[回归分析]]，&lt;br /&gt;
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* 天气的[[系综预报]]，&lt;br /&gt;
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* [[神经网络]]动力学，&lt;br /&gt;
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* 博弈论和经济学中的有界理性[[潜在对策]]。&lt;br /&gt;
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== History 历史==&lt;br /&gt;
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In 1738, Swiss physicist and mathematician [[Daniel Bernoulli]] published ''Hydrodynamica'' which laid the basis for the [[kinetic theory of gases]]. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as [[heat]] is simply the kinetic energy of their motion.&amp;lt;ref name=&amp;quot;uffink&amp;quot;/&amp;gt;&lt;br /&gt;
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In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.&lt;br /&gt;
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1738年，瑞士的物理学家和数学家丹尼尔·伯努利发表了《水动力学》 ，这本书奠定了&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;气体动力学理论 Kinetic Theory of Gases&amp;lt;/font&amp;gt;的基础。在这项工作中，伯努利假定气体是由大量向各个方向运动的分子组成的，它们对表面的影响导致了我们感觉到的气体压力，而我们感受到的热仅仅是它们运动的动能，这一点直到今天仍在沿用。&lt;br /&gt;
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In 1859, after reading a paper on the diffusion of molecules by [[Rudolf Clausius]], Scottish physicist [[James Clerk Maxwell]] formulated the [[Maxwell distribution]] of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.&amp;lt;ref&amp;gt;See:&lt;br /&gt;
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In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.&amp;lt;ref&amp;gt;See:&lt;br /&gt;
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1859年，在阅读了 Rudolf Clausius 的一篇关于分子扩散的论文之后，苏格兰物理学家詹姆斯·克拉克·麦克斯韦提出了分子速度的麦克斯韦-玻尔兹曼分布，它给出了在一个特定范围内具有某种速度的分子的比例。这是物理学里第一个统计定律。麦克斯韦还提出了第一个力学论点，即分子碰撞必然导致温度的平衡，从而趋向平衡。五年之后，也就是1864年，维也纳的年轻学生路德维希·玻尔兹曼(Ludwig Boltzmann)偶然发现了麦克斯韦尔的论文，并花了大半辈子的时间来进一步研究这一课题。&lt;br /&gt;
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*  Maxwell, J.C. (1860) [https://books.google.com/books?id=-YU7AQAAMAAJ&amp;amp;pg=PA19#v=onepage&amp;amp;q&amp;amp;f=false &amp;quot;Illustrations of the dynamical theory of gases. Part I.  On the motions and collisions of perfectly elastic spheres,&amp;quot;] ''Philosophical Magazine'', 4th series, '''19''' :  19–32. &lt;br /&gt;
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*  Maxwell, J.C. (1860) [https://books.google.com/books?id=DIc7AQAAMAAJ&amp;amp;pg=PA21#v=onepage&amp;amp;q&amp;amp;f=false &amp;quot;Illustrations of the dynamical theory of gases. Part II.  On the process of diffusion of two or more kinds of moving particles among one another,&amp;quot;] ''Philosophical Magazine'', 4th series, '''20''' :  21–37.&amp;lt;/ref&amp;gt; This was the first-ever statistical law in physics.&amp;lt;ref&amp;gt;{{cite book |last = Mahon |first = Basil |title=The Man Who Changed Everything – the Life of James Clerk Maxwell |location=Hoboken, NJ |publisher=Wiley |year=2003 |isbn=978-0-470-86171-4 |oclc=52358254}}&amp;lt;/ref&amp;gt; Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.&amp;lt;ref&amp;gt;{{cite journal | last = Gyenis | first = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}&amp;lt;/ref&amp;gt; Five years later, in 1864, [[Ludwig Boltzmann]], a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.&lt;br /&gt;
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Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 ''Lectures on Gas Theory''.&amp;lt;ref&amp;gt;{{cite book |title = Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems |journal=Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Edited by Ebeling Werner &amp;amp; Sokolov Igor M. Published by World Scientific Press |volume=8 |last1=Ebeling |first1=Werner |last2=Sokolov |first2=Igor M. |year=2005 |isbn=978-90-277-1674-3 |pages=3–12 |url = https://books.google.com/books?id=KUjFHbid8A0C|bibcode=2005stst.book.....E |doi=10.1142/2012 |series = Series on Advances in Statistical Mechanics }} (section 1.2)&amp;lt;/ref&amp;gt; Boltzmann's original papers on the statistical interpretation of thermodynamics, the [[H-theorem]], [[transport theory (statistical physics)|transport theory]], [[thermal equilibrium]], the [[equation of state]] of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his [[H-theorem|''H''-theorem]].&lt;br /&gt;
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Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory. Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.&lt;br /&gt;
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统计力学是19世纪70年代由玻尔兹曼的工作创立的，其中大部分在他1896年的气体理论演讲中集结出版。在维也纳学院和其他学会的会议记录中，玻尔兹曼关于热力学的统计解释、 &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;H-定理H-theorem&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;输运理论 Transport Theory&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热平衡Thermal Equilibrium&amp;lt;/font&amp;gt;、气体的状态方程以及类似主题的原始论文占据了大约2000页。玻尔兹曼引入了平衡系综的概念，并用他的H-定理第一次研究了非平衡态统计力学。&lt;br /&gt;
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The term &amp;quot;statistical mechanics&amp;quot; was coined by the American mathematical physicist [[Josiah Willard Gibbs|J. Willard Gibbs]] in 1884.&amp;lt;ref&amp;gt;J. W. Gibbs, &amp;quot;On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics.&amp;quot; Proceedings of the American Association for the Advancement of Science, '''33''', 57-58 (1884). Reproduced in ''The Scientific Papers of J. Willard Gibbs, Vol II'' (1906), [https://archive.org/stream/scientificpapers02gibbuoft#page/16/mode/2up pp.&amp;amp;nbsp;16].&amp;lt;/ref&amp;gt;{{NoteTag|1 = According to Gibbs, the term &amp;quot;statistical&amp;quot;, in the context of mechanics, i.e. statistical mechanics, was first used by the Scottish physicist [[James Clerk Maxwell]] in 1871. From: J. Clerk Maxwell, ''Theory of Heat'' (London, England: Longmans, Green, and Co., 1871), [https://books.google.com/books?id=DqAAAAAAMAAJ&amp;amp;pg=PA309 p.&amp;amp;nbsp;309]: &amp;quot;In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus.&amp;quot;}} &amp;quot;Probabilistic mechanics&amp;quot; might today seem a more appropriate term, but &amp;quot;statistical mechanics&amp;quot; is firmly entrenched.&amp;lt;ref&amp;gt;{{cite book |title = The enigma of probability and physics |last=Mayants |first=Lazar |year=1984 |publisher=Springer |isbn=978-90-277-1674-3 |page=174 |url = https://books.google.com/books?id=zmwEfXUdBJ8C&amp;amp;pg=PA174 }}&amp;lt;/ref&amp;gt; Shortly before his death, Gibbs published in 1902 ''[[Elementary Principles in Statistical Mechanics]]'', a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt; Gibbs' methods were initially derived in the framework [[classical mechanics]], however they were of such generality that they were found to adapt easily to the later [[quantum mechanics]], and still form the foundation of statistical mechanics to this day.&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&lt;br /&gt;
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The term &amp;quot;statistical mechanics&amp;quot; was coined by the American mathematical physicist J. Willard Gibbs in 1884. &amp;quot;Probabilistic mechanics&amp;quot; might today seem a more appropriate term, but &amp;quot;statistical mechanics&amp;quot; is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.&lt;br /&gt;
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1884年，美国数学物理学家约西亚·威拉德·吉布斯首创了“统计力学”一词。在今天看来，“概率力学”似乎是一个更合适的术语，但“统计力学”却根深蒂固。在吉布斯去世前不久，他于1902年在《统计力学》出版了《基本原理》一书，这本书正式确定了统计力学是解决所有力学系统——宏观的或微观的、气态的或非气态的——的一种完全通用的方法。吉布斯的方法最初是在经典力学的框架下产生的，然而它们是如此的普遍，以至于人们发现它们很容易适应后来的量子力学，直到今天仍然是统计力学的基础。&lt;br /&gt;
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== See also 其他可见==&lt;br /&gt;
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{{Div col}}&lt;br /&gt;
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{{Books-inline|Fundamentals of Statistical Mechanics}}&lt;br /&gt;
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* [[Thermodynamics]]: [[Non-equilibrium thermodynamics|non-equilibrium]], [[Chemical thermodynamics|chemical]]&lt;br /&gt;
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* [[Mechanics]]: [[Classical mechanics|classical]], [[Quantum mechanics|quantum]]&lt;br /&gt;
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* [[Probability]], [[Statistical ensemble (mathematical physics)|statistical ensemble]]&lt;br /&gt;
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* Numerical methods: [[Monte Carlo method]], [[molecular dynamics]]&lt;br /&gt;
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* [[Statistical physics]]&lt;br /&gt;
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* [[Quantum statistical mechanics]]&lt;br /&gt;
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* [[List of notable textbooks in statistical mechanics]]&lt;br /&gt;
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* [[List of publications in physics#Statistical mechanics|List of important publications in statistical mechanics]]&lt;br /&gt;
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{{Books-inline|统计力学基础}}&lt;br /&gt;
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* [[热力学]]: [[非平衡态热力学]], [[化学热力学]]&lt;br /&gt;
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* [[力学]]: [[经典力学]], [[量子力学]]&lt;br /&gt;
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* [[概率]], [[统计系综（数学物理）]]&lt;br /&gt;
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* 数值方法: [[蒙特卡洛方法]], [[分子动力学]]&lt;br /&gt;
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* [[统计物理]]&lt;br /&gt;
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* [[量子统计力学]]&lt;br /&gt;
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* [[统计力学著名教科书列表]]&lt;br /&gt;
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* [[统计力学重要文献列表]]&lt;br /&gt;
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== Notes ==&lt;br /&gt;
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== References ==&lt;br /&gt;
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{{Reflist&lt;br /&gt;
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参考文献&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;gibbs&amp;quot;&amp;gt;{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York |title-link=Elementary Principles in Statistical Mechanics }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;tolman&amp;quot;&amp;gt;{{cite book | last=Tolman |first=R. C. | author-link = Richard C. Tolman | year=1938 | title = The Principles of Statistical Mechanics | publisher=[[Dover Publications]] | isbn = 9780486638966 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;balescu&amp;quot;&amp;gt;{{cite book | isbn = 9780471046004 | title = Equilibrium and Non-Equilibrium Statistical Mechanics |last = Balescu |first = Radu | author-link = Radu Balescu | year = 1975 | publisher = John Wiley &amp;amp; Sons }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;uffink&amp;quot;&amp;gt;J. Uffink, &amp;quot;[http://philsci-archive.pitt.edu/2691/1/UffinkFinal.pdf Compendium of the foundations of classical statistical physics.]&amp;quot; (2006)&amp;lt;/ref&amp;gt;&lt;br /&gt;
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== External links ==&lt;br /&gt;
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{{Commons category|Statistical mechanics}}&lt;br /&gt;
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* [http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics] article by Lawrence Sklar for the [[Stanford Encyclopedia of Philosophy]].&lt;br /&gt;
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* [http://www.sklogwiki.org/ Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials.]  SklogWiki is particularly orientated towards liquids and soft condensed matter.&lt;br /&gt;
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* [https://web.archive.org/web/20060616002812/http://history.hyperjeff.net/statmech.html Statistical Thermodynamics] - Historical Timeline&lt;br /&gt;
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* [http://farside.ph.utexas.edu/teaching/sm1/statmech.pdf Thermodynamics and Statistical Mechanics] by Richard Fitzpatrick&lt;br /&gt;
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* [https://arxiv.org/abs/1107.0568 Lecture Notes in Statistical Mechanics and Mesoscopics] by Doron Cohen&lt;br /&gt;
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* {{YouTube |id = H1Zbp6__uNw&amp;amp;list=PLB72416C707D85AB0&amp;amp;index=1 |title = Videos of lecture series in statistical mechanics }} taught by [[Leonard Susskind]].&lt;br /&gt;
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* Vu-Quoc, L., [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008. this wiki site is down; see [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 this article in the web archive on 2012 April 28].&lt;br /&gt;
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[[Category:Concepts in physics]]&lt;br /&gt;
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Category:Concepts in physics&lt;br /&gt;
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分类: 物理概念&lt;br /&gt;
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[[Category:Physics]]&lt;br /&gt;
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Category:Physics&lt;br /&gt;
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类别: 物理学&lt;br /&gt;
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[[Category:Thermodynamics]]&lt;br /&gt;
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Category:Thermodynamics&lt;br /&gt;
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分类: 热力学&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Statistical mechanics]]. Its edit history can be viewed at [[统计力学/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
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	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E7%BB%9F%E8%AE%A1%E5%8A%9B%E5%AD%A6&amp;diff=12951</id>
		<title>统计力学</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E7%BB%9F%E8%AE%A1%E5%8A%9B%E5%AD%A6&amp;diff=12951"/>
		<updated>2020-08-18T11:59:56Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;已由佳欣初步翻译&lt;br /&gt;
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{{Short description|Physics of large number of particles' statistical behavior}}&lt;br /&gt;
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{{Statistical mechanics}}&lt;br /&gt;
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'''Statistical mechanics''' is one of the pillars of modern [[physics]]. It is necessary for the fundamental study of any physical system that has many [[degrees of freedom]]. The approach is based on [[Statistics|statistical]] methods, [[probability theory]] and the [[Microscopic scale|microscopic]] physical laws.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;{{NoteTag|The term ''statistical mechanics'' is sometimes used to refer to only ''statistical thermodynamics''. This article takes the broader view. By some definitions, ''[[statistical physics]]'' is an even broader term which statistically studies any type of physical system, but is often taken to be synonymous with statistical mechanics.}}&lt;br /&gt;
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Statistical mechanics is one of the pillars of modern physics. It is necessary for the fundamental study of any physical system that has many degrees of freedom. The approach is based on statistical methods, probability theory and the microscopic physical laws.&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计力学 Statistical mechanics&amp;lt;/font&amp;gt;是现代物理学的支柱之一。对于具有多个&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;自由度 Degrees of Freedom&amp;lt;/font&amp;gt;的物理系统的基础研究，统计力学是不可或缺的。统计力学的方法是基于&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计学方法 Statistical Methods&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;概率论 Probability Theory&amp;lt;/font&amp;gt;和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微观物理定律 Microscopic Physical Laws&amp;lt;/font&amp;gt;。&lt;br /&gt;
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It can be used to explain the [[thermodynamics|thermodynamic]] behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as '''statistical thermodynamics''' or '''equilibrium statistical mechanics'''.&lt;br /&gt;
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It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.&lt;br /&gt;
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统计力学一个分支可以用来解释大系统的热力学行为。该分支完善和扩展了经典热力学，被称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计热力学 Statistical Thermodynamics&amp;lt;/font&amp;gt;或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;平衡态统计力学 Equilibrium Statistical Mechanics&amp;lt;/font&amp;gt;。&lt;br /&gt;
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Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as [[heat capacity]]) to microscopic behavior, whereas, in [[classical thermodynamics]], the only available option would be to measure and tabulate such quantities for various materials.&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt;&lt;br /&gt;
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Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.&lt;br /&gt;
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统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热容 Heat Capacity&amp;lt;/font&amp;gt;)与微观行为联系起来。而在&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典热力学 Classical Thermodynamics&amp;lt;/font&amp;gt;中，唯一可行的选择就是测量和列出各种材料的热力学量。&lt;br /&gt;
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Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as '''non-equilibrium statistical mechanics''' (sometimes called '''statistical dynamics''') deals with the issue of microscopically modelling the speed of [[irreversible process]]es that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The [[fluctuation–dissipation theorem]] is the basic knowledge obtained from applying [[non-equilibrium statistical mechanics]] to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.&lt;br /&gt;
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Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics (sometimes called statistical dynamics) deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.&lt;br /&gt;
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统计力学也可以用来研究非平衡的系统。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;非平衡统计力学 non-equilibrium statistical mechanics&amp;lt;/font&amp;gt;(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;涨落-耗散定理 Fluctuation–Dissipation Theorem&amp;lt;/font&amp;gt;是人们从非平衡态统计力学中获得的基本知识，它是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。&lt;br /&gt;
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== Principles: mechanics and ensembles 原理：力学和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;系综 Ensembles&amp;lt;/font&amp;gt;==&lt;br /&gt;
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{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}&lt;br /&gt;
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In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts:&lt;br /&gt;
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In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:&lt;br /&gt;
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在物理学中，有两种力学被广泛研究: &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典力学 Classical Mechanics&amp;lt;/font&amp;gt;和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;量子力学 Quantum Mechanics&amp;lt;/font&amp;gt;。对于这两种力学，标准的数学方法与两个概念有关&lt;br /&gt;
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# The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics).&lt;br /&gt;
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The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).&lt;br /&gt;
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力学系统在给定时间内的完整状态，用数学表示为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;相空间 Phase Space&amp;lt;/font&amp;gt;中的点(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;纯量子态矢量 Pure Quantum State Vector&amp;lt;/font&amp;gt;(量子力学)。&lt;br /&gt;
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# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)&lt;br /&gt;
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An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)&lt;br /&gt;
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一个运动方程描述状态在时间上的演化: &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;哈密尔顿方程 Hamilton's Equations&amp;lt;/font&amp;gt;(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;含时薛定谔方程 Time-dependent Schrödinger Equation&amp;lt;/font&amp;gt;(量子力学)&lt;br /&gt;
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Using these two concepts, the state at any other time, past or future, can in principle be calculated.&lt;br /&gt;
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Using these two concepts, the state at any other time, past or future, can in principle be calculated.&lt;br /&gt;
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使用这两个概念，系统在任何时间的状态，无论过去或未来，原则上都可以计算出来。&lt;br /&gt;
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There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.&lt;br /&gt;
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There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.&lt;br /&gt;
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然而，这些定律与日常生活经验之间存在着脱节。因为对于在人类尺度上进行过程(例如化学反应)，我们没有必要(甚至在理论上也不可能)在微观层面上准确地知道每个分子所在的位置及其速度。统计力学通过增加一些对于系统状态的不确定性，填补了力学定律和人类不完全知识的实践经验之间的脱节。&lt;br /&gt;
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Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the [[Statistical ensemble (mathematical physics)|statistical ensemble]], which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a [[probability distribution]] over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a [[phase space]] with [[canonical coordinates]]. In quantum statistical mechanics, the ensemble is a probability distribution over pure states,{{NoteTag|The probabilities in quantum statistical mechanics should not be confused with [[quantum superposition]]. While a quantum ensemble can contain states with quantum superpositions, a single quantum state cannot be used to represent an ensemble.}} and can be compactly summarized as a [[density matrix]].&lt;br /&gt;
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Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix.&lt;br /&gt;
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普通力学只考虑单一状态的行为，而统计力学引入了&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计系综 Statistical Ensemble&amp;lt;/font&amp;gt;，它是系统在各种状态下的大量虚拟、独立的拷贝的集合。系综是一个覆盖系统所有可能状态的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;概率分布 Probability Distribution&amp;lt;/font&amp;gt;。在经典的统计力学中，系综是相点上的概率分布(与普通力学中的单相点相反) ，通常表现为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;正则坐标 Canonical Coordinates&amp;lt;/font&amp;gt;下相空间中的分布。在量子统计力学中，系综是纯态上的概率分布，可以简单地概括为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;密度矩阵 Density Matrix&amp;lt;/font&amp;gt;。 &lt;br /&gt;
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As is usual for probabilities, the ensemble can be interpreted in different ways:&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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As is usual for probabilities, the ensemble can be interpreted in different ways:&lt;br /&gt;
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与通常的概率一样，系综可以用不同的方式来解释:&lt;br /&gt;
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* an ensemble can be taken to represent the various possible states that a ''single system'' could be in ([[epistemic probability]], a form of knowledge), or&lt;br /&gt;
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* the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner ([[empirical probability]]), in the limit of an infinite number of trials.&lt;br /&gt;
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* 系综可以表示&amp;quot;单个系统&amp;quot;的所有可能状态（&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;认识概率 Epistemic Probability&amp;lt;/font&amp;gt;，知识的一种形式），或者&lt;br /&gt;
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* 系综的元素可以理解为在无限次试验的极限下，在类似但不完全受控的独立系统中，重复进行实验得到的系统的状态（&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经验概率 Empirical Probability&amp;lt;/font&amp;gt;）。&lt;br /&gt;
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These two meanings are equivalent for many purposes, and will be used interchangeably in this article.&lt;br /&gt;
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These two meanings are equivalent for many purposes, and will be used interchangeably in this article.&lt;br /&gt;
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这两种意义在很多情况下是等价的，在本文中可以互换使用。&lt;br /&gt;
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However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the [[Liouville's theorem (Hamiltonian)|Liouville equation]] (classical mechanics) or the [[von Neumann equation]] (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.&lt;br /&gt;
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However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.&lt;br /&gt;
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然而，这种概率是可被解释的，系综中的每个随时间演化的状态都可以由运动方程给出。因此，系综本身(状态的概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;刘维尔方程 Liouville Equation&amp;lt;/font&amp;gt;(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;冯·诺依曼方程 Von Neumann Equation&amp;lt;/font&amp;gt;(量子力学)给出。这些方程是简单地通过应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统的概率随时间演化过程中是守恒的。&lt;br /&gt;
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One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as ''equilibrium ensembles'' and their condition is known as ''statistical equilibrium''. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state.{{NoteTag|Statistical equilibrium should not be confused with ''[[mechanical equilibrium]]''. The latter occurs when a mechanical system has completely ceased to evolve even on a microscopic scale, due to being in a state with a perfect balancing of forces. Statistical equilibrium generally involves states that are very far from mechanical equilibrium.}} The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.&lt;br /&gt;
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One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.&lt;br /&gt;
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有一种特殊的系综是不随时间演化的。这样的系综称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;平衡系综 Equilibrium Ensembles&amp;lt;/font&amp;gt;，它们的状态称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计平衡 Statistical Equilibrium&amp;lt;/font&amp;gt;。如果对于每个状态，无论是未来还是过去，该系综都包含在内，并且其概率等于处于该状态的概率，则出现统计平衡的情况。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学研究更一般情况下的可以随时间演化的系综，以及（或）非孤立系统的系综。&lt;br /&gt;
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== Statistical thermodynamics 统计热力学==&lt;br /&gt;
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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the [[classical thermodynamics]] of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in [[thermodynamic equilibrium]], and the microscopic behaviours and motions occurring inside the material.&lt;br /&gt;
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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.&lt;br /&gt;
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统计热力学(也称为平衡态统计力学)的主要目标是根据组成某材料的粒子的性质和它们之间的相互作用，推导出材料的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典热力学 Classical Thermodynamics&amp;lt;/font&amp;gt;。换句话说，统计热力学提供了热力学平衡态中物质的宏观性质与物质内部微观行为和运动之间的联系。&lt;br /&gt;
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Whereas statistical mechanics proper involves dynamics, here the attention is focussed on ''statistical equilibrium'' (steady state). Statistical equilibrium does not mean that the particles have stopped moving ([[mechanical equilibrium]]), rather, only that the ensemble is not evolving.&lt;br /&gt;
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Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.&lt;br /&gt;
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然而统计力学本身就涉及到动态变化，此时的关注点集中在统计平衡(稳态)上。统计平衡并不意味着粒子已经停止运动(力学平衡) ，相反，只是系综没有进化。&lt;br /&gt;
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=== Fundamental postulate 基本假设===&lt;br /&gt;
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A [[sufficient condition|sufficient]] (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).&lt;br /&gt;
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孤立系统统计平衡的一个充分(但不是必要)条件是，其概率分布只是某些守恒量(总能量、总粒子数等)的函数。&lt;br /&gt;
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There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt; Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.&lt;br /&gt;
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There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.&lt;br /&gt;
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有许多不同的平衡系综可以考虑，但只有一些适用于热力学。为了说明为什么给定系统的系综具有这样或那样的形式，还需要一些额外的假设。&lt;br /&gt;
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A common approach found in many textbooks is to take the ''equal a priori probability postulate''.&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt; This postulate states that&lt;br /&gt;
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A common approach found in many textbooks is to take the equal a priori probability postulate. This postulate states that&lt;br /&gt;
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在许多教科书中常见的一种方法是采用&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;先验概率相等假设 Equal A Priori Probability Postulate&amp;lt;/font&amp;gt;。这个假设表明&lt;br /&gt;
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''For an isolated system with an exactly known energy and exactly known composition, the system can be found with ''equal probability'' in any [[microstate (statistical mechanics)|microstate]] consistent with that knowledge.''&lt;br /&gt;
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For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge.&lt;br /&gt;
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对于一个已知精确能量和组成的孤立系统，可以在任何符合条件的微观状态下以等概率找到该系统。&lt;br /&gt;
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The equal a priori probability postulate therefore provides a motivation for the [[microcanonical ensemble]] described below. There are various arguments in favour of the equal a priori probability postulate:&lt;br /&gt;
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The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:&lt;br /&gt;
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因此，先验概率相等假设为下面描述的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微正则系综 microcanonical ensemble&amp;lt;/font&amp;gt;提供了一个动力。有各种各样的论据支持先验概率相等假设:&lt;br /&gt;
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* [[Ergodic hypothesis]]: An ergodic system is one that evolves over time to explore &amp;quot;all accessible&amp;quot; states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.&lt;br /&gt;
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* [[Principle of indifference]]: In the absence of any further information, we can only assign equal probabilities to each compatible situation.&lt;br /&gt;
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* [[Maximum entropy thermodynamics|Maximum information entropy]]: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest [[Gibbs entropy]] ([[information entropy]]).&amp;lt;ref&amp;gt;{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;各态历经假设 Ergodic hypothesis&amp;lt;/font&amp;gt;]]:各态历经系统是一种随着时间的演化而探索“所有可到达”状态的系统:所有具有相同能量和组成的状态。在各态历经系统中，微正则系综是唯一可能的具有固定能量的平衡系综。这种方法的适用性有限，因为大多数系统不是各态历经的。&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;无差别原则 Principle of Indifference&amp;lt;/font&amp;gt;]]: 在没有更多信息的情况下，我们只能对每一个相容的情况分配相等的概率。&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;最大热力学熵|最大信息熵 Maximum Entropy Thermodynamics|Maximum Information Entropy&amp;lt;/font&amp;gt;]]: 无差异原则的一个更详细的版本表明，正确的系综是与已知信息兼容且具有最大[[吉布斯熵]] ([[信息熵]])的系综。&amp;lt;ref&amp;gt;{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Other fundamental postulates for statistical mechanics have also been proposed.&amp;lt;ref name=&amp;quot;uffink&amp;quot;/&amp;gt;&lt;br /&gt;
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Other fundamental postulates for statistical mechanics have also been proposed.&lt;br /&gt;
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其他关于统计力学的基本假设也有被提出。&lt;br /&gt;
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===Three thermodynamic ensembles 三种热力学系综===&lt;br /&gt;
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{{main|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}&lt;br /&gt;
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There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt; These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.&lt;br /&gt;
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There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.&lt;br /&gt;
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对于任何有限体积的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;孤立系统 Isolated System&amp;lt;/font&amp;gt;，可以定义三种简单形式的平衡系综。这些是统计热力学中最经常讨论的系综。在宏观极限(定义如下) ，它们都与经典热力学有对应。&lt;br /&gt;
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; [[Microcanonical ensemble]]&lt;br /&gt;
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Microcanonical Ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微正则系综 Microcanonical Ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.&lt;br /&gt;
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describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.&lt;br /&gt;
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描述了一个具有精确给定能量和固定成分(精确数量的粒子)的系统。微正则系综中，与能量和组成相一致的每个可能状态的概率是相等的。&lt;br /&gt;
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; [[Canonical ensemble]]&lt;br /&gt;
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Canonical ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;正则系综 Canonical Ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system of fixed composition that is in [[thermal equilibrium]]{{NoteTag|The transitive thermal equilibrium (as in, &amp;quot;X is thermal equilibrium with Y&amp;quot;) used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.}} with a [[heat bath]] of a precise [[thermodynamic temperature|temperature]]. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.&lt;br /&gt;
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describes a system of fixed composition that is in thermal equilibrium with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.&lt;br /&gt;
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描述了一个固定成分的系统，这个系统与一个精确温度的热浴形成热平衡。正则系综包含能量不同但组成完全相同的状态; 根据总能量的不同，系综中不同的状态被赋予不同的概率。&lt;br /&gt;
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; [[Grand canonical ensemble]]&lt;br /&gt;
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Grand canonical ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;巨正则系综 Grand Canonical Ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise [[chemical potential]]s for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.&lt;br /&gt;
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describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.&lt;br /&gt;
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描述了一个具有非固定成分(不确定粒子数)在热库中处于热力学和化学平衡的系统。热库具有精确的温度，各种类型的粒子具有精确的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;化学势 Chemical Potential&amp;lt;/font&amp;gt;。巨正则系综包含不同能量和状态的大量粒子; 根据总能量和粒子数的不同，系综中不同状态的概率也不同。&lt;br /&gt;
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For systems containing many particles (the [[thermodynamic limit]]), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used.&amp;lt;ref name=&amp;quot;Reif&amp;quot;&amp;gt;{{cite book | last = Reif | first = F. | title = Fundamentals of Statistical and Thermal Physics | publisher = McGraw–Hill | year = 1965 | isbn = 9780070518001 | page = [https://archive.org/details/fundamentalsofst00fred/page/227 227] | url-access = registration | url = https://archive.org/details/fundamentalsofst00fred/page/227 }}&amp;lt;/ref&amp;gt; The Gibbs theorem about equivalence of ensembles&amp;lt;ref&amp;gt;{{cite journal |doi=10.1007/s10955-015-1212-2|title=Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels|journal=Journal of Statistical Physics|volume=159|issue=5|pages=987–1016|year=2015|last1=Touchette|first1=Hugo|arxiv=1403.6608}}&amp;lt;/ref&amp;gt; was developed into the theory of [[concentration of measure]] phenomenon,&amp;lt;ref&amp;gt;{{cite book |doi=10.1090/surv/089|title=The Concentration of Measure Phenomenon|volume=89|series=Mathematical Surveys and Monographs|year=2005|isbn=9780821837924|last1=Ledoux|first1=Michel}}.&amp;lt;/ref&amp;gt; which has applications in many areas of science, from functional analysis to methods of [[artificial intelligence]] and [[big data]] technology.&amp;lt;ref&amp;gt;{{cite journal |doi=10.1098/rsta.2017.0237|pmc=5869543|title=Blessing of dimensionality: Mathematical foundations of the statistical physics of data|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=376|issue=2118|pages=20170237|year=2018|last1=Gorban|first1=A. N.|last2=Tyukin|first2=I. Y.}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.&lt;br /&gt;
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对于包含大量粒子的系统(&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热力学极限 Thermodynamic Limit&amp;lt;/font&amp;gt;) ，上面列出的三种系综都倾向于体现出相同的行为。因此，使用哪种系综只是一个简单的数学问题。发展成为(&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;测度现象集中理论 Concentration of Measure Phenomenon&amp;lt;/font&amp;gt;系综等价的吉布斯定理，在从函数分析到人工智能和大数据技术等许多科学领域都有广泛的应用。&lt;br /&gt;
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Important cases where the thermodynamic ensembles ''do not'' give identical results include:&lt;br /&gt;
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Important cases where the thermodynamic ensembles do not give identical results include:&lt;br /&gt;
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热力学系综不能给出相同结果的重要例子包括:&lt;br /&gt;
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* Microscopic systems.&lt;br /&gt;
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* Large systems at a phase transition.&lt;br /&gt;
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* Large systems with long-range interactions.&lt;br /&gt;
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* 微观系统&lt;br /&gt;
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* 处于相变的宏观系统&lt;br /&gt;
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* 长程关联的宏观系统&lt;br /&gt;
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In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&lt;br /&gt;
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In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.&lt;br /&gt;
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在这些情况下必须选择正确的热力学系综，因为这些系综之间不仅在涨落的大小方面有可观测的差异，而且在平均量方面都有可观察的差异，如粒子数的分布。正确的系综是对应于该系统的准备和表征的方式ーー换句话说，系综反映我们对系统的认知。&lt;br /&gt;
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:{| class=&amp;quot;wikitable sortable mw-collapsible mw-collapsed&amp;quot;&lt;br /&gt;
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{| class=&amp;quot;wikitable sortable mw-collapsible mw-collapsed&amp;quot;&lt;br /&gt;
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{ | class“ wikable sortable mw-collable mw-collapse”&lt;br /&gt;
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|+&lt;br /&gt;
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|+&lt;br /&gt;
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|+&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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! rowspan=&amp;quot;2&amp;quot;|&lt;br /&gt;
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! rowspan=&amp;quot;2&amp;quot;|&lt;br /&gt;
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!Rowspan“2” | &lt;br /&gt;
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! colspan=&amp;quot;3&amp;quot;| Thermodynamic ensembles&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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! colspan=&amp;quot;3&amp;quot;| Thermodynamic ensembles&lt;br /&gt;
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!Colspan“3” | 热力学系数&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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! [[Microcanonical ensemble|Microcanonical]]&lt;br /&gt;
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! Microcanonical&lt;br /&gt;
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!微正则化&lt;br /&gt;
&lt;br /&gt;
! [[Canonical ensemble|Canonical]]&lt;br /&gt;
&lt;br /&gt;
! Canonical&lt;br /&gt;
&lt;br /&gt;
!典范&lt;br /&gt;
&lt;br /&gt;
! [[Grand canonical ensemble|Grand canonical]]&lt;br /&gt;
&lt;br /&gt;
! Grand canonical&lt;br /&gt;
&lt;br /&gt;
!巨正典&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Fixed variables&lt;br /&gt;
&lt;br /&gt;
! Fixed variables&lt;br /&gt;
&lt;br /&gt;
!固定变量&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;{{math|''N, E, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
中心，中心&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;{{math|''N, T, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
中心，中心&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;{{math|''μ, T, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
中心，中心&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Microscopic features&lt;br /&gt;
&lt;br /&gt;
! Microscopic features&lt;br /&gt;
&lt;br /&gt;
!微观特征&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;Number of [[Microstate (statistical mechanics)|microstates]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt; W &amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Canonical partition function]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;Z = \sum_k e^{- E_k / k_B T}&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Grand partition function]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\mathcal Z = \sum_k e^{ -(E_k - \mu N_k) /k_B T}&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Macroscopic function&lt;br /&gt;
&lt;br /&gt;
! Macroscopic function&lt;br /&gt;
&lt;br /&gt;
!宏观功能&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Boltzmann entropy]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;S =  k_B \log W&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Helmholtz free energy]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;F = - k_B T \log Z&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Grand potential]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\Omega =- k_B T \log \mathcal Z &amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
=== Calculation Methods 计算方法===&lt;br /&gt;
&lt;br /&gt;
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.&lt;br /&gt;
&lt;br /&gt;
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.&lt;br /&gt;
&lt;br /&gt;
一旦计算出一个系统的特征状态函数，该系统就被“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是对最一般的(和现实的)情况进行精确的求解实在是太复杂了。存在各种方法来近似真实的系综，并且计算平均量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
====Exact 精确解====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are some cases which allow exact solutions.&lt;br /&gt;
&lt;br /&gt;
There are some cases which allow exact solutions.&lt;br /&gt;
&lt;br /&gt;
有些情况可以得到精确解。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics).&lt;br /&gt;
&lt;br /&gt;
* Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably, idealized gases of non-interacting particles have this property, allowing exact derivations of [[Maxwell–Boltzmann statistics]], [[Fermi–Dirac statistics]], and [[Bose–Einstein statistics]].&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few [[toy model]]s.&amp;lt;ref&amp;gt;{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}&amp;lt;/ref&amp;gt; Some examples include the [[Bethe ansatz]], [[square-lattice Ising model]] in zero field, [[hard hexagon model]].&lt;br /&gt;
&lt;br /&gt;
* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。&lt;br /&gt;
&lt;br /&gt;
* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;麦克斯韦–玻尔兹曼统计 Maxwell–Boltzmann Statistics&amp;lt;/font&amp;gt;]]， [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;费米-狄拉克统计 Fermi–Dirac Statistics&amp;lt;/font&amp;gt;]]，和[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;波色-爱因斯坦统计 Bose–Einstein Statistics&amp;lt;/font&amp;gt;]]。&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。&amp;lt;ref&amp;gt;{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}&amp;lt;/ref&amp;gt; 一些例子包括[[Bethe ansatz]]，零场下的[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt; 二维格点伊辛模型 Square-Lattice Ising Model&amp;lt;/font&amp;gt;]]，[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;硬六边形模型 Hard Hexagon Model&amp;lt;/font&amp;gt;]]。&lt;br /&gt;
&lt;br /&gt;
====Monte Carlo 蒙特卡罗方法====&lt;br /&gt;
&lt;br /&gt;
{{main|Monte Carlo method}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
One approximate approach that is particularly well suited to computers is the [[Monte Carlo method]], which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.&lt;br /&gt;
&lt;br /&gt;
One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.&lt;br /&gt;
&lt;br /&gt;
一个特别适合于计算机的近似方法是&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;蒙特卡罗方法 Monte Carlo Method&amp;lt;/font&amp;gt;，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态可构成系统全部状态集的代表样本，就可以得到近似的特征函数。随着随机样本数量的增加，误差可以降低到任意低的水平。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* The [[Metropolis–Hastings algorithm]] is a classic Monte Carlo method which was initially used to sample the canonical ensemble.&lt;br /&gt;
&lt;br /&gt;
* [[Path integral Monte Carlo]], also used to sample the canonical ensemble.&lt;br /&gt;
&lt;br /&gt;
* [[Metropolis–Hastings 算法]] 是一种经典的蒙特卡罗算法，最初用于正则系综的采样。&lt;br /&gt;
&lt;br /&gt;
* [[路径积分蒙特卡罗方法]] 也可以用于正则系综的采样。&lt;br /&gt;
&lt;br /&gt;
==== Other 其他====&lt;br /&gt;
&lt;br /&gt;
* For rarefied non-ideal gases, approaches such as the [[cluster expansion]] use [[perturbation theory]] to include the effect of weak interactions, leading to a [[virial expansion]].&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* For dense fluids, another approximate approach is based on reduced distribution functions, in particular the [[radial distribution function]].&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* [[Molecular dynamics]] computer simulations can be used to calculate [[microcanonical ensemble]] averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions.&lt;br /&gt;
&lt;br /&gt;
* Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.&lt;br /&gt;
&lt;br /&gt;
* 对于稀薄的非理想气体，[[团簇膨胀 cluster expansion]]等方法使用[[微扰理论 perturbation theory]]来涵盖弱相互作用的影响，得到[[维里展开 virial expansion]]。&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是[[径向分布函数 radial distribution function]]。&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* [[分子动力学 Molecular dynamics]] 计算机模拟可以用来计算各态历经系统中的[[微正则系综]]平均。对于包含与随机热浴有连接的系统，他们还可以在正则和巨正则条件下建模。&lt;br /&gt;
&lt;br /&gt;
* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。&lt;br /&gt;
&lt;br /&gt;
== Non-equilibrium statistical mechanics 非平衡态统计力学==&lt;br /&gt;
&lt;br /&gt;
{{see also|Non-equilibrium thermodynamics}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:&lt;br /&gt;
&lt;br /&gt;
There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:&lt;br /&gt;
&lt;br /&gt;
有许多令人感兴趣的物理现象都涉及到失去平衡的准热力学过程，例如:&lt;br /&gt;
&lt;br /&gt;
* [[Thermal conduction|heat transport by the internal motions in a material]], driven by a temperature imbalance,&lt;br /&gt;
&lt;br /&gt;
* [[Electrical conduction|electric currents carried by the motion of charges in a conductor]], driven by a voltage imbalance,&lt;br /&gt;
&lt;br /&gt;
* spontaneous [[chemical reaction]]s driven by a decrease in free energy,&lt;br /&gt;
&lt;br /&gt;
* [[friction]], [[dissipation]], [[quantum decoherence]],&lt;br /&gt;
&lt;br /&gt;
* systems being pumped by external forces ([[optical pumping]], etc.),&lt;br /&gt;
&lt;br /&gt;
* and irreversible processes in general.&lt;br /&gt;
&lt;br /&gt;
* [[热传导|材料的内部运动来传导热量]]，由热度不平衡来驱动，&lt;br /&gt;
&lt;br /&gt;
* [[导电|导体内电荷的运动产生电流]]，由电压不平衡来驱动，&lt;br /&gt;
&lt;br /&gt;
* 自发的 [[化学反应]]，由自由能的下降驱动，&lt;br /&gt;
&lt;br /&gt;
* [[摩擦]]，[[耗散]]，[[量子退相干]，&lt;br /&gt;
&lt;br /&gt;
* 由外力泵送的系统([光泵]]等)&lt;br /&gt;
&lt;br /&gt;
* 以及一般意义的可逆过程&lt;br /&gt;
&lt;br /&gt;
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)&lt;br /&gt;
&lt;br /&gt;
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)&lt;br /&gt;
&lt;br /&gt;
所有这些过程都是以特征速率随时间发生，这些速率对于工程来说非常重要。非平衡态统计力学研究领域关注的是在微观水平上理解这些非平衡过程。(统计热力学只能用来计算在外部不平衡被消除，整体回归到平衡状态之后的最终结果。)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as [[Liouville's theorem (Hamiltonian)|Liouville's equation]] or its quantum equivalent, the [[von Neumann equation]]. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's [[Gibbs entropy]] is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.&lt;br /&gt;
&lt;br /&gt;
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.&lt;br /&gt;
&lt;br /&gt;
原则上，非平衡态统计力学在数学上可以是精确的: 孤立系统的系综根据确定性方程随时间演化，如刘维尔方程或其量子等价、冯·诺依曼方程。这些方程是将运动力学方程独立应用于系综中每个状态的结果。不幸的是，这些系综演化方程继承了底层动力学运动的大部分复杂性，因此很难得到精确解。此外，系综演化方程是完全可逆的，不会破坏信息(系综的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;吉布斯熵 Gibbs entropy&amp;lt;/font&amp;gt;被保留)。为了在模拟不可逆过程中取得进展，除了概率和可逆力学外，还必须考虑其他因素。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.&lt;br /&gt;
&lt;br /&gt;
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.&lt;br /&gt;
&lt;br /&gt;
因此，非平衡力学是一个活跃的理论研究领域，因为这些额外假设的有效范围仍将继续探索。在下面的小节中描述了一些方法。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
=== Stochastic methods 随机方法===&lt;br /&gt;
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One approach to non-equilibrium statistical mechanics is to incorporate [[stochastic]] (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from [[Black hole information paradox|hypothetical situations involving black holes]], a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as [[Chaos theory|chaotic]] or [[pseudorandom]] influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.&lt;br /&gt;
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One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.&lt;br /&gt;
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处理非平衡态统计力学的一个方法是将随机行为引入系统。随机行为可以破坏系综中包含的信息。虽然这在技术上是不准确的(除了涉及黑洞的假设情况外，黑洞系统本身不会导致信息丢失) ，但这种随机性是为了反映出，随着时间的推移，感兴趣的信息会在系统内部转化为微妙的相关性，或者系统与环境之间的相关性。这些关联表现为对感兴趣的变量的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;混沌 Chaos&amp;lt;/font&amp;gt;或伪随机的影响。用适当的随机性取代这些相关性，计算可以变得容易得多。&lt;br /&gt;
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|1 = ''[[Boltzmann transport equation]]'': An early form of stochastic mechanics appeared even before the term &amp;quot;statistical mechanics&amp;quot; had been coined, in studies of [[kinetic theory of gases|kinetic theory]]. [[James Clerk Maxwell]] had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. [[Ludwig Boltzmann]] subsequently showed that, by taking this [[molecular chaos]] for granted as a complete randomization, the motions of particles in a gas would follow a simple [[Boltzmann transport equation]] that would rapidly restore a gas to an equilibrium state (see [[H-theorem]]).&lt;br /&gt;
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|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term &amp;quot;statistical mechanics&amp;quot; had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;玻尔兹曼输运方程 Boltzmann Transport Equation&amp;lt;/font&amp;gt;: 在动力学理论研究中，早期的随机力学甚至在“统计力学”一词被创造之前就已经出现了。詹姆斯·克拉克·麦克斯韦已经证明分子碰撞会导致气体内部明显的混沌运动。路德维希·玻尔兹曼随后证明，如果把这种分子混沌理所当然地看作是一种完全的随机化，那么气体中粒子的运动将遵循一个简单的玻尔兹曼输运方程，这个方程将使气体迅速恢复到平衡状态(见H-定理)。&lt;br /&gt;
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The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the &amp;quot;interesting&amp;quot; information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped [[semiconductor]]s (in [[transistor]]s), where the electrons are indeed analogous to a rarefied gas.&lt;br /&gt;
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The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the &amp;quot;interesting&amp;quot; information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas.&lt;br /&gt;
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玻耳兹曼输运方程及其相关方法是非平衡态统计力学的重要工具，因为它们极其简单。这些近似方法在“感兴趣的”信息立即(在一次碰撞之后)变成微妙关联的系统中非常有效，这种关联本质上限制它们为稀薄气体。玻耳兹曼输运方程被发现在模拟轻掺杂半导体(晶体管)的电子输运中非常有用，其中的电子确实类似于稀薄气体。&lt;br /&gt;
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A quantum technique related in theme is the [[random phase approximation]].&lt;br /&gt;
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A quantum technique related in theme is the random phase approximation.&lt;br /&gt;
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一个与主题相关的量子技术是&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;随机相位近似 Random Phase Approximation&amp;lt;/font&amp;gt;。&lt;br /&gt;
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|2 = ''[[BBGKY hierarchy]]'':&lt;br /&gt;
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|2 = BBGKY hierarchy:&lt;br /&gt;
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2 BBGKY 层级:&lt;br /&gt;
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In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The [[BBGKY hierarchy]] (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.&lt;br /&gt;
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In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.&lt;br /&gt;
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在液体和稠密气体中，不能在一次碰撞后立即丢掉粒子之间的关联。BBGKY 层级结构(&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;层级结构 Bogoliubov-Born-Green-Kirkwood-Yvon&amp;lt;/font&amp;gt;)提供了一种推导玻尔兹曼型方程的方法，但也可以将它们扩展到稀薄气体情况之外，包括在几次碰撞之后的相关性。&lt;br /&gt;
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|3 = ''[[Keldysh formalism]]'' (a.k.a. NEGF—non-equilibrium Green functions):&lt;br /&gt;
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|3 = Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions):&lt;br /&gt;
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| 3 Keldysh 公式。NEGF—非平衡态格林函数):&lt;br /&gt;
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A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic [[quantum transport]] calculations.&lt;br /&gt;
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A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic quantum transport calculations.&lt;br /&gt;
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人们在 Keldysh 公式中发明了包含随机动力学的量子方法，这种方法常用于电子量子输运计算。&lt;br /&gt;
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|4 = Stochastic [[Liouville's theorem (Hamiltonian)|Liouville equation]]&lt;br /&gt;
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|4 = Stochastic Liouville equation&lt;br /&gt;
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| 4随机 刘维尔方程&lt;br /&gt;
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=== Near-equilibrium methods 近平衡态方法 ===&lt;br /&gt;
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Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in [[linear response theory]]. A remarkable result, as formalized by the [[fluctuation-dissipation theorem]], is that the response of a system when near equilibrium is precisely related to the [[Statistical fluctuations|fluctuations]] that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or &amp;quot;know&amp;quot; how it came to be away from equilibrium.&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;{{rp|664}}&lt;br /&gt;
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Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or &amp;quot;know&amp;quot; how it came to be away from equilibrium.&lt;br /&gt;
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非平衡态统计力学模型处理的另一类重要的系统，是对平衡态仅有非常轻微扰动的系统。在很小的扰动下，响应可以用&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;线性响应理论 Linear Response Theory&amp;lt;/font&amp;gt;进行分析。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;涨落-耗散定理 Fluctuation-Dissipation Theorem&amp;lt;/font&amp;gt;是其中一个重要的结果，近平衡态系统的响应与系统总体平衡时的涨落准确相关。从本质上讲，一个系统如果稍微偏离平衡，无论是由于外力还是由于涨落，都会以同样的方式向平衡方向弛豫。因为这个系统无法区分偏离和回归，也“不知道”它是如何偏离平衡的。&lt;br /&gt;
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This provides an indirect avenue for obtaining numbers such as [[Ohm's law|ohmic conductivity]] and [[thermal conductivity]] by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.&lt;br /&gt;
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This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.&lt;br /&gt;
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这提供了一个间接的方法，通过从平衡态统计力学中提取结果来获得诸如欧姆电导率和热导率之类的物理量。由于平衡态统计力学在数学上有很好的定义，而且(在某些情况下)更易于计算，因此在近平衡态统计力学中，涨落-耗散关联可以成为一种方便的计算捷径。&lt;br /&gt;
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A few of the theoretical tools used to make this connection include:&lt;br /&gt;
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A few of the theoretical tools used to make this connection include:&lt;br /&gt;
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用于建立这种关联的一些理论工具包括:&lt;br /&gt;
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* [[Fluctuation–dissipation theorem]]&lt;br /&gt;
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* [[Onsager reciprocal relations]]&lt;br /&gt;
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* [[Green–Kubo relations]]&lt;br /&gt;
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* [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker formalism]]&lt;br /&gt;
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* [[Mori–Zwanzig formalism]]&lt;br /&gt;
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* [[涨落-耗散定理]]&lt;br /&gt;
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* [[昂萨格倒易关系]]&lt;br /&gt;
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* [[格林-库伯关系]]&lt;br /&gt;
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* [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker 公式]]&lt;br /&gt;
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* [[Mori–Zwanzig 公式]]&lt;br /&gt;
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=== Hybrid methods 组合方法===&lt;br /&gt;
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An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects ([[weak localization]], [[conductance fluctuations]]) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic [[dephasing]] by interactions between various electrons by use of the Keldysh method.&amp;lt;ref&amp;gt;{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid =  | pmc = |bibcode = 1982JPhC...15.7367A }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid =  | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.&lt;br /&gt;
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一种先进的方法使用了随机方法和线性响应理论的结合。例如，计算电子系统电导中的量子相干效应(弱局域化，电导涨落)的一种方法是使用 Green-Kubo 关系，包括随机退相的各种电子之间的相互作用，使用Keldysh 方法。&lt;br /&gt;
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==Applications outside thermodynamics==&lt;br /&gt;
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The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:&lt;br /&gt;
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The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:&lt;br /&gt;
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系综也可以用来分析系统状态有不确定性的一般机械系统。在下列情况中也会使用系综:&lt;br /&gt;
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* [[propagation of uncertainty]] over time,&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt;&lt;br /&gt;
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* [[regression analysis]] of gravitational [[orbit]]s,&lt;br /&gt;
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* [[ensemble forecasting]] of weather,&lt;br /&gt;
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* dynamics of [[neural networks]],&lt;br /&gt;
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* bounded-rational [[potential game]]s in game theory and economics.&lt;br /&gt;
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* 不确定性随时间传播，&lt;br /&gt;
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* 引力[[轨道]]的[[回归分析]]，&lt;br /&gt;
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* 天气的[[系综预报]]，&lt;br /&gt;
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* [[神经网络]]动力学，&lt;br /&gt;
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* 博弈论和经济学中的有界理性[[潜在对策]]。&lt;br /&gt;
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== History 历史==&lt;br /&gt;
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In 1738, Swiss physicist and mathematician [[Daniel Bernoulli]] published ''Hydrodynamica'' which laid the basis for the [[kinetic theory of gases]]. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as [[heat]] is simply the kinetic energy of their motion.&amp;lt;ref name=&amp;quot;uffink&amp;quot;/&amp;gt;&lt;br /&gt;
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In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.&lt;br /&gt;
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1738年，瑞士的物理学家和数学家丹尼尔·伯努利发表了《水动力学》 ，这本书奠定了&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;气体动力学理论 Kinetic Theory of Gases&amp;lt;/font&amp;gt;的基础。在这项工作中，伯努利假定气体是由大量向各个方向运动的分子组成的，它们对表面的影响导致了我们感觉到的气体压力，而我们感受到的热仅仅是它们运动的动能，这一点直到今天仍在沿用。&lt;br /&gt;
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In 1859, after reading a paper on the diffusion of molecules by [[Rudolf Clausius]], Scottish physicist [[James Clerk Maxwell]] formulated the [[Maxwell distribution]] of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.&amp;lt;ref&amp;gt;See:&lt;br /&gt;
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In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.&amp;lt;ref&amp;gt;See:&lt;br /&gt;
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1859年，在阅读了 Rudolf Clausius 的一篇关于分子扩散的论文之后，苏格兰物理学家詹姆斯·克拉克·麦克斯韦提出了分子速度的麦克斯韦-玻尔兹曼分布，它给出了在一个特定范围内具有某种速度的分子的比例。这是物理学里第一个统计定律。麦克斯韦还提出了第一个力学论点，即分子碰撞必然导致温度的平衡，从而趋向平衡。五年之后，也就是1864年，维也纳的年轻学生路德维希·玻尔兹曼(Ludwig Boltzmann)偶然发现了麦克斯韦尔的论文，并花了大半辈子的时间来进一步研究这一课题。&lt;br /&gt;
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*  Maxwell, J.C. (1860) [https://books.google.com/books?id=-YU7AQAAMAAJ&amp;amp;pg=PA19#v=onepage&amp;amp;q&amp;amp;f=false &amp;quot;Illustrations of the dynamical theory of gases. Part I.  On the motions and collisions of perfectly elastic spheres,&amp;quot;] ''Philosophical Magazine'', 4th series, '''19''' :  19–32. &lt;br /&gt;
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*  Maxwell, J.C. (1860) [https://books.google.com/books?id=DIc7AQAAMAAJ&amp;amp;pg=PA21#v=onepage&amp;amp;q&amp;amp;f=false &amp;quot;Illustrations of the dynamical theory of gases. Part II.  On the process of diffusion of two or more kinds of moving particles among one another,&amp;quot;] ''Philosophical Magazine'', 4th series, '''20''' :  21–37.&amp;lt;/ref&amp;gt; This was the first-ever statistical law in physics.&amp;lt;ref&amp;gt;{{cite book |last = Mahon |first = Basil |title=The Man Who Changed Everything – the Life of James Clerk Maxwell |location=Hoboken, NJ |publisher=Wiley |year=2003 |isbn=978-0-470-86171-4 |oclc=52358254}}&amp;lt;/ref&amp;gt; Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.&amp;lt;ref&amp;gt;{{cite journal | last = Gyenis | first = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}&amp;lt;/ref&amp;gt; Five years later, in 1864, [[Ludwig Boltzmann]], a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.&lt;br /&gt;
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Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 ''Lectures on Gas Theory''.&amp;lt;ref&amp;gt;{{cite book |title = Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems |journal=Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Edited by Ebeling Werner &amp;amp; Sokolov Igor M. Published by World Scientific Press |volume=8 |last1=Ebeling |first1=Werner |last2=Sokolov |first2=Igor M. |year=2005 |isbn=978-90-277-1674-3 |pages=3–12 |url = https://books.google.com/books?id=KUjFHbid8A0C|bibcode=2005stst.book.....E |doi=10.1142/2012 |series = Series on Advances in Statistical Mechanics }} (section 1.2)&amp;lt;/ref&amp;gt; Boltzmann's original papers on the statistical interpretation of thermodynamics, the [[H-theorem]], [[transport theory (statistical physics)|transport theory]], [[thermal equilibrium]], the [[equation of state]] of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his [[H-theorem|''H''-theorem]].&lt;br /&gt;
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Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory. Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.&lt;br /&gt;
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统计力学是19世纪70年代由玻尔兹曼的工作创立的，其中大部分在他1896年的气体理论演讲中集结出版。在维也纳学院和其他学会的会议记录中，玻尔兹曼关于热力学的统计解释、 &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;H-定理H-theorem&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;输运理论 Transport Theory&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热平衡Thermal Equilibrium&amp;lt;/font&amp;gt;、气体的状态方程以及类似主题的原始论文占据了大约2000页。玻尔兹曼引入了平衡系综的概念，并用他的H-定理第一次研究了非平衡态统计力学。&lt;br /&gt;
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The term &amp;quot;statistical mechanics&amp;quot; was coined by the American mathematical physicist [[Josiah Willard Gibbs|J. Willard Gibbs]] in 1884.&amp;lt;ref&amp;gt;J. W. Gibbs, &amp;quot;On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics.&amp;quot; Proceedings of the American Association for the Advancement of Science, '''33''', 57-58 (1884). Reproduced in ''The Scientific Papers of J. Willard Gibbs, Vol II'' (1906), [https://archive.org/stream/scientificpapers02gibbuoft#page/16/mode/2up pp.&amp;amp;nbsp;16].&amp;lt;/ref&amp;gt;{{NoteTag|1 = According to Gibbs, the term &amp;quot;statistical&amp;quot;, in the context of mechanics, i.e. statistical mechanics, was first used by the Scottish physicist [[James Clerk Maxwell]] in 1871. From: J. Clerk Maxwell, ''Theory of Heat'' (London, England: Longmans, Green, and Co., 1871), [https://books.google.com/books?id=DqAAAAAAMAAJ&amp;amp;pg=PA309 p.&amp;amp;nbsp;309]: &amp;quot;In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus.&amp;quot;}} &amp;quot;Probabilistic mechanics&amp;quot; might today seem a more appropriate term, but &amp;quot;statistical mechanics&amp;quot; is firmly entrenched.&amp;lt;ref&amp;gt;{{cite book |title = The enigma of probability and physics |last=Mayants |first=Lazar |year=1984 |publisher=Springer |isbn=978-90-277-1674-3 |page=174 |url = https://books.google.com/books?id=zmwEfXUdBJ8C&amp;amp;pg=PA174 }}&amp;lt;/ref&amp;gt; Shortly before his death, Gibbs published in 1902 ''[[Elementary Principles in Statistical Mechanics]]'', a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt; Gibbs' methods were initially derived in the framework [[classical mechanics]], however they were of such generality that they were found to adapt easily to the later [[quantum mechanics]], and still form the foundation of statistical mechanics to this day.&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&lt;br /&gt;
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The term &amp;quot;statistical mechanics&amp;quot; was coined by the American mathematical physicist J. Willard Gibbs in 1884. &amp;quot;Probabilistic mechanics&amp;quot; might today seem a more appropriate term, but &amp;quot;statistical mechanics&amp;quot; is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.&lt;br /&gt;
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1884年，美国数学物理学家约西亚·威拉德·吉布斯首创了“统计力学”一词。在今天看来，“概率力学”似乎是一个更合适的术语，但“统计力学”却根深蒂固。在吉布斯去世前不久，他于1902年在《统计力学》出版了《基本原理》一书，这本书正式确定了统计力学是解决所有力学系统——宏观的或微观的、气态的或非气态的——的一种完全通用的方法。吉布斯的方法最初是在经典力学的框架下产生的，然而它们是如此的普遍，以至于人们发现它们很容易适应后来的量子力学，直到今天仍然是统计力学的基础。&lt;br /&gt;
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== See also 其他可见==&lt;br /&gt;
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{{Books-inline|Fundamentals of Statistical Mechanics}}&lt;br /&gt;
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* [[Thermodynamics]]: [[Non-equilibrium thermodynamics|non-equilibrium]], [[Chemical thermodynamics|chemical]]&lt;br /&gt;
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* [[Mechanics]]: [[Classical mechanics|classical]], [[Quantum mechanics|quantum]]&lt;br /&gt;
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* [[Probability]], [[Statistical ensemble (mathematical physics)|statistical ensemble]]&lt;br /&gt;
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* Numerical methods: [[Monte Carlo method]], [[molecular dynamics]]&lt;br /&gt;
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* [[Statistical physics]]&lt;br /&gt;
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* [[Quantum statistical mechanics]]&lt;br /&gt;
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* [[List of notable textbooks in statistical mechanics]]&lt;br /&gt;
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* [[List of publications in physics#Statistical mechanics|List of important publications in statistical mechanics]]&lt;br /&gt;
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{{Books-inline|统计力学基础}}&lt;br /&gt;
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* [[热力学]]: [[非平衡态热力学]], [[化学热力学]]&lt;br /&gt;
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* [[力学]]: [[经典力学]], [[量子力学]]&lt;br /&gt;
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* [[概率]], [[统计系综（数学物理）]]&lt;br /&gt;
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* 数值方法: [[蒙特卡洛方法]], [[分子动力学]]&lt;br /&gt;
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* [[统计物理]]&lt;br /&gt;
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* [[量子统计力学]]&lt;br /&gt;
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* [[统计力学著名教科书列表]]&lt;br /&gt;
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* [[统计力学重要文献列表]]&lt;br /&gt;
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== Notes ==&lt;br /&gt;
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== References ==&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;gibbs&amp;quot;&amp;gt;{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York |title-link=Elementary Principles in Statistical Mechanics }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;tolman&amp;quot;&amp;gt;{{cite book | last=Tolman |first=R. C. | author-link = Richard C. Tolman | year=1938 | title = The Principles of Statistical Mechanics | publisher=[[Dover Publications]] | isbn = 9780486638966 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;balescu&amp;quot;&amp;gt;{{cite book | isbn = 9780471046004 | title = Equilibrium and Non-Equilibrium Statistical Mechanics |last = Balescu |first = Radu | author-link = Radu Balescu | year = 1975 | publisher = John Wiley &amp;amp; Sons }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;uffink&amp;quot;&amp;gt;J. Uffink, &amp;quot;[http://philsci-archive.pitt.edu/2691/1/UffinkFinal.pdf Compendium of the foundations of classical statistical physics.]&amp;quot; (2006)&amp;lt;/ref&amp;gt;&lt;br /&gt;
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== External links ==&lt;br /&gt;
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{{Commons category|Statistical mechanics}}&lt;br /&gt;
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* [http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics] article by Lawrence Sklar for the [[Stanford Encyclopedia of Philosophy]].&lt;br /&gt;
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* [http://www.sklogwiki.org/ Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials.]  SklogWiki is particularly orientated towards liquids and soft condensed matter.&lt;br /&gt;
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* [https://web.archive.org/web/20060616002812/http://history.hyperjeff.net/statmech.html Statistical Thermodynamics] - Historical Timeline&lt;br /&gt;
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* [http://farside.ph.utexas.edu/teaching/sm1/statmech.pdf Thermodynamics and Statistical Mechanics] by Richard Fitzpatrick&lt;br /&gt;
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* [https://arxiv.org/abs/1107.0568 Lecture Notes in Statistical Mechanics and Mesoscopics] by Doron Cohen&lt;br /&gt;
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* {{YouTube |id = H1Zbp6__uNw&amp;amp;list=PLB72416C707D85AB0&amp;amp;index=1 |title = Videos of lecture series in statistical mechanics }} taught by [[Leonard Susskind]].&lt;br /&gt;
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* Vu-Quoc, L., [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008. this wiki site is down; see [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 this article in the web archive on 2012 April 28].&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Statistical mechanics]]. Its edit history can be viewed at [[统计力学/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
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		<title>统计力学</title>
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		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;已由佳欣初步翻译&lt;br /&gt;
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{{Short description|Physics of large number of particles' statistical behavior}}&lt;br /&gt;
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{{Statistical mechanics}}&lt;br /&gt;
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'''Statistical mechanics''' is one of the pillars of modern [[physics]]. It is necessary for the fundamental study of any physical system that has many [[degrees of freedom]]. The approach is based on [[Statistics|statistical]] methods, [[probability theory]] and the [[Microscopic scale|microscopic]] physical laws.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;{{NoteTag|The term ''statistical mechanics'' is sometimes used to refer to only ''statistical thermodynamics''. This article takes the broader view. By some definitions, ''[[statistical physics]]'' is an even broader term which statistically studies any type of physical system, but is often taken to be synonymous with statistical mechanics.}}&lt;br /&gt;
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Statistical mechanics is one of the pillars of modern physics. It is necessary for the fundamental study of any physical system that has many degrees of freedom. The approach is based on statistical methods, probability theory and the microscopic physical laws.&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计力学 Statistical mechanics&amp;lt;/font&amp;gt;是现代物理学的支柱之一。对于具有多个&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;自由度 Degrees of Freedom&amp;lt;/font&amp;gt;的物理系统的基础研究，统计力学是不可或缺的。统计力学的方法是基于&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计学方法 Statistical Methods&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;概率论 Probability Theory&amp;lt;/font&amp;gt;和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微观物理定律 Microscopic Physical Laws&amp;lt;/font&amp;gt;。&lt;br /&gt;
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It can be used to explain the [[thermodynamics|thermodynamic]] behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as '''statistical thermodynamics''' or '''equilibrium statistical mechanics'''.&lt;br /&gt;
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It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.&lt;br /&gt;
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统计力学一个分支可以用来解释大系统的热力学行为。该分支完善和扩展了经典热力学，被称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计热力学 Statistical Thermodynamics&amp;lt;/font&amp;gt;或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;平衡态统计力学 Equilibrium Statistical Mechanics&amp;lt;/font&amp;gt;。&lt;br /&gt;
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Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as [[heat capacity]]) to microscopic behavior, whereas, in [[classical thermodynamics]], the only available option would be to measure and tabulate such quantities for various materials.&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt;&lt;br /&gt;
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Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.&lt;br /&gt;
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统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热容 Heat Capacity&amp;lt;/font&amp;gt;)与微观行为联系起来。而在&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典热力学 Classical Thermodynamics&amp;lt;/font&amp;gt;中，唯一可行的选择就是测量和列出各种材料的热力学量。&lt;br /&gt;
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Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as '''non-equilibrium statistical mechanics''' (sometimes called '''statistical dynamics''') deals with the issue of microscopically modelling the speed of [[irreversible process]]es that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The [[fluctuation–dissipation theorem]] is the basic knowledge obtained from applying [[non-equilibrium statistical mechanics]] to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.&lt;br /&gt;
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Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics (sometimes called statistical dynamics) deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.&lt;br /&gt;
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统计力学也可以用来研究非平衡的系统。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;非平衡统计力学 non-equilibrium statistical mechanics&amp;lt;/font&amp;gt;(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;涨落-耗散定理 Fluctuation–Dissipation Theorem&amp;lt;/font&amp;gt;是人们从非平衡态统计力学中获得的基本知识，它是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。&lt;br /&gt;
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== Principles: mechanics and ensembles 原理：力学和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;系综 Ensembles&amp;lt;/font&amp;gt;==&lt;br /&gt;
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In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts:&lt;br /&gt;
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In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:&lt;br /&gt;
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在物理学中，有两种力学被广泛研究: &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典力学 Classical Mechanics&amp;lt;/font&amp;gt;和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;量子力学 Quantum Mechanics&amp;lt;/font&amp;gt;。对于这两种力学，标准的数学方法与两个概念有关&lt;br /&gt;
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# The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics).&lt;br /&gt;
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The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).&lt;br /&gt;
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力学系统在给定时间内的完整状态，用数学表示为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;相空间 Phase Space&amp;lt;/font&amp;gt;中的点(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;纯量子态矢量 Pure Quantum State Vector&amp;lt;/font&amp;gt;(量子力学)。&lt;br /&gt;
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# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)&lt;br /&gt;
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An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)&lt;br /&gt;
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一个运动方程描述状态在时间上的演化: &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;哈密尔顿方程 Hamilton's Equations&amp;lt;/font&amp;gt;(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;含时薛定谔方程 Time-dependent Schrödinger Equation&amp;lt;/font&amp;gt;(量子力学)&lt;br /&gt;
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Using these two concepts, the state at any other time, past or future, can in principle be calculated.&lt;br /&gt;
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Using these two concepts, the state at any other time, past or future, can in principle be calculated.&lt;br /&gt;
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使用这两个概念，系统在任何时间的状态，无论过去或未来，原则上都可以计算出来。&lt;br /&gt;
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There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.&lt;br /&gt;
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There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.&lt;br /&gt;
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然而，这些定律与日常生活经验之间存在着脱节。因为对于在人类尺度上进行过程(例如化学反应)，我们没有必要(甚至在理论上也不可能)在微观层面上准确地知道每个分子所在的位置及其速度。统计力学通过增加一些对于系统状态的不确定性，填补了力学定律和人类不完全知识的实践经验之间的脱节。&lt;br /&gt;
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Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the [[Statistical ensemble (mathematical physics)|statistical ensemble]], which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a [[probability distribution]] over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a [[phase space]] with [[canonical coordinates]]. In quantum statistical mechanics, the ensemble is a probability distribution over pure states,{{NoteTag|The probabilities in quantum statistical mechanics should not be confused with [[quantum superposition]]. While a quantum ensemble can contain states with quantum superpositions, a single quantum state cannot be used to represent an ensemble.}} and can be compactly summarized as a [[density matrix]].&lt;br /&gt;
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Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix.&lt;br /&gt;
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普通力学只考虑单一状态的行为，而统计力学引入了&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计系综 Statistical Ensemble&amp;lt;/font&amp;gt;，它是系统在各种状态下的大量虚拟、独立的拷贝的集合。系综是一个覆盖系统所有可能状态的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;概率分布 Probability Distribution&amp;lt;/font&amp;gt;。在经典的统计力学中，系综是相点上的概率分布(与普通力学中的单相点相反) ，通常表现为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;正则坐标 Canonical Coordinates&amp;lt;/font&amp;gt;下相空间中的分布。在量子统计力学中，系综是纯态上的概率分布，可以简单地概括为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;密度矩阵 Density Matrix&amp;lt;/font&amp;gt;。 &lt;br /&gt;
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As is usual for probabilities, the ensemble can be interpreted in different ways:&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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As is usual for probabilities, the ensemble can be interpreted in different ways:&lt;br /&gt;
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与通常的概率一样，系综可以用不同的方式来解释:&lt;br /&gt;
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* an ensemble can be taken to represent the various possible states that a ''single system'' could be in ([[epistemic probability]], a form of knowledge), or&lt;br /&gt;
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* the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner ([[empirical probability]]), in the limit of an infinite number of trials.&lt;br /&gt;
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* 系综可以表示&amp;quot;单个系统&amp;quot;的所有可能状态（&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;认识概率 Epistemic Probability&amp;lt;/font&amp;gt;，知识的一种形式），或者&lt;br /&gt;
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* 系综的元素可以理解为在无限次试验的极限下，在类似但不完全受控的独立系统中，重复进行实验得到的系统的状态（&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经验概率 Empirical Probability&amp;lt;/font&amp;gt;）。&lt;br /&gt;
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These two meanings are equivalent for many purposes, and will be used interchangeably in this article.&lt;br /&gt;
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These two meanings are equivalent for many purposes, and will be used interchangeably in this article.&lt;br /&gt;
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这两种意义在很多情况下是等价的，在本文中可以互换使用。&lt;br /&gt;
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However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the [[Liouville's theorem (Hamiltonian)|Liouville equation]] (classical mechanics) or the [[von Neumann equation]] (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.&lt;br /&gt;
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However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.&lt;br /&gt;
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然而，这种概率是可被解释的，系综中的每个随时间演化的状态都可以由运动方程给出。因此，系综本身(状态的概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;刘维尔方程 Liouville Equation&amp;lt;/font&amp;gt;(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;冯·诺依曼方程 Von Neumann Equation&amp;lt;/font&amp;gt;(量子力学)给出。这些方程是简单地通过应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统的概率随时间演化过程中是守恒的。&lt;br /&gt;
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One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as ''equilibrium ensembles'' and their condition is known as ''statistical equilibrium''. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state.{{NoteTag|Statistical equilibrium should not be confused with ''[[mechanical equilibrium]]''. The latter occurs when a mechanical system has completely ceased to evolve even on a microscopic scale, due to being in a state with a perfect balancing of forces. Statistical equilibrium generally involves states that are very far from mechanical equilibrium.}} The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.&lt;br /&gt;
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One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.&lt;br /&gt;
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有一种特殊的系综是不随时间演化的。这样的系综称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;平衡系综 Equilibrium Ensembles&amp;lt;/font&amp;gt;，它们的状态称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计平衡 Statistical Equilibrium&amp;lt;/font&amp;gt;。如果对于每个状态，无论是未来还是过去，该系综都包含在内，并且其概率等于处于该状态的概率，则出现统计平衡的情况。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学研究更一般情况下的可以随时间演化的系综，以及（或）非孤立系统的系综。&lt;br /&gt;
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== Statistical thermodynamics 统计热力学==&lt;br /&gt;
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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the [[classical thermodynamics]] of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in [[thermodynamic equilibrium]], and the microscopic behaviours and motions occurring inside the material.&lt;br /&gt;
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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.&lt;br /&gt;
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统计热力学(也称为平衡态统计力学)的主要目标是根据组成某材料的粒子的性质和它们之间的相互作用，推导出材料的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典热力学 Classical Thermodynamics&amp;lt;/font&amp;gt;。换句话说，统计热力学提供了热力学平衡态中物质的宏观性质与物质内部微观行为和运动之间的联系。&lt;br /&gt;
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Whereas statistical mechanics proper involves dynamics, here the attention is focussed on ''statistical equilibrium'' (steady state). Statistical equilibrium does not mean that the particles have stopped moving ([[mechanical equilibrium]]), rather, only that the ensemble is not evolving.&lt;br /&gt;
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Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.&lt;br /&gt;
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然而统计力学本身就涉及到动态变化，此时的关注点集中在统计平衡(稳态)上。统计平衡并不意味着粒子已经停止运动(力学平衡) ，相反，只是系综没有进化。&lt;br /&gt;
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=== Fundamental postulate 基本假设===&lt;br /&gt;
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A [[sufficient condition|sufficient]] (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).&lt;br /&gt;
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孤立系统统计平衡的一个充分(但不是必要)条件是，其概率分布只是某些守恒量(总能量、总粒子数等)的函数。&lt;br /&gt;
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There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt; Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.&lt;br /&gt;
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There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.&lt;br /&gt;
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有许多不同的平衡系综可以考虑，但只有一些适用于热力学。为了说明为什么给定系统的系综具有这样或那样的形式，还需要一些额外的假设。&lt;br /&gt;
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A common approach found in many textbooks is to take the ''equal a priori probability postulate''.&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt; This postulate states that&lt;br /&gt;
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A common approach found in many textbooks is to take the equal a priori probability postulate. This postulate states that&lt;br /&gt;
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在许多教科书中常见的一种方法是采用&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;先验概率相等假设 Equal A Priori Probability Postulate&amp;lt;/font&amp;gt;。这个假设表明&lt;br /&gt;
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''For an isolated system with an exactly known energy and exactly known composition, the system can be found with ''equal probability'' in any [[microstate (statistical mechanics)|microstate]] consistent with that knowledge.''&lt;br /&gt;
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For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge.&lt;br /&gt;
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对于一个已知精确能量和组成的孤立系统，可以在任何符合条件的微观状态下以等概率找到该系统。&lt;br /&gt;
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The equal a priori probability postulate therefore provides a motivation for the [[microcanonical ensemble]] described below. There are various arguments in favour of the equal a priori probability postulate:&lt;br /&gt;
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The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:&lt;br /&gt;
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因此，先验概率相等假设为下面描述的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微正则系综 microcanonical ensemble&amp;lt;/font&amp;gt;提供了一个动力。有各种各样的论据支持先验概率相等假设:&lt;br /&gt;
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* [[Ergodic hypothesis]]: An ergodic system is one that evolves over time to explore &amp;quot;all accessible&amp;quot; states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.&lt;br /&gt;
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* [[Principle of indifference]]: In the absence of any further information, we can only assign equal probabilities to each compatible situation.&lt;br /&gt;
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* [[Maximum entropy thermodynamics|Maximum information entropy]]: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest [[Gibbs entropy]] ([[information entropy]]).&amp;lt;ref&amp;gt;{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;各态历经假设 Ergodic hypothesis&amp;lt;/font&amp;gt;]]:各态历经系统是一种随着时间的演化而探索“所有可到达”状态的系统:所有具有相同能量和组成的状态。在各态历经系统中，微正则系综是唯一可能的具有固定能量的平衡系综。这种方法的适用性有限，因为大多数系统不是各态历经的。&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;无差别原则 Principle of Indifference&amp;lt;/font&amp;gt;]]: 在没有更多信息的情况下，我们只能对每一个相容的情况分配相等的概率。&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;最大热力学熵|最大信息熵 Maximum Entropy Thermodynamics|Maximum Information Entropy&amp;lt;/font&amp;gt;]]: 无差异原则的一个更详细的版本表明，正确的系综是与已知信息兼容且具有最大[[吉布斯熵]] ([[信息熵]])的系综。&amp;lt;ref&amp;gt;{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Other fundamental postulates for statistical mechanics have also been proposed.&amp;lt;ref name=&amp;quot;uffink&amp;quot;/&amp;gt;&lt;br /&gt;
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Other fundamental postulates for statistical mechanics have also been proposed.&lt;br /&gt;
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其他关于统计力学的基本假设也有被提出。&lt;br /&gt;
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===Three thermodynamic ensembles 三种热力学系综===&lt;br /&gt;
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{{main|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}&lt;br /&gt;
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There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt; These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.&lt;br /&gt;
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There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.&lt;br /&gt;
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对于任何有限体积的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;孤立系统 Isolated System&amp;lt;/font&amp;gt;，可以定义三种简单形式的平衡系综。这些是统计热力学中最经常讨论的系综。在宏观极限(定义如下) ，它们都与经典热力学有对应。&lt;br /&gt;
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; [[Microcanonical ensemble]]&lt;br /&gt;
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Microcanonical ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微正则系综 Microcanonical ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.&lt;br /&gt;
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describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.&lt;br /&gt;
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描述了一个具有精确给定能量和固定成分(精确数量的粒子)的系统。微正则系综中，与能量和组成相一致的每个可能状态的概率是相等的。&lt;br /&gt;
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; [[Canonical ensemble]]&lt;br /&gt;
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Canonical ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;正则系综 Canonical ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system of fixed composition that is in [[thermal equilibrium]]{{NoteTag|The transitive thermal equilibrium (as in, &amp;quot;X is thermal equilibrium with Y&amp;quot;) used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.}} with a [[heat bath]] of a precise [[thermodynamic temperature|temperature]]. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.&lt;br /&gt;
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describes a system of fixed composition that is in thermal equilibrium with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.&lt;br /&gt;
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描述了一个固定成分的系统，这个系统与一个精确温度的热浴形成热平衡。正则系综包含能量不同但组成完全相同的状态; 根据总能量的不同，系综中不同的状态被赋予不同的概率。&lt;br /&gt;
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; [[Grand canonical ensemble]]&lt;br /&gt;
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Grand canonical ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;巨正则系综 Grand canonical ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise [[chemical potential]]s for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.&lt;br /&gt;
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describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.&lt;br /&gt;
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描述了一个具有非固定成分(不确定粒子数)的系统，在热库中处于热力学和化学平衡。热库具有精确的温度，对各种类型的粒子具有精确的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;化学势 Chemical Potential&amp;lt;/font&amp;gt;。巨正则系综包含不同能量和粒子数的状态; 根据总能量和粒子数的不同，系综中不同状态的概率也不同。&lt;br /&gt;
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For systems containing many particles (the [[thermodynamic limit]]), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used.&amp;lt;ref name=&amp;quot;Reif&amp;quot;&amp;gt;{{cite book | last = Reif | first = F. | title = Fundamentals of Statistical and Thermal Physics | publisher = McGraw–Hill | year = 1965 | isbn = 9780070518001 | page = [https://archive.org/details/fundamentalsofst00fred/page/227 227] | url-access = registration | url = https://archive.org/details/fundamentalsofst00fred/page/227 }}&amp;lt;/ref&amp;gt; The Gibbs theorem about equivalence of ensembles&amp;lt;ref&amp;gt;{{cite journal |doi=10.1007/s10955-015-1212-2|title=Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels|journal=Journal of Statistical Physics|volume=159|issue=5|pages=987–1016|year=2015|last1=Touchette|first1=Hugo|arxiv=1403.6608}}&amp;lt;/ref&amp;gt; was developed into the theory of [[concentration of measure]] phenomenon,&amp;lt;ref&amp;gt;{{cite book |doi=10.1090/surv/089|title=The Concentration of Measure Phenomenon|volume=89|series=Mathematical Surveys and Monographs|year=2005|isbn=9780821837924|last1=Ledoux|first1=Michel}}.&amp;lt;/ref&amp;gt; which has applications in many areas of science, from functional analysis to methods of [[artificial intelligence]] and [[big data]] technology.&amp;lt;ref&amp;gt;{{cite journal |doi=10.1098/rsta.2017.0237|pmc=5869543|title=Blessing of dimensionality: Mathematical foundations of the statistical physics of data|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=376|issue=2118|pages=20170237|year=2018|last1=Gorban|first1=A. N.|last2=Tyukin|first2=I. Y.}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.&lt;br /&gt;
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对于包含大量粒子的系统(&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热力学极限 Thermodynamic Limit&amp;lt;/font&amp;gt;) ，上面列出的三种系综都倾向于给出相同的行为。因此，使用哪种系综只是一个简单的数学问题。关于系综等价的吉布斯定理被发展成为(&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;测度现象集中理论 Concentration of Measure Phenomenon&amp;lt;/font&amp;gt;，在从函数分析到人工智能和大数据技术等许多科学领域都有广泛的应用。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Important cases where the thermodynamic ensembles ''do not'' give identical results include:&lt;br /&gt;
&lt;br /&gt;
Important cases where the thermodynamic ensembles do not give identical results include:&lt;br /&gt;
&lt;br /&gt;
热力学系综不能给出相同结果的重要情况包括:&lt;br /&gt;
&lt;br /&gt;
* Microscopic systems.&lt;br /&gt;
&lt;br /&gt;
* Large systems at a phase transition.&lt;br /&gt;
&lt;br /&gt;
* Large systems with long-range interactions.&lt;br /&gt;
&lt;br /&gt;
* 微观系统&lt;br /&gt;
&lt;br /&gt;
* 处于相变的宏观系统&lt;br /&gt;
&lt;br /&gt;
* 长程关联的宏观系统&lt;br /&gt;
&lt;br /&gt;
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.&lt;br /&gt;
&lt;br /&gt;
在这些情况下必须选择正确的热力学系综，因为这些系综之间不仅在涨落的大小方面有可观测的差异，而且在平均量方面都有可观察的差异，如粒子数的分布。正确的系综是对应于该系统的制备和表征的方式ーー换句话说，系综反映我们对系统的认知。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:{| class=&amp;quot;wikitable sortable mw-collapsible mw-collapsed&amp;quot;&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable sortable mw-collapsible mw-collapsed&amp;quot;&lt;br /&gt;
&lt;br /&gt;
{ | class“ wikable sortable mw-collable mw-collapse”&lt;br /&gt;
&lt;br /&gt;
|+&lt;br /&gt;
&lt;br /&gt;
|+&lt;br /&gt;
&lt;br /&gt;
|+&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! rowspan=&amp;quot;2&amp;quot;|&lt;br /&gt;
&lt;br /&gt;
! rowspan=&amp;quot;2&amp;quot;|&lt;br /&gt;
&lt;br /&gt;
!Rowspan“2” | &lt;br /&gt;
&lt;br /&gt;
! colspan=&amp;quot;3&amp;quot;| Thermodynamic ensembles&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
! colspan=&amp;quot;3&amp;quot;| Thermodynamic ensembles&lt;br /&gt;
&lt;br /&gt;
!Colspan“3” | 热力学系数&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! [[Microcanonical ensemble|Microcanonical]]&lt;br /&gt;
&lt;br /&gt;
! Microcanonical&lt;br /&gt;
&lt;br /&gt;
!微正则化&lt;br /&gt;
&lt;br /&gt;
! [[Canonical ensemble|Canonical]]&lt;br /&gt;
&lt;br /&gt;
! Canonical&lt;br /&gt;
&lt;br /&gt;
!典范&lt;br /&gt;
&lt;br /&gt;
! [[Grand canonical ensemble|Grand canonical]]&lt;br /&gt;
&lt;br /&gt;
! Grand canonical&lt;br /&gt;
&lt;br /&gt;
!巨正典&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Fixed variables&lt;br /&gt;
&lt;br /&gt;
! Fixed variables&lt;br /&gt;
&lt;br /&gt;
!固定变量&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;{{math|''N, E, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
中心，中心&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;{{math|''N, T, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
中心，中心&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;{{math|''μ, T, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
中心，中心&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Microscopic features&lt;br /&gt;
&lt;br /&gt;
! Microscopic features&lt;br /&gt;
&lt;br /&gt;
!微观特征&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;Number of [[Microstate (statistical mechanics)|microstates]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt; W &amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Canonical partition function]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;Z = \sum_k e^{- E_k / k_B T}&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Grand partition function]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\mathcal Z = \sum_k e^{ -(E_k - \mu N_k) /k_B T}&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Macroscopic function&lt;br /&gt;
&lt;br /&gt;
! Macroscopic function&lt;br /&gt;
&lt;br /&gt;
!宏观功能&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Boltzmann entropy]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;S =  k_B \log W&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Helmholtz free energy]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;F = - k_B T \log Z&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Grand potential]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\Omega =- k_B T \log \mathcal Z &amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
=== Calculation methods 计算方法===&lt;br /&gt;
&lt;br /&gt;
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.&lt;br /&gt;
&lt;br /&gt;
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.&lt;br /&gt;
&lt;br /&gt;
一旦计算出一个系统的特征状态函数，该系统就被“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是最一般的(和现实的)情况对于一个精确的解来说太复杂了。存在各种方法来近似真实的系综，并且计算平均量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
====Exact 精确解====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are some cases which allow exact solutions.&lt;br /&gt;
&lt;br /&gt;
There are some cases which allow exact solutions.&lt;br /&gt;
&lt;br /&gt;
有些情况可以得到精确解。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics).&lt;br /&gt;
&lt;br /&gt;
* Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably, idealized gases of non-interacting particles have this property, allowing exact derivations of [[Maxwell–Boltzmann statistics]], [[Fermi–Dirac statistics]], and [[Bose–Einstein statistics]].&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few [[toy model]]s.&amp;lt;ref&amp;gt;{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}&amp;lt;/ref&amp;gt; Some examples include the [[Bethe ansatz]], [[square-lattice Ising model]] in zero field, [[hard hexagon model]].&lt;br /&gt;
&lt;br /&gt;
* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。&lt;br /&gt;
&lt;br /&gt;
* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;麦克斯韦–玻尔兹曼统计 Maxwell–Boltzmann statistics&amp;lt;/font&amp;gt;]]， [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;费米-狄拉克统计 Fermi–Dirac statistics&amp;lt;/font&amp;gt;]]，和[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;波色-爱因斯坦统计 Bose–Einstein statistics&amp;lt;/font&amp;gt;]]。&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。&amp;lt;ref&amp;gt;{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}&amp;lt;/ref&amp;gt; 一些例子包括[[Bethe ansatz]]，零场下[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt; 二维格点伊辛模型 square-lattice Ising model&amp;lt;/font&amp;gt;]]，[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;硬六边形模型 hard hexagon model&amp;lt;/font&amp;gt;]]。&lt;br /&gt;
&lt;br /&gt;
====Monte Carlo 蒙特卡罗方法====&lt;br /&gt;
&lt;br /&gt;
{{main|Monte Carlo method}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
One approximate approach that is particularly well suited to computers is the [[Monte Carlo method]], which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.&lt;br /&gt;
&lt;br /&gt;
One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.&lt;br /&gt;
&lt;br /&gt;
一个特别适合于计算机的近似方法是&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;蒙特卡罗方法 Monte Carlo method&amp;lt;/font&amp;gt;，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态构成系统全部状态集的代表样本，就可以得到近似的特征函数。随着随机样本数量的增加，误差可以降低到任意低的水平。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* The [[Metropolis–Hastings algorithm]] is a classic Monte Carlo method which was initially used to sample the canonical ensemble.&lt;br /&gt;
&lt;br /&gt;
* [[Path integral Monte Carlo]], also used to sample the canonical ensemble.&lt;br /&gt;
&lt;br /&gt;
* [[Metropolis–Hastings 算法]] 是一种经典的蒙特卡罗算法，最初用于正则系综的采样。&lt;br /&gt;
&lt;br /&gt;
* [[路径积分蒙特卡罗方法]] 也可以用于正则系综的采样。&lt;br /&gt;
&lt;br /&gt;
==== Other 其他====&lt;br /&gt;
&lt;br /&gt;
* For rarefied non-ideal gases, approaches such as the [[cluster expansion]] use [[perturbation theory]] to include the effect of weak interactions, leading to a [[virial expansion]].&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* For dense fluids, another approximate approach is based on reduced distribution functions, in particular the [[radial distribution function]].&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* [[Molecular dynamics]] computer simulations can be used to calculate [[microcanonical ensemble]] averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions.&lt;br /&gt;
&lt;br /&gt;
* Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.&lt;br /&gt;
&lt;br /&gt;
* 对于稀薄的非理想气体，[[团簇膨胀 cluster expansion]]等方法使用[[微扰理论 perturbation theory]]来涵盖弱相互作用的影响，得到[[维里展开 virial expansion]]。&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是[[径向分布函数 radial distribution function]]。&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* [[分子动力学 Molecular dynamics]] 计算机模拟可以用来计算各态历经系统中的[[微正则系综]]平均。对于包含与随机热浴的连接的系统，他们还可以在正则和巨正则条件下建模。&lt;br /&gt;
&lt;br /&gt;
* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。&lt;br /&gt;
&lt;br /&gt;
== Non-equilibrium statistical mechanics 非平衡态统计力学==&lt;br /&gt;
&lt;br /&gt;
{{see also|Non-equilibrium thermodynamics}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:&lt;br /&gt;
&lt;br /&gt;
There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:&lt;br /&gt;
&lt;br /&gt;
有许多令人感兴趣的物理现象都涉及到失去平衡的准热力学过程，例如:&lt;br /&gt;
&lt;br /&gt;
* [[Thermal conduction|heat transport by the internal motions in a material]], driven by a temperature imbalance,&lt;br /&gt;
&lt;br /&gt;
* [[Electrical conduction|electric currents carried by the motion of charges in a conductor]], driven by a voltage imbalance,&lt;br /&gt;
&lt;br /&gt;
* spontaneous [[chemical reaction]]s driven by a decrease in free energy,&lt;br /&gt;
&lt;br /&gt;
* [[friction]], [[dissipation]], [[quantum decoherence]],&lt;br /&gt;
&lt;br /&gt;
* systems being pumped by external forces ([[optical pumping]], etc.),&lt;br /&gt;
&lt;br /&gt;
* and irreversible processes in general.&lt;br /&gt;
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* [[热传导|材料的内部运动来传导热量]]，由热度不平衡来驱动，&lt;br /&gt;
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* [[导电|导体内电荷的运动产生电流]]，由电压不平衡来驱动，&lt;br /&gt;
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* 自发的 [[化学反应]]，由自由能的下降驱动，&lt;br /&gt;
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* [[摩擦]]，[[耗散]]，[[量子退相干]，&lt;br /&gt;
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* 由外力泵送的系统([光泵]]等)&lt;br /&gt;
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* 以及一般意义的可逆过程&lt;br /&gt;
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All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)&lt;br /&gt;
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All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)&lt;br /&gt;
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所有这些过程都是以特征速率随时间发生的，这些速率对于工程来说非常重要。非平衡态统计力学研究领域关注的是在微观水平上理解这些非平衡过程。(统计热力学只能用来计算在外部不平衡被消除，整体回归到平衡状态之后的最终结果。)&lt;br /&gt;
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In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as [[Liouville's theorem (Hamiltonian)|Liouville's equation]] or its quantum equivalent, the [[von Neumann equation]]. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's [[Gibbs entropy]] is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.&lt;br /&gt;
&lt;br /&gt;
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.&lt;br /&gt;
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原则上，非平衡态统计力学在数学上可以是精确的: 孤立系统的系综根据确定性方程随时间演化，如刘维尔方程或其量子等价、冯·诺依曼方程。这些方程是将运动力学方程独立应用于系综中每个状态的结果。不幸的是，这些系综演化方程继承了潜在动力学运动的大部分复杂性，因此很难得到精确解。此外，系综演化方程是完全可逆的，不会破坏信息(系综的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;吉布斯熵 Gibbs entropy&amp;lt;/font&amp;gt;被保留)。为了在模拟不可逆过程中取得进展，除了概率和可逆力学外，还必须考虑其他因素。&lt;br /&gt;
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Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.&lt;br /&gt;
&lt;br /&gt;
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.&lt;br /&gt;
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因此，非平衡力学是一个活跃的理论研究领域，因为这些额外假设的有效范围仍将继续探索。在下面的小节中描述了一些方法。&lt;br /&gt;
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=== Stochastic methods 随机方法===&lt;br /&gt;
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One approach to non-equilibrium statistical mechanics is to incorporate [[stochastic]] (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from [[Black hole information paradox|hypothetical situations involving black holes]], a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as [[Chaos theory|chaotic]] or [[pseudorandom]] influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.&lt;br /&gt;
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One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.&lt;br /&gt;
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处理非平衡态统计力学的一个方法是将随机行为引入系统。随机行为可以破坏系综中包含的信息。虽然这在技术上是不准确的(除了涉及黑洞的假设情况外，黑洞系统本身不会导致信息丢失) ，但这种随机性是为了反映出，随着时间的推移，感兴趣的信息会在系统内部转化为微妙的相关性，或者系统与环境之间的相关性。这些关联表现为对感兴趣的变量的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;混沌 Chaos&amp;lt;/font&amp;gt;或伪随机的影响。用适当的随机性取代这些相关性，计算可以变得容易得多。&lt;br /&gt;
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{{unordered list&lt;br /&gt;
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{{unordered list&lt;br /&gt;
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{无序列表&lt;br /&gt;
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|1 = ''[[Boltzmann transport equation]]'': An early form of stochastic mechanics appeared even before the term &amp;quot;statistical mechanics&amp;quot; had been coined, in studies of [[kinetic theory of gases|kinetic theory]]. [[James Clerk Maxwell]] had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. [[Ludwig Boltzmann]] subsequently showed that, by taking this [[molecular chaos]] for granted as a complete randomization, the motions of particles in a gas would follow a simple [[Boltzmann transport equation]] that would rapidly restore a gas to an equilibrium state (see [[H-theorem]]).&lt;br /&gt;
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|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term &amp;quot;statistical mechanics&amp;quot; had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;玻尔兹曼输运方程 Boltzmann transport equation&amp;lt;/font&amp;gt;: 在动力学理论研究中，早期的随机力学形式甚至在“统计力学”一词被创造之前就已经出现了。詹姆斯·克拉克·麦克斯韦已经证明分子碰撞会导致气体内部明显的混沌运动。路德维希·玻尔兹曼随后证明，如果把这种分子混沌理所当然地看作是一种完全的随机化，那么气体中粒子的运动将遵循一个简单的玻尔兹曼输运方程，这个方程将使气体迅速恢复到平衡状态(见H-定理)。&lt;br /&gt;
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The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the &amp;quot;interesting&amp;quot; information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped [[semiconductor]]s (in [[transistor]]s), where the electrons are indeed analogous to a rarefied gas.&lt;br /&gt;
&lt;br /&gt;
The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the &amp;quot;interesting&amp;quot; information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas.&lt;br /&gt;
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玻耳兹曼输运方程及其相关方法是非平衡态统计力学的重要工具，因为它们极其简单。这些近似方法在“感兴趣的”信息立即(在一次碰撞之后)变成微妙关联的系统中非常有效，这种关联本质上限制它们为稀薄气体。玻耳兹曼输运方程被发现在模拟轻掺杂半导体(晶体管)的电子输运中非常有用，其中的电子确实类似于稀薄气体。&lt;br /&gt;
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A quantum technique related in theme is the [[random phase approximation]].&lt;br /&gt;
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A quantum technique related in theme is the random phase approximation.&lt;br /&gt;
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一个与主题相关的量子技术是随机相位近似。&lt;br /&gt;
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|2 = ''[[BBGKY hierarchy]]'':&lt;br /&gt;
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|2 = BBGKY hierarchy:&lt;br /&gt;
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2 BBGKY 层级:&lt;br /&gt;
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In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The [[BBGKY hierarchy]] (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.&lt;br /&gt;
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In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.&lt;br /&gt;
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在液体和稠密气体中，不能在一次碰撞后立即丢掉粒子之间的关联。BBGKY 层级结构(Bogoliubov-Born-Green-Kirkwood-Yvon 层级结构)提供了一种推导玻尔兹曼型方程的方法，但也可以将它们扩展到稀薄气体情况之外，包括在几次碰撞之后的相关性。&lt;br /&gt;
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|3 = ''[[Keldysh formalism]]'' (a.k.a. NEGF—non-equilibrium Green functions):&lt;br /&gt;
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|3 = Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions):&lt;br /&gt;
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| 3 Keldysh 公式。NEGF—非平衡态格林函数):&lt;br /&gt;
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A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic [[quantum transport]] calculations.&lt;br /&gt;
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A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic quantum transport calculations.&lt;br /&gt;
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人们在 Keldysh 公式中发明了包含随机动力学的量子方法，这种方法常用于电子量子输运计算。&lt;br /&gt;
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|4 = Stochastic [[Liouville's theorem (Hamiltonian)|Liouville equation]]&lt;br /&gt;
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|4 = Stochastic Liouville equation&lt;br /&gt;
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| 4随机 刘维尔方程&lt;br /&gt;
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}}&lt;br /&gt;
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=== Near-equilibrium methods 近平衡态方法 ===&lt;br /&gt;
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Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in [[linear response theory]]. A remarkable result, as formalized by the [[fluctuation-dissipation theorem]], is that the response of a system when near equilibrium is precisely related to the [[Statistical fluctuations|fluctuations]] that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or &amp;quot;know&amp;quot; how it came to be away from equilibrium.&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;{{rp|664}}&lt;br /&gt;
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Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or &amp;quot;know&amp;quot; how it came to be away from equilibrium.&lt;br /&gt;
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非平衡态统计力学模型处理的另一类重要的系统，是对平衡态仅有非常轻微扰动的系统。在很小的扰动下，响应可以用&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;线性响应理论 linear response theory&amp;lt;/font&amp;gt;进行分析。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;涨落-耗散定理 fluctuation-dissipation theorem&amp;lt;/font&amp;gt;是其中一个重要的结果，近平衡态系统的响应与系统总体平衡时的涨落准确相关。从本质上讲，一个系统如果稍微偏离平衡，无论是由于外力还是由于涨落，都会以同样的方式向平衡方向弛豫。因为这个系统无法区分偏离和回归，也“不知道”它是如何偏离平衡的。&lt;br /&gt;
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This provides an indirect avenue for obtaining numbers such as [[Ohm's law|ohmic conductivity]] and [[thermal conductivity]] by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.&lt;br /&gt;
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This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.&lt;br /&gt;
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这提供了一个间接的方法，通过从平衡态统计力学中提取结果来获得诸如欧姆电导率和热导率之类的物理量。由于平衡态统计力学在数学上有很好的定义，而且(在某些情况下)更易于计算，因此在近平衡态统计力学中，涨落-耗散关联可以成为一种方便的计算捷径。&lt;br /&gt;
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A few of the theoretical tools used to make this connection include:&lt;br /&gt;
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A few of the theoretical tools used to make this connection include:&lt;br /&gt;
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用于建立这种关联的一些理论工具包括:&lt;br /&gt;
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* [[Fluctuation–dissipation theorem]]&lt;br /&gt;
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* [[Onsager reciprocal relations]]&lt;br /&gt;
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* [[Green–Kubo relations]]&lt;br /&gt;
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* [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker formalism]]&lt;br /&gt;
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* [[Mori–Zwanzig formalism]]&lt;br /&gt;
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* [[涨落-耗散定理]]&lt;br /&gt;
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* [[昂萨格倒易关系]]&lt;br /&gt;
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* [[格林-库伯关系]]&lt;br /&gt;
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* [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker 公式]]&lt;br /&gt;
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* [[Mori–Zwanzig 公式]]&lt;br /&gt;
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=== Hybrid methods 组合方法===&lt;br /&gt;
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An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects ([[weak localization]], [[conductance fluctuations]]) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic [[dephasing]] by interactions between various electrons by use of the Keldysh method.&amp;lt;ref&amp;gt;{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid =  | pmc = |bibcode = 1982JPhC...15.7367A }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid =  | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.&lt;br /&gt;
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一种先进的方法使用了随机方法和线性响应理论的结合。例如，计算电子系统电导中的量子相干效应(弱局域化，电导涨落)的一种方法是使用 Green-Kubo 关系，包括随机退相的各种电子之间的相互作用，使用Keldysh 方法。&lt;br /&gt;
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==Applications outside thermodynamics==&lt;br /&gt;
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The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:&lt;br /&gt;
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The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:&lt;br /&gt;
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系综也可以用来分析系统状态有不确定性的一般机械系统。在下列情况中也会使用系综:&lt;br /&gt;
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* [[propagation of uncertainty]] over time,&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt;&lt;br /&gt;
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* [[regression analysis]] of gravitational [[orbit]]s,&lt;br /&gt;
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* [[ensemble forecasting]] of weather,&lt;br /&gt;
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* dynamics of [[neural networks]],&lt;br /&gt;
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* bounded-rational [[potential game]]s in game theory and economics.&lt;br /&gt;
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* 不确定性随时间传播，&lt;br /&gt;
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* 引力[[轨道]]的[[回归分析]]，&lt;br /&gt;
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* 天气的[[系综预报]]，&lt;br /&gt;
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* [[神经网络]]动力学，&lt;br /&gt;
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* 博弈论和经济学中的有界理性[[潜在对策]]。&lt;br /&gt;
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== History 历史==&lt;br /&gt;
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In 1738, Swiss physicist and mathematician [[Daniel Bernoulli]] published ''Hydrodynamica'' which laid the basis for the [[kinetic theory of gases]]. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as [[heat]] is simply the kinetic energy of their motion.&amp;lt;ref name=&amp;quot;uffink&amp;quot;/&amp;gt;&lt;br /&gt;
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In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.&lt;br /&gt;
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1738年，瑞士的物理学家和数学家丹尼尔·伯努利发表了《水动力学》 ，这本书奠定了&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;气体动力学理论 kinetic theory of gases&amp;lt;/font&amp;gt;的基础。在这项工作中，伯努利假定气体是由大量向各个方向运动的分子组成的，它们对表面的影响导致了我们感觉到的气体压力，而我们感受到的热仅仅是它们运动的动能，这一点直到今天仍在沿用。&lt;br /&gt;
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In 1859, after reading a paper on the diffusion of molecules by [[Rudolf Clausius]], Scottish physicist [[James Clerk Maxwell]] formulated the [[Maxwell distribution]] of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.&amp;lt;ref&amp;gt;See:&lt;br /&gt;
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In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.&amp;lt;ref&amp;gt;See:&lt;br /&gt;
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1859年，在阅读了 Rudolf Clausius 的一篇关于分子扩散的论文之后，苏格兰物理学家詹姆斯·克拉克·麦克斯韦提出了分子速度的麦克斯韦-玻尔兹曼分布，它给出了在一个特定范围内具有某种速度的分子的比例。这是物理学里第一个统计定律。麦克斯韦还提出了第一个力学论点，即分子碰撞必然导致温度的平衡，从而趋向平衡。五年之后，也就是1864年，维也纳的年轻学生路德维希·玻尔兹曼(Ludwig Boltzmann)偶然发现了麦克斯韦尔的论文，并花了大半辈子的时间来进一步研究这一课题。&lt;br /&gt;
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*  Maxwell, J.C. (1860) [https://books.google.com/books?id=-YU7AQAAMAAJ&amp;amp;pg=PA19#v=onepage&amp;amp;q&amp;amp;f=false &amp;quot;Illustrations of the dynamical theory of gases. Part I.  On the motions and collisions of perfectly elastic spheres,&amp;quot;] ''Philosophical Magazine'', 4th series, '''19''' :  19–32. &lt;br /&gt;
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*  Maxwell, J.C. (1860) [https://books.google.com/books?id=DIc7AQAAMAAJ&amp;amp;pg=PA21#v=onepage&amp;amp;q&amp;amp;f=false &amp;quot;Illustrations of the dynamical theory of gases. Part II.  On the process of diffusion of two or more kinds of moving particles among one another,&amp;quot;] ''Philosophical Magazine'', 4th series, '''20''' :  21–37.&amp;lt;/ref&amp;gt; This was the first-ever statistical law in physics.&amp;lt;ref&amp;gt;{{cite book |last = Mahon |first = Basil |title=The Man Who Changed Everything – the Life of James Clerk Maxwell |location=Hoboken, NJ |publisher=Wiley |year=2003 |isbn=978-0-470-86171-4 |oclc=52358254}}&amp;lt;/ref&amp;gt; Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.&amp;lt;ref&amp;gt;{{cite journal | last = Gyenis | first = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}&amp;lt;/ref&amp;gt; Five years later, in 1864, [[Ludwig Boltzmann]], a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.&lt;br /&gt;
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Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 ''Lectures on Gas Theory''.&amp;lt;ref&amp;gt;{{cite book |title = Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems |journal=Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Edited by Ebeling Werner &amp;amp; Sokolov Igor M. Published by World Scientific Press |volume=8 |last1=Ebeling |first1=Werner |last2=Sokolov |first2=Igor M. |year=2005 |isbn=978-90-277-1674-3 |pages=3–12 |url = https://books.google.com/books?id=KUjFHbid8A0C|bibcode=2005stst.book.....E |doi=10.1142/2012 |series = Series on Advances in Statistical Mechanics }} (section 1.2)&amp;lt;/ref&amp;gt; Boltzmann's original papers on the statistical interpretation of thermodynamics, the [[H-theorem]], [[transport theory (statistical physics)|transport theory]], [[thermal equilibrium]], the [[equation of state]] of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his [[H-theorem|''H''-theorem]].&lt;br /&gt;
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Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory. Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.&lt;br /&gt;
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统计力学是19世纪70年代由玻尔兹曼的工作创立的，其中大部分在他1896年的气体理论演讲中集结出版。在维也纳学院和其他学会的会议记录中，玻尔兹曼关于热力学的统计解释、 &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;H-定理H-theorem&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;输运理论 transport theory&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热平衡thermal equilibrium&amp;lt;/font&amp;gt;、气体的状态方程以及类似主题的原始论文占据了大约2000页。玻尔兹曼引入了平衡系综的概念，并用他的H-定理第一次研究了非平衡态统计力学。&lt;br /&gt;
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The term &amp;quot;statistical mechanics&amp;quot; was coined by the American mathematical physicist [[Josiah Willard Gibbs|J. Willard Gibbs]] in 1884.&amp;lt;ref&amp;gt;J. W. Gibbs, &amp;quot;On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics.&amp;quot; Proceedings of the American Association for the Advancement of Science, '''33''', 57-58 (1884). Reproduced in ''The Scientific Papers of J. Willard Gibbs, Vol II'' (1906), [https://archive.org/stream/scientificpapers02gibbuoft#page/16/mode/2up pp.&amp;amp;nbsp;16].&amp;lt;/ref&amp;gt;{{NoteTag|1 = According to Gibbs, the term &amp;quot;statistical&amp;quot;, in the context of mechanics, i.e. statistical mechanics, was first used by the Scottish physicist [[James Clerk Maxwell]] in 1871. From: J. Clerk Maxwell, ''Theory of Heat'' (London, England: Longmans, Green, and Co., 1871), [https://books.google.com/books?id=DqAAAAAAMAAJ&amp;amp;pg=PA309 p.&amp;amp;nbsp;309]: &amp;quot;In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus.&amp;quot;}} &amp;quot;Probabilistic mechanics&amp;quot; might today seem a more appropriate term, but &amp;quot;statistical mechanics&amp;quot; is firmly entrenched.&amp;lt;ref&amp;gt;{{cite book |title = The enigma of probability and physics |last=Mayants |first=Lazar |year=1984 |publisher=Springer |isbn=978-90-277-1674-3 |page=174 |url = https://books.google.com/books?id=zmwEfXUdBJ8C&amp;amp;pg=PA174 }}&amp;lt;/ref&amp;gt; Shortly before his death, Gibbs published in 1902 ''[[Elementary Principles in Statistical Mechanics]]'', a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt; Gibbs' methods were initially derived in the framework [[classical mechanics]], however they were of such generality that they were found to adapt easily to the later [[quantum mechanics]], and still form the foundation of statistical mechanics to this day.&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&lt;br /&gt;
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The term &amp;quot;statistical mechanics&amp;quot; was coined by the American mathematical physicist J. Willard Gibbs in 1884. &amp;quot;Probabilistic mechanics&amp;quot; might today seem a more appropriate term, but &amp;quot;statistical mechanics&amp;quot; is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.&lt;br /&gt;
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1884年，美国数学物理学家约西亚·威拉德·吉布斯首创了“统计力学”一词。在今天看来，“概率力学”似乎是一个更合适的术语，但“统计力学”却根深蒂固。在吉布斯去世前不久，他于1902年在《统计力学》出版了《基本原理》一书，这本书正式确定了统计力学是解决所有力学系统——宏观的或微观的、气态的或非气态的——的一种完全通用的方法。吉布斯的方法最初是在经典力学的框架下产生的，然而它们是如此的普遍，以至于人们发现它们很容易适应后来的量子力学，直到今天仍然是统计力学的基础。&lt;br /&gt;
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== See also 其他可见==&lt;br /&gt;
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{{Books-inline|Fundamentals of Statistical Mechanics}}&lt;br /&gt;
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* [[Thermodynamics]]: [[Non-equilibrium thermodynamics|non-equilibrium]], [[Chemical thermodynamics|chemical]]&lt;br /&gt;
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* [[Mechanics]]: [[Classical mechanics|classical]], [[Quantum mechanics|quantum]]&lt;br /&gt;
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* [[Probability]], [[Statistical ensemble (mathematical physics)|statistical ensemble]]&lt;br /&gt;
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* Numerical methods: [[Monte Carlo method]], [[molecular dynamics]]&lt;br /&gt;
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* [[Statistical physics]]&lt;br /&gt;
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* [[Quantum statistical mechanics]]&lt;br /&gt;
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* [[List of notable textbooks in statistical mechanics]]&lt;br /&gt;
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* [[List of publications in physics#Statistical mechanics|List of important publications in statistical mechanics]]&lt;br /&gt;
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{{Books-inline|统计力学基础}}&lt;br /&gt;
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* [[热力学]]: [[非平衡态热力学]], [[化学热力学]]&lt;br /&gt;
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* [[力学]]: [[经典力学]], [[量子力学]]&lt;br /&gt;
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* [[概率]], [[统计系综（数学物理）]]&lt;br /&gt;
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* 数值方法: [[蒙特卡洛方法]], [[分子动力学]]&lt;br /&gt;
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* [[统计物理]]&lt;br /&gt;
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* [[量子统计力学]]&lt;br /&gt;
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* [[统计力学著名教科书列表]]&lt;br /&gt;
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* [[统计力学重要文献列表]]&lt;br /&gt;
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== Notes ==&lt;br /&gt;
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== References ==&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;gibbs&amp;quot;&amp;gt;{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York |title-link=Elementary Principles in Statistical Mechanics }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;tolman&amp;quot;&amp;gt;{{cite book | last=Tolman |first=R. C. | author-link = Richard C. Tolman | year=1938 | title = The Principles of Statistical Mechanics | publisher=[[Dover Publications]] | isbn = 9780486638966 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;balescu&amp;quot;&amp;gt;{{cite book | isbn = 9780471046004 | title = Equilibrium and Non-Equilibrium Statistical Mechanics |last = Balescu |first = Radu | author-link = Radu Balescu | year = 1975 | publisher = John Wiley &amp;amp; Sons }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;uffink&amp;quot;&amp;gt;J. Uffink, &amp;quot;[http://philsci-archive.pitt.edu/2691/1/UffinkFinal.pdf Compendium of the foundations of classical statistical physics.]&amp;quot; (2006)&amp;lt;/ref&amp;gt;&lt;br /&gt;
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== External links ==&lt;br /&gt;
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{{Commons category|Statistical mechanics}}&lt;br /&gt;
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* [http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics] article by Lawrence Sklar for the [[Stanford Encyclopedia of Philosophy]].&lt;br /&gt;
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* [http://www.sklogwiki.org/ Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials.]  SklogWiki is particularly orientated towards liquids and soft condensed matter.&lt;br /&gt;
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* [https://web.archive.org/web/20060616002812/http://history.hyperjeff.net/statmech.html Statistical Thermodynamics] - Historical Timeline&lt;br /&gt;
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* [http://farside.ph.utexas.edu/teaching/sm1/statmech.pdf Thermodynamics and Statistical Mechanics] by Richard Fitzpatrick&lt;br /&gt;
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* [https://arxiv.org/abs/1107.0568 Lecture Notes in Statistical Mechanics and Mesoscopics] by Doron Cohen&lt;br /&gt;
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* {{YouTube |id = H1Zbp6__uNw&amp;amp;list=PLB72416C707D85AB0&amp;amp;index=1 |title = Videos of lecture series in statistical mechanics }} taught by [[Leonard Susskind]].&lt;br /&gt;
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* Vu-Quoc, L., [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008. this wiki site is down; see [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 this article in the web archive on 2012 April 28].&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Statistical mechanics]]. Its edit history can be viewed at [[统计力学/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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		<author><name>Ryan</name></author>
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		<title>统计力学</title>
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{{Short description|Physics of large number of particles' statistical behavior}}&lt;br /&gt;
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{{Statistical mechanics}}&lt;br /&gt;
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'''Statistical mechanics''' is one of the pillars of modern [[physics]]. It is necessary for the fundamental study of any physical system that has many [[degrees of freedom]]. The approach is based on [[Statistics|statistical]] methods, [[probability theory]] and the [[Microscopic scale|microscopic]] physical laws.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;{{NoteTag|The term ''statistical mechanics'' is sometimes used to refer to only ''statistical thermodynamics''. This article takes the broader view. By some definitions, ''[[statistical physics]]'' is an even broader term which statistically studies any type of physical system, but is often taken to be synonymous with statistical mechanics.}}&lt;br /&gt;
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Statistical mechanics is one of the pillars of modern physics. It is necessary for the fundamental study of any physical system that has many degrees of freedom. The approach is based on statistical methods, probability theory and the microscopic physical laws.&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计力学 Statistical mechanics&amp;lt;/font&amp;gt;是现代物理学的支柱之一。对于具有多个&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;自由度 Degrees of Freedom&amp;lt;/font&amp;gt;的物理系统的基础研究，统计力学是不可或缺的。统计力学的方法是基于&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计学方法 Statistical Methods&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;概率论 Probability Theory&amp;lt;/font&amp;gt;和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微观物理定律 Microscopic Physical Laws&amp;lt;/font&amp;gt;。&lt;br /&gt;
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It can be used to explain the [[thermodynamics|thermodynamic]] behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as '''statistical thermodynamics''' or '''equilibrium statistical mechanics'''.&lt;br /&gt;
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It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.&lt;br /&gt;
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统计力学一个分支可以用来解释大系统的热力学行为。该分支完善和扩展了经典热力学，被称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计热力学 Statistical Thermodynamics&amp;lt;/font&amp;gt;或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;平衡态统计力学 Equilibrium Statistical Mechanics&amp;lt;/font&amp;gt;。&lt;br /&gt;
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Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as [[heat capacity]]) to microscopic behavior, whereas, in [[classical thermodynamics]], the only available option would be to measure and tabulate such quantities for various materials.&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt;&lt;br /&gt;
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Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.&lt;br /&gt;
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统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热容 Heat Capacity&amp;lt;/font&amp;gt;)与微观行为联系起来。而在&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典热力学 Classical Thermodynamics&amp;lt;/font&amp;gt;中，唯一可行的选择就是测量和列出各种材料的热力学量。&lt;br /&gt;
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Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as '''non-equilibrium statistical mechanics''' (sometimes called '''statistical dynamics''') deals with the issue of microscopically modelling the speed of [[irreversible process]]es that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The [[fluctuation–dissipation theorem]] is the basic knowledge obtained from applying [[non-equilibrium statistical mechanics]] to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.&lt;br /&gt;
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Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics (sometimes called statistical dynamics) deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.&lt;br /&gt;
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统计力学也可以用来研究非平衡的系统。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;非平衡统计力学 non-equilibrium statistical mechanics&amp;lt;/font&amp;gt;(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;涨落-耗散定理 Fluctuation–Dissipation Theorem&amp;lt;/font&amp;gt;是人们从非平衡态统计力学中获得的基本知识，它是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。&lt;br /&gt;
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== Principles: mechanics and ensembles 原理：力学和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;系综 Ensembles&amp;lt;/font&amp;gt;==&lt;br /&gt;
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{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}&lt;br /&gt;
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In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts:&lt;br /&gt;
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In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:&lt;br /&gt;
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在物理学中，有两种力学被广泛研究: &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典力学 Classical Mechanics&amp;lt;/font&amp;gt;和&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;量子力学 Quantum Mechanics&amp;lt;/font&amp;gt;。对于这两种力学，标准的数学方法与两个概念有关&lt;br /&gt;
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# The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics).&lt;br /&gt;
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The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).&lt;br /&gt;
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力学系统在给定时间内的完整状态，用数学表示为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;相空间 Phase Space&amp;lt;/font&amp;gt;中的点(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;纯量子态矢量 Pure Quantum State Vector&amp;lt;/font&amp;gt;(量子力学)。&lt;br /&gt;
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# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)&lt;br /&gt;
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An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)&lt;br /&gt;
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一个运动方程描述状态在时间上的演化: &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;哈密尔顿方程 Hamilton's Equations&amp;lt;/font&amp;gt;(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;含时薛定谔方程 Time-dependent Schrödinger Equation&amp;lt;/font&amp;gt;(量子力学)&lt;br /&gt;
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Using these two concepts, the state at any other time, past or future, can in principle be calculated.&lt;br /&gt;
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Using these two concepts, the state at any other time, past or future, can in principle be calculated.&lt;br /&gt;
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使用这两个概念，系统在任何时间的状态，无论过去或未来，原则上都可以计算出来。&lt;br /&gt;
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There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.&lt;br /&gt;
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There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.&lt;br /&gt;
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然而，这些定律与日常生活经验之间存在着脱节。因为对于在人类尺度上进行过程(例如化学反应)，我们没有必要(甚至在理论上也不可能)在微观层面上准确地知道每个分子所在的位置及其速度。统计力学通过增加一些对于系统状态的不确定性，填补了力学定律和人类不完全知识的实践经验之间的脱节。&lt;br /&gt;
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Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the [[Statistical ensemble (mathematical physics)|statistical ensemble]], which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a [[probability distribution]] over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a [[phase space]] with [[canonical coordinates]]. In quantum statistical mechanics, the ensemble is a probability distribution over pure states,{{NoteTag|The probabilities in quantum statistical mechanics should not be confused with [[quantum superposition]]. While a quantum ensemble can contain states with quantum superpositions, a single quantum state cannot be used to represent an ensemble.}} and can be compactly summarized as a [[density matrix]].&lt;br /&gt;
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Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix.&lt;br /&gt;
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普通力学只考虑单一状态的行为，而统计力学引入了&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计系综 Statistical Ensemble&amp;lt;/font&amp;gt;，它是系统在各种状态下的大量虚拟、独立的拷贝的集合。系综是一个覆盖系统所有可能状态的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;概率分布 Probability Distribution&amp;lt;/font&amp;gt;。在经典的统计力学中，系综是相点上的概率分布(与普通力学中的单相点相反) ，通常表现为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;正则坐标 Canonical Coordinates&amp;lt;/font&amp;gt;下相空间中的分布。在量子统计力学中，系综是纯态上的概率分布，可以简单地概括为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;密度矩阵 Density Matrix&amp;lt;/font&amp;gt;。 &lt;br /&gt;
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As is usual for probabilities, the ensemble can be interpreted in different ways:&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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As is usual for probabilities, the ensemble can be interpreted in different ways:&lt;br /&gt;
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与通常的概率一样，系综可以用不同的方式来解释:&lt;br /&gt;
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* an ensemble can be taken to represent the various possible states that a ''single system'' could be in ([[epistemic probability]], a form of knowledge), or&lt;br /&gt;
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* the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner ([[empirical probability]]), in the limit of an infinite number of trials.&lt;br /&gt;
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* 系综可以表示&amp;quot;单个系统&amp;quot;的所有可能状态（&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;认识概率 Epistemic Probability&amp;lt;/font&amp;gt;，知识的一种形式），或者&lt;br /&gt;
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* 系综的元素可以理解为在无限次试验的极限下，在类似但不完全受控的独立系统中，重复进行实验得到的系统的状态（&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经验概率 Empirical Probability&amp;lt;/font&amp;gt;）。&lt;br /&gt;
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These two meanings are equivalent for many purposes, and will be used interchangeably in this article.&lt;br /&gt;
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These two meanings are equivalent for many purposes, and will be used interchangeably in this article.&lt;br /&gt;
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这两种意义在很多情况下是等价的，在本文中可以互换使用。&lt;br /&gt;
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However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the [[Liouville's theorem (Hamiltonian)|Liouville equation]] (classical mechanics) or the [[von Neumann equation]] (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.&lt;br /&gt;
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However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.&lt;br /&gt;
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然而，这种概率是可被解释的，系综中的每个随时间演化的状态都可以由运动方程给出。因此，系综本身(状态的概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;刘维尔方程 Liouville Equation&amp;lt;/font&amp;gt;(经典力学)或&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;冯·诺依曼方程 Von Neumann Equation&amp;lt;/font&amp;gt;(量子力学)给出。这些方程是简单地通过应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统的概率随时间演化过程中是守恒的。&lt;br /&gt;
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One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as ''equilibrium ensembles'' and their condition is known as ''statistical equilibrium''. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state.{{NoteTag|Statistical equilibrium should not be confused with ''[[mechanical equilibrium]]''. The latter occurs when a mechanical system has completely ceased to evolve even on a microscopic scale, due to being in a state with a perfect balancing of forces. Statistical equilibrium generally involves states that are very far from mechanical equilibrium.}} The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.&lt;br /&gt;
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One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.&lt;br /&gt;
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有一种特殊的系综是不随时间演化的。这样的系综称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;平衡系综 Equilibrium Ensembles&amp;lt;/font&amp;gt;，它们的状态称为&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;统计平衡 Statistical Equilibrium&amp;lt;/font&amp;gt;。如果对于每个状态，无论是未来还是过去，该系综都包含在内，并且其概率等于处于该状态的概率，则出现统计平衡的情况。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学研究更一般情况下的可以随时间演化的系综，以及（或）非孤立系统的系综。&lt;br /&gt;
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== Statistical thermodynamics 统计热力学==&lt;br /&gt;
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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the [[classical thermodynamics]] of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in [[thermodynamic equilibrium]], and the microscopic behaviours and motions occurring inside the material.&lt;br /&gt;
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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.&lt;br /&gt;
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统计热力学(也称为平衡态统计力学)的主要目标是根据组成某材料的粒子的性质和它们之间的相互作用，推导出材料的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;经典热力学 Classical Thermodynamics&amp;lt;/font&amp;gt;。换句话说，统计热力学提供了热力学平衡态中物质的宏观性质与物质内部微观行为和运动之间的联系。&lt;br /&gt;
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Whereas statistical mechanics proper involves dynamics, here the attention is focussed on ''statistical equilibrium'' (steady state). Statistical equilibrium does not mean that the particles have stopped moving ([[mechanical equilibrium]]), rather, only that the ensemble is not evolving.&lt;br /&gt;
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Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.&lt;br /&gt;
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然而统计力学本身就涉及到动态变化，此时的关注点集中在统计平衡(稳态)上。统计平衡并不意味着粒子已经停止运动(力学平衡) ，相反，只是系综没有进化。&lt;br /&gt;
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=== Fundamental postulate 基本假设===&lt;br /&gt;
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A [[sufficient condition|sufficient]] (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
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A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).&lt;br /&gt;
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孤立系统统计平衡的一个充分(但不是必要)条件是，其概率分布只是某些守恒量(总能量、总粒子数等)的函数。&lt;br /&gt;
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There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt; Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.&lt;br /&gt;
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There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.&lt;br /&gt;
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有许多不同的平衡系综可以考虑，但只有一些适用于热力学。为了说明为什么给定系统的系综具有这样或那样的形式，还需要一些额外的假设。&lt;br /&gt;
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A common approach found in many textbooks is to take the ''equal a priori probability postulate''.&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt; This postulate states that&lt;br /&gt;
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A common approach found in many textbooks is to take the equal a priori probability postulate. This postulate states that&lt;br /&gt;
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在许多教科书中常见的一种方法是采用&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;先验概率相等假设 Equal A Priori Probability Postulate&amp;lt;/font&amp;gt;。这个假设表明&lt;br /&gt;
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''For an isolated system with an exactly known energy and exactly known composition, the system can be found with ''equal probability'' in any [[microstate (statistical mechanics)|microstate]] consistent with that knowledge.''&lt;br /&gt;
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For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge.&lt;br /&gt;
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对于一个已知精确能量和组成的孤立系统，可以在任何符合条件的微观状态下以等概率找到该系统。&lt;br /&gt;
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The equal a priori probability postulate therefore provides a motivation for the [[microcanonical ensemble]] described below. There are various arguments in favour of the equal a priori probability postulate:&lt;br /&gt;
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The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:&lt;br /&gt;
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因此，先验概率相等假设为下面描述的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微正则系综 microcanonical ensemble&amp;lt;/font&amp;gt;提供了一个动力。有各种各样的论据支持先验概率相等假设:&lt;br /&gt;
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* [[Ergodic hypothesis]]: An ergodic system is one that evolves over time to explore &amp;quot;all accessible&amp;quot; states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.&lt;br /&gt;
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* [[Principle of indifference]]: In the absence of any further information, we can only assign equal probabilities to each compatible situation.&lt;br /&gt;
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* [[Maximum entropy thermodynamics|Maximum information entropy]]: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest [[Gibbs entropy]] ([[information entropy]]).&amp;lt;ref&amp;gt;{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;各态历经假设 Ergodic hypothesis&amp;lt;/font&amp;gt;]]:各态历经系统是一种随着时间的演化而探索“所有可到达”状态的系统:所有具有相同能量和组成的状态。在各态历经系统中，微正则系综是唯一可能的具有固定能量的平衡系综。这种方法的适用性有限，因为大多数系统不是各态历经的。&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;无差别原则 Principle of Indifference&amp;lt;/font&amp;gt;]]: 在没有更多信息的情况下，我们只能对每一个相容的情况分配相等的概率。&lt;br /&gt;
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* [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;最大热力学熵|最大信息熵 Maximum Entropy Thermodynamics|Maximum Information Entropy&amp;lt;/font&amp;gt;]]: 无差异原则的一个更详细的版本表明，正确的系综是与已知信息兼容且具有最大[[吉布斯熵]] ([[信息熵]])的系综。&amp;lt;ref&amp;gt;{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Other fundamental postulates for statistical mechanics have also been proposed.&amp;lt;ref name=&amp;quot;uffink&amp;quot;/&amp;gt;&lt;br /&gt;
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Other fundamental postulates for statistical mechanics have also been proposed.&lt;br /&gt;
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其他关于统计力学的基本假设也有被提出。&lt;br /&gt;
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===Three thermodynamic ensembles 三种热力学系综===&lt;br /&gt;
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{{main|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}&lt;br /&gt;
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There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt; These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.&lt;br /&gt;
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There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.&lt;br /&gt;
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对于任何有限体积的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;孤立系统 isolated system&amp;lt;/font&amp;gt;，可以定义三种简单形式的平衡系综。这些是统计热力学中最经常讨论的系综。在宏观极限(定义如下) ，它们都与经典热力学有对应。&lt;br /&gt;
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; [[Microcanonical ensemble]]&lt;br /&gt;
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 Microcanonical ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;微正则系综 Microcanonical ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.&lt;br /&gt;
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 describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.&lt;br /&gt;
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描述了一个具有精确给定的能量和固定成分(精确数量的粒子)的系统。微正则系综中，与能量和组成相一致的每个可能状态的概率是相等的。&lt;br /&gt;
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; [[Canonical ensemble]]&lt;br /&gt;
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 Canonical ensemble&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;正则系综 Canonical ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
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: describes a system of fixed composition that is in [[thermal equilibrium]]{{NoteTag|The transitive thermal equilibrium (as in, &amp;quot;X is thermal equilibrium with Y&amp;quot;) used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.}} with a [[heat bath]] of a precise [[thermodynamic temperature|temperature]]. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.&lt;br /&gt;
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 describes a system of fixed composition that is in thermal equilibrium with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.&lt;br /&gt;
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描述了一个固定成分的系统，这个系统与一个精确温度的热浴形成热平衡。正则系综包含能量不同但组成完全相同的状态; 根据总能量的不同，系综中不同的状态被赋予不同的概率。&lt;br /&gt;
&lt;br /&gt;
; [[Grand canonical ensemble]]&lt;br /&gt;
&lt;br /&gt;
 Grand canonical ensemble&lt;br /&gt;
&lt;br /&gt;
&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;巨正则系综 Grand canonical ensemble&amp;lt;/font&amp;gt;&lt;br /&gt;
&lt;br /&gt;
: describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise [[chemical potential]]s for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.&lt;br /&gt;
&lt;br /&gt;
 describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.&lt;br /&gt;
&lt;br /&gt;
描述了一个具有非固定成分(不确定粒子数)的系统，在热库中处于热力学和化学平衡。热库具有精确的温度，对各种类型的粒子具有精确的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;化学势 chemical potential&amp;lt;/font&amp;gt;。巨正则系综包含不同能量和粒子数的状态; 根据总能量和粒子数的不同，系综中不同状态的概率也不同。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
For systems containing many particles (the [[thermodynamic limit]]), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used.&amp;lt;ref name=&amp;quot;Reif&amp;quot;&amp;gt;{{cite book | last = Reif | first = F. | title = Fundamentals of Statistical and Thermal Physics | publisher = McGraw–Hill | year = 1965 | isbn = 9780070518001 | page = [https://archive.org/details/fundamentalsofst00fred/page/227 227] | url-access = registration | url = https://archive.org/details/fundamentalsofst00fred/page/227 }}&amp;lt;/ref&amp;gt; The Gibbs theorem about equivalence of ensembles&amp;lt;ref&amp;gt;{{cite journal |doi=10.1007/s10955-015-1212-2|title=Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels|journal=Journal of Statistical Physics|volume=159|issue=5|pages=987–1016|year=2015|last1=Touchette|first1=Hugo|arxiv=1403.6608}}&amp;lt;/ref&amp;gt; was developed into the theory of [[concentration of measure]] phenomenon,&amp;lt;ref&amp;gt;{{cite book |doi=10.1090/surv/089|title=The Concentration of Measure Phenomenon|volume=89|series=Mathematical Surveys and Monographs|year=2005|isbn=9780821837924|last1=Ledoux|first1=Michel}}.&amp;lt;/ref&amp;gt; which has applications in many areas of science, from functional analysis to methods of [[artificial intelligence]] and [[big data]] technology.&amp;lt;ref&amp;gt;{{cite journal |doi=10.1098/rsta.2017.0237|pmc=5869543|title=Blessing of dimensionality: Mathematical foundations of the statistical physics of data|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=376|issue=2118|pages=20170237|year=2018|last1=Gorban|first1=A. N.|last2=Tyukin|first2=I. Y.}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.&lt;br /&gt;
&lt;br /&gt;
对于包含大量粒子的系统(&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热力学极限 thermodynamic limit&amp;lt;/font&amp;gt;) ，上面列出的三种系综都倾向于给出相同的行为。因此，使用哪种系综只是一个简单的数学方便问题。关于系综等价的吉布斯定理被发展成为测度现象集中理论，在从函数分析到人工智能和大数据技术等许多科学领域都有广泛的应用。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Important cases where the thermodynamic ensembles ''do not'' give identical results include:&lt;br /&gt;
&lt;br /&gt;
Important cases where the thermodynamic ensembles do not give identical results include:&lt;br /&gt;
&lt;br /&gt;
热力学系综不能给出相同结果的重要情况包括:&lt;br /&gt;
&lt;br /&gt;
* Microscopic systems.&lt;br /&gt;
&lt;br /&gt;
* Large systems at a phase transition.&lt;br /&gt;
&lt;br /&gt;
* Large systems with long-range interactions.&lt;br /&gt;
&lt;br /&gt;
* 微观系统&lt;br /&gt;
&lt;br /&gt;
* 相变处的宏观系统&lt;br /&gt;
&lt;br /&gt;
* 长程关联的宏观系统&lt;br /&gt;
&lt;br /&gt;
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.&lt;br /&gt;
&lt;br /&gt;
在这些情况下必须选择正确的热力学系综，因为这些系综之间不仅在涨落的大小方面有可观测的差异，而且在平均量方面都有可观察的差异，如粒子数的分布。正确的系综是对应于该系统的制备和表征的方式ーー换句话说，系综反映我们对系统的认知。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:{| class=&amp;quot;wikitable sortable mw-collapsible mw-collapsed&amp;quot;&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable sortable mw-collapsible mw-collapsed&amp;quot;&lt;br /&gt;
&lt;br /&gt;
{ | class“ wikable sortable mw-collable mw-collapse”&lt;br /&gt;
&lt;br /&gt;
|+&lt;br /&gt;
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|+&lt;br /&gt;
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|+&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
&lt;br /&gt;
! rowspan=&amp;quot;2&amp;quot;|&lt;br /&gt;
&lt;br /&gt;
! rowspan=&amp;quot;2&amp;quot;|&lt;br /&gt;
&lt;br /&gt;
!Rowspan“2” | &lt;br /&gt;
&lt;br /&gt;
! colspan=&amp;quot;3&amp;quot;| Thermodynamic ensembles&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
! colspan=&amp;quot;3&amp;quot;| Thermodynamic ensembles&lt;br /&gt;
&lt;br /&gt;
!Colspan“3” | 热力学系数&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! [[Microcanonical ensemble|Microcanonical]]&lt;br /&gt;
&lt;br /&gt;
! Microcanonical&lt;br /&gt;
&lt;br /&gt;
!微正则化&lt;br /&gt;
&lt;br /&gt;
! [[Canonical ensemble|Canonical]]&lt;br /&gt;
&lt;br /&gt;
! Canonical&lt;br /&gt;
&lt;br /&gt;
!典范&lt;br /&gt;
&lt;br /&gt;
! [[Grand canonical ensemble|Grand canonical]]&lt;br /&gt;
&lt;br /&gt;
! Grand canonical&lt;br /&gt;
&lt;br /&gt;
!巨正典&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Fixed variables&lt;br /&gt;
&lt;br /&gt;
! Fixed variables&lt;br /&gt;
&lt;br /&gt;
!固定变量&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;{{math|''N, E, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
中心，中心&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;{{math|''N, T, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
中心，中心&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;{{math|''μ, T, V''}}&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;center&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
中心，中心&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Microscopic features&lt;br /&gt;
&lt;br /&gt;
! Microscopic features&lt;br /&gt;
&lt;br /&gt;
!微观特征&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;Number of [[Microstate (statistical mechanics)|microstates]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt; W &amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Canonical partition function]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;Z = \sum_k e^{- E_k / k_B T}&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
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&amp;lt;/div&amp;gt;&lt;br /&gt;
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/ div&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Grand partition function]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\mathcal Z = \sum_k e^{ -(E_k - \mu N_k) /k_B T}&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
/ div&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Macroscopic function&lt;br /&gt;
&lt;br /&gt;
! Macroscopic function&lt;br /&gt;
&lt;br /&gt;
!宏观功能&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Boltzmann entropy]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;S =  k_B \log W&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
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&amp;lt;/div&amp;gt;&lt;br /&gt;
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/ div&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“ div class”“ plainlist”&lt;br /&gt;
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*&amp;lt;center&amp;gt;[[Helmholtz free energy]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;F = - k_B T \log Z&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
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&amp;lt;/div&amp;gt;&lt;br /&gt;
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/ div&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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| &amp;lt;div class=&amp;quot;plainlist&amp;quot;&amp;gt;&lt;br /&gt;
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“ div class”“ plainlist”&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;[[Grand potential]]&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\Omega =- k_B T \log \mathcal Z &amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
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&amp;lt;/div&amp;gt;&lt;br /&gt;
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/ div&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|}&lt;br /&gt;
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|}&lt;br /&gt;
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|}&lt;br /&gt;
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=== Calculation methods 计算方法===&lt;br /&gt;
&lt;br /&gt;
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.&lt;br /&gt;
&lt;br /&gt;
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.&lt;br /&gt;
&lt;br /&gt;
一旦计算出一个系统的特征状态函数，该系统就“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是最一般的(和现实的)情况对于一个精确的解来说太复杂了。存在各种方法来近似真实的系综，并且计算平均量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
====Exact 精确解====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are some cases which allow exact solutions.&lt;br /&gt;
&lt;br /&gt;
There are some cases which allow exact solutions.&lt;br /&gt;
&lt;br /&gt;
有些情况可以得到精确解。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics).&lt;br /&gt;
&lt;br /&gt;
* Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably, idealized gases of non-interacting particles have this property, allowing exact derivations of [[Maxwell–Boltzmann statistics]], [[Fermi–Dirac statistics]], and [[Bose–Einstein statistics]].&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few [[toy model]]s.&amp;lt;ref&amp;gt;{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}&amp;lt;/ref&amp;gt; Some examples include the [[Bethe ansatz]], [[square-lattice Ising model]] in zero field, [[hard hexagon model]].&lt;br /&gt;
&lt;br /&gt;
* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。&lt;br /&gt;
&lt;br /&gt;
* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;麦克斯韦–玻尔兹曼统计 Maxwell–Boltzmann statistics&amp;lt;/font&amp;gt;]]， [[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;费米-狄拉克统计 Fermi–Dirac statistics&amp;lt;/font&amp;gt;]]，和[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;波色-爱因斯坦统计 Bose–Einstein statistics&amp;lt;/font&amp;gt;]]。&amp;lt;ref name=&amp;quot;tolman&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。&amp;lt;ref&amp;gt;{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}&amp;lt;/ref&amp;gt; 一些例子包括[[Bethe ansatz]]，零场下[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt; 二维格点伊辛模型 square-lattice Ising model&amp;lt;/font&amp;gt;]]，[[&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;硬六边形模型 hard hexagon model&amp;lt;/font&amp;gt;]]。&lt;br /&gt;
&lt;br /&gt;
====Monte Carlo 蒙特卡罗方法====&lt;br /&gt;
&lt;br /&gt;
{{main|Monte Carlo method}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
One approximate approach that is particularly well suited to computers is the [[Monte Carlo method]], which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.&lt;br /&gt;
&lt;br /&gt;
One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.&lt;br /&gt;
&lt;br /&gt;
一个特别适合于计算机的近似方法是&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;蒙特卡罗方法 Monte Carlo method&amp;lt;/font&amp;gt;，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态构成系统全部状态集的代表样本，就可以得到近似的特征函数。随着随机样本数量的增加，误差可以降低到任意低的水平。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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* The [[Metropolis–Hastings algorithm]] is a classic Monte Carlo method which was initially used to sample the canonical ensemble.&lt;br /&gt;
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* [[Path integral Monte Carlo]], also used to sample the canonical ensemble.&lt;br /&gt;
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* [[Metropolis–Hastings 算法]] 是一种经典的蒙特卡罗算法，最初用于正则系综的采样。&lt;br /&gt;
&lt;br /&gt;
* [[路径积分蒙特卡罗方法]] 也可以用于正则系综的采样。&lt;br /&gt;
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==== Other 其他====&lt;br /&gt;
&lt;br /&gt;
* For rarefied non-ideal gases, approaches such as the [[cluster expansion]] use [[perturbation theory]] to include the effect of weak interactions, leading to a [[virial expansion]].&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* For dense fluids, another approximate approach is based on reduced distribution functions, in particular the [[radial distribution function]].&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* [[Molecular dynamics]] computer simulations can be used to calculate [[microcanonical ensemble]] averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions.&lt;br /&gt;
&lt;br /&gt;
* Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.&lt;br /&gt;
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* 对于稀薄的非理想气体，[[团簇膨胀 cluster expansion]]等方法使用[[微扰理论 perturbation theory]]来涵盖弱相互作用的影响，得到[[维里展开 virial expansion]]。&amp;lt;ref name=&amp;quot;balescu&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是[[径向分布函数 radial distribution function]]。&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;&lt;br /&gt;
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* [[分子动力学 Molecular dynamics]] 计算机模拟可以用来计算各态历经系统中的[[微正则系综]]平均。对于包含与随机热浴的连接的系统，他们还可以在正则和巨正则条件下建模。&lt;br /&gt;
&lt;br /&gt;
* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。&lt;br /&gt;
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== Non-equilibrium statistical mechanics 非平衡态统计力学==&lt;br /&gt;
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{{see also|Non-equilibrium thermodynamics}}&lt;br /&gt;
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There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:&lt;br /&gt;
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There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:&lt;br /&gt;
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有许多令人感兴趣的物理现象都涉及到失去平衡的准热力学过程，例如:&lt;br /&gt;
&lt;br /&gt;
* [[Thermal conduction|heat transport by the internal motions in a material]], driven by a temperature imbalance,&lt;br /&gt;
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* [[Electrical conduction|electric currents carried by the motion of charges in a conductor]], driven by a voltage imbalance,&lt;br /&gt;
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* spontaneous [[chemical reaction]]s driven by a decrease in free energy,&lt;br /&gt;
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* [[friction]], [[dissipation]], [[quantum decoherence]],&lt;br /&gt;
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* systems being pumped by external forces ([[optical pumping]], etc.),&lt;br /&gt;
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* and irreversible processes in general.&lt;br /&gt;
&lt;br /&gt;
* [[热传导|材料的内部运动来传导热量]]，由热度不平衡来驱动，&lt;br /&gt;
&lt;br /&gt;
* [[导电|导体内电荷的运动产生电流]]，由电压不平衡来驱动，&lt;br /&gt;
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* 自发的 [[化学反应]]，由自由能的下降驱动，&lt;br /&gt;
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* [[摩擦]]，[[耗散]]，[[量子退相干]，&lt;br /&gt;
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* 由外力泵送的系统([光泵]]等)&lt;br /&gt;
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* 以及一般意义的可逆过程&lt;br /&gt;
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All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)&lt;br /&gt;
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All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)&lt;br /&gt;
&lt;br /&gt;
所有这些过程都是以特征速率随时间发生的，这些速率对于工程来说非常重要。非平衡态统计力学研究领域关注的是在微观水平上理解这些非平衡过程。(统计热力学只能用来计算在外部不平衡被消除，整体回归到平衡状态之后的最终结果。)&lt;br /&gt;
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In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as [[Liouville's theorem (Hamiltonian)|Liouville's equation]] or its quantum equivalent, the [[von Neumann equation]]. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's [[Gibbs entropy]] is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.&lt;br /&gt;
&lt;br /&gt;
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.&lt;br /&gt;
&lt;br /&gt;
原则上，非平衡态统计力学在数学上可以是精确的: 孤立系统的系综根据确定性方程随时间演化，如刘维尔方程或其量子等价、冯·诺依曼方程。这些方程是将运动力学方程独立应用于系综中每个状态的结果。不幸的是，这些系综演化方程继承了潜在动力学运动的大部分复杂性，因此很难得到精确解。此外，系综演化方程是完全可逆的，不会破坏信息(系综的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;吉布斯熵 Gibbs entropy&amp;lt;/font&amp;gt;被保留)。为了在模拟不可逆过程中取得进展，除了概率和可逆力学外，还必须考虑其他因素。&lt;br /&gt;
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Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.&lt;br /&gt;
&lt;br /&gt;
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.&lt;br /&gt;
&lt;br /&gt;
因此，非平衡力学是一个活跃的理论研究领域，因为这些额外假设的有效范围仍将继续探索。在下面的小节中描述了一些方法。&lt;br /&gt;
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=== Stochastic methods 随机方法===&lt;br /&gt;
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One approach to non-equilibrium statistical mechanics is to incorporate [[stochastic]] (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from [[Black hole information paradox|hypothetical situations involving black holes]], a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as [[Chaos theory|chaotic]] or [[pseudorandom]] influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.&lt;br /&gt;
&lt;br /&gt;
One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.&lt;br /&gt;
&lt;br /&gt;
处理非平衡态统计力学的一个方法是将随机行为引入系统。随机行为可以破坏系综中包含的信息。虽然这在技术上是不准确的(除了涉及黑洞的假设情况外，黑洞系统本身不会导致信息丢失) ，但这种随机性是为了反映出，随着时间的推移，感兴趣的信息会在系统内部转化为微妙的相关性，或者系统与环境之间的相关性。这些关联表现为对感兴趣的变量的&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;混沌 Chaos&amp;lt;/font&amp;gt;或伪随机的影响。用适当的随机性取代这些相关性，计算可以变得容易得多。&lt;br /&gt;
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{{unordered list&lt;br /&gt;
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{{unordered list&lt;br /&gt;
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{无序列表&lt;br /&gt;
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|1 = ''[[Boltzmann transport equation]]'': An early form of stochastic mechanics appeared even before the term &amp;quot;statistical mechanics&amp;quot; had been coined, in studies of [[kinetic theory of gases|kinetic theory]]. [[James Clerk Maxwell]] had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. [[Ludwig Boltzmann]] subsequently showed that, by taking this [[molecular chaos]] for granted as a complete randomization, the motions of particles in a gas would follow a simple [[Boltzmann transport equation]] that would rapidly restore a gas to an equilibrium state (see [[H-theorem]]).&lt;br /&gt;
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|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term &amp;quot;statistical mechanics&amp;quot; had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).&lt;br /&gt;
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&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;玻尔兹曼输运方程 Boltzmann transport equation&amp;lt;/font&amp;gt;: 在动力学理论研究中，早期的随机力学形式甚至在“统计力学”一词被创造之前就已经出现了。詹姆斯·克拉克·麦克斯韦已经证明分子碰撞会导致气体内部明显的混沌运动。路德维希·玻尔兹曼随后证明，如果把这种分子混沌理所当然地看作是一种完全的随机化，那么气体中粒子的运动将遵循一个简单的玻尔兹曼输运方程，这个方程将使气体迅速恢复到平衡状态(见H-定理)。&lt;br /&gt;
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The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the &amp;quot;interesting&amp;quot; information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped [[semiconductor]]s (in [[transistor]]s), where the electrons are indeed analogous to a rarefied gas.&lt;br /&gt;
&lt;br /&gt;
The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the &amp;quot;interesting&amp;quot; information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas.&lt;br /&gt;
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玻耳兹曼输运方程及其相关方法是非平衡态统计力学的重要工具，因为它们极其简单。这些近似方法在“感兴趣的”信息立即(在一次碰撞之后)变成微妙关联的系统中非常有效，这种关联本质上限制它们为稀薄气体。玻耳兹曼输运方程被发现在模拟轻掺杂半导体(晶体管)的电子输运中非常有用，其中的电子确实类似于稀薄气体。&lt;br /&gt;
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A quantum technique related in theme is the [[random phase approximation]].&lt;br /&gt;
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A quantum technique related in theme is the random phase approximation.&lt;br /&gt;
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一个与主题相关的量子技术是随机相位近似。&lt;br /&gt;
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|2 = ''[[BBGKY hierarchy]]'':&lt;br /&gt;
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|2 = BBGKY hierarchy:&lt;br /&gt;
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2 BBGKY 层级:&lt;br /&gt;
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In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The [[BBGKY hierarchy]] (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.&lt;br /&gt;
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In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.&lt;br /&gt;
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在液体和稠密气体中，不能在一次碰撞后立即丢掉粒子之间的关联。BBGKY 层级结构(Bogoliubov-Born-Green-Kirkwood-Yvon 层级结构)提供了一种推导玻尔兹曼型方程的方法，但也可以将它们扩展到稀薄气体情况之外，包括在几次碰撞之后的相关性。&lt;br /&gt;
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|3 = ''[[Keldysh formalism]]'' (a.k.a. NEGF—non-equilibrium Green functions):&lt;br /&gt;
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|3 = Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions):&lt;br /&gt;
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| 3 Keldysh 公式。NEGF—非平衡态格林函数):&lt;br /&gt;
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A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic [[quantum transport]] calculations.&lt;br /&gt;
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A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic quantum transport calculations.&lt;br /&gt;
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人们在 Keldysh 公式中发明了包含随机动力学的量子方法，这种方法常用于电子量子输运计算。&lt;br /&gt;
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|4 = Stochastic [[Liouville's theorem (Hamiltonian)|Liouville equation]]&lt;br /&gt;
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|4 = Stochastic Liouville equation&lt;br /&gt;
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| 4随机 刘维尔方程&lt;br /&gt;
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}}&lt;br /&gt;
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=== Near-equilibrium methods 近平衡态方法 ===&lt;br /&gt;
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Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in [[linear response theory]]. A remarkable result, as formalized by the [[fluctuation-dissipation theorem]], is that the response of a system when near equilibrium is precisely related to the [[Statistical fluctuations|fluctuations]] that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or &amp;quot;know&amp;quot; how it came to be away from equilibrium.&amp;lt;ref name=&amp;quot;balescu&amp;quot;/&amp;gt;{{rp|664}}&lt;br /&gt;
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Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or &amp;quot;know&amp;quot; how it came to be away from equilibrium.&lt;br /&gt;
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非平衡态统计力学模型处理的另一类重要的系统，是对平衡态仅有非常轻微扰动的系统。在很小的扰动下，响应可以用&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;线性响应理论 linear response theory&amp;lt;/font&amp;gt;进行分析。&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;涨落-耗散定理 fluctuation-dissipation theorem&amp;lt;/font&amp;gt;是其中一个重要的结果，近平衡态系统的响应与系统总体平衡时的涨落准确相关。从本质上讲，一个系统如果稍微偏离平衡，无论是由于外力还是由于涨落，都会以同样的方式向平衡方向弛豫。因为这个系统无法区分偏离和回归，也“不知道”它是如何偏离平衡的。&lt;br /&gt;
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This provides an indirect avenue for obtaining numbers such as [[Ohm's law|ohmic conductivity]] and [[thermal conductivity]] by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.&lt;br /&gt;
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This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.&lt;br /&gt;
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这提供了一个间接的方法，通过从平衡态统计力学中提取结果来获得诸如欧姆电导率和热导率之类的物理量。由于平衡态统计力学在数学上有很好的定义，而且(在某些情况下)更易于计算，因此在近平衡态统计力学中，涨落-耗散关联可以成为一种方便的计算捷径。&lt;br /&gt;
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A few of the theoretical tools used to make this connection include:&lt;br /&gt;
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A few of the theoretical tools used to make this connection include:&lt;br /&gt;
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用于建立这种关联的一些理论工具包括:&lt;br /&gt;
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* [[Fluctuation–dissipation theorem]]&lt;br /&gt;
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* [[Onsager reciprocal relations]]&lt;br /&gt;
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* [[Green–Kubo relations]]&lt;br /&gt;
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* [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker formalism]]&lt;br /&gt;
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* [[Mori–Zwanzig formalism]]&lt;br /&gt;
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* [[涨落-耗散定理]]&lt;br /&gt;
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* [[昂萨格倒易关系]]&lt;br /&gt;
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* [[格林-库伯关系]]&lt;br /&gt;
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* [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker 公式]]&lt;br /&gt;
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* [[Mori–Zwanzig 公式]]&lt;br /&gt;
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=== Hybrid methods 组合方法===&lt;br /&gt;
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An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects ([[weak localization]], [[conductance fluctuations]]) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic [[dephasing]] by interactions between various electrons by use of the Keldysh method.&amp;lt;ref&amp;gt;{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid =  | pmc = |bibcode = 1982JPhC...15.7367A }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid =  | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.&lt;br /&gt;
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一种先进的方法使用了随机方法和线性响应理论的结合。例如，计算电子系统电导中的量子相干效应(弱局域化，电导涨落)的一种方法是使用 Green-Kubo 关系，包括随机退相的各种电子之间的相互作用，使用Keldysh 方法。&lt;br /&gt;
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==Applications outside thermodynamics==&lt;br /&gt;
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The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:&lt;br /&gt;
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The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:&lt;br /&gt;
&lt;br /&gt;
系综也可以用来分析系统状态有不确定性的一般机械系统。在下列情况中也会使用系综:&lt;br /&gt;
&lt;br /&gt;
* [[propagation of uncertainty]] over time,&amp;lt;ref name=&amp;quot;gibbs&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* [[regression analysis]] of gravitational [[orbit]]s,&lt;br /&gt;
&lt;br /&gt;
* [[ensemble forecasting]] of weather,&lt;br /&gt;
&lt;br /&gt;
* dynamics of [[neural networks]],&lt;br /&gt;
&lt;br /&gt;
* bounded-rational [[potential game]]s in game theory and economics.&lt;br /&gt;
&lt;br /&gt;
* 不确定性随时间传播，&lt;br /&gt;
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* 引力[[轨道]]的[[回归分析]]，&lt;br /&gt;
&lt;br /&gt;
* 天气的[[系综预报]]，&lt;br /&gt;
&lt;br /&gt;
* [[神经网络]]动力学，&lt;br /&gt;
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* 博弈论和经济学中的有界理性[[潜在对策]]。&lt;br /&gt;
&lt;br /&gt;
== History 历史==&lt;br /&gt;
&lt;br /&gt;
In 1738, Swiss physicist and mathematician [[Daniel Bernoulli]] published ''Hydrodynamica'' which laid the basis for the [[kinetic theory of gases]]. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as [[heat]] is simply the kinetic energy of their motion.&amp;lt;ref name=&amp;quot;uffink&amp;quot;/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.&lt;br /&gt;
&lt;br /&gt;
1738年，瑞士的物理学家和数学家丹尼尔·伯努利发表了《水动力学》 ，这本书奠定了&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;气体动力学理论 kinetic theory of gases&amp;lt;/font&amp;gt;的基础。在这项工作中，伯努利假定气体是由大量向各个方向运动的分子组成的，它们对表面的影响导致了我们感觉到的气体压力，而我们感受到的热仅仅是它们运动的动能，这一点直到今天仍在沿用。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In 1859, after reading a paper on the diffusion of molecules by [[Rudolf Clausius]], Scottish physicist [[James Clerk Maxwell]] formulated the [[Maxwell distribution]] of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.&amp;lt;ref&amp;gt;See:&lt;br /&gt;
&lt;br /&gt;
In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.&amp;lt;ref&amp;gt;See:&lt;br /&gt;
&lt;br /&gt;
1859年，在阅读了 Rudolf Clausius 的一篇关于分子扩散的论文之后，苏格兰物理学家詹姆斯·克拉克·麦克斯韦提出了分子速度的麦克斯韦-玻尔兹曼分布，它给出了在一个特定范围内具有某种速度的分子的比例。这是物理学里第一个统计定律。麦克斯韦还提出了第一个力学论点，即分子碰撞必然导致温度的平衡，从而趋向平衡。五年之后，也就是1864年，维也纳的年轻学生路德维希·玻尔兹曼(Ludwig Boltzmann)偶然发现了麦克斯韦尔的论文，并花了大半辈子的时间来进一步研究这一课题。&lt;br /&gt;
&lt;br /&gt;
*  Maxwell, J.C. (1860) [https://books.google.com/books?id=-YU7AQAAMAAJ&amp;amp;pg=PA19#v=onepage&amp;amp;q&amp;amp;f=false &amp;quot;Illustrations of the dynamical theory of gases. Part I.  On the motions and collisions of perfectly elastic spheres,&amp;quot;] ''Philosophical Magazine'', 4th series, '''19''' :  19–32. &lt;br /&gt;
&lt;br /&gt;
*  Maxwell, J.C. (1860) [https://books.google.com/books?id=DIc7AQAAMAAJ&amp;amp;pg=PA21#v=onepage&amp;amp;q&amp;amp;f=false &amp;quot;Illustrations of the dynamical theory of gases. Part II.  On the process of diffusion of two or more kinds of moving particles among one another,&amp;quot;] ''Philosophical Magazine'', 4th series, '''20''' :  21–37.&amp;lt;/ref&amp;gt; This was the first-ever statistical law in physics.&amp;lt;ref&amp;gt;{{cite book |last = Mahon |first = Basil |title=The Man Who Changed Everything – the Life of James Clerk Maxwell |location=Hoboken, NJ |publisher=Wiley |year=2003 |isbn=978-0-470-86171-4 |oclc=52358254}}&amp;lt;/ref&amp;gt; Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.&amp;lt;ref&amp;gt;{{cite journal | last = Gyenis | first = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}&amp;lt;/ref&amp;gt; Five years later, in 1864, [[Ludwig Boltzmann]], a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 ''Lectures on Gas Theory''.&amp;lt;ref&amp;gt;{{cite book |title = Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems |journal=Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Edited by Ebeling Werner &amp;amp; Sokolov Igor M. Published by World Scientific Press |volume=8 |last1=Ebeling |first1=Werner |last2=Sokolov |first2=Igor M. |year=2005 |isbn=978-90-277-1674-3 |pages=3–12 |url = https://books.google.com/books?id=KUjFHbid8A0C|bibcode=2005stst.book.....E |doi=10.1142/2012 |series = Series on Advances in Statistical Mechanics }} (section 1.2)&amp;lt;/ref&amp;gt; Boltzmann's original papers on the statistical interpretation of thermodynamics, the [[H-theorem]], [[transport theory (statistical physics)|transport theory]], [[thermal equilibrium]], the [[equation of state]] of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his [[H-theorem|''H''-theorem]].&lt;br /&gt;
&lt;br /&gt;
Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory. Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.&lt;br /&gt;
&lt;br /&gt;
统计力学是19世纪70年代由玻尔兹曼的工作创立的，其中大部分在他1896年的气体理论演讲中集结出版。在维也纳学院和其他学会的会议记录中，玻尔兹曼关于热力学的统计解释、 &amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;H-定理H-theorem&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;输运理论 transport theory&amp;lt;/font&amp;gt;、&amp;lt;font color=&amp;quot;#FF8000&amp;quot;&amp;gt;热平衡thermal equilibrium&amp;lt;/font&amp;gt;、气体的状态方程以及类似主题的原始论文占据了大约2000页。玻尔兹曼引入了平衡系综的概念，并用他的H-定理第一次研究了非平衡态统计力学。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The term &amp;quot;statistical mechanics&amp;quot; was coined by the American mathematical physicist [[Josiah Willard Gibbs|J. Willard Gibbs]] in 1884.&amp;lt;ref&amp;gt;J. W. Gibbs, &amp;quot;On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics.&amp;quot; Proceedings of the American Association for the Advancement of Science, '''33''', 57-58 (1884). Reproduced in ''The Scientific Papers of J. Willard Gibbs, Vol II'' (1906), [https://archive.org/stream/scientificpapers02gibbuoft#page/16/mode/2up pp.&amp;amp;nbsp;16].&amp;lt;/ref&amp;gt;{{NoteTag|1 = According to Gibbs, the term &amp;quot;statistical&amp;quot;, in the context of mechanics, i.e. statistical mechanics, was first used by the Scottish physicist [[James Clerk Maxwell]] in 1871. From: J. Clerk Maxwell, ''Theory of Heat'' (London, England: Longmans, Green, and Co., 1871), [https://books.google.com/books?id=DqAAAAAAMAAJ&amp;amp;pg=PA309 p.&amp;amp;nbsp;309]: &amp;quot;In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus.&amp;quot;}} &amp;quot;Probabilistic mechanics&amp;quot; might today seem a more appropriate term, but &amp;quot;statistical mechanics&amp;quot; is firmly entrenched.&amp;lt;ref&amp;gt;{{cite book |title = The enigma of probability and physics |last=Mayants |first=Lazar |year=1984 |publisher=Springer |isbn=978-90-277-1674-3 |page=174 |url = https://books.google.com/books?id=zmwEfXUdBJ8C&amp;amp;pg=PA174 }}&amp;lt;/ref&amp;gt; Shortly before his death, Gibbs published in 1902 ''[[Elementary Principles in Statistical Mechanics]]'', a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous.&amp;lt;ref name=&amp;quot;gibbs&amp;quot; /&amp;gt; Gibbs' methods were initially derived in the framework [[classical mechanics]], however they were of such generality that they were found to adapt easily to the later [[quantum mechanics]], and still form the foundation of statistical mechanics to this day.&amp;lt;ref name=&amp;quot;tolman&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The term &amp;quot;statistical mechanics&amp;quot; was coined by the American mathematical physicist J. Willard Gibbs in 1884. &amp;quot;Probabilistic mechanics&amp;quot; might today seem a more appropriate term, but &amp;quot;statistical mechanics&amp;quot; is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.&lt;br /&gt;
&lt;br /&gt;
1884年，美国数学物理学家约西亚·威拉德·吉布斯首创了“统计力学”一词。在今天看来，“概率力学”似乎是一个更合适的术语，但“统计力学”却根深蒂固。在吉布斯去世前不久，他于1902年在《统计力学》出版了《基本原理》一书，这本书正式确定了统计力学是解决所有力学系统——宏观的或微观的、气态的或非气态的——的一种完全通用的方法。吉布斯的方法最初是在经典力学的框架下产生的，然而它们是如此的普遍，以至于人们发现它们很容易适应后来的量子力学，直到今天仍然是统计力学的基础。&lt;br /&gt;
&lt;br /&gt;
== See also 其他可见==&lt;br /&gt;
&lt;br /&gt;
{{Div col}}&lt;br /&gt;
&lt;br /&gt;
{{Books-inline|Fundamentals of Statistical Mechanics}}&lt;br /&gt;
&lt;br /&gt;
* [[Thermodynamics]]: [[Non-equilibrium thermodynamics|non-equilibrium]], [[Chemical thermodynamics|chemical]]&lt;br /&gt;
&lt;br /&gt;
* [[Mechanics]]: [[Classical mechanics|classical]], [[Quantum mechanics|quantum]]&lt;br /&gt;
&lt;br /&gt;
* [[Probability]], [[Statistical ensemble (mathematical physics)|statistical ensemble]]&lt;br /&gt;
&lt;br /&gt;
* Numerical methods: [[Monte Carlo method]], [[molecular dynamics]]&lt;br /&gt;
&lt;br /&gt;
* [[Statistical physics]]&lt;br /&gt;
&lt;br /&gt;
* [[Quantum statistical mechanics]]&lt;br /&gt;
&lt;br /&gt;
* [[List of notable textbooks in statistical mechanics]]&lt;br /&gt;
&lt;br /&gt;
* [[List of publications in physics#Statistical mechanics|List of important publications in statistical mechanics]]&lt;br /&gt;
&lt;br /&gt;
{{Div col end}}&lt;br /&gt;
&lt;br /&gt;
{{Div col}}&lt;br /&gt;
&lt;br /&gt;
{{Books-inline|统计力学基础}}&lt;br /&gt;
&lt;br /&gt;
* [[热力学]]: [[非平衡态热力学]], [[化学热力学]]&lt;br /&gt;
&lt;br /&gt;
* [[力学]]: [[经典力学]], [[量子力学]]&lt;br /&gt;
&lt;br /&gt;
* [[概率]], [[统计系综（数学物理）]]&lt;br /&gt;
&lt;br /&gt;
* 数值方法: [[蒙特卡洛方法]], [[分子动力学]]&lt;br /&gt;
&lt;br /&gt;
* [[统计物理]]&lt;br /&gt;
&lt;br /&gt;
* [[量子统计力学]]&lt;br /&gt;
&lt;br /&gt;
* [[统计力学著名教科书列表]]&lt;br /&gt;
&lt;br /&gt;
* [[统计力学重要文献列表]]&lt;br /&gt;
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{{Div col end}}&lt;br /&gt;
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== Notes ==&lt;br /&gt;
&lt;br /&gt;
{{NoteFoot}}&lt;br /&gt;
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== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist&lt;br /&gt;
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{{Reflist&lt;br /&gt;
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{通货再膨胀&lt;br /&gt;
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|refs =&lt;br /&gt;
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参考文献&lt;br /&gt;
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&amp;lt;ref name=&amp;quot;gibbs&amp;quot;&amp;gt;{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York |title-link=Elementary Principles in Statistical Mechanics }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;tolman&amp;quot;&amp;gt;{{cite book | last=Tolman |first=R. C. | author-link = Richard C. Tolman | year=1938 | title = The Principles of Statistical Mechanics | publisher=[[Dover Publications]] | isbn = 9780486638966 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;balescu&amp;quot;&amp;gt;{{cite book | isbn = 9780471046004 | title = Equilibrium and Non-Equilibrium Statistical Mechanics |last = Balescu |first = Radu | author-link = Radu Balescu | year = 1975 | publisher = John Wiley &amp;amp; Sons }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;ref name=&amp;quot;uffink&amp;quot;&amp;gt;J. Uffink, &amp;quot;[http://philsci-archive.pitt.edu/2691/1/UffinkFinal.pdf Compendium of the foundations of classical statistical physics.]&amp;quot; (2006)&amp;lt;/ref&amp;gt;&lt;br /&gt;
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}}&lt;br /&gt;
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}}&lt;br /&gt;
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== External links ==&lt;br /&gt;
&lt;br /&gt;
{{Commons category|Statistical mechanics}}&lt;br /&gt;
&lt;br /&gt;
* [http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics] article by Lawrence Sklar for the [[Stanford Encyclopedia of Philosophy]].&lt;br /&gt;
&lt;br /&gt;
* [http://www.sklogwiki.org/ Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials.]  SklogWiki is particularly orientated towards liquids and soft condensed matter.&lt;br /&gt;
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* [https://web.archive.org/web/20060616002812/http://history.hyperjeff.net/statmech.html Statistical Thermodynamics] - Historical Timeline&lt;br /&gt;
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* [http://farside.ph.utexas.edu/teaching/sm1/statmech.pdf Thermodynamics and Statistical Mechanics] by Richard Fitzpatrick&lt;br /&gt;
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* [https://arxiv.org/abs/1107.0568 Lecture Notes in Statistical Mechanics and Mesoscopics] by Doron Cohen&lt;br /&gt;
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* {{YouTube |id = H1Zbp6__uNw&amp;amp;list=PLB72416C707D85AB0&amp;amp;index=1 |title = Videos of lecture series in statistical mechanics }} taught by [[Leonard Susskind]].&lt;br /&gt;
&lt;br /&gt;
* Vu-Quoc, L., [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008. this wiki site is down; see [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 this article in the web archive on 2012 April 28].&lt;br /&gt;
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{{Statistical mechanics topics}}&lt;br /&gt;
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[[Category:Concepts in physics]]&lt;br /&gt;
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Category:Concepts in physics&lt;br /&gt;
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分类: 物理概念&lt;br /&gt;
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[[Category:Physics]]&lt;br /&gt;
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Category:Physics&lt;br /&gt;
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类别: 物理学&lt;br /&gt;
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[[Category:Statistical mechanics| ]]&lt;br /&gt;
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[[Category:Thermodynamics]]&lt;br /&gt;
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Category:Thermodynamics&lt;br /&gt;
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分类: 热力学&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Statistical mechanics]]. Its edit history can be viewed at [[统计力学/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12760</id>
		<title>度</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12760"/>
		<updated>2020-08-16T14:38:04Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
[[File:UndirectedDegrees (Loop).svg|thumb|&lt;br /&gt;
图1：A graph with a loop having vertices labeled by degree 内含自环按度标记的图]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In [[graph theory]], the '''degree''' (or '''valency''') of a [[vertex (graph theory)|vertex]] of a [[Graph (discrete mathematics)|graph]] is the number of [[edge (graph theory)|edges]] that are [[incidence (graph)|incident]] to the vertex, and in a [[multigraph]], [[loop (graph theory)|loop]]s are counted twice.&amp;lt;ref&amp;gt;Diestel p.5&amp;lt;/ref&amp;gt; The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The '''maximum degree''' of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the '''minimum degree''' of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
&lt;br /&gt;
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The maximum degree of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the minimum degree of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''中，图中顶点的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度 Degree&amp;lt;/font&amp;gt;'''(或价)是入射到该顶点的边的数量。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;多重图 Multigraph&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;自环 Loops&amp;lt;/font&amp;gt;'''会被计算两次。顶点的度数可表示为&amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt;或&amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;。一个图&amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;的最大度值表示为&amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;，最小度值表示为 &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;。在右侧的多重图中，最大度值为5，最小度值为0。&lt;br /&gt;
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In a [[regular graph]], every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A [[complete graph]] (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
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In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;正则图 Regular Graph&amp;lt;/font&amp;gt;'''中，每个顶点都具有相同的度值，因此我们把该度值可以称之为图的度值。一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全图 Complete Graph&amp;lt;/font&amp;gt;'''(表示为&amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;，其中&amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt;是图中顶点的数目)是一种特殊的正则图，其中所有顶点都有最大度值，&amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;。&lt;br /&gt;
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==Handshaking lemma==&lt;br /&gt;
握手引理&lt;br /&gt;
{{main|Handshaking lemma}}&lt;br /&gt;
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The '''degree sum formula''' states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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The degree sum formula states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度和公式 Degree Sum Formula&amp;lt;/font&amp;gt;'''表明，给定一个图&amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree  sum formula) is known as the [[handshaking lemma]]. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
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The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
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该公式表明，在任何无向图中，拥有奇数度值的顶点的个数是偶数。这个陈述(以及度和公式)被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;握手引理 Handshaking Lemma&amp;lt;/font&amp;gt;'''。该名字来自一个有趣的数学问题，即证明无论该群体内有多少人，与奇数个人握过手的人数是偶数。&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）标题涉及到的专业名词 要么标注一下标题要么标注一下正文~&lt;br /&gt;
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==Degree sequence==&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度序列 Degree Sequence&amp;lt;/font&amp;gt;'''&lt;br /&gt;
[[File:Conjugate-dessins.svg|thumb|200px|&lt;br /&gt;
图2：Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1).&lt;br /&gt;
两个具有相同度序列的非同构图]]&lt;br /&gt;
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The '''degree sequence''' of an undirected graph is the non-increasing sequence of its vertex degrees;&amp;lt;ref&amp;gt;Diestel p.216&amp;lt;/ref&amp;gt; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a [[graph invariant]] so [[Graph isomorphism|isomorphic graphs]] have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
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The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
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无向图的度序列是指将其各顶点度值按非递增方式排序，对于上述图是(5,3,3,2,2,1,0)。度序列是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图不变量 Graph Invariant&amp;lt;/font&amp;gt;'''，因此'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;同构图 Non-isomorphic Graphs&amp;lt;/font&amp;gt;'''（两个图中顶点的度值都相同，但形状不同）具有相同的度序列。然而，度序列通常不能唯一地标识一个图，在某些情况下，非同构图具有相同的度序列。&lt;br /&gt;
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The '''degree sequence problem'''  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a '''graphic''' or '''graphical sequence'''. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a [[matching (graph theory)|matching]], and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
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The degree sequence problem  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度序列问题 Degree Sequence Problem&amp;lt;/font&amp;gt;'''，是指寻找具有以非增方式排列给定正整数的度序列的局部或全部图的问题。(序列尾部的零可能会被忽略，因为通过向图中添加适当数量的孤立顶点就可以轻松实现序列尾部不断加零。)一个序列是某个图的度序列，即一个度序列问题有解时，该序列称为图形序列。由于度和公式的存在，任何具有奇数和的序列，如(3,3,1) ，都不会是图的度序列。反之亦然: 如果一个序列和是偶数，它就是重图的度序列。构造这样一个图是很简单: 匹配奇数度值的顶点并成对连接起来，然后剩余的偶数度值顶点都连出一条边指向自己。&lt;br /&gt;
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The question of whether a given degree sequence can be realized by a [[simple graph]] is more challenging. This problem is also called [[graph realization problem]] and can either be solved by the [[Erdős–Gallai theorem]] or the [[Havel–Hakimi algorithm]]. &lt;br /&gt;
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The question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm. &lt;br /&gt;
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一个给定的度序列是否可以用一个简单的图进行表示，是一个具有挑战性的问题。这个问题也称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图实现 Graph Realization Problem&amp;lt;/font&amp;gt;'''问题，可以用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Erdős–Gallai定理 Erdős–Gallai Theoreme&amp;lt;/font&amp;gt;'''解决，也可以用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Havel-Hakimi算法 Havel–Hakimi Algorithm&amp;lt;/font&amp;gt;'''解决。&lt;br /&gt;
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The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of [[graph enumeration]].&lt;br /&gt;
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The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of graph enumeration.&lt;br /&gt;
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基于给定度序列寻找或估计可能的图的个数，是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图枚举 Graph Enumeration&amp;lt;/font&amp;gt;'''领域中的一个问题。&lt;br /&gt;
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More generally, the '''degree sequence''' of a [[hypergraph]] is the non-increasing sequence of its vertex degrees. A sequence is '''&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic''' if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in [[Time complexity|polynomial time]] for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the [[Erdős–Gallai theorem]] but is [[NP-completeness|NP-complete]] for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 &amp;lt;ref&amp;gt;{{Cite journal|last=Deza|first=Antoine|last2=Levin|first2=Asaf|last3=Meesum|first3=Syed M.|last4=Onn|first4=Shmuel|date=January 2018|title=Optimization over Degree Sequences|url=https://epubs.siam.org/doi/10.1137/17M1134482|journal=SIAM Journal on Discrete Mathematics|language=en|volume=32|issue=3|pages=2067–2079|doi=10.1137/17M1134482|issn=0895-4801|arxiv=1706.03951}}&amp;lt;/ref&amp;gt;).&lt;br /&gt;
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More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in polynomial time for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the Erdős–Gallai theorem but is NP-complete for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 ).&lt;br /&gt;
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一般来说，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;超图 Hypergraph&amp;lt;/font&amp;gt;'''的度序列是其顶点度的非递增序列。一个序列是&amp;lt;math&amp;gt;k-graphic&amp;lt;/math&amp;gt;，如果它是一些&amp;lt;math&amp;gt;k-uniform&amp;lt;/math&amp;gt;超图的度序列。特别是&amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic序列是图形。决定一个给定的序列是否是&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic可以在 &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt;的多项式时间内通过Erdős–Gallai定理实现，但是当&amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt;时该问题可转化为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;NP完全问题 NP-complete&amp;lt;/font&amp;gt;'''(Deza et al. ，2018)。&lt;br /&gt;
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==Special values==&lt;br /&gt;
特殊值&lt;br /&gt;
[[File:Depth-first-tree.png|thumb|&lt;br /&gt;
图3：An undirected graph with leaf nodes 4, 5, 6, 7, 10, 11, and 12 一个具有叶节点的无定向图]]&lt;br /&gt;
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*A vertex with degree 0 is called an [[isolated vertex]].&lt;br /&gt;
度值为0的顶点成为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;孤立顶点 Isolated Vertex&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of [[tree (graph theory)|tree]]s in graph theory and especially [[tree (data structure)|tree]]s as [[data structure]]s.&lt;br /&gt;
度值为1的顶点称为叶顶点或尾顶点，该顶点的入射边称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;悬挂边 Pendant Edge&amp;lt;/font&amp;gt;'''。在右侧的图中，{3,5}就是一个悬挂边。在图论中，该术语主要在研究'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;树 Tree&amp;lt;/font&amp;gt;'''时使用，特别是具有树形结构的数据。&lt;br /&gt;
* A vertex with degree ''n''&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1  in a graph on ''n'' vertices is called a [[dominating vertex]].&lt;br /&gt;
在有n个顶点的图中，度值为n-1的顶点叫作'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;主导顶点 Dominating Vertex&amp;lt;/font&amp;gt;'''&lt;br /&gt;
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==Global properties==&lt;br /&gt;
全局属性&lt;br /&gt;
*If each vertex of the graph has the same degree&amp;amp;nbsp;''k'' the graph is called a [[regular graph|''k''-regular graph]] and the graph itself is said to have degree&amp;amp;nbsp;''k''. Similarly, a [[bipartite graph]] in which every two vertices on the same side of the bipartition as each other have the same degree is called a [[biregular graph]].&lt;br /&gt;
如果一个图中的所有顶点的度值都为k，该图被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;''k''-正则图 ''k''-regular graph&amp;lt;/font&amp;gt;'''，该图的度值也为k。同样的，同侧每两个顶点都具有相同度值的二分图叫作'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二分正则图 Biregular Graph&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
*An undirected, connected graph has an [[Eulerian path]] if and only if it has either 0 or 2 vertices of odd degree. If it has 0 vertices of odd degree, the Eulerian path is an Eulerian circuit.&lt;br /&gt;
当且仅当具有0或2个奇数度的顶点时，无向连通图才具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;欧拉路径 Eulerian Path&amp;lt;/font&amp;gt;'''。 如果它具有0个奇数度的顶点，则欧拉路径为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;欧拉回路 Eulerian Circuit&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
*A directed graph is a [[pseudoforest]] if and only if every vertex has outdegree at most&amp;amp;nbsp;1. A [[functional graph]] is a special case of a pseudoforest in which every vertex has outdegree exactly&amp;amp;nbsp;1.&lt;br /&gt;
有向图是当且仅当每个顶点的度值最大为1时才是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;伪森林 Pseudoforest&amp;lt;/font&amp;gt;'''。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;功能图 Functional Graph&amp;lt;/font&amp;gt;'''是伪森林的特例，其中每个顶点的度数都恰好为1。&lt;br /&gt;
*By [[Brooks' theorem]], any graph other than a clique or an odd cycle has [[chromatic number]] at most&amp;amp;nbsp;Δ, and by [[Vizing's theorem]] any graph has [[chromatic index]] at most Δ&amp;amp;nbsp;+&amp;amp;nbsp;1.&lt;br /&gt;
根据布鲁克斯定理，除团簇或奇数循环外，任何图的色度数最大为Δ，而根据维辛定理，任何图的色度指数最大为Δ+ 1。&lt;br /&gt;
*A [[Degeneracy (graph theory)|''k''-degenerate graph]] is a graph in which each subgraph has a vertex of degree at most ''k''.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;k简并图 K-Degenerate Graph &amp;lt;/font&amp;gt;'''是其中每个子图最多具有度值为k的顶点的图。&lt;br /&gt;
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==See also==&lt;br /&gt;
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*[[Indegree]], [[outdegree]] for [[digraph (mathematics)|digraph]]s&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向图的入度和出度 Indegree,Outdegree For Digraph&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*[[Degree distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度分布 Degree Distribution&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*[[bipartite graph|degree sequence]] for bipartite graphs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二分图的度序列 Degree Sequence For Bipartite Graphs &amp;lt;/font&amp;gt;'''&lt;br /&gt;
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==Notes==&lt;br /&gt;
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{{reflist}}&lt;br /&gt;
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==References==&lt;br /&gt;
&lt;br /&gt;
*{{Citation | last1=Diestel | first1=Reinhard | title=Graph Theory | url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-3-540-26183-4 | year=2005 }}.&lt;br /&gt;
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| 页数 = 477-480&lt;br /&gt;
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| url = http://eudml.org/doc/19050&lt;br /&gt;
&lt;br /&gt;
| url = http://eudml.org/doc/19050&lt;br /&gt;
&lt;br /&gt;
Http://eudml.org/doc/19050&lt;br /&gt;
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}}&lt;br /&gt;
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最后 = Hakimi | first = S.l. | authorlink = S.l. Hakimi&lt;br /&gt;
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 | journal = Journal of the [[Society for Industrial and Applied Mathematics]]&lt;br /&gt;
&lt;br /&gt;
 | journal = Journal of the Society for Industrial and Applied Mathematics&lt;br /&gt;
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工业与应用数学学会杂志&lt;br /&gt;
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 | mr = 0148049&lt;br /&gt;
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 | mr = 0148049&lt;br /&gt;
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0148049先生&lt;br /&gt;
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 | pages = 496–506&lt;br /&gt;
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 | pages = 496–506&lt;br /&gt;
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| 页数 = 496-506&lt;br /&gt;
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 | title = On realizability of a set of integers as degrees of the vertices of a linear graph. I&lt;br /&gt;
&lt;br /&gt;
 | title = On realizability of a set of integers as degrees of the vertices of a linear graph. I&lt;br /&gt;
&lt;br /&gt;
| title = 一组整数作为线性图顶点度的可实现性。一&lt;br /&gt;
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 | volume = 10&lt;br /&gt;
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10&lt;br /&gt;
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 | year = 1962}}.&lt;br /&gt;
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 | year = 1962}}.&lt;br /&gt;
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1962}}.&lt;br /&gt;
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*{{citation&lt;br /&gt;
&lt;br /&gt;
 | last1 = Sierksma | first1 = Gerard&lt;br /&gt;
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 | last1 = Sierksma | first1 = Gerard&lt;br /&gt;
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1 = 杰拉德&lt;br /&gt;
&lt;br /&gt;
 | last2 = Hoogeveen | first2 = Han&lt;br /&gt;
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 | last2 = Hoogeveen | first2 = Han&lt;br /&gt;
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 | last2 = Hoogeveen | first2 = Han&lt;br /&gt;
&lt;br /&gt;
 | doi = 10.1002/jgt.3190150209&lt;br /&gt;
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 | doi = 10.1002/jgt.3190150209&lt;br /&gt;
&lt;br /&gt;
10.1002/jgt. 3190150209&lt;br /&gt;
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 | issue = 2&lt;br /&gt;
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 | issue = 2&lt;br /&gt;
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2&lt;br /&gt;
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 | journal = [[Journal of Graph Theory]]&lt;br /&gt;
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 | journal = Journal of Graph Theory&lt;br /&gt;
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| Journal = Journal of Graph Theory&lt;br /&gt;
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 | mr = 1106533&lt;br /&gt;
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 | mr = 1106533&lt;br /&gt;
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1106533先生&lt;br /&gt;
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 | pages = 223–231&lt;br /&gt;
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 | pages = 223–231&lt;br /&gt;
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| pages = 223-231&lt;br /&gt;
&lt;br /&gt;
 | title = Seven criteria for integer sequences being graphic&lt;br /&gt;
&lt;br /&gt;
 | title = Seven criteria for integer sequences being graphic&lt;br /&gt;
&lt;br /&gt;
| title = 整数序列为图形的七个条件&lt;br /&gt;
&lt;br /&gt;
 | volume = 15&lt;br /&gt;
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 | volume = 15&lt;br /&gt;
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15&lt;br /&gt;
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 | year = 1991}}.&lt;br /&gt;
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 | year = 1991}}.&lt;br /&gt;
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1991}}.&lt;br /&gt;
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[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree (graph theory)]]. Its edit history can be viewed at [[度/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12759</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12759"/>
		<updated>2020-08-16T14:37:26Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图 Graphs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络 Networks&amp;lt;/font&amp;gt;'''的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P''(''k'')定义为网络中度值为''k''的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;累积度分布函数 Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（随机选择一个节点，其度值小于k的概率）的形式表示，或是用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;互补累积度分布函数 Complementary Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机图 Random Graph&amp;lt;/font&amp;gt;'''，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项分布 Binomial Distribution&amp;lt;/font&amp;gt;'''的度值为''k'':&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(如果平均度\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt;保持不变，也会出现有限节点的泊松分布)。然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
γ是一个常数。这种网络被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks&amp;lt;/font&amp;gt;'''，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究表明，尽管大多数观测到的网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;肥尾度分布 Fat-Tailed Degree Distributions&amp;lt;/font&amp;gt;'''，但它们无标度分布的特点并不明显。&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）同样 标题为专业名词的  可以把正文中对应的专业名词进行标注标橙&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;超额度分布 Excess Degree Distribution&amp;lt;/font&amp;gt;'''是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的其传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;,通过选择一个节点(随机或非随机)跟踪它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; 个邻居的概率不是由&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;.给出的原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt;的真实概率是&amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;，它被称为该节点的超额度。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;配置模型 Configuration Model&amp;lt;/font&amp;gt;'''中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;友谊悖论 Friendship Paradox&amp;lt;/font&amp;gt;'''。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生成函数方法 Generating Functions Method&amp;lt;/font&amp;gt;'''&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|url=https://link.aps.org/doi/10.1103/PhysRevE.64.026118|journal=Physical Review E|language=en|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|issn=1063-651X|doi-access=free}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; 的值也可得出通过 &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;的导数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻性。然后，可以很容易地进行计算依据&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和它的导数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的。进一步扩展，&amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th的邻居的分布是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, 以&amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 迭代到 &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 函数本身。第一个邻居的平均数量&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; 第二个邻居的平均数量是:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向网络的度分布 Degree Distribution For Directed Networks&amp;lt;/font&amp;gt;'''&lt;br /&gt;
&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; 和一些出度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt;分别指代的是指向该节点的边的数量和从该节点指出的边的数量。如果 &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; 是一个随机选择的具有入出度的节点的可能性。那么分配给这个生成函数的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;联合概率分布  Joint Probability Distribution&amp;lt;/font&amp;gt;'''可以被写成两个变量 &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt;是网络中节点的平均度(内部和外部)&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
使用函数&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。可以将&amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt;表示离开节点的边的数量:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;，或者像之前表示的&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;从一个随机选择的节点上可达到的第二邻居的平均数是&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. 这些是从随机选择的节点所达到第一和第二邻居的数量，因为这些方程显然是在&amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt;和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;上对称的。&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;复杂网络 Complex Network&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-free Network&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机网络 Random Graph&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;结构截止值 Structural Cut-off&amp;lt;/font&amp;gt;'''&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）该部分也为专业名词 需要进行标橙&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
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&lt;br /&gt;
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2002RvMP... 74... 47A&lt;br /&gt;
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}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
2 = Mendes，J.f.&lt;br /&gt;
&lt;br /&gt;
  | title = Evolution of networks&lt;br /&gt;
&lt;br /&gt;
  | title = Evolution of networks&lt;br /&gt;
&lt;br /&gt;
| title = 网络的进化&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
| 杂志 = 物理进展&lt;br /&gt;
&lt;br /&gt;
  | volume = 51&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
51&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
| pages = 1079-1187&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
  | year = 2002&lt;br /&gt;
&lt;br /&gt;
2002年&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
  | doi = 10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
第四期&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
2002/adphy. . 51.1079 d }&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
纽曼&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
| first = m.E.j.&lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
| title = 复杂网络的结构和功能&lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
| journal = SIAM Review&lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
45&lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
| 页 = 167-256&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
2003年&lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
| doi = 10.1137/S003614450342480&lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
2&lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516&lt;br /&gt;
&lt;br /&gt;
 |bibcode=2003SIAMR..45..167N &lt;br /&gt;
&lt;br /&gt;
 |bibcode=2003SIAMR..45..167N &lt;br /&gt;
&lt;br /&gt;
| bibcode = 2003siaml. . 45. . 167 n&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
* {{cite book &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
复杂网络: 结构、健壮性和功能&lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
作者: Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
| 作者链接 = Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
   &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
2010年&lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
剑桥大学出版社&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph invariants]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph invariants&lt;br /&gt;
&lt;br /&gt;
类别: 图形不变量&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12758</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12758"/>
		<updated>2020-08-16T14:35:41Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图 Graphs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络 Networks&amp;lt;/font&amp;gt;'''的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P''(''k'')定义为网络中度值为''k''的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;累积度分布函数 Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（随机选择一个节点，其度值小于k的概率）的形式表示，或是用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;互补累积度分布函数 Complementary Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机图 Random Graph&amp;lt;/font&amp;gt;'''，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项分布 Binomial Distribution&amp;lt;/font&amp;gt;'''的度值为''k'':&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(如果平均度\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt;保持不变，也会出现有限节点的泊松分布)。然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
γ是一个常数。这种网络被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks&amp;lt;/font&amp;gt;'''，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究表明，尽管大多数观测到的网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;肥尾度分布 Fat-Tailed Degree Distributions&amp;lt;/font&amp;gt;'''，但它们无标度分布的特点并不明显。&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）同样 标题为专业名词的  可以把正文中对应的专业名词进行标注标橙&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;超额度分布 Excess Degree Distribution&amp;lt;/font&amp;gt;'''是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的其传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;,通过选择一个节点(随机或非随机)跟踪它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; 个邻居的概率不是由&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;.给出的原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt;的真实概率是&amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;，它被称为该节点的超额度。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;配置模型 Configuration Model&amp;lt;/font&amp;gt;'''中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;友谊悖论 Friendship Paradox&amp;lt;/font&amp;gt;'''。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生成函数方法 Generating Functions Method&amp;lt;/font&amp;gt;'''&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|url=https://link.aps.org/doi/10.1103/PhysRevE.64.026118|journal=Physical Review E|language=en|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|issn=1063-651X|doi-access=free}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; 的值也可得出通过 &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;的导数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻性。然后，可以很容易地进行计算依据&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和它的导数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的。进一步扩展，&amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th的邻居的分布是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, 以&amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 迭代到 &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 函数本身。第一个邻居的平均数量&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; 第二个邻居的平均数量是:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
有向网络的度分布&lt;br /&gt;
&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; 和一些出度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt;分别指代的是指向该节点的边的数量和从该节点指出的边的数量。如果 &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; 是一个随机选择的具有入出度的节点的可能性。那么分配给这个生成函数的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;联合概率分布  Joint Probability Distribution&amp;lt;/font&amp;gt;'''可以被写成两个变量 &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt;是网络中节点的平均度(内部和外部)&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
使用函数&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。可以将&amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt;表示离开节点的边的数量:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;，或者像之前表示的&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;从一个随机选择的节点上可达到的第二邻居的平均数是&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. 这些是从随机选择的节点所达到第一和第二邻居的数量，因为这些方程显然是在&amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt;和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;上对称的。&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;复杂网络 Complex Network&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-free Network&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机网络 Random Graph&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;结构截止值 Structural Cut-off&amp;lt;/font&amp;gt;'''&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）该部分也为专业名词 需要进行标橙&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
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&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
| 杂志 = 物理进展&lt;br /&gt;
&lt;br /&gt;
  | volume = 51&lt;br /&gt;
&lt;br /&gt;
  | volume = 51&lt;br /&gt;
&lt;br /&gt;
51&lt;br /&gt;
&lt;br /&gt;
  | pages = 1079–1187&lt;br /&gt;
&lt;br /&gt;
  | pages = 1079–1187&lt;br /&gt;
&lt;br /&gt;
| pages = 1079-1187&lt;br /&gt;
&lt;br /&gt;
  | year = 2002&lt;br /&gt;
&lt;br /&gt;
  | year = 2002&lt;br /&gt;
&lt;br /&gt;
2002年&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | doi = 10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
  | doi = 10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
第四期&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
2002/adphy. . 51.1079 d }&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
&lt;br /&gt;
 |last=Newman &lt;br /&gt;
&lt;br /&gt;
 |last=Newman &lt;br /&gt;
&lt;br /&gt;
纽曼&lt;br /&gt;
&lt;br /&gt;
 |first=M. E. J. &lt;br /&gt;
&lt;br /&gt;
 |first=M. E. J. &lt;br /&gt;
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| first = m.E.j.&lt;br /&gt;
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 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
| title = 复杂网络的结构和功能&lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
| journal = SIAM Review&lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
45&lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
| 页 = 167-256&lt;br /&gt;
&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
2003年&lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
| doi = 10.1137/S003614450342480&lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
2&lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516&lt;br /&gt;
&lt;br /&gt;
 |bibcode=2003SIAMR..45..167N &lt;br /&gt;
&lt;br /&gt;
 |bibcode=2003SIAMR..45..167N &lt;br /&gt;
&lt;br /&gt;
| bibcode = 2003siaml. . 45. . 167 n&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
* {{cite book &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
复杂网络: 结构、健壮性和功能&lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
作者: Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
| 作者链接 = Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
   &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
2010年&lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
剑桥大学出版社&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
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}}&lt;br /&gt;
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}}&lt;br /&gt;
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&lt;br /&gt;
[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph invariants]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph invariants&lt;br /&gt;
&lt;br /&gt;
类别: 图形不变量&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12757</id>
		<title>度</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12757"/>
		<updated>2020-08-16T14:34:16Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
[[File:UndirectedDegrees (Loop).svg|thumb|&lt;br /&gt;
图1：A graph with a loop having vertices labeled by degree 内含自环按度标记的图]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In [[graph theory]], the '''degree''' (or '''valency''') of a [[vertex (graph theory)|vertex]] of a [[Graph (discrete mathematics)|graph]] is the number of [[edge (graph theory)|edges]] that are [[incidence (graph)|incident]] to the vertex, and in a [[multigraph]], [[loop (graph theory)|loop]]s are counted twice.&amp;lt;ref&amp;gt;Diestel p.5&amp;lt;/ref&amp;gt; The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The '''maximum degree''' of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the '''minimum degree''' of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
&lt;br /&gt;
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The maximum degree of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the minimum degree of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''中，图中顶点的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度 Degree&amp;lt;/font&amp;gt;'''(或价)是入射到该顶点的边的数量。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;多重图 Multigraph&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;自环 Loops&amp;lt;/font&amp;gt;'''会被计算两次。顶点的度数可表示为&amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt;或&amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;。一个图&amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;的最大度值表示为&amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;，最小度值表示为 &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;。在右侧的多重图中，最大度值为5，最小度值为0。&lt;br /&gt;
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In a [[regular graph]], every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A [[complete graph]] (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;正则图 Regular Graph&amp;lt;/font&amp;gt;'''中，每个顶点都具有相同的度值，因此我们把该度值可以称之为图的度值。一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全图 Complete Graph&amp;lt;/font&amp;gt;'''(表示为&amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;，其中&amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt;是图中顶点的数目)是一种特殊的正则图，其中所有顶点都有最大度值，&amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;。&lt;br /&gt;
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==Handshaking lemma==&lt;br /&gt;
握手引理&lt;br /&gt;
{{main|Handshaking lemma}}&lt;br /&gt;
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The '''degree sum formula''' states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
The degree sum formula states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度和公式 Degree Sum Formula&amp;lt;/font&amp;gt;'''表明，给定一个图&amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree  sum formula) is known as the [[handshaking lemma]]. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
&lt;br /&gt;
The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
&lt;br /&gt;
该公式表明，在任何无向图中，拥有奇数度值的顶点的个数是偶数。这个陈述(以及度和公式)被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;握手引理 Handshaking Lemma&amp;lt;/font&amp;gt;'''。该名字来自一个有趣的数学问题，即证明无论该群体内有多少人，与奇数个人握过手的人数是偶数。&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）标题涉及到的专业名词 要么标注一下标题要么标注一下正文~&lt;br /&gt;
&lt;br /&gt;
==Degree sequence==&lt;br /&gt;
度序列&lt;br /&gt;
[[File:Conjugate-dessins.svg|thumb|200px|&lt;br /&gt;
图2：Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1).&lt;br /&gt;
两个具有相同度序列的非同构图]]&lt;br /&gt;
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The '''degree sequence''' of an undirected graph is the non-increasing sequence of its vertex degrees;&amp;lt;ref&amp;gt;Diestel p.216&amp;lt;/ref&amp;gt; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a [[graph invariant]] so [[Graph isomorphism|isomorphic graphs]] have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
&lt;br /&gt;
The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
&lt;br /&gt;
无向图的度序列是指将其各顶点度值按非递增方式排序，对于上述图是(5,3,3,2,2,1,0)。度序列是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图不变量 Graph Invariant&amp;lt;/font&amp;gt;'''，因此'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;同构图 Non-isomorphic Graphs&amp;lt;/font&amp;gt;'''（两个图中顶点的度值都相同，但形状不同）具有相同的度序列。然而，度序列通常不能唯一地标识一个图，在某些情况下，非同构图具有相同的度序列。&lt;br /&gt;
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The '''degree sequence problem'''  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a '''graphic''' or '''graphical sequence'''. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a [[matching (graph theory)|matching]], and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
&lt;br /&gt;
The degree sequence problem  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度序列问题 Degree Sequence Problem&amp;lt;/font&amp;gt;'''，是指寻找具有以非增方式排列给定正整数的度序列的局部或全部图的问题。(序列尾部的零可能会被忽略，因为通过向图中添加适当数量的孤立顶点就可以轻松实现序列尾部不断加零。)一个序列是某个图的度序列，即一个度序列问题有解时，该序列称为图形序列。由于度和公式的存在，任何具有奇数和的序列，如(3,3,1) ，都不会是图的度序列。反之亦然: 如果一个序列和是偶数，它就是重图的度序列。构造这样一个图是很简单: 匹配奇数度值的顶点并成对连接起来，然后剩余的偶数度值顶点都连出一条边指向自己。&lt;br /&gt;
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The question of whether a given degree sequence can be realized by a [[simple graph]] is more challenging. This problem is also called [[graph realization problem]] and can either be solved by the [[Erdős–Gallai theorem]] or the [[Havel–Hakimi algorithm]]. &lt;br /&gt;
&lt;br /&gt;
The question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm. &lt;br /&gt;
&lt;br /&gt;
一个给定的度序列是否可以用一个简单的图进行表示，是一个具有挑战性的问题。这个问题也称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图实现 Graph Realization Problem&amp;lt;/font&amp;gt;'''问题，可以用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Erdős–Gallai定理 Erdős–Gallai Theoreme&amp;lt;/font&amp;gt;'''解决，也可以用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Havel-Hakimi算法 Havel–Hakimi Algorithm&amp;lt;/font&amp;gt;'''解决。&lt;br /&gt;
&lt;br /&gt;
The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of [[graph enumeration]].&lt;br /&gt;
&lt;br /&gt;
The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of graph enumeration.&lt;br /&gt;
&lt;br /&gt;
基于给定度序列寻找或估计可能的图的个数，是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图枚举 Graph Enumeration&amp;lt;/font&amp;gt;'''领域中的一个问题。&lt;br /&gt;
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&lt;br /&gt;
&lt;br /&gt;
More generally, the '''degree sequence''' of a [[hypergraph]] is the non-increasing sequence of its vertex degrees. A sequence is '''&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic''' if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in [[Time complexity|polynomial time]] for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the [[Erdős–Gallai theorem]] but is [[NP-completeness|NP-complete]] for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 &amp;lt;ref&amp;gt;{{Cite journal|last=Deza|first=Antoine|last2=Levin|first2=Asaf|last3=Meesum|first3=Syed M.|last4=Onn|first4=Shmuel|date=January 2018|title=Optimization over Degree Sequences|url=https://epubs.siam.org/doi/10.1137/17M1134482|journal=SIAM Journal on Discrete Mathematics|language=en|volume=32|issue=3|pages=2067–2079|doi=10.1137/17M1134482|issn=0895-4801|arxiv=1706.03951}}&amp;lt;/ref&amp;gt;).&lt;br /&gt;
&lt;br /&gt;
More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in polynomial time for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the Erdős–Gallai theorem but is NP-complete for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 ).&lt;br /&gt;
&lt;br /&gt;
一般来说，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;超图 Hypergraph&amp;lt;/font&amp;gt;'''的度序列是其顶点度的非递增序列。一个序列是&amp;lt;math&amp;gt;k-graphic&amp;lt;/math&amp;gt;，如果它是一些&amp;lt;math&amp;gt;k-uniform&amp;lt;/math&amp;gt;超图的度序列。特别是&amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic序列是图形。决定一个给定的序列是否是&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic可以在 &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt;的多项式时间内通过Erdős–Gallai定理实现，但是当&amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt;时该问题可转化为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;NP完全问题 NP-complete&amp;lt;/font&amp;gt;'''(Deza et al. ，2018)。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Special values==&lt;br /&gt;
特殊值&lt;br /&gt;
[[File:Depth-first-tree.png|thumb|&lt;br /&gt;
图3：An undirected graph with leaf nodes 4, 5, 6, 7, 10, 11, and 12 一个具有叶节点的无定向图]]&lt;br /&gt;
&lt;br /&gt;
*A vertex with degree 0 is called an [[isolated vertex]].&lt;br /&gt;
度值为0的顶点成为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;孤立顶点 Isolated Vertex&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of [[tree (graph theory)|tree]]s in graph theory and especially [[tree (data structure)|tree]]s as [[data structure]]s.&lt;br /&gt;
度值为1的顶点称为叶顶点或尾顶点，该顶点的入射边称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;悬挂边 Pendant Edge&amp;lt;/font&amp;gt;'''。在右侧的图中，{3,5}就是一个悬挂边。在图论中，该术语主要在研究'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;树 Tree&amp;lt;/font&amp;gt;'''时使用，特别是具有树形结构的数据。&lt;br /&gt;
* A vertex with degree ''n''&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1  in a graph on ''n'' vertices is called a [[dominating vertex]].&lt;br /&gt;
在有n个顶点的图中，度值为n-1的顶点叫作'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;主导顶点 Dominating Vertex&amp;lt;/font&amp;gt;'''&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Global properties==&lt;br /&gt;
全局属性&lt;br /&gt;
*If each vertex of the graph has the same degree&amp;amp;nbsp;''k'' the graph is called a [[regular graph|''k''-regular graph]] and the graph itself is said to have degree&amp;amp;nbsp;''k''. Similarly, a [[bipartite graph]] in which every two vertices on the same side of the bipartition as each other have the same degree is called a [[biregular graph]].&lt;br /&gt;
如果一个图中的所有顶点的度值都为k，该图被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;''k''-正则图 ''k''-regular graph&amp;lt;/font&amp;gt;'''，该图的度值也为k。同样的，同侧每两个顶点都具有相同度值的二分图叫作'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二分正则图 Biregular Graph&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
*An undirected, connected graph has an [[Eulerian path]] if and only if it has either 0 or 2 vertices of odd degree. If it has 0 vertices of odd degree, the Eulerian path is an Eulerian circuit.&lt;br /&gt;
当且仅当具有0或2个奇数度的顶点时，无向连通图才具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;欧拉路径 Eulerian Path&amp;lt;/font&amp;gt;'''。 如果它具有0个奇数度的顶点，则欧拉路径为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;欧拉回路 Eulerian Circuit&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
*A directed graph is a [[pseudoforest]] if and only if every vertex has outdegree at most&amp;amp;nbsp;1. A [[functional graph]] is a special case of a pseudoforest in which every vertex has outdegree exactly&amp;amp;nbsp;1.&lt;br /&gt;
有向图是当且仅当每个顶点的度值最大为1时才是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;伪森林 Pseudoforest&amp;lt;/font&amp;gt;'''。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;功能图 Functional Graph&amp;lt;/font&amp;gt;'''是伪森林的特例，其中每个顶点的度数都恰好为1。&lt;br /&gt;
*By [[Brooks' theorem]], any graph other than a clique or an odd cycle has [[chromatic number]] at most&amp;amp;nbsp;Δ, and by [[Vizing's theorem]] any graph has [[chromatic index]] at most Δ&amp;amp;nbsp;+&amp;amp;nbsp;1.&lt;br /&gt;
根据布鲁克斯定理，除团簇或奇数循环外，任何图的色度数最大为Δ，而根据维辛定理，任何图的色度指数最大为Δ+ 1。&lt;br /&gt;
*A [[Degeneracy (graph theory)|''k''-degenerate graph]] is a graph in which each subgraph has a vertex of degree at most ''k''.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;k简并图 K-Degenerate Graph &amp;lt;/font&amp;gt;'''是其中每个子图最多具有度值为k的顶点的图。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==See also==&lt;br /&gt;
&lt;br /&gt;
*[[Indegree]], [[outdegree]] for [[digraph (mathematics)|digraph]]s&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向图的入度和出度 Indegree,Outdegree For Digraph&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*[[Degree distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度分布 Degree Distribution&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*[[bipartite graph|degree sequence]] for bipartite graphs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二分图的度序列 Degree Sequence For Bipartite Graphs &amp;lt;/font&amp;gt;'''&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Notes==&lt;br /&gt;
&lt;br /&gt;
{{reflist}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&lt;br /&gt;
*{{Citation | last1=Diestel | first1=Reinhard | title=Graph Theory | url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-3-540-26183-4 | year=2005 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=P.|last1=Erdős|author1-link=Paul Erdős|first2=T.|last2=Gallai|author2-link=Tibor Gallai|title=Gráfok előírt fokszámú pontokkal|language=Hungarian|journal=Matematikai Lapok|volume=11|year=1960|pages=264&amp;amp;ndash;274|url=http://www.renyi.hu/~p_erdos/1961-05.pdf}}.&lt;br /&gt;
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&lt;br /&gt;
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15&lt;br /&gt;
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1991}}.&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree (graph theory)]]. Its edit history can be viewed at [[度/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12756</id>
		<title>度</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12756"/>
		<updated>2020-08-16T14:32:18Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
[[File:UndirectedDegrees (Loop).svg|thumb|&lt;br /&gt;
图1：A graph with a loop having vertices labeled by degree 内含自环按度标记的图]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In [[graph theory]], the '''degree''' (or '''valency''') of a [[vertex (graph theory)|vertex]] of a [[Graph (discrete mathematics)|graph]] is the number of [[edge (graph theory)|edges]] that are [[incidence (graph)|incident]] to the vertex, and in a [[multigraph]], [[loop (graph theory)|loop]]s are counted twice.&amp;lt;ref&amp;gt;Diestel p.5&amp;lt;/ref&amp;gt; The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The '''maximum degree''' of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the '''minimum degree''' of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
&lt;br /&gt;
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The maximum degree of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the minimum degree of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''中，图中顶点的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度 Degree&amp;lt;/font&amp;gt;'''(或价)是入射到该顶点的边的数量。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;多重图 Multigraph&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;自环 Loops&amp;lt;/font&amp;gt;'''会被计算两次。顶点的度数可表示为&amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt;或&amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;。一个图&amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;的最大度值表示为&amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;，最小度值表示为 &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;。在右侧的多重图中，最大度值为5，最小度值为0。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In a [[regular graph]], every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A [[complete graph]] (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;正则图 Regular Graph&amp;lt;/font&amp;gt;'''中，每个顶点都具有相同的度值，因此我们把该度值可以称之为图的度值。一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全图 Complete Graph&amp;lt;/font&amp;gt;'''(表示为&amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;，其中&amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt;是图中顶点的数目)是一种特殊的正则图，其中所有顶点都有最大度值，&amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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==Handshaking lemma==&lt;br /&gt;
握手引理&lt;br /&gt;
{{main|Handshaking lemma}}&lt;br /&gt;
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The '''degree sum formula''' states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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The degree sum formula states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度和公式 Degree Sum Formula&amp;lt;/font&amp;gt;'''表明，给定一个图&amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree  sum formula) is known as the [[handshaking lemma]]. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
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The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
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该公式表明，在任何无向图中，拥有奇数度值的顶点的个数是偶数。这个陈述(以及度和公式)被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;握手引理 Handshaking Lemma&amp;lt;/font&amp;gt;'''。该名字来自一个有趣的数学问题，即证明无论该群体内有多少人，与奇数个人握过手的人数是偶数。&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）标题涉及到的专业名词 要么标注一下标题要么标注一下正文~&lt;br /&gt;
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==Degree sequence==&lt;br /&gt;
度序列&lt;br /&gt;
[[File:Conjugate-dessins.svg|thumb|200px|&lt;br /&gt;
图2：Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1).&lt;br /&gt;
两个具有相同度序列的非同构图]]&lt;br /&gt;
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The '''degree sequence''' of an undirected graph is the non-increasing sequence of its vertex degrees;&amp;lt;ref&amp;gt;Diestel p.216&amp;lt;/ref&amp;gt; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a [[graph invariant]] so [[Graph isomorphism|isomorphic graphs]] have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
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The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
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无向图的度序列是指将其各顶点度值按非递增方式排序，对于上述图是(5,3,3,2,2,1,0)。度序列是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图不变量 Graph Invariant&amp;lt;/font&amp;gt;'''，因此'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;同构图 Non-isomorphic Graphs&amp;lt;/font&amp;gt;'''（两个图中顶点的度值都相同，但形状不同）具有相同的度序列。然而，度序列通常不能唯一地标识一个图，在某些情况下，非同构图具有相同的度序列。&lt;br /&gt;
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The '''degree sequence problem'''  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a '''graphic''' or '''graphical sequence'''. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a [[matching (graph theory)|matching]], and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
&lt;br /&gt;
The degree sequence problem  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度序列问题 Degree Sequence Problem&amp;lt;/font&amp;gt;'''，是指寻找具有以非增方式排列给定正整数的度序列的局部或全部图的问题。(序列尾部的零可能会被忽略，因为通过向图中添加适当数量的孤立顶点就可以轻松实现序列尾部不断加零。)一个序列是某个图的度序列，即一个度序列问题有解时，该序列称为图形序列。由于度和公式的存在，任何具有奇数和的序列，如(3,3,1) ，都不会是图的度序列。反之亦然: 如果一个序列和是偶数，它就是重图的度序列。构造这样一个图是很简单: 匹配奇数度值的顶点并成对连接起来，然后剩余的偶数度值顶点都连出一条边指向自己。&lt;br /&gt;
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The question of whether a given degree sequence can be realized by a [[simple graph]] is more challenging. This problem is also called [[graph realization problem]] and can either be solved by the [[Erdős–Gallai theorem]] or the [[Havel–Hakimi algorithm]]. &lt;br /&gt;
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The question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm. &lt;br /&gt;
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一个给定的度序列是否可以用一个简单的图进行表示，是一个具有挑战性的问题。这个问题也称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图实现 Graph Realization Problem&amp;lt;/font&amp;gt;'''问题，可以用 Erdős–Gallai定理解决，也可以用Havel-Hakimi算法解决。&lt;br /&gt;
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The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of [[graph enumeration]].&lt;br /&gt;
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The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of graph enumeration.&lt;br /&gt;
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基于给定度序列寻找或估计可能的图的个数，是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图枚举 Graph Enumeration&amp;lt;/font&amp;gt;'''领域中的一个问题。&lt;br /&gt;
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More generally, the '''degree sequence''' of a [[hypergraph]] is the non-increasing sequence of its vertex degrees. A sequence is '''&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic''' if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in [[Time complexity|polynomial time]] for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the [[Erdős–Gallai theorem]] but is [[NP-completeness|NP-complete]] for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 &amp;lt;ref&amp;gt;{{Cite journal|last=Deza|first=Antoine|last2=Levin|first2=Asaf|last3=Meesum|first3=Syed M.|last4=Onn|first4=Shmuel|date=January 2018|title=Optimization over Degree Sequences|url=https://epubs.siam.org/doi/10.1137/17M1134482|journal=SIAM Journal on Discrete Mathematics|language=en|volume=32|issue=3|pages=2067–2079|doi=10.1137/17M1134482|issn=0895-4801|arxiv=1706.03951}}&amp;lt;/ref&amp;gt;).&lt;br /&gt;
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More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in polynomial time for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the Erdős–Gallai theorem but is NP-complete for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 ).&lt;br /&gt;
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一般来说，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;超图 Hypergraph&amp;lt;/font&amp;gt;'''的度序列是其顶点度的非递增序列。一个序列是&amp;lt;math&amp;gt;k-graphic&amp;lt;/math&amp;gt;，如果它是一些&amp;lt;math&amp;gt;k-uniform&amp;lt;/math&amp;gt;超图的度序列。特别是&amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic序列是图形。决定一个给定的序列是否是&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic可以在 &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt;的多项式时间内通过Erdős–Gallai定理实现，但是当&amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt;时该问题可转化为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;NP完全问题 NP-complete&amp;lt;/font&amp;gt;'''(Deza et al. ，2018)。&lt;br /&gt;
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==Special values==&lt;br /&gt;
特殊值&lt;br /&gt;
[[File:Depth-first-tree.png|thumb|&lt;br /&gt;
图3：An undirected graph with leaf nodes 4, 5, 6, 7, 10, 11, and 12 一个具有叶节点的无定向图]]&lt;br /&gt;
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*A vertex with degree 0 is called an [[isolated vertex]].&lt;br /&gt;
度值为0的顶点成为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;孤立顶点 Isolated Vertex&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of [[tree (graph theory)|tree]]s in graph theory and especially [[tree (data structure)|tree]]s as [[data structure]]s.&lt;br /&gt;
度值为1的顶点称为叶顶点或尾顶点，该顶点的入射边称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;悬挂边 Pendant Edge&amp;lt;/font&amp;gt;'''。在右侧的图中，{3,5}就是一个悬挂边。在图论中，该术语主要在研究'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;树 Tree&amp;lt;/font&amp;gt;'''时使用，特别是具有树形结构的数据。&lt;br /&gt;
* A vertex with degree ''n''&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1  in a graph on ''n'' vertices is called a [[dominating vertex]].&lt;br /&gt;
在有n个顶点的图中，度值为n-1的顶点叫作'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;主导顶点 Dominating Vertex&amp;lt;/font&amp;gt;'''&lt;br /&gt;
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==Global properties==&lt;br /&gt;
全局属性&lt;br /&gt;
*If each vertex of the graph has the same degree&amp;amp;nbsp;''k'' the graph is called a [[regular graph|''k''-regular graph]] and the graph itself is said to have degree&amp;amp;nbsp;''k''. Similarly, a [[bipartite graph]] in which every two vertices on the same side of the bipartition as each other have the same degree is called a [[biregular graph]].&lt;br /&gt;
如果一个图中的所有顶点的度值都为k，该图被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;''k''-正则图 ''k''-regular graph&amp;lt;/font&amp;gt;'''，该图的度值也为k。同样的，同侧每两个顶点都具有相同度值的二分图叫作'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二分正则图 Biregular Graph&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
*An undirected, connected graph has an [[Eulerian path]] if and only if it has either 0 or 2 vertices of odd degree. If it has 0 vertices of odd degree, the Eulerian path is an Eulerian circuit.&lt;br /&gt;
当且仅当具有0或2个奇数度的顶点时，无向连通图才具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;欧拉路径 Eulerian Path&amp;lt;/font&amp;gt;'''。 如果它具有0个奇数度的顶点，则欧拉路径为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;欧拉回路 Eulerian Circuit&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
*A directed graph is a [[pseudoforest]] if and only if every vertex has outdegree at most&amp;amp;nbsp;1. A [[functional graph]] is a special case of a pseudoforest in which every vertex has outdegree exactly&amp;amp;nbsp;1.&lt;br /&gt;
有向图是当且仅当每个顶点的度值最大为1时才是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;伪森林 Pseudoforest&amp;lt;/font&amp;gt;'''。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;功能图 Functional Graph&amp;lt;/font&amp;gt;'''是伪森林的特例，其中每个顶点的度数都恰好为1。&lt;br /&gt;
*By [[Brooks' theorem]], any graph other than a clique or an odd cycle has [[chromatic number]] at most&amp;amp;nbsp;Δ, and by [[Vizing's theorem]] any graph has [[chromatic index]] at most Δ&amp;amp;nbsp;+&amp;amp;nbsp;1.&lt;br /&gt;
根据布鲁克斯定理，除团簇或奇数循环外，任何图的色度数最大为Δ，而根据维辛定理，任何图的色度指数最大为Δ+ 1。&lt;br /&gt;
*A [[Degeneracy (graph theory)|''k''-degenerate graph]] is a graph in which each subgraph has a vertex of degree at most ''k''.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;k简并图 K-Degenerate Graph &amp;lt;/font&amp;gt;'''是其中每个子图最多具有度值为k的顶点的图。&lt;br /&gt;
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==See also==&lt;br /&gt;
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*[[Indegree]], [[outdegree]] for [[digraph (mathematics)|digraph]]s&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向图的入度和出度 Indegree,Outdegree For Digraph&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*[[Degree distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度分布 Degree Distribution&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*[[bipartite graph|degree sequence]] for bipartite graphs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二分图的度序列 Degree Sequence For Bipartite Graphs &amp;lt;/font&amp;gt;'''&lt;br /&gt;
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==Notes==&lt;br /&gt;
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{{reflist}}&lt;br /&gt;
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==References==&lt;br /&gt;
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 | doi = 10.1002/jgt.3190150209&lt;br /&gt;
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 | doi = 10.1002/jgt.3190150209&lt;br /&gt;
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10.1002/jgt. 3190150209&lt;br /&gt;
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 | issue = 2&lt;br /&gt;
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 | journal = [[Journal of Graph Theory]]&lt;br /&gt;
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 | journal = Journal of Graph Theory&lt;br /&gt;
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| Journal = Journal of Graph Theory&lt;br /&gt;
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 | mr = 1106533&lt;br /&gt;
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 | mr = 1106533&lt;br /&gt;
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1106533先生&lt;br /&gt;
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 | pages = 223–231&lt;br /&gt;
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 | pages = 223–231&lt;br /&gt;
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| pages = 223-231&lt;br /&gt;
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 | title = Seven criteria for integer sequences being graphic&lt;br /&gt;
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 | title = Seven criteria for integer sequences being graphic&lt;br /&gt;
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| title = 整数序列为图形的七个条件&lt;br /&gt;
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 | volume = 15&lt;br /&gt;
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15&lt;br /&gt;
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 | year = 1991}}.&lt;br /&gt;
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 | year = 1991}}.&lt;br /&gt;
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1991}}.&lt;br /&gt;
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[[Category:Graph theory]]&lt;br /&gt;
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Category:Graph theory&lt;br /&gt;
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范畴: 图论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree (graph theory)]]. Its edit history can be viewed at [[度/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12755</id>
		<title>度</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12755"/>
		<updated>2020-08-16T14:26:36Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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[[File:UndirectedDegrees (Loop).svg|thumb|&lt;br /&gt;
图1：A graph with a loop having vertices labeled by degree 内含自环按度标记的图]]&lt;br /&gt;
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In [[graph theory]], the '''degree''' (or '''valency''') of a [[vertex (graph theory)|vertex]] of a [[Graph (discrete mathematics)|graph]] is the number of [[edge (graph theory)|edges]] that are [[incidence (graph)|incident]] to the vertex, and in a [[multigraph]], [[loop (graph theory)|loop]]s are counted twice.&amp;lt;ref&amp;gt;Diestel p.5&amp;lt;/ref&amp;gt; The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The '''maximum degree''' of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the '''minimum degree''' of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
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In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The maximum degree of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the minimum degree of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''中，图中顶点的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度 Degree&amp;lt;/font&amp;gt;'''(或价)是入射到该顶点的边的数量。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;多重图 Multigraph&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;自环 Loops&amp;lt;/font&amp;gt;'''会被计算两次。顶点的度数可表示为&amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt;或&amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;。一个图&amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;的最大度值表示为&amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;，最小度值表示为 &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;。在右侧的多重图中，最大度值为5，最小度值为0。&lt;br /&gt;
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In a [[regular graph]], every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A [[complete graph]] (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
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In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;正则图 Regular Graph&amp;lt;/font&amp;gt;'''中，每个顶点都具有相同的度值，因此我们把该度值可以称之为图的度值。一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全图 Complete Graph&amp;lt;/font&amp;gt;'''(表示为&amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;，其中&amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt;是图中顶点的数目)是一种特殊的正则图，其中所有顶点都有最大度值，&amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;。&lt;br /&gt;
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==Handshaking lemma==&lt;br /&gt;
握手引理&lt;br /&gt;
{{main|Handshaking lemma}}&lt;br /&gt;
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The '''degree sum formula''' states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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The degree sum formula states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度和公式 Degree Sum Formula&amp;lt;/font&amp;gt;'''表明，给定一个图&amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree  sum formula) is known as the [[handshaking lemma]]. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
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The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
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该公式表明，在任何无向图中，拥有奇数度值的顶点的个数是偶数。这个陈述(以及度和公式)被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;握手引理 Handshaking Lemma&amp;lt;/font&amp;gt;'''。该名字来自一个有趣的数学问题，即证明无论该群体内有多少人，与奇数个人握过手的人数是偶数。&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）标题涉及到的专业名词 要么标注一下标题要么标注一下正文~&lt;br /&gt;
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==Degree sequence==&lt;br /&gt;
度序列&lt;br /&gt;
[[File:Conjugate-dessins.svg|thumb|200px|&lt;br /&gt;
图2：Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1).&lt;br /&gt;
两个具有相同度序列的非同构图]]&lt;br /&gt;
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The '''degree sequence''' of an undirected graph is the non-increasing sequence of its vertex degrees;&amp;lt;ref&amp;gt;Diestel p.216&amp;lt;/ref&amp;gt; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a [[graph invariant]] so [[Graph isomorphism|isomorphic graphs]] have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
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The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
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无向图的度序列是指将其各顶点度值按非递增方式排序，对于上述图是(5,3,3,2,2,1,0)。度序列是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图不变量 Graph Invariant&amp;lt;/font&amp;gt;'''，因此'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;同构图 Non-isomorphic Graphs&amp;lt;/font&amp;gt;'''（两个图中顶点的度值都相同，但形状不同）具有相同的度序列。然而，度序列通常不能唯一地标识一个图，在某些情况下，非同构图具有相同的度序列。&lt;br /&gt;
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The '''degree sequence problem'''  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a '''graphic''' or '''graphical sequence'''. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a [[matching (graph theory)|matching]], and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
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The degree sequence problem  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度序列问题 Degree Sequence Problem&amp;lt;/font&amp;gt;'''，是指寻找具有以非增方式排列给定正整数的度序列的局部或全部图的问题。(序列尾部的零可能会被忽略，因为通过向图中添加适当数量的孤立顶点就可以轻松实现序列尾部不断加零。)一个序列是某个图的度序列，即一个度序列问题有解时，该序列称为图形序列。由于度和公式的存在，任何具有奇数和的序列，如(3,3,1) ，都不会是图的度序列。反之亦然: 如果一个序列和是偶数，它就是重图的度序列。构造这样一个图是很简单: 匹配奇数度值的顶点并成对连接起来，然后剩余的偶数度值顶点都连出一条边指向自己。&lt;br /&gt;
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The question of whether a given degree sequence can be realized by a [[simple graph]] is more challenging. This problem is also called [[graph realization problem]] and can either be solved by the [[Erdős–Gallai theorem]] or the [[Havel–Hakimi algorithm]]. &lt;br /&gt;
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The question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm. &lt;br /&gt;
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一个给定的度序列是否可以用一个简单的图进行表示，是一个具有挑战性的问题。这个问题也称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图实现 Graph Realization Problem&amp;lt;/font&amp;gt;'''问题，可以用 Erdős–Gallai定理解决，也可以用Havel-Hakimi算法解决。&lt;br /&gt;
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The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of [[graph enumeration]].&lt;br /&gt;
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The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of graph enumeration.&lt;br /&gt;
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基于给定度序列寻找或估计可能的图的个数，是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图枚举 Graph Enumeration&amp;lt;/font&amp;gt;'''领域中的一个问题。&lt;br /&gt;
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More generally, the '''degree sequence''' of a [[hypergraph]] is the non-increasing sequence of its vertex degrees. A sequence is '''&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic''' if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in [[Time complexity|polynomial time]] for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the [[Erdős–Gallai theorem]] but is [[NP-completeness|NP-complete]] for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 &amp;lt;ref&amp;gt;{{Cite journal|last=Deza|first=Antoine|last2=Levin|first2=Asaf|last3=Meesum|first3=Syed M.|last4=Onn|first4=Shmuel|date=January 2018|title=Optimization over Degree Sequences|url=https://epubs.siam.org/doi/10.1137/17M1134482|journal=SIAM Journal on Discrete Mathematics|language=en|volume=32|issue=3|pages=2067–2079|doi=10.1137/17M1134482|issn=0895-4801|arxiv=1706.03951}}&amp;lt;/ref&amp;gt;).&lt;br /&gt;
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More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in polynomial time for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the Erdős–Gallai theorem but is NP-complete for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 ).&lt;br /&gt;
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一般来说，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;超图 Hypergraph&amp;lt;/font&amp;gt;'''的度序列是其顶点度的非递增序列。一个序列是&amp;lt;math&amp;gt;k-graphic&amp;lt;/math&amp;gt;，如果它是一些&amp;lt;math&amp;gt;k-uniform&amp;lt;/math&amp;gt;超图的度序列。特别是&amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic序列是图形。决定一个给定的序列是否是&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic可以在 &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt;的多项式时间内通过Erdős–Gallai定理实现，但是当&amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt;时该问题可转化为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;NP完全问题 NP-complete&amp;lt;/font&amp;gt;'''(Deza et al. ，2018)。&lt;br /&gt;
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==Special values==&lt;br /&gt;
特殊值&lt;br /&gt;
[[File:Depth-first-tree.png|thumb|&lt;br /&gt;
图3：An undirected graph with leaf nodes 4, 5, 6, 7, 10, 11, and 12 一个具有叶节点的无定向图]]&lt;br /&gt;
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*A vertex with degree 0 is called an [[isolated vertex]].&lt;br /&gt;
度值为0的顶点成为孤立顶点&lt;br /&gt;
*A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of [[tree (graph theory)|tree]]s in graph theory and especially [[tree (data structure)|tree]]s as [[data structure]]s.&lt;br /&gt;
度值为1的顶点称为叶顶点或尾顶点，该顶点的入射边称为悬挂边。在右侧的图中，{3,5}就是一个悬挂边。在图论中，该术语主要在研究“树”时使用，特别是具有树形结构的数据。&lt;br /&gt;
* A vertex with degree ''n''&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1  in a graph on ''n'' vertices is called a [[dominating vertex]].&lt;br /&gt;
在有n个顶点的图中，度值为n-1的顶点叫作主导顶点&lt;br /&gt;
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==Global properties==&lt;br /&gt;
全局属性&lt;br /&gt;
*If each vertex of the graph has the same degree&amp;amp;nbsp;''k'' the graph is called a [[regular graph|''k''-regular graph]] and the graph itself is said to have degree&amp;amp;nbsp;''k''. Similarly, a [[bipartite graph]] in which every two vertices on the same side of the bipartition as each other have the same degree is called a [[biregular graph]].&lt;br /&gt;
如果一个图中的所有顶点的度值都为k，该图被称为k-正则图，该图的度值也为k。同样的，同侧每两个顶点都具有相同度值的二分图叫作二分正则图。&lt;br /&gt;
*An undirected, connected graph has an [[Eulerian path]] if and only if it has either 0 or 2 vertices of odd degree. If it has 0 vertices of odd degree, the Eulerian path is an Eulerian circuit.&lt;br /&gt;
当且仅当具有0或2个奇数度的顶点时，无向连通图才具有欧拉路径。 如果它具有0个奇数度的顶点，则欧拉路径为欧拉回路。&lt;br /&gt;
*A directed graph is a [[pseudoforest]] if and only if every vertex has outdegree at most&amp;amp;nbsp;1. A [[functional graph]] is a special case of a pseudoforest in which every vertex has outdegree exactly&amp;amp;nbsp;1.&lt;br /&gt;
有向图是当且仅当每个顶点的度值最大为1时才是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;伪森林 Pseudoforest&amp;lt;/font&amp;gt;'''。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;功能图 Functional Graph&amp;lt;/font&amp;gt;'''是伪森林的特例，其中每个顶点的度数都恰好为1。&lt;br /&gt;
*By [[Brooks' theorem]], any graph other than a clique or an odd cycle has [[chromatic number]] at most&amp;amp;nbsp;Δ, and by [[Vizing's theorem]] any graph has [[chromatic index]] at most Δ&amp;amp;nbsp;+&amp;amp;nbsp;1.&lt;br /&gt;
根据布鲁克斯定理，除团簇或奇数循环外，任何图的色度数最大为Δ，而根据维辛定理，任何图的色度指数最大为Δ+ 1。&lt;br /&gt;
*A [[Degeneracy (graph theory)|''k''-degenerate graph]] is a graph in which each subgraph has a vertex of degree at most ''k''.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;k简并图 K-Degenerate Graph &amp;lt;/font&amp;gt;'''是其中每个子图最多具有度值为k的顶点的图。&lt;br /&gt;
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==See also==&lt;br /&gt;
&lt;br /&gt;
*[[Indegree]], [[outdegree]] for [[digraph (mathematics)|digraph]]s&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有向图的入度和出度 Indegree,Outdegree For Digraph&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*[[Degree distribution]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度分布 Degree Distribution&amp;lt;/font&amp;gt;'''&lt;br /&gt;
*[[bipartite graph|degree sequence]] for bipartite graphs&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二分图的度序列 Degree Sequence For Bipartite Graphs &amp;lt;/font&amp;gt;'''&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Notes==&lt;br /&gt;
&lt;br /&gt;
{{reflist}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
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&lt;br /&gt;
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2&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Journal of Graph Theory]]&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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| title = 整数序列为图形的七个条件&lt;br /&gt;
&lt;br /&gt;
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15&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
1991}}.&lt;br /&gt;
&lt;br /&gt;
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[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree (graph theory)]]. Its edit history can be viewed at [[度/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12754</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12754"/>
		<updated>2020-08-16T14:22:51Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图 Graphs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络 Networks&amp;lt;/font&amp;gt;'''的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P''(''k'')定义为网络中度值为''k''的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;累积度分布函数 Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（随机选择一个节点，其度值小于k的概率）的形式表示，或是用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;互补累积度分布函数 Complementary Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机图 Random Graph&amp;lt;/font&amp;gt;'''，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项分布 Binomial Distribution&amp;lt;/font&amp;gt;'''的度值为''k'':&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(如果平均度\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt;保持不变，也会出现有限节点的泊松分布)。然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
γ是一个常数。这种网络被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks&amp;lt;/font&amp;gt;'''，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究表明，尽管大多数观测到的网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;肥尾度分布 Fat-Tailed Degree Distributions&amp;lt;/font&amp;gt;'''，但它们无标度分布的特点并不明显。&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）同样 标题为专业名词的  可以把正文中对应的专业名词进行标注标橙&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;超额度分布 Excess Degree Distribution&amp;lt;/font&amp;gt;'''是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的其传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;,通过选择一个节点(随机或非随机)跟踪它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; 个邻居的概率不是由&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;.给出的原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt;的真实概率是&amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;，它被称为该节点的超额度。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;配置模型 Configuration Model&amp;lt;/font&amp;gt;'''中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;友谊悖论 Friendship Paradox&amp;lt;/font&amp;gt;'''。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
生成函数方法&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|url=https://link.aps.org/doi/10.1103/PhysRevE.64.026118|journal=Physical Review E|language=en|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|issn=1063-651X|doi-access=free}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; 的值也可得出通过 &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;的导数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻性。然后，可以很容易地进行计算依据&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和它的导数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的。进一步扩展，&amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th的邻居的分布是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, 以&amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 迭代到 &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 函数本身。第一个邻居的平均数量&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; 第二个邻居的平均数量是:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
有向网络的度分布&lt;br /&gt;
&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; 和一些出度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt;分别指代的是指向该节点的边的数量和从该节点指出的边的数量。如果 &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; 是一个随机选择的具有入出度的节点的可能性。那么分配给这个生成函数的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;联合概率分布  Joint Probability Distribution&amp;lt;/font&amp;gt;'''可以被写成两个变量 &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt;是网络中节点的平均度(内部和外部)&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
使用函数&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。可以将&amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt;表示离开节点的边的数量:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;，或者像之前表示的&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;从一个随机选择的节点上可达到的第二邻居的平均数是&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. 这些是从随机选择的节点所达到第一和第二邻居的数量，因为这些方程显然是在&amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt;和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;上对称的。&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;复杂网络 Complex Network&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-free Network&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机网络 Random Graph&amp;lt;/font&amp;gt;'''&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;结构截止值 Structural Cut-off&amp;lt;/font&amp;gt;'''&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）该部分也为专业名词 需要进行标橙&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
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* {{cite journal&lt;br /&gt;
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| title = 复杂网络的结构和功能&lt;br /&gt;
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* {{cite book &lt;br /&gt;
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| 作者链接 = Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
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2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
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  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
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Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
剑桥大学出版社&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph invariants]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph invariants&lt;br /&gt;
&lt;br /&gt;
类别: 图形不变量&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12720</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12720"/>
		<updated>2020-08-16T11:33:46Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图 Graphs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络 Networks&amp;lt;/font&amp;gt;'''的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
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&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P''(''k'')定义为网络中度值为''k''的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;累积度分布函数 Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（随机选择一个节点，其度值小于k的概率）的形式表示，或是用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;互补累积度分布函数 Complementary Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机图 Random Graph&amp;lt;/font&amp;gt;'''，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项分布 Binomial Distribution&amp;lt;/font&amp;gt;'''的度值为''k'':&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(如果平均度\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt;保持不变，也会出现有限节点的泊松分布)。然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
 where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
γ是一个常数。这种网络被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks&amp;lt;/font&amp;gt;'''，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究表明，尽管大多数观测到的网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;肥尾度分布 Fat-Tailed Degree Distributions&amp;lt;/font&amp;gt;'''，但它们无标度分布的特点并不明显。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
超额度分布是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的其传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;,通过选择一个节点(随机或非随机)跟踪它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; 个邻居的概率不是由&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;.给出的原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt;的真实概率是&amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;，它被称为该节点的超额度。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;配置模型 Configuration Model&amp;lt;/font&amp;gt;'''中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;友谊悖论 Friendship Paradox&amp;lt;/font&amp;gt;'''。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
生成函数方法&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|url=https://link.aps.org/doi/10.1103/PhysRevE.64.026118|journal=Physical Review E|language=en|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|issn=1063-651X|doi-access=free}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; 的值也可得出通过 &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;的导数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻性。然后，可以很容易地进行计算依据&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和它的导数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的。进一步扩展，&amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th的邻居的分布是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, 以&amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 迭代到 &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 函数本身。第一个邻居的平均数量&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; 第二个邻居的平均数量是:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
有向网络的度分布&lt;br /&gt;
&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; 和一些出度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt;分别指代的是指向该节点的边的数量和从该节点指出的边的数量。如果 &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; 是一个随机选择的具有入出度的节点的可能性。那么分配给这个生成函数的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;联合概率分布  Joint Probability Distribution&amp;lt;/font&amp;gt;'''可以被写成两个变量 &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt;是网络中节点的平均度(内部和外部)&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
使用函数&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。可以将&amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt;表示离开节点的边的数量:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;，或者像之前表示的&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;从一个随机选择的节点上可达到的第二邻居的平均数是&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. 这些是从随机选择的节点所达到第一和第二邻居的数量，因为这些方程显然是在&amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt;和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;上对称的。&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
图理论&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
无标度网络&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
随机网络&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
结构截止值&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
&lt;br /&gt;
  | last = Albert&lt;br /&gt;
&lt;br /&gt;
  | last = Albert&lt;br /&gt;
&lt;br /&gt;
| last = Albert&lt;br /&gt;
&lt;br /&gt;
  | first = R.&lt;br /&gt;
&lt;br /&gt;
  | first = R.&lt;br /&gt;
&lt;br /&gt;
| 第一 = r。&lt;br /&gt;
&lt;br /&gt;
  |author2=Barabasi, A.-L.&lt;br /&gt;
&lt;br /&gt;
  |author2=Barabasi, A.-L.&lt;br /&gt;
&lt;br /&gt;
2 = Barabasi，A.-L.&lt;br /&gt;
&lt;br /&gt;
  | title = Statistical mechanics of complex networks&lt;br /&gt;
&lt;br /&gt;
  | title = Statistical mechanics of complex networks&lt;br /&gt;
&lt;br /&gt;
| title = 复杂网络的统计力学&lt;br /&gt;
&lt;br /&gt;
  | journal = Reviews of Modern Physics&lt;br /&gt;
&lt;br /&gt;
  | journal = Reviews of Modern Physics&lt;br /&gt;
&lt;br /&gt;
| 杂志 = 现代物理学评论&lt;br /&gt;
&lt;br /&gt;
  | volume = 74&lt;br /&gt;
&lt;br /&gt;
  | volume = 74&lt;br /&gt;
&lt;br /&gt;
74&lt;br /&gt;
&lt;br /&gt;
  | pages = 47–97&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
| 页数 = 47-97&lt;br /&gt;
&lt;br /&gt;
  | year = 2002&lt;br /&gt;
&lt;br /&gt;
  | year = 2002&lt;br /&gt;
&lt;br /&gt;
2002年&lt;br /&gt;
&lt;br /&gt;
  | issue = 1&lt;br /&gt;
&lt;br /&gt;
  | issue = 1&lt;br /&gt;
&lt;br /&gt;
1&lt;br /&gt;
&lt;br /&gt;
 | arxiv = cond-mat/0106096&lt;br /&gt;
&lt;br /&gt;
 | arxiv = cond-mat/0106096&lt;br /&gt;
&lt;br /&gt;
 | arxiv = cond-mat/0106096&lt;br /&gt;
&lt;br /&gt;
  | doi = 10.1103/RevModPhys.74.47&lt;br /&gt;
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&lt;br /&gt;
10.1103/RevModPhys. 74.47&lt;br /&gt;
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| bibcode=2002RvMP...74...47A&lt;br /&gt;
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| bibcode=2002RvMP...74...47A&lt;br /&gt;
&lt;br /&gt;
2002RvMP... 74... 47A&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
&lt;br /&gt;
  | last = Dorogovtsev&lt;br /&gt;
&lt;br /&gt;
  | last = Dorogovtsev&lt;br /&gt;
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2 = Mendes，J.f.&lt;br /&gt;
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  | title = Evolution of networks&lt;br /&gt;
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  | title = Evolution of networks&lt;br /&gt;
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| title = 网络的进化&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
| 杂志 = 物理进展&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
51&lt;br /&gt;
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| pages = 1079-1187&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
2002年&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
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&lt;br /&gt;
10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
第四期&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
2002/adphy. . 51.1079 d }&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
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&lt;br /&gt;
 |last=Newman &lt;br /&gt;
&lt;br /&gt;
纽曼&lt;br /&gt;
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&lt;br /&gt;
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| first = m.E.j.&lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
| title = 复杂网络的结构和功能&lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
| journal = SIAM Review&lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
45&lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
| 页 = 167-256&lt;br /&gt;
&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
2003年&lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
| doi = 10.1137/S003614450342480&lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
2&lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516&lt;br /&gt;
&lt;br /&gt;
 |bibcode=2003SIAMR..45..167N &lt;br /&gt;
&lt;br /&gt;
 |bibcode=2003SIAMR..45..167N &lt;br /&gt;
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| bibcode = 2003siaml. . 45. . 167 n&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
* {{cite book &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
复杂网络: 结构、健壮性和功能&lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
作者: Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
| 作者链接 = Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
   &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
2010年&lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
剑桥大学出版社&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph invariants]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph invariants&lt;br /&gt;
&lt;br /&gt;
类别: 图形不变量&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12719</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12719"/>
		<updated>2020-08-16T11:32:09Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图 Graphs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络 Networks&amp;lt;/font&amp;gt;'''的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P''(''k'')定义为网络中度值为''k''的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;累积度分布函数 Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（随机选择一个节点，其度值小于k的概率）的形式表示，或是用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;互补累积度分布函数 Complementary Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机图 Random Graph&amp;lt;/font&amp;gt;'''，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项分布 Binomial Distribution&amp;lt;/font&amp;gt;'''的度值为''k'':&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(如果平均度\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt;保持不变，也会出现有限节点的泊松分布)。然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
 where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
γ是一个常数。这种网络被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks&amp;lt;/font&amp;gt;'''，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究表明，尽管大多数观测到的网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;肥尾度分布 Fat-Tailed Degree Distributions&amp;lt;/font&amp;gt;'''，但它们无标度分布的特点并不明显。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
超额度分布是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的其传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;,通过选择一个节点(随机或非随机)跟踪它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; 个邻居的概率不是由&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;.给出的原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt;的真实概率是&amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;，它被称为该节点的超额度。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;配置模型 Configuration Model&amp;lt;/font&amp;gt;'''中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;友谊悖论 Friendship Paradox&amp;lt;/font&amp;gt;'''。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
生成函数方法&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|url=https://link.aps.org/doi/10.1103/PhysRevE.64.026118|journal=Physical Review E|language=en|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|issn=1063-651X|doi-access=free}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; 的值也可得出通过 &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;的导数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻性。然后，可以很容易地进行计算依据&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和它的导数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的。进一步扩展，&amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th的邻居的分布是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, 以&amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 迭代到 &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 函数本身。第一个邻居的平均数量&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; 第二个邻居的平均数量是:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
有向网络的度分布&lt;br /&gt;
&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; 和一些出度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt;分别指代的是指向该节点的边的数量和从该节点指出的边的数量。如果 &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; 是一个随机选择的具有入出度的节点的可能性。那么分配给这个生成函数的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;联合概率分布  Joint Probability Distribution&amp;lt;/font&amp;gt;'''可以被写成两个变量 &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt;是网络中节点的平均度(内部和外部)&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
使用函数&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。可以将&amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt;表示离开节点的边的数量:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;，或者像之前表示的&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;从一个随机选择的节点上可达到的第二邻居的平均数是&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. 这些是从随机选择的节点所达到第一和第二邻居的数量，因为这些方程显然是在&amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt;和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;上对称的。&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
图理论&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
无标度网络&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
随机网络&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
结构截止值&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
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&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
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| 作者链接 = Shlomo Havlin&lt;br /&gt;
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   &lt;br /&gt;
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  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
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  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
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2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
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  | year= 2010 &lt;br /&gt;
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  | year= 2010 &lt;br /&gt;
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2010年&lt;br /&gt;
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  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
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  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
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Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
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  | publisher= Cambridge University Press&lt;br /&gt;
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  | publisher= Cambridge University Press&lt;br /&gt;
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剑桥大学出版社&lt;br /&gt;
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}}&lt;br /&gt;
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}}&lt;br /&gt;
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}}&lt;br /&gt;
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[[Category:Graph theory]]&lt;br /&gt;
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Category:Graph theory&lt;br /&gt;
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范畴: 图论&lt;br /&gt;
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[[Category:Graph invariants]]&lt;br /&gt;
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Category:Graph invariants&lt;br /&gt;
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类别: 图形不变量&lt;br /&gt;
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[[Category:Network theory]]&lt;br /&gt;
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Category:Network theory&lt;br /&gt;
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范畴: 网络理论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12718</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12718"/>
		<updated>2020-08-16T11:25:39Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图 Graphs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络 Networks&amp;lt;/font&amp;gt;'''的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P''(''k'')定义为网络中度值为''k''的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;累积度分布函数 Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（随机选择一个节点，其度值小于k的概率）的形式表示，或是用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;互补累积度分布函数 Complementary Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
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== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机图 Random Graph&amp;lt;/font&amp;gt;'''，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项分布 Binomial Distribution&amp;lt;/font&amp;gt;'''的度值为''k'':&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &lt;br /&gt;
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&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(如果平均度\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt;保持不变，也会出现有限节点的泊松分布)。然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
 where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
γ是一个常数。这种网络被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks&amp;lt;/font&amp;gt;'''，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究表明，尽管大多数观测到的网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;肥尾度分布 Fat-Tailed Degree Distributions&amp;lt;/font&amp;gt;'''，但它们无标度分布的特点并不明显。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
超额度分布是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的其传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;,通过选择一个节点(随机或非随机)跟踪它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; 个邻居的概率不是由&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;.给出的原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt;的真实概率是&amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;，它被称为该节点的超额度。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;配置模型 Configuration Model&amp;lt;/font&amp;gt;'''中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;友谊悖论 Friendship Paradox&amp;lt;/font&amp;gt;'''。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
生成函数方法&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|url=https://link.aps.org/doi/10.1103/PhysRevE.64.026118|journal=Physical Review E|language=en|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|issn=1063-651X|doi-access=free}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; 的值也可得出通过 &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;的导数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻性。然后，可以很容易地进行计算依据&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和它的导数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的。进一步扩展，&amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th的邻居的分布是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, 以&amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 迭代到 &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 函数本身。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
第一个邻居的平均数量&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; 第二个邻居的平均数量是:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
有向网络的度分布&lt;br /&gt;
&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{&lt;br /&gt;
in}&lt;br /&gt;
&amp;lt;/math&amp;gt; 和一些出度&amp;lt;math&amp;gt;&lt;br /&gt;
k_{out}&lt;br /&gt;
&amp;lt;/math&amp;gt;分别指代的是指向该节点的边的数量和从该节点指出的边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
如果 &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; 是一个随机选择的具有入出度的节点的可能性。那么分配给这个生成函数的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;联合概率分布  Joint Probability Distribution&amp;lt;/font&amp;gt;'''可以被写成两个变量 &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c &lt;br /&gt;
&amp;lt;/math&amp;gt;是网络中节点的平均度(内部和外部)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
使用函数&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。可以将&amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt;表示离开节点的边的数量:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;，或者像之前表示的&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
从一个随机选择的节点上可达到的第二邻居的平均数是&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. :&lt;br /&gt;
&lt;br /&gt;
这些是从随机选择的节点所达到第一和第二邻居的数量，因为这些方程显然是在&amp;lt;math&amp;gt;&lt;br /&gt;
x&lt;br /&gt;
&amp;lt;/math&amp;gt;和 &amp;lt;math&amp;gt;&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;上对称的。&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
图理论&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
无标度网络&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
随机网络&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
结构截止值&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
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| 杂志 = 现代物理学评论&lt;br /&gt;
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2002RvMP... 74... 47A&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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* {{cite journal&lt;br /&gt;
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&lt;br /&gt;
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2 = Mendes，J.f.&lt;br /&gt;
&lt;br /&gt;
  | title = Evolution of networks&lt;br /&gt;
&lt;br /&gt;
  | title = Evolution of networks&lt;br /&gt;
&lt;br /&gt;
| title = 网络的进化&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
| 杂志 = 物理进展&lt;br /&gt;
&lt;br /&gt;
  | volume = 51&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
51&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
2002年&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
第四期&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
2002/adphy. . 51.1079 d }&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
 |last=Newman &lt;br /&gt;
&lt;br /&gt;
纽曼&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
 |first=M. E. J. &lt;br /&gt;
&lt;br /&gt;
| first = m.E.j.&lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
| title = 复杂网络的结构和功能&lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
| journal = SIAM Review&lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
45&lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
| 页 = 167-256&lt;br /&gt;
&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
2003年&lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
| doi = 10.1137/S003614450342480&lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
2&lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516&lt;br /&gt;
&lt;br /&gt;
 |bibcode=2003SIAMR..45..167N &lt;br /&gt;
&lt;br /&gt;
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| bibcode = 2003siaml. . 45. . 167 n&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
* {{cite book &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
复杂网络: 结构、健壮性和功能&lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
作者: Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
| 作者链接 = Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
   &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
2010年&lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
剑桥大学出版社&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph invariants]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph invariants&lt;br /&gt;
&lt;br /&gt;
类别: 图形不变量&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12713</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12713"/>
		<updated>2020-08-16T11:03:28Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图 Graphs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络 Networks&amp;lt;/font&amp;gt;'''的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P''(''k'')定义为网络中度值为''k''的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;累积度分布函数 Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（随机选择一个节点，其度值小于k的概率）的形式表示，或是用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;互补累积度分布函数 Complementary Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机图 Random Graph&amp;lt;/font&amp;gt;'''，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项分布 Binomial Distribution&amp;lt;/font&amp;gt;'''的度值为''k'':&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(如果平均度&amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt;保持不变，也会出现有限节点的泊松分布)。然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
γ是一个常数。这种网络被称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;无标度网络 Scale-Free Networks&amp;lt;/font&amp;gt;'''，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究表明，尽管大多数观测到的网络具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;肥尾度分布 Fat-Tailed Degree Distributions&amp;lt;/font&amp;gt;'''，但它们无标度分布的特点并不明显。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
超额度分布是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的其传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
通过选择一个节点(随机或非随机)跟踪它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt; 个邻居的概率不是由&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;.给出的&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree&lt;br /&gt;
&lt;br /&gt;
The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt;math&amp;gt;&lt;br /&gt;
k&lt;br /&gt;
&amp;lt;/math&amp;gt;的真实概率是&amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
which is called the excess degree of that node. In the configuration model, which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as:&lt;br /&gt;
&lt;br /&gt;
它被称为该节点的超额度。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;配置模型 Configuration Model&amp;lt;/font&amp;gt;'''中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;友谊悖论 Friendship Paradox&amp;lt;/font&amp;gt;'''。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
生成函数方法&lt;br /&gt;
&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network,&lt;br /&gt;
&lt;br /&gt;
Generating functions can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt;math&amp;gt;&lt;br /&gt;
P(k)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
q(k)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
 respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
 respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt; 也可得出通过 &amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;的导数:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻，可以很容易地计算依据 &amp;lt; math &amp;gt; G0(x)&amp;lt;/math&amp;gt;和它的导数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
And in general:&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For Poisson-distributed random networks, such as the ER graph, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &lt;br /&gt;
&lt;br /&gt;
that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数&amp;lt;math&amp;gt;&lt;br /&gt;
G_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt; 和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
继续扩展，&amp;lt;math&amp;gt;&lt;br /&gt;
m&lt;br /&gt;
&amp;lt;/math&amp;gt;-th的邻居的分布&lt;br /&gt;
&lt;br /&gt;
m&lt;br /&gt;
&lt;br /&gt;
m&lt;br /&gt;
&lt;br /&gt;
M&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&amp;lt;/math&amp;gt;, 以&amp;lt;math&amp;gt;&lt;br /&gt;
m-1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 迭代到 &amp;lt;math&amp;gt;&lt;br /&gt;
G_1 &lt;br /&gt;
&amp;lt;/math&amp;gt; 函数本身。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
第一个邻居的平均数量&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
 and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
 and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
第二个邻居的平均数量是:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
有向网络的度分布&lt;br /&gt;
&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度k{in}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
和一些出度k{out}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
分别指代的是指向该节点的边的数量和从该节点指出的边的数量。&lt;br /&gt;
&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&lt;br /&gt;
P (k _ { in } ，k _ { out }) &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果 &amp;lt;math&amp;gt;&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&amp;lt;/math&amp;gt; 是一个随机选择的具有入出度的节点的可能性&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this joint probability distribution can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
那么分配给这个生成函数的联合概率分布可以被写成两个变量 &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;c &lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;c &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
C是网络中节点的平均度(内部和外部)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
使用函数&amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&lt;br /&gt;
0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
可以将&amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&amp;lt;/math&amp;gt;定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&lt;br /&gt;
1(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&amp;lt;/math&amp;gt;可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&amp;lt;/math&amp;gt; 和 &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&lt;br /&gt;
&lt;br /&gt;
离开这样一个节点的数字:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量math&amp;gt;&lt;br /&gt;
c&lt;br /&gt;
&amp;lt;/math&amp;gt;，&amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
c&lt;br /&gt;
&lt;br /&gt;
c&lt;br /&gt;
&lt;br /&gt;
C&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
或者像之前表示的&amp;lt;math&amp;gt;&lt;br /&gt;
c_1&lt;br /&gt;
&amp;lt;/math&amp;gt;是&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
从一个随机选择的节点上可达到的第二邻居的平均数是&amp;lt;math&amp;gt;&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&amp;lt;/math&amp;gt;. :&lt;br /&gt;
&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这些是从随机选择的节点达到第一和第二邻居的数量，因为这些方程显然是在x和y上对称的 &lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
X&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
Y&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
数学。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
图理论&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
无标度网络&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
随机网络&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
结构截止值&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
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&lt;br /&gt;
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* {{cite book &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
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  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
复杂网络: 结构、健壮性和功能&lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
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作者: Shlomo Havlin&lt;br /&gt;
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  | author-link= Shlomo Havlin &lt;br /&gt;
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  | author-link= Shlomo Havlin &lt;br /&gt;
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| 作者链接 = Shlomo Havlin&lt;br /&gt;
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   &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
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  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
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2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
2010年&lt;br /&gt;
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  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
剑桥大学出版社&lt;br /&gt;
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}}&lt;br /&gt;
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}}&lt;br /&gt;
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}}&lt;br /&gt;
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[[Category:Graph theory]]&lt;br /&gt;
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Category:Graph theory&lt;br /&gt;
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范畴: 图论&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph invariants]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph invariants&lt;br /&gt;
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类别: 图形不变量&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12684</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12684"/>
		<updated>2020-08-16T09:29:10Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图 Graphs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络 Networks&amp;lt;/font&amp;gt;'''的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P''(''k'')定义为网络中度值为''k''的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;累积度分布函数 Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（随机选择一个节点，其度值小于k的概率）的形式表示，或是用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;互补累积度分布函数 Complementary Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机图 Random Graph&amp;lt;/font&amp;gt;'''，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项分布 Binomial Distribution&amp;lt;/font&amp;gt;'''的度值为''k'':&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
(如果平均度 &amp;lt; math &amp;gt; langle k rangle = p (n-1) &amp;lt;/math &amp;gt; 保持不变，也会出现在有限节点的泊松分布)。&lt;br /&gt;
然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布: &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;,γ是一个常数。这种网络被称为无标度网络，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究声称，尽管大多数观测到的网络具有肥尾度分布，但它们偏离了无标度分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
超额度分布是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;P(k)&amp;lt;/math &amp;gt;，&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
通过选择一个节点(随机或非随机)并转到它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt; math &amp;gt; k&amp;lt;/math&amp;gt;个邻居的概率&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
K&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
不是由 &amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;给出的&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt; math &amp;gt;k&amp;lt;/math&amp;gt;的真实概率&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
K&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
是&amp;lt;math&amp;gt;q(k)&amp;lt;/math &amp;gt;&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
Q (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the excess degree of that node. In the configuration model, which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as:&lt;br /&gt;
&lt;br /&gt;
它被称为该节点的超额度。在配置模型中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&lt;br /&gt;
Q (k) = frac { k + 1}{ langle k rangle } p (k + 1) ,&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的友谊悖论。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&lt;br /&gt;
1 right tarrow { langle k ^ 2 rangle }-2{ langle k rangle } &amp;gt; 0&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
生成函数方法&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Generating functions can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt; math &amp;gt;P(k)&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;q(k)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
Q (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&lt;br /&gt;
1(x) = textstyle sum { k } displaystyle q (k) x ^ k = textstyle sum { k } displastyle frac { k }{ langle k rangle } p (k) x ^ { k-1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
G_1(x)&lt;br /&gt;
&lt;br /&gt;
G_1(x)&lt;br /&gt;
&lt;br /&gt;
G _ 1(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G_0(x)&amp;lt;/math &amp;gt; 也可以从 &amp;lt; math &amp;gt; 的导数得到&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
G_0(x)&lt;br /&gt;
&lt;br /&gt;
G _ 0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
(/math)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&lt;br /&gt;
1(x) = frac { g’ _ 0(x)}{ g’ _ 0(1)}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&lt;br /&gt;
= frac {1}{ k! }{ operatorname { d } ^ k! g over operatorname { d } ! x ^ k } bigvert { x = 0}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻，可以很容易地计算依据 &amp;lt; math &amp;gt; G0(x)&amp;lt;/math&amp;gt;和它的导数:&lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G _ 0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&lt;br /&gt;
{ langle k rangle } = g’ _ 0(1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&lt;br /&gt;
{\langle k^2 \rangle} = G_0(1) + G'_0(1) &lt;br /&gt;
&lt;br /&gt;
{ langle k ^ 2 rangle } = g _ 0(1) + g’ _ 0(1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
And in general:&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&lt;br /&gt;
{ langle k ^ m rangle } = Biggl [{ bigg (operatorname { x }{ operatorname { d } ! over operatorname { dx } ！[} Biggl) ^ m } g _ 0(x) Biggl ] _ { x = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For Poisson-distributed random networks, such as the ER graph, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&amp;lt; math &amp;gt; G1(x) = G0(x)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&lt;br /&gt;
G1(x) = g0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数 &amp;lt; math &amp;gt; G0(x)&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的&lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G _ 0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&lt;br /&gt;
G _ 0(g _ 1(x))&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
继续扩展，&amp;lt; math &amp;gt; m-th&amp;lt;/math&amp;gt;的邻居的分布&lt;br /&gt;
&lt;br /&gt;
m&lt;br /&gt;
&lt;br /&gt;
m&lt;br /&gt;
&lt;br /&gt;
M&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
这个邻居是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&lt;br /&gt;
G _ 0 bigl (g _ 1(... g _ 1(x) ...) bigr)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ] ，with &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
m-1 &lt;br /&gt;
&lt;br /&gt;
m-1 &lt;br /&gt;
&lt;br /&gt;
M-1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
函数的迭代&lt;br /&gt;
&lt;br /&gt;
G_1 &lt;br /&gt;
&lt;br /&gt;
G_1 &lt;br /&gt;
&lt;br /&gt;
1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|url=https://link.aps.org/doi/10.1103/PhysRevE.64.026118|journal=Physical Review E|language=en|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|issn=1063-651X|doi-access=free}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&lt;br /&gt;
&lt;br /&gt;
自我作用。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
第一个邻居的平均数量c1是&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
C _ 1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ] ，is &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&lt;br /&gt;
{ langle k rangle } = { dG _ 0(x) over dx } | { x = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
第二个邻居的平均数量是:&lt;br /&gt;
&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G_0(1)&lt;br /&gt;
&lt;br /&gt;
C2 = biggl [{ d over dx } g0 big (g1(x) big)] _ { x = 1} = g1’(1) g’0 big (g1(1) big) = g1’(1) g’0(1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
有向网络的度分布&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度k{in}&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
这是一个很好的例子&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
和一些出度k{out}&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
分别指代的是指向该节点的边的数量和从该节点指出的边的数量。&lt;br /&gt;
&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&lt;br /&gt;
P (k _ { in } ，k _ { out }) &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果 &amp;lt; math &amp;gt; P(k_{in}, k_{out})&amp;lt;/math&amp;gt;是一个随机选择的具有入出度的节点的可能性&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
这是一个很好的例子&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
开始吧&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“数学”和“学位”&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
我不知道你在说什么&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this joint probability distribution can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
那么分配给这个生成函数的联合概率分布可以被写成两个变量 &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
X&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
Y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
&lt;br /&gt;
{ partial mathcal { g } over partial x } vert _ { x，y = 1} = { partial mathcal { g } over partial y } vert _ { x，y = 1} = c,&lt;br /&gt;
&lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
c &lt;br /&gt;
&lt;br /&gt;
c &lt;br /&gt;
&lt;br /&gt;
C&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
C是网络中节点的平均度(内部和外部)&lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
使用函数 &amp;lt; math &amp;gt; \mathcal{G}(x,y)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math &amp;gt; ，如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&lt;br /&gt;
0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
可以将 &amp;lt;/math &amp;gt; 定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&lt;br /&gt;
1(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math &amp;gt; 可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&lt;br /&gt;
0(y)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&lt;br /&gt;
1(y)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&lt;br /&gt;
&lt;br /&gt;
离开这样一个节点的数字:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&lt;br /&gt;
0(x) = mathcal { g }(x，1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&lt;br /&gt;
1(x) = frac {1}{ c }{ partial mathcal { g } over partial x } vert _ { y = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&lt;br /&gt;
0(y) = mathcal { g }(1，y)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&lt;br /&gt;
1(y) = frac {1}{ c }{ partial mathcal { g } over partial y } vert _ { x = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量C，&amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
c&lt;br /&gt;
&lt;br /&gt;
c&lt;br /&gt;
&lt;br /&gt;
C&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
或者像之前表示的C1&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
C _ 1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ] ，is &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
{ partial mathcal { g } over partial x } biggl vert _ { x，y = 1} = { partial mathcal { g } over partial y } biggl vert _ { x，y = 1}&lt;br /&gt;
&lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
从一个随机选择的节点上可达到的第二邻居的平均数是:&lt;br /&gt;
&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这些也是从这些随机节点可以达到第一和第二邻居的数量，因为这些方程显然是在x和y上对称的 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
X&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
Y&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
数学。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
图理论&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
无标度网络&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
随机网络&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
结构截止值&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
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| 杂志 = 现代物理学评论&lt;br /&gt;
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2002RvMP... 74... 47A&lt;br /&gt;
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* {{cite journal&lt;br /&gt;
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&lt;br /&gt;
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2 = Mendes，J.f.&lt;br /&gt;
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  | title = Evolution of networks&lt;br /&gt;
&lt;br /&gt;
  | title = Evolution of networks&lt;br /&gt;
&lt;br /&gt;
| title = 网络的进化&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
| 杂志 = 物理进展&lt;br /&gt;
&lt;br /&gt;
  | volume = 51&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
51&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
| pages = 1079-1187&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
2002年&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | arxiv = cond-mat/0106144&lt;br /&gt;
&lt;br /&gt;
  | doi = 10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
  | doi = 10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
第四期&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
2002/adphy. . 51.1079 d }&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
 |last=Newman &lt;br /&gt;
&lt;br /&gt;
纽曼&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
| first = m.E.j.&lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
| title = 复杂网络的结构和功能&lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
| journal = SIAM Review&lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
45&lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
| 页 = 167-256&lt;br /&gt;
&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
2003年&lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
| doi = 10.1137/S003614450342480&lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
2&lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516&lt;br /&gt;
&lt;br /&gt;
 |bibcode=2003SIAMR..45..167N &lt;br /&gt;
&lt;br /&gt;
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| bibcode = 2003siaml. . 45. . 167 n&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
* {{cite book &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
复杂网络: 结构、健壮性和功能&lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
作者: Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
| 作者链接 = Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
   &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
2010年&lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
剑桥大学出版社&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph invariants]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph invariants&lt;br /&gt;
&lt;br /&gt;
类别: 图形不变量&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12683</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12683"/>
		<updated>2020-08-16T09:27:47Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图 Graphs&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络 Networks&amp;lt;/font&amp;gt;'''的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P''(''k'')定义为网络中度值为''k''的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;累积度分布函数 Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（随机选择一个节点，其度值小于k的概率）的形式表示，或是用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;互补累积度分布函数 Complementary Cumulative Degree Distribution&amp;lt;/font&amp;gt;'''（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;随机图 Random Graph&amp;lt;/font&amp;gt;'''，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;二项分布 Binomial Distribution&amp;lt;/font&amp;gt;'''的度值为''k'':&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
(如果平均度 &amp;lt; math &amp;gt; langle k rangle = p (n-1) &amp;lt;/math &amp;gt; 保持不变，也会出现在有限节点的泊松分布)。&lt;br /&gt;
然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布: &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;,γ是一个常数。这种网络被称为无标度网络，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究声称，尽管大多数观测到的网络具有肥尾度分布，但它们偏离了无标度分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
超额度分布是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;P(k)&amp;lt;/math &amp;gt;，&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
通过选择一个节点(随机或非随机)并转到它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt; math &amp;gt; k&amp;lt;/math&amp;gt;个邻居的概率&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
K&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
不是由 &amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;给出的&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt; math &amp;gt;k&amp;lt;/math&amp;gt;的真实概率&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
K&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
是&amp;lt;math&amp;gt;q(k)&amp;lt;/math &amp;gt;&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
Q (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the excess degree of that node. In the configuration model, which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as:&lt;br /&gt;
&lt;br /&gt;
它被称为该节点的超额度。在配置模型中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&lt;br /&gt;
Q (k) = frac { k + 1}{ langle k rangle } p (k + 1) ,&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的友谊悖论。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&lt;br /&gt;
1 right tarrow { langle k ^ 2 rangle }-2{ langle k rangle } &amp;gt; 0&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
生成函数方法&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Generating functions can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt; math &amp;gt;P(k)&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;q(k)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
Q (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&lt;br /&gt;
1(x) = textstyle sum { k } displaystyle q (k) x ^ k = textstyle sum { k } displastyle frac { k }{ langle k rangle } p (k) x ^ { k-1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
G_1(x)&lt;br /&gt;
&lt;br /&gt;
G_1(x)&lt;br /&gt;
&lt;br /&gt;
G _ 1(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G_0(x)&amp;lt;/math &amp;gt; 也可以从 &amp;lt; math &amp;gt; 的导数得到&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
G_0(x)&lt;br /&gt;
&lt;br /&gt;
G _ 0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
(/math)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&lt;br /&gt;
1(x) = frac { g’ _ 0(x)}{ g’ _ 0(1)}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&lt;br /&gt;
= frac {1}{ k! }{ operatorname { d } ^ k! g over operatorname { d } ! x ^ k } bigvert { x = 0}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻，可以很容易地计算依据 &amp;lt; math &amp;gt; G0(x)&amp;lt;/math&amp;gt;和它的导数:&lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G _ 0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&lt;br /&gt;
{ langle k rangle } = g’ _ 0(1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&lt;br /&gt;
{\langle k^2 \rangle} = G_0(1) + G'_0(1) &lt;br /&gt;
&lt;br /&gt;
{ langle k ^ 2 rangle } = g _ 0(1) + g’ _ 0(1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
And in general:&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&lt;br /&gt;
{ langle k ^ m rangle } = Biggl [{ bigg (operatorname { x }{ operatorname { d } ! over operatorname { dx } ！[} Biggl) ^ m } g _ 0(x) Biggl ] _ { x = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For Poisson-distributed random networks, such as the ER graph, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&amp;lt; math &amp;gt; G1(x) = G0(x)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&lt;br /&gt;
G1(x) = g0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数 &amp;lt; math &amp;gt; G0(x)&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的&lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G _ 0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&lt;br /&gt;
G _ 0(g _ 1(x))&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
继续扩展，&amp;lt; math &amp;gt; m-th&amp;lt;/math&amp;gt;的邻居的分布&lt;br /&gt;
&lt;br /&gt;
m&lt;br /&gt;
&lt;br /&gt;
m&lt;br /&gt;
&lt;br /&gt;
M&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
这个邻居是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&lt;br /&gt;
G _ 0 bigl (g _ 1(... g _ 1(x) ...) bigr)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ] ，with &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
m-1 &lt;br /&gt;
&lt;br /&gt;
m-1 &lt;br /&gt;
&lt;br /&gt;
M-1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
函数的迭代&lt;br /&gt;
&lt;br /&gt;
G_1 &lt;br /&gt;
&lt;br /&gt;
G_1 &lt;br /&gt;
&lt;br /&gt;
1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|url=https://link.aps.org/doi/10.1103/PhysRevE.64.026118|journal=Physical Review E|language=en|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|issn=1063-651X|doi-access=free}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&lt;br /&gt;
&lt;br /&gt;
自我作用。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
第一个邻居的平均数量c1是&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
C _ 1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ] ，is &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&lt;br /&gt;
{ langle k rangle } = { dG _ 0(x) over dx } | { x = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
第二个邻居的平均数量是:&lt;br /&gt;
&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G_0(1)&lt;br /&gt;
&lt;br /&gt;
C2 = biggl [{ d over dx } g0 big (g1(x) big)] _ { x = 1} = g1’(1) g’0 big (g1(1) big) = g1’(1) g’0(1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
有向网络的度分布&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度k{in}&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
这是一个很好的例子&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
和一些出度k{out}&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
分别指代的是指向该节点的边的数量和从该节点指出的边的数量。&lt;br /&gt;
&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&lt;br /&gt;
P (k _ { in } ，k _ { out }) &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果 &amp;lt; math &amp;gt; P(k_{in}, k_{out})&amp;lt;/math&amp;gt;是一个随机选择的具有入出度的节点的可能性&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
这是一个很好的例子&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
开始吧&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“数学”和“学位”&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
我不知道你在说什么&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this joint probability distribution can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
那么分配给这个生成函数的联合概率分布可以被写成两个变量 &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
X&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
Y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
&lt;br /&gt;
{ partial mathcal { g } over partial x } vert _ { x，y = 1} = { partial mathcal { g } over partial y } vert _ { x，y = 1} = c,&lt;br /&gt;
&lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
c &lt;br /&gt;
&lt;br /&gt;
c &lt;br /&gt;
&lt;br /&gt;
C&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
C是网络中节点的平均度(内部和外部)&lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
使用函数 &amp;lt; math &amp;gt; \mathcal{G}(x,y)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math &amp;gt; ，如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&lt;br /&gt;
0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
可以将 &amp;lt;/math &amp;gt; 定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&lt;br /&gt;
1(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math &amp;gt; 可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&lt;br /&gt;
0(y)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&lt;br /&gt;
1(y)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&lt;br /&gt;
&lt;br /&gt;
离开这样一个节点的数字:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&lt;br /&gt;
0(x) = mathcal { g }(x，1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&lt;br /&gt;
1(x) = frac {1}{ c }{ partial mathcal { g } over partial x } vert _ { y = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&lt;br /&gt;
0(y) = mathcal { g }(1，y)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&lt;br /&gt;
1(y) = frac {1}{ c }{ partial mathcal { g } over partial y } vert _ { x = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量C，&amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
c&lt;br /&gt;
&lt;br /&gt;
c&lt;br /&gt;
&lt;br /&gt;
C&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
或者像之前表示的C1&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
C _ 1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ] ，is &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
{ partial mathcal { g } over partial x } biggl vert _ { x，y = 1} = { partial mathcal { g } over partial y } biggl vert _ { x，y = 1}&lt;br /&gt;
&lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
从一个随机选择的节点上可达到的第二邻居的平均数是:&lt;br /&gt;
&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这些也是从这些随机节点可以达到第一和第二邻居的数量，因为这些方程显然是在x和y上对称的 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
X&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
Y&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
数学。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
图理论&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
无标度网络&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
随机网络&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
结构截止值&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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* {{cite journal&lt;br /&gt;
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&lt;br /&gt;
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| 杂志 = 物理进展&lt;br /&gt;
&lt;br /&gt;
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51&lt;br /&gt;
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2002年&lt;br /&gt;
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10.1080/00018730110112519&lt;br /&gt;
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|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
2002/adphy. . 51.1079 d }&lt;br /&gt;
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* {{cite journal&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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| title = 复杂网络的结构和功能&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
| journal = SIAM Review&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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 |issue=2 &lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
2&lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516&lt;br /&gt;
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| bibcode = 2003siaml. . 45. . 167 n&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
* {{cite book &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
复杂网络: 结构、健壮性和功能&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
作者: Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
| 作者链接 = Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
   &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
2010年&lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
剑桥大学出版社&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph invariants]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph invariants&lt;br /&gt;
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类别: 图形不变量&lt;br /&gt;
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[[Category:Network theory]]&lt;br /&gt;
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Category:Network theory&lt;br /&gt;
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范畴: 网络理论&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12680</id>
		<title>度</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12680"/>
		<updated>2020-08-16T09:17:47Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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[[File:UndirectedDegrees (Loop).svg|thumb|&lt;br /&gt;
图1：A graph with a loop having vertices labeled by degree 内含自环按度标记的图]]&lt;br /&gt;
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In [[graph theory]], the '''degree''' (or '''valency''') of a [[vertex (graph theory)|vertex]] of a [[Graph (discrete mathematics)|graph]] is the number of [[edge (graph theory)|edges]] that are [[incidence (graph)|incident]] to the vertex, and in a [[multigraph]], [[loop (graph theory)|loop]]s are counted twice.&amp;lt;ref&amp;gt;Diestel p.5&amp;lt;/ref&amp;gt; The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The '''maximum degree''' of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the '''minimum degree''' of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
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In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The maximum degree of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the minimum degree of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''中，图中顶点的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度 Degree&amp;lt;/font&amp;gt;'''(或价)是入射到该顶点的边的数量。在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;多重图 Multigraph&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;自环 Loops&amp;lt;/font&amp;gt;'''会被计算两次。顶点的度数可表示为&amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt;或&amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;。一个图&amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;的最大度值表示为&amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;，最小度值表示为 &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;。在右侧的多重图中，最大度值为5，最小度值为0。&lt;br /&gt;
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In a [[regular graph]], every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A [[complete graph]] (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
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In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
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在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;正则图 Regular Graph&amp;lt;/font&amp;gt;'''中，每个顶点都具有相同的度值，因此我们把该度值可以称之为图的度值。一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;完全图 Complete Graph&amp;lt;/font&amp;gt;'''(表示为&amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;，其中&amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt;是图中顶点的数目)是一种特殊的正则图，其中所有顶点都有最大度值，&amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;。&lt;br /&gt;
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==Handshaking lemma==&lt;br /&gt;
握手引理&lt;br /&gt;
{{main|Handshaking lemma}}&lt;br /&gt;
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The '''degree sum formula''' states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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The degree sum formula states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度和公式 Degree Sum Formula&amp;lt;/font&amp;gt;'''表明，给定一个图&amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
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The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree  sum formula) is known as the [[handshaking lemma]]. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
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The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
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该公式表明，在任何无向图中，拥有奇数度值的顶点的个数是偶数。这个陈述(以及度和公式)被称为握手引理。该名字来自一个有趣的数学问题，即证明无论该群体内有多少人，与奇数个人握过手的人数是偶数。&lt;br /&gt;
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==Degree sequence==&lt;br /&gt;
度序列&lt;br /&gt;
[[File:Conjugate-dessins.svg|thumb|200px|&lt;br /&gt;
图2：Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1).&lt;br /&gt;
两个具有相同度序列的非同构图]]&lt;br /&gt;
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The '''degree sequence''' of an undirected graph is the non-increasing sequence of its vertex degrees;&amp;lt;ref&amp;gt;Diestel p.216&amp;lt;/ref&amp;gt; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a [[graph invariant]] so [[Graph isomorphism|isomorphic graphs]] have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
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The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
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无向图的度序列是指将其各顶点度值按非递增方式排序，对于上述图是(5,3,3,2,2,1,0)。度序列是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图不变量 Graph Invariant&amp;lt;/font&amp;gt;'''，因此'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;同构图 Non-isomorphic Graphs&amp;lt;/font&amp;gt;'''（两个图中顶点的度值都相同，但形状不同）具有相同的度序列。然而，度序列通常不能唯一地标识一个图，在某些情况下，非同构图具有相同的度序列。&lt;br /&gt;
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The '''degree sequence problem'''  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a '''graphic''' or '''graphical sequence'''. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a [[matching (graph theory)|matching]], and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
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The degree sequence problem  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度序列问题 Degree Sequence Problem&amp;lt;/font&amp;gt;'''，是指寻找具有以非增方式排列给定正整数的度序列的局部或全部图的问题。(序列尾部的零可能会被忽略，因为通过向图中添加适当数量的孤立顶点就可以轻松实现序列尾部不断加零。)一个序列是某个图的度序列，即一个度序列问题有解时，该序列称为图形序列。由于度和公式的存在，任何具有奇数和的序列，如(3,3,1) ，都不会是图的度序列。反之亦然: 如果一个序列和是偶数，它就是重图的度序列。构造这样一个图是很简单: 匹配奇数度值的顶点并成对连接起来，然后剩余的偶数度值顶点都连出一条边指向自己。&lt;br /&gt;
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The question of whether a given degree sequence can be realized by a [[simple graph]] is more challenging. This problem is also called [[graph realization problem]] and can either be solved by the [[Erdős–Gallai theorem]] or the [[Havel–Hakimi algorithm]]. &lt;br /&gt;
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The question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm. &lt;br /&gt;
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一个给定的度序列是否可以用一个简单的图进行表示，是一个具有挑战性的问题。这个问题也称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图实现 Graph Realization Problem&amp;lt;/font&amp;gt;'''问题，可以用 Erdős–Gallai定理解决，也可以用Havel-Hakimi算法解决。&lt;br /&gt;
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The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of [[graph enumeration]].&lt;br /&gt;
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The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of graph enumeration.&lt;br /&gt;
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基于给定度序列寻找或估计可能的图的个数，是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图枚举 Graph Enumeration&amp;lt;/font&amp;gt;'''领域中的一个问题。&lt;br /&gt;
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More generally, the '''degree sequence''' of a [[hypergraph]] is the non-increasing sequence of its vertex degrees. A sequence is '''&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic''' if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in [[Time complexity|polynomial time]] for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the [[Erdős–Gallai theorem]] but is [[NP-completeness|NP-complete]] for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 &amp;lt;ref&amp;gt;{{Cite journal|last=Deza|first=Antoine|last2=Levin|first2=Asaf|last3=Meesum|first3=Syed M.|last4=Onn|first4=Shmuel|date=January 2018|title=Optimization over Degree Sequences|url=https://epubs.siam.org/doi/10.1137/17M1134482|journal=SIAM Journal on Discrete Mathematics|language=en|volume=32|issue=3|pages=2067–2079|doi=10.1137/17M1134482|issn=0895-4801|arxiv=1706.03951}}&amp;lt;/ref&amp;gt;).&lt;br /&gt;
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More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in polynomial time for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the Erdős–Gallai theorem but is NP-complete for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 ).&lt;br /&gt;
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一般来说，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;超图 Hypergraph&amp;lt;/font&amp;gt;'''的度序列是其顶点度的非递增序列。一个序列是&amp;lt;math&amp;gt;k-graphic&amp;lt;/math&amp;gt;，如果它是一些&amp;lt;math&amp;gt;k-uniform&amp;lt;/math&amp;gt;超图的度序列。特别是&amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic序列是图形。决定一个给定的序列是否是&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic可以在 &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt;的多项式时间内通过Erdős–Gallai定理实现，但是当&amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt;时该问题可转化为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;NP完全问题 NP-complete&amp;lt;/font&amp;gt;'''(Deza et al. ，2018)。&lt;br /&gt;
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==Special values==&lt;br /&gt;
特殊值&lt;br /&gt;
[[File:Depth-first-tree.png|thumb|&lt;br /&gt;
图3：An undirected graph with leaf nodes 4, 5, 6, 7, 10, 11, and 12 一个具有叶节点的无定向图]]&lt;br /&gt;
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*A vertex with degree 0 is called an [[isolated vertex]].&lt;br /&gt;
度值为0的顶点成为孤立顶点&lt;br /&gt;
*A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of [[tree (graph theory)|tree]]s in graph theory and especially [[tree (data structure)|tree]]s as [[data structure]]s.&lt;br /&gt;
度值为1的顶点称为叶顶点或尾顶点，该顶点的入射边称为悬挂边。在右侧的图中，{3,5}就是一个悬挂边。在图论中，该术语主要在研究“树”时使用，特别是具有树形结构的数据。&lt;br /&gt;
* A vertex with degree ''n''&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1  in a graph on ''n'' vertices is called a [[dominating vertex]].&lt;br /&gt;
在有n个顶点的图中，度值为n-1的顶点叫作主导顶点&lt;br /&gt;
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==Global properties==&lt;br /&gt;
全局属性&lt;br /&gt;
*If each vertex of the graph has the same degree&amp;amp;nbsp;''k'' the graph is called a [[regular graph|''k''-regular graph]] and the graph itself is said to have degree&amp;amp;nbsp;''k''. Similarly, a [[bipartite graph]] in which every two vertices on the same side of the bipartition as each other have the same degree is called a [[biregular graph]].&lt;br /&gt;
如果一个图中的所有顶点的度值都为k，该图被称为k-正则图，该图的度值也为k。同样的，同侧每两个顶点都具有相同度值的二分图叫作二分正则图。&lt;br /&gt;
*An undirected, connected graph has an [[Eulerian path]] if and only if it has either 0 or 2 vertices of odd degree. If it has 0 vertices of odd degree, the Eulerian path is an Eulerian circuit.&lt;br /&gt;
当且仅当具有0或2个奇数度的顶点时，无向连通图才具有欧拉路径。 如果它具有0个奇数度的顶点，则欧拉路径为欧拉回路。&lt;br /&gt;
*A directed graph is a [[pseudoforest]] if and only if every vertex has outdegree at most&amp;amp;nbsp;1. A [[functional graph]] is a special case of a pseudoforest in which every vertex has outdegree exactly&amp;amp;nbsp;1.&lt;br /&gt;
有向图是当且仅当每个顶点的度值最大为1时才是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;伪森林 Pseudoforest&amp;lt;/font&amp;gt;'''。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;功能图 Functional Graph&amp;lt;/font&amp;gt;'''是伪森林的特例，其中每个顶点的度数都恰好为1。&lt;br /&gt;
*By [[Brooks' theorem]], any graph other than a clique or an odd cycle has [[chromatic number]] at most&amp;amp;nbsp;Δ, and by [[Vizing's theorem]] any graph has [[chromatic index]] at most Δ&amp;amp;nbsp;+&amp;amp;nbsp;1.&lt;br /&gt;
根据布鲁克斯定理，除团簇或奇数循环外，任何图的色度数最大为Δ，而根据维辛定理，任何图的色度指数最大为Δ+ 1。&lt;br /&gt;
*A [[Degeneracy (graph theory)|''k''-degenerate graph]] is a graph in which each subgraph has a vertex of degree at most ''k''.&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;k简并图 K-Degenerate Graph &amp;lt;/font&amp;gt;'''是其中每个子图最多具有度值为k的顶点的图。&lt;br /&gt;
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==See also==&lt;br /&gt;
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*[[Indegree]], [[outdegree]] for [[digraph (mathematics)|digraph]]s&lt;br /&gt;
有向图的入度和出度&lt;br /&gt;
*[[Degree distribution]]&lt;br /&gt;
度分布&lt;br /&gt;
*[[bipartite graph|degree sequence]] for bipartite graphs&lt;br /&gt;
二分图的度序列&lt;br /&gt;
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==Notes==&lt;br /&gt;
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{{reflist}}&lt;br /&gt;
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==References==&lt;br /&gt;
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*{{Citation | last1=Diestel | first1=Reinhard | title=Graph Theory | url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-3-540-26183-4 | year=2005 }}.&lt;br /&gt;
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*{{citation|first1=P.|last1=Erdős|author1-link=Paul Erdős|first2=T.|last2=Gallai|author2-link=Tibor Gallai|title=Gráfok előírt fokszámú pontokkal|language=Hungarian|journal=Matematikai Lapok|volume=11|year=1960|pages=264&amp;amp;ndash;274|url=http://www.renyi.hu/~p_erdos/1961-05.pdf}}.&lt;br /&gt;
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*{{citation&lt;br /&gt;
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| authorlink =V. J. Havel&lt;br /&gt;
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| authorlink =V. J. Havel&lt;br /&gt;
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| authorlink = v.J · 哈维尔&lt;br /&gt;
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| year = 1955&lt;br /&gt;
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| year = 1955&lt;br /&gt;
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1955年&lt;br /&gt;
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| title = A remark on the existence of finite graphs&lt;br /&gt;
&lt;br /&gt;
| title = A remark on the existence of finite graphs&lt;br /&gt;
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| title = 关于有限图存在性的一个注记&lt;br /&gt;
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| language = Czech&lt;br /&gt;
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| language = Czech&lt;br /&gt;
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| language = Czech&lt;br /&gt;
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| journal = Časopis pro pěstování matematiky&lt;br /&gt;
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| journal = Časopis pro pěstování matematiky&lt;br /&gt;
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| journal = Časopis pro pěstování matematiky&lt;br /&gt;
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| volume = 80&lt;br /&gt;
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| volume = 80&lt;br /&gt;
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80&lt;br /&gt;
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| pages = 477–480&lt;br /&gt;
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| pages = 477–480&lt;br /&gt;
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| 页数 = 477-480&lt;br /&gt;
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| url = http://eudml.org/doc/19050&lt;br /&gt;
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| url = http://eudml.org/doc/19050&lt;br /&gt;
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Http://eudml.org/doc/19050&lt;br /&gt;
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}}&lt;br /&gt;
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}}&lt;br /&gt;
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}}&lt;br /&gt;
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*{{citation&lt;br /&gt;
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最后 = Hakimi | first = S.l. | authorlink = S.l. Hakimi&lt;br /&gt;
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 | journal = Journal of the [[Society for Industrial and Applied Mathematics]]&lt;br /&gt;
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工业与应用数学学会杂志&lt;br /&gt;
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 | mr = 0148049&lt;br /&gt;
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0148049先生&lt;br /&gt;
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 | pages = 496–506&lt;br /&gt;
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 | pages = 496–506&lt;br /&gt;
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| 页数 = 496-506&lt;br /&gt;
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 | title = On realizability of a set of integers as degrees of the vertices of a linear graph. I&lt;br /&gt;
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 | title = On realizability of a set of integers as degrees of the vertices of a linear graph. I&lt;br /&gt;
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| title = 一组整数作为线性图顶点度的可实现性。一&lt;br /&gt;
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 | volume = 10&lt;br /&gt;
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10&lt;br /&gt;
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 | year = 1962}}.&lt;br /&gt;
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 | year = 1962}}.&lt;br /&gt;
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1962}}.&lt;br /&gt;
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*{{citation&lt;br /&gt;
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 | last1 = Sierksma | first1 = Gerard&lt;br /&gt;
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 | last1 = Sierksma | first1 = Gerard&lt;br /&gt;
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1 = 杰拉德&lt;br /&gt;
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 | last2 = Hoogeveen | first2 = Han&lt;br /&gt;
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 | last2 = Hoogeveen | first2 = Han&lt;br /&gt;
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 | doi = 10.1002/jgt.3190150209&lt;br /&gt;
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 | doi = 10.1002/jgt.3190150209&lt;br /&gt;
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10.1002/jgt. 3190150209&lt;br /&gt;
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 | issue = 2&lt;br /&gt;
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 | issue = 2&lt;br /&gt;
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2&lt;br /&gt;
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 | journal = [[Journal of Graph Theory]]&lt;br /&gt;
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 | journal = Journal of Graph Theory&lt;br /&gt;
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| Journal = Journal of Graph Theory&lt;br /&gt;
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 | mr = 1106533&lt;br /&gt;
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 | mr = 1106533&lt;br /&gt;
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1106533先生&lt;br /&gt;
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 | title = Seven criteria for integer sequences being graphic&lt;br /&gt;
&lt;br /&gt;
 | title = Seven criteria for integer sequences being graphic&lt;br /&gt;
&lt;br /&gt;
| title = 整数序列为图形的七个条件&lt;br /&gt;
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 | volume = 15&lt;br /&gt;
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 | volume = 15&lt;br /&gt;
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15&lt;br /&gt;
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1991}}.&lt;br /&gt;
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[[Category:Graph theory]]&lt;br /&gt;
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Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree (graph theory)]]. Its edit history can be viewed at [[度/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12673</id>
		<title>度分布</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6%E5%88%86%E5%B8%83&amp;diff=12673"/>
		<updated>2020-08-16T08:51:35Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
{{Network Science}}&lt;br /&gt;
&lt;br /&gt;
In the study of [[Graph (discrete mathematics)|graphs]] and [[complex network|networks]], the [[degree (graph theory)|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.&lt;br /&gt;
&lt;br /&gt;
在图和网络的研究中，网络中节点的度是它与其他节点的连接数，而度的分布就是整个网络中这些度的概率分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Definition==&lt;br /&gt;
定义&lt;br /&gt;
The [[degree (graph theory)|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)|connectivity]]) is the number of connections or [[Edge (graph theory)#Graph|edges]] the node has to other nodes. If a network is [[directed graph|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.&lt;br /&gt;
&lt;br /&gt;
网络中一个节点的度(有时会被错误地指代为连通性)是该节点与其他节点的连接或边的数量。如果一个网络是有向的，这意味着边沿着一个方向从一个节点指向另一个节点。那么节点们就会有两个不同的度，一个度表示传入边的数量，另一个度表示传出边的数量。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''.  Thus if there are ''n'' nodes in total in a network and ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt; of them have degree ''k'', we have ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''.&lt;br /&gt;
&lt;br /&gt;
The degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k.  Thus if there are n nodes in total in a network and n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; of them have degree k, we have P(k) = n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;/n.&lt;br /&gt;
&lt;br /&gt;
将网络的度分布''P(k)''定义为网络中度值为k的所有节点与总节点数量的分数比值，如果一个网络中有''n''个节点，且其中''n&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt;''个节点的度值为''k''，则 ''P''(''k'') = ''n''&amp;lt;sub&amp;gt;''k''&amp;lt;/sub&amp;gt;/''n''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree smaller than ''k'', or even the ''complementary cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k'' (1 - ''C'') if one considers ''C'' as the ''cumulative degree distribution'';  i.e. the complement of ''C''.&lt;br /&gt;
&lt;br /&gt;
The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution;  i.e. the complement of C.&lt;br /&gt;
&lt;br /&gt;
同样的信息有时也以累积度分布函数（随机选择一个节点，其度值小于k的概率）的形式表示，或是用互补累积度分布（如果把''C''看作累积度分布，该函数则为度大于或等于''k'' (1 - ''C'')的节点比例）的形式表示，与积累度分布互补。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Observed degree distributions ==&lt;br /&gt;
观察度分布&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) [[random graph]], in which each of ''n'' nodes is independently connected (or not) with probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees ''k'':&lt;br /&gt;
&lt;br /&gt;
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.  The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:&lt;br /&gt;
&lt;br /&gt;
度分布在研究真实网络(如互联网和社会网络)和理论网络中都非常重要。最简单的网络模型，例如(Erdős–Rényi 模型)随机图，其中每''n''个节点都以概率''p'' (或1 − ''p'')独立连接(或不独立连接) ，它的二项分布的度值为k:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&lt;br /&gt;
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},&lt;br /&gt;
&lt;br /&gt;
选择 k } p ^ k (1-p) ^ { n-1-k } ,&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly [[Skewness|right-skewed]], meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks were argued to have degree distributions that approximately follow a [[power law]]: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
(or Poisson in the limit of large n, if the average degree &amp;lt;math&amp;gt;\langle k\rangle=p(n-1)&amp;lt;/math&amp;gt; is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as &amp;quot;hubs&amp;quot;, have high degree. Some networks, notably the Internet, the world wide web, and some social networks were argued to have degree distributions that approximately follow a power law: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
(如果平均度 &amp;lt; math &amp;gt; langle k rangle = p (n-1) &amp;lt;/math &amp;gt; 保持不变，也会出现在有限节点的泊松分布)。&lt;br /&gt;
然而，现实世界中的大多数网络的度分布与上述分布非常不同。大多数节点是高度右倾的，这意味着大多数节点的度值较低，但少数节点，即所谓的“枢纽” ，度值较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有近似遵循幂定律的幂律分布: &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&lt;br /&gt;
P(k)\sim k^{-\gamma}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties&amp;lt;ref name=&amp;quot;BA&amp;quot;&amp;gt;{{cite journal | last=Barabási | first=Albert-László | last2=Albert | first2=Réka | title=Emergence of Scaling in Random Networks | journal=Science | volume=286 | issue=5439 | date=1999-10-15 | issn=0036-8075 | doi=10.1126/science.286.5439.509 | pages=509–512| pmid=10521342 | arxiv=cond-mat/9910332 | bibcode=1999Sci...286..509B }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;AB&amp;quot;&amp;gt;{{cite journal | last=Albert | first=Réka | last2=Barabási | first2=Albert-László | title=Topology of Evolving Networks: Local Events and Universality | journal=Physical Review Letters | volume=85 | issue=24 | date=2000-12-11 | issn=0031-9007 | doi=10.1103/physrevlett.85.5234 | pages=5234–5237| pmid=11102229 | arxiv=cond-mat/0005085 | bibcode=2000PhRvL..85.5234A | hdl=2047/d20000695 | url=https://repository.library.northeastern.edu/files/neu:331099/fulltext.pdf }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Doro&amp;quot;&amp;gt;{{cite journal | last=Dorogovtsev | first=S. N. | last2=Mendes | first2=J. F. F. | last3=Samukhin | first3=A. N. | title=Size-dependent degree distribution of a scale-free growing network | journal=Physical Review E | volume=63 | issue=6 | date=2001-05-21 | issn=1063-651X | doi=10.1103/physreve.63.062101 | page=062101| pmid=11415146 |arxiv=cond-mat/0011115| bibcode=2001PhRvE..63f2101D }}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;PSY&amp;quot;&amp;gt;{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|year=2018|first=Angelica |last=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |volume=371|pages=1–12|doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P}}&amp;lt;/ref&amp;gt;. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have [[Fat-tailed distribution|fat-tailed degree distributions]], they deviate from being [[Scale-free network|scale-free]].&amp;lt;ref&amp;gt;{{Cite journal|last=Holme|first=Petter|date=2019-03-04|title=Rare and everywhere: Perspectives on scale-free networks|journal=Nature Communications|language=en|volume=10|issue=1|pages=1016|doi=10.1038/s41467-019-09038-8|issn=2041-1723|pmc=6399274|pmid=30833568|bibcode=2019NatCo..10.1016H}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, where γ is a constant. Such networks are called scale-free networks and have attracted particular attention for their structural and dynamical properties. However, recently, there have been some researches based on real-world data sets claiming despite the fact that most of the observed networks have fat-tailed degree distributions, they deviate from being scale-free. &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;,γ是一个常数。这种网络被称为无标度网络，因其结构和动力学性质而引起人们的特别关注。然而，最近有一些基于真实数据的研究声称，尽管大多数观测到的网络具有肥尾度分布，但它们偏离了无标度分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Excess degree distribution ==&lt;br /&gt;
超额度分布&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.&amp;lt;ref name=&amp;quot;:0&amp;quot;&amp;gt;{{Cite book|last=Newman|first=Mark|url=http://www.oxfordscholarship.com/view/10.1093/oso/9780198805090.001.0001/oso-9780198805090|title=Networks|date=2018-10-18|publisher=Oxford University Press|isbn=978-0-19-880509-0|volume=1|language=en|doi=10.1093/oso/9780198805090.001.0001}}&amp;lt;/ref&amp;gt; In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.&lt;br /&gt;
&lt;br /&gt;
超额度分布是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的传出链接的分布。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Suppose a network has a degree distribution &lt;br /&gt;
&lt;br /&gt;
假设一个网络具有度分布&amp;lt;math&amp;gt;P(k)&amp;lt;/math &amp;gt;，&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, by selecting one node (randomly or not) and going to one of its neighbors (assuming to have one neighbor at least), then the probability of that node to have &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
通过选择一个节点(随机或非随机)并转到它的一个邻居(假设至少有一个邻居) ，那么该节点具有&amp;lt; math &amp;gt; k&amp;lt;/math&amp;gt;个邻居的概率&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
K&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; neighbors is not given by &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
不是由 &amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;给出的&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. The reason is that, whenever some node is selected in a heterogeneous network, it is more probable to reach the hobs by following one of the existing neighbors of that node. The true probability of such nodes to have degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度&amp;lt; math &amp;gt;k&amp;lt;/math&amp;gt;的真实概率&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
k&lt;br /&gt;
&lt;br /&gt;
K&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
是&amp;lt;math&amp;gt;q(k)&amp;lt;/math &amp;gt;&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
Q (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the ''excess degree'' of that node. In the [[configuration model]], which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which is called the excess degree of that node. In the configuration model, which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as:&lt;br /&gt;
&lt;br /&gt;
它被称为该节点的超额度。在配置模型中，忽略节点之间的相关性，并假定每个节点以相同的概率连接到网络中的其他任何节点，超额度分布表示为:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&lt;br /&gt;
q(k) =  \frac{k+1}{\langle k \rangle}P(k+1),&lt;br /&gt;
&lt;br /&gt;
Q (k) = frac { k + 1}{ langle k rangle } p (k + 1) ,&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the [[friendship paradox]]. It can be shown that a network can have a [[giant component]], if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:&lt;br /&gt;
&lt;br /&gt;
这里&amp;lt;math&amp;gt;{\langle k \rangle}&amp;lt;/math &amp;gt; 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的友谊悖论。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&lt;br /&gt;
\sum_k kq(k) &amp;gt; 1 \Rightarrow  {\langle k^2 \rangle}-2{\langle k \rangle}&amp;gt;0 &lt;br /&gt;
&lt;br /&gt;
1 right tarrow { langle k ^ 2 rangle }-2{ langle k rangle } &amp;gt; 0&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the [[configuration model]] and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.&lt;br /&gt;
&lt;br /&gt;
记住，最后两个方程只适用于配置模型，要推导出实词网络的超额度分布，还应考虑度相关性。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== The Generating Functions Method ==&lt;br /&gt;
生成函数方法&lt;br /&gt;
[[Probability-generating function|Generating functions]] can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Generating functions can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
生成函数可以用来计算随机网络的不同性质。给定某些网络的度分布和超度分布，&amp;lt; math &amp;gt;P(k)&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;q(k)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
q(k)&lt;br /&gt;
&lt;br /&gt;
Q (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; respectively, it is possible to write two power series in the following forms:&lt;br /&gt;
&lt;br /&gt;
可以以下列形式写出两个幂级数:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&lt;br /&gt;
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&lt;br /&gt;
G_1(x) = \textstyle \sum_{k} \displaystyle q(k)x^k = \textstyle \sum_{k} \displaystyle \frac{k}{\langle k \rangle}P(k)x^{k-1} &lt;br /&gt;
&lt;br /&gt;
1(x) = textstyle sum { k } displaystyle q (k) x ^ k = textstyle sum { k } displastyle frac { k }{ langle k rangle } p (k) x ^ { k-1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
G_1(x)&lt;br /&gt;
&lt;br /&gt;
G_1(x)&lt;br /&gt;
&lt;br /&gt;
G _ 1(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can also be obtained from derivatives of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G_0(x)&amp;lt;/math &amp;gt; 也可以从 &amp;lt; math &amp;gt; 的导数得到&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
G_0(x)&lt;br /&gt;
&lt;br /&gt;
G _ 0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
(/math)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&lt;br /&gt;
G_1(x) = \frac{G'_0(x)}{G'_0(1)} &lt;br /&gt;
&lt;br /&gt;
1(x) = frac { g’ _ 0(x)}{ g’ _ 0(1)}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If we know the generating function for a probability distribution &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果我们知道生成函数的一个概率分布，然后我们就得到&amp;lt;math&amp;gt;P(k)&amp;lt;/math&amp;gt;的值通过鉴别：&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then we can recover the values of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P(k)&lt;br /&gt;
&lt;br /&gt;
P (k)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; by differentiating:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&lt;br /&gt;
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}&lt;br /&gt;
&lt;br /&gt;
= frac {1}{ k! }{ operatorname { d } ^ k! g over operatorname { d } ! x ^ k } bigvert { x = 0}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some properties, e.g. the moments, can be easily calculated from &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
一些性质，例如，时刻，可以很容易地计算依据 &amp;lt; math &amp;gt; G0(x)&amp;lt;/math&amp;gt;和它的导数:&lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G _ 0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and its derivatives:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = G'_0(1)&lt;br /&gt;
&lt;br /&gt;
{ langle k rangle } = g’ _ 0(1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
*&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{\langle k^2 \rangle} = G''_0(1) + G'_0(1) &lt;br /&gt;
&lt;br /&gt;
{\langle k^2 \rangle} = G_0(1) + G'_0(1) &lt;br /&gt;
&lt;br /&gt;
{ langle k ^ 2 rangle } = g _ 0(1) + g’ _ 0(1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
And in general&amp;lt;ref name=&amp;quot;:0&amp;quot; /&amp;gt;:&lt;br /&gt;
&lt;br /&gt;
And in general:&lt;br /&gt;
&lt;br /&gt;
一般来说:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&lt;br /&gt;
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}&lt;br /&gt;
&lt;br /&gt;
{ langle k ^ m rangle } = Biggl [{ bigg (operatorname { x }{ operatorname { d } ! over operatorname { dx } ！[} Biggl) ^ m } g _ 0(x) Biggl ] _ { x = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For Poisson-distributed random networks, such as the ER graph, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
对于泊松分布的随机网络，如 ER 图，&amp;lt; math &amp;gt; G1(x) = G0(x)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&lt;br /&gt;
G_1(x) = G_0(x) &lt;br /&gt;
&lt;br /&gt;
G1(x) = g0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, that is the reason why the theory of random networks of this type is especially simple. The probability distributions for the 1st and 2nd-nearest neighbors are generated by the functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这就是为什么这种类型的随机网络理论特别简单的原因。第一和第二近邻的概率分布是由函数 &amp;lt; math &amp;gt; G0(x)&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;G0(G1(x))&amp;lt;/math&amp;gt;生成的&lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G_0(x) &lt;br /&gt;
&lt;br /&gt;
G _ 0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&lt;br /&gt;
G_0(G_1(x)) &lt;br /&gt;
&lt;br /&gt;
G _ 0(g _ 1(x))&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. By extension, the distribution of &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
继续扩展，&amp;lt; math &amp;gt; m-th&amp;lt;/math&amp;gt;的邻居的分布&lt;br /&gt;
&lt;br /&gt;
m&lt;br /&gt;
&lt;br /&gt;
m&lt;br /&gt;
&lt;br /&gt;
M&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;-th neighbors is generated by:&lt;br /&gt;
&lt;br /&gt;
这个邻居是由以下几个因素产生的:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&lt;br /&gt;
G_0\bigl(G_1(...G_1(x)...)\bigr) &lt;br /&gt;
&lt;br /&gt;
G _ 0 bigl (g _ 1(... g _ 1(x) ...) bigr)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, with &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ] ，with &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
m-1 &lt;br /&gt;
&lt;br /&gt;
m-1 &lt;br /&gt;
&lt;br /&gt;
M-1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; iterations of the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
函数的迭代&lt;br /&gt;
&lt;br /&gt;
G_1 &lt;br /&gt;
&lt;br /&gt;
G_1 &lt;br /&gt;
&lt;br /&gt;
1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&amp;lt;ref name=&amp;quot;:1&amp;quot;&amp;gt;{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|url=https://link.aps.org/doi/10.1103/PhysRevE.64.026118|journal=Physical Review E|language=en|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|issn=1063-651X|doi-access=free}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; acting on itself.&lt;br /&gt;
&lt;br /&gt;
自我作用。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
第一个邻居的平均数量c1是&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
C _ 1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ] ，is &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&lt;br /&gt;
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}&lt;br /&gt;
&lt;br /&gt;
{ langle k rangle } = { dG _ 0(x) over dx } | { x = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors is: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
第二个邻居的平均数量是:&lt;br /&gt;
&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G''_0(1)&lt;br /&gt;
&lt;br /&gt;
c_2 = \biggl[ {d\over dx}G_0\big(G_1(x)\big)\biggl]_{x=1} = G_1'(1)G'_0\big(G_1(1)\big) =  G_1'(1)G'_0(1) = G_0(1)&lt;br /&gt;
&lt;br /&gt;
C2 = biggl [{ d over dx } g0 big (g1(x) big)] _ { x = 1} = g1’(1) g’0 big (g1(1) big) = g1’(1) g’0(1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Degree distribution for directed networks ==&lt;br /&gt;
有向网络的度分布&lt;br /&gt;
[[File:Enwiki-degree-distribution.png|thumb|right|320px|&lt;br /&gt;
图1：In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales) 维基百科超链接图(对数尺度)的入/出度分布]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In a directed network, each node has some in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
在有向网络中，每个节点都有一些入度k{in}&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
这是一个很好的例子&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and some out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
和一些出度k{out}&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; which are the number of links which have run into and out of that node respectfully. If &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
分别指代的是指向该节点的边的数量和从该节点指出的边的数量。&lt;br /&gt;
&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&lt;br /&gt;
P(k_{in}, k_{out})&lt;br /&gt;
&lt;br /&gt;
P (k _ { in } ，k _ { out }) &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the probability that a randomly chosen node has in-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
如果 &amp;lt; math &amp;gt; P(k_{in}, k_{out})&amp;lt;/math&amp;gt;是一个随机选择的具有入出度的节点的可能性&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
k_{&lt;br /&gt;
&lt;br /&gt;
这是一个很好的例子&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
in}&lt;br /&gt;
&lt;br /&gt;
开始吧&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and out-degree &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
“数学”和“学位”&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
k_{out}&lt;br /&gt;
&lt;br /&gt;
我不知道你在说什么&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this [[joint probability distribution]] can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; then the generating function assigned to this joint probability distribution can be written with two valuables &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
那么分配给这个生成函数的联合概率分布可以被写成两个变量 &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt;和&amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
X&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
Y&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; as:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y) =  \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; &lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
Since every link in a directed network must leave some node and enter another, the net average number of links entering&lt;br /&gt;
&lt;br /&gt;
由于有向网络中的每个链路必须离开某个节点并进入另一个节点，因此进入一个节点的链路的净平均数是零。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
a node is zero. Therefore, &lt;br /&gt;
&lt;br /&gt;
所以,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0 &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
which implies that, the generation function must satisfy:&lt;br /&gt;
&lt;br /&gt;
这意味着，生成函数必须满足:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
《数学》&lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,&lt;br /&gt;
&lt;br /&gt;
{ partial mathcal { g } over partial x } vert _ { x，y = 1} = { partial mathcal { g } over partial y } vert _ { x，y = 1} = c,&lt;br /&gt;
&lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
c &lt;br /&gt;
&lt;br /&gt;
c &lt;br /&gt;
&lt;br /&gt;
C&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; is the mean degree (both in and out) of the nodes in the network; &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
C是网络中节点的平均度(内部和外部)&lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&lt;br /&gt;
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c. &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Using the function &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
使用函数 &amp;lt; math &amp;gt; \mathcal{G}(x,y)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&lt;br /&gt;
\mathcal{G}(x,y)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math &amp;gt; ，如前所述，我们可以再次找到入/出度分布和入/出超量度分布的生成函数。&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) &lt;br /&gt;
&lt;br /&gt;
0(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; can be defined as generating functions for the number of arriving links at a randomly chosen node, and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
可以将 &amp;lt;/math &amp;gt; 定义为一个随机选择的节点上到达的链接数的生成函数，以及 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x)&lt;br /&gt;
&lt;br /&gt;
1(x)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math &amp;gt; 可以定义为按照随机选择的链接到达一个节点的到达链接数。我们也可以定义生成函数 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y)&lt;br /&gt;
&lt;br /&gt;
0(y)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y)&lt;br /&gt;
&lt;br /&gt;
1(y)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; for the number leaving such a node:&lt;br /&gt;
&lt;br /&gt;
离开这样一个节点的数字:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&lt;br /&gt;
G^{in}_0(x) = \mathcal{G}(x,1)&lt;br /&gt;
&lt;br /&gt;
0(x) = mathcal { g }(x，1)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&lt;br /&gt;
G^{in}_1(x) =  \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1} &lt;br /&gt;
&lt;br /&gt;
1(x) = frac {1}{ c }{ partial mathcal { g } over partial x } vert _ { y = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&lt;br /&gt;
G^{out}_0(y) = \mathcal{G}(1,y)&lt;br /&gt;
&lt;br /&gt;
0(y) = mathcal { g }(1，y)&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
* &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&lt;br /&gt;
G^{out}_1(y) =  \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1} &lt;br /&gt;
&lt;br /&gt;
1(y) = frac {1}{ c }{ partial mathcal { g } over partial y } vert _ { x = 1}&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
数学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Here, the average number of 1st neighbors, &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这里，第一个邻居的平均数量C，&amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
c&lt;br /&gt;
&lt;br /&gt;
c&lt;br /&gt;
&lt;br /&gt;
C&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, or as previously introduced as &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
或者像之前表示的C1&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
c_1&lt;br /&gt;
&lt;br /&gt;
C _ 1&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;, is &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ] ，is &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
 {\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} =  {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
{ partial mathcal { g } over partial x } biggl vert _ { x，y = 1} = { partial mathcal { g } over partial y } biggl vert _ { x，y = 1}&lt;br /&gt;
&lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and the average number of 2nd neighbors reachable from a randomly chosen node is given by: &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
从一个随机选择的节点上可达到的第二邻居的平均数是:&lt;br /&gt;
&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;. These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
这些也是从这些随机节点可以达到第一和第二邻居的数量，因为这些方程显然是在x和y上对称的 &amp;lt; math &amp;gt; &lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
x&lt;br /&gt;
&lt;br /&gt;
X&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[ math ]和[ math ]&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
y&lt;br /&gt;
&lt;br /&gt;
Y&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&amp;lt;ref name=&amp;quot;:1&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
数学。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [[Graph theory]]&lt;br /&gt;
图理论&lt;br /&gt;
* [[Complex network]]&lt;br /&gt;
复杂网络&lt;br /&gt;
* [[Scale-free network]]&lt;br /&gt;
无标度网络&lt;br /&gt;
* [[Random graph]]&lt;br /&gt;
随机网络&lt;br /&gt;
* [[Structural cut-off]]&lt;br /&gt;
结构截止值&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
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&lt;br /&gt;
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* {{cite journal&lt;br /&gt;
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| title = 网络的进化&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
  | journal = Advances in Physics&lt;br /&gt;
&lt;br /&gt;
| 杂志 = 物理进展&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
51&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
2002年&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
10.1080/00018730110112519&lt;br /&gt;
&lt;br /&gt;
  | issue = 4&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
第四期&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
|bibcode = 2002AdPhy..51.1079D }}&lt;br /&gt;
&lt;br /&gt;
2002/adphy. . 51.1079 d }&lt;br /&gt;
&lt;br /&gt;
* {{cite journal&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
纽曼&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
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| first = m.E.j.&lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
 |title=The structure and function of complex networks &lt;br /&gt;
&lt;br /&gt;
| title = 复杂网络的结构和功能&lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
 |journal=SIAM Review &lt;br /&gt;
&lt;br /&gt;
| journal = SIAM Review&lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
 |volume=45 &lt;br /&gt;
&lt;br /&gt;
45&lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
 |pages=167–256 &lt;br /&gt;
&lt;br /&gt;
| 页 = 167-256&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
 |year=2003 &lt;br /&gt;
&lt;br /&gt;
2003年&lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
 |doi=10.1137/S003614450342480 &lt;br /&gt;
&lt;br /&gt;
| doi = 10.1137/S003614450342480&lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
 |issue=2 &lt;br /&gt;
&lt;br /&gt;
2&lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516 &lt;br /&gt;
&lt;br /&gt;
 |arxiv=cond-mat/0303516&lt;br /&gt;
&lt;br /&gt;
 |bibcode=2003SIAMR..45..167N &lt;br /&gt;
&lt;br /&gt;
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| bibcode = 2003siaml. . 45. . 167 n&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
* {{cite book &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
  | title= Complex Networks: Structure, Robustness and Function &lt;br /&gt;
&lt;br /&gt;
复杂网络: 结构、健壮性和功能&lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
作者: Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
  | author-link= Shlomo Havlin &lt;br /&gt;
&lt;br /&gt;
| 作者链接 = Shlomo Havlin&lt;br /&gt;
&lt;br /&gt;
   &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
  | author2= Reuven Cohen| last-author-amp= yes &lt;br /&gt;
&lt;br /&gt;
2 = Reuven Cohen | last-author-amp = yes&lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
  | year= 2010 &lt;br /&gt;
&lt;br /&gt;
2010年&lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
  | url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php &lt;br /&gt;
&lt;br /&gt;
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
  | publisher= Cambridge University Press&lt;br /&gt;
&lt;br /&gt;
剑桥大学出版社&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
&lt;br /&gt;
[[Category:Graph invariants]]&lt;br /&gt;
&lt;br /&gt;
Category:Graph invariants&lt;br /&gt;
&lt;br /&gt;
类别: 图形不变量&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12581</id>
		<title>度</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E5%BA%A6&amp;diff=12581"/>
		<updated>2020-08-15T09:16:29Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
[[File:UndirectedDegrees (Loop).svg|thumb|&lt;br /&gt;
图1：A graph with a loop having vertices labeled by degree 内含自环按度标记的图]]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In [[graph theory]], the '''degree''' (or '''valency''') of a [[vertex (graph theory)|vertex]] of a [[Graph (discrete mathematics)|graph]] is the number of [[edge (graph theory)|edges]] that are [[incidence (graph)|incident]] to the vertex, and in a [[multigraph]], [[loop (graph theory)|loop]]s are counted twice.&amp;lt;ref&amp;gt;Diestel p.5&amp;lt;/ref&amp;gt; The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The '''maximum degree''' of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the '''minimum degree''' of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
&lt;br /&gt;
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt; or &amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;. The maximum degree of a graph &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt;\Delta(G)&amp;lt;/math&amp;gt;, and the minimum degree of a graph, denoted by &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;, are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.&lt;br /&gt;
&lt;br /&gt;
在图论中，图中顶点的度(或价)是入射到该顶点的边的数量。在多重图中，自环会被计算两次。顶点的度数可表示为&amp;lt;math&amp;gt;\deg(v)&amp;lt;/math&amp;gt;或&amp;lt;math&amp;gt;\deg v&amp;lt;/math&amp;gt;。一个图&amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt;的最大度值表示为 &amp;lt; math &amp;gt; Delta (g) &amp;lt;/math &amp;gt; ，最小度值表示为 &amp;lt;math&amp;gt;\delta(G)&amp;lt;/math&amp;gt;。在右侧的多重图中，最大度值为5，最小度值为0。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In a [[regular graph]], every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A [[complete graph]] (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted &amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, &amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
在正则图中，每个顶点都具有相同的度值，因此我们可以称之为该图的度值。一个完全图(表示为&amp;lt;math&amp;gt;K_n&amp;lt;/math&amp;gt;，其中&amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt;是图中顶点的数目)是一种特殊的正则图，其中所有顶点都有最大度值，&amp;lt;math&amp;gt;n-1&amp;lt;/math&amp;gt;。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Handshaking lemma==&lt;br /&gt;
握手引理&lt;br /&gt;
{{main|Handshaking lemma}}&lt;br /&gt;
&lt;br /&gt;
The '''degree sum formula''' states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
The degree sum formula states that, given a graph &amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
度和公式表明，给定一个图&amp;lt;math&amp;gt;G=(V, E)&amp;lt;/math&amp;gt;,&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\sum_{v \in V} \deg(v) = 2|E|\, .&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree  sum formula) is known as the [[handshaking lemma]]. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
&lt;br /&gt;
The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.&lt;br /&gt;
&lt;br /&gt;
该公式表明，在任何无向图中，拥有奇数度值的顶点的个数是偶数。这个陈述(以及度和公式)被称为握手引理。该名字来自一个著名的数学问题，即证明无论该群体内有多少人，与奇数人握过手的人数是偶数。&lt;br /&gt;
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==Degree sequence==&lt;br /&gt;
度序列&lt;br /&gt;
[[File:Conjugate-dessins.svg|thumb|200px|&lt;br /&gt;
图2：Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1).]]&lt;br /&gt;
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The '''degree sequence''' of an undirected graph is the non-increasing sequence of its vertex degrees;&amp;lt;ref&amp;gt;Diestel p.216&amp;lt;/ref&amp;gt; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a [[graph invariant]] so [[Graph isomorphism|isomorphic graphs]] have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
&lt;br /&gt;
The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.&lt;br /&gt;
&lt;br /&gt;
无向图的度序列是指将其各顶点度值按非递增方式排序，对于上述图是(5,3,3,2,2,1,0)。度序列是图的不变量，因此同构图（两个图中顶点的度值都相同，但形状不同）具有相同的度序列。然而，度序列通常不能唯一地标识一个图，在某些情况下，非同构图具有相同的度序列。&lt;br /&gt;
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The '''degree sequence problem'''  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a '''graphic''' or '''graphical sequence'''. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a [[matching (graph theory)|matching]], and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
&lt;br /&gt;
The degree sequence problem  is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3,&amp;amp;nbsp;3,&amp;amp;nbsp;1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self-loops.&lt;br /&gt;
&lt;br /&gt;
度序列问题，是指寻找具有以非增方式排列给定正整数的度序列的局部或全部图的问题。(序列尾部的零可能会被忽略，因为通过向图中添加适当数量的孤立顶点就可以轻松实现序列尾部不断加零。)一个序列是某个图的度序列，即一个度序列问题有解时，该序列称为图形序列。由于度和公式的存在，任何具有奇数和的序列，如(3,3,1) ，都不会是图的度序列。反之亦然: 如果一个序列有偶数和，它就是重图的度序列。构造这样一个图是很简单: 通过匹配将奇数度值的顶点成对连接起来，然后剩余的偶数度值顶点都连出一条边指向自己。&lt;br /&gt;
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The question of whether a given degree sequence can be realized by a [[simple graph]] is more challenging. This problem is also called [[graph realization problem]] and can either be solved by the [[Erdős–Gallai theorem]] or the [[Havel–Hakimi algorithm]]. &lt;br /&gt;
&lt;br /&gt;
The question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm. &lt;br /&gt;
&lt;br /&gt;
一个给定的度序列是否可以用一个简单的图进行表示，是一个具有挑战性的问题。这个问题也称为图实现问题，可以用 Erdős–Gallai定理解决，也可以用Havel-Hakimi算法解决。&lt;br /&gt;
&lt;br /&gt;
The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of [[graph enumeration]].&lt;br /&gt;
&lt;br /&gt;
The problem of finding or estimating the  number of graphs with a given degree sequence is a problem from the field of graph enumeration.&lt;br /&gt;
&lt;br /&gt;
基于给定度序列寻找或估计可能的图的个数，是图枚举领域中的一个问题。&lt;br /&gt;
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More generally, the '''degree sequence''' of a [[hypergraph]] is the non-increasing sequence of its vertex degrees. A sequence is '''&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic''' if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in [[Time complexity|polynomial time]] for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the [[Erdős–Gallai theorem]] but is [[NP-completeness|NP-complete]] for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 &amp;lt;ref&amp;gt;{{Cite journal|last=Deza|first=Antoine|last2=Levin|first2=Asaf|last3=Meesum|first3=Syed M.|last4=Onn|first4=Shmuel|date=January 2018|title=Optimization over Degree Sequences|url=https://epubs.siam.org/doi/10.1137/17M1134482|journal=SIAM Journal on Discrete Mathematics|language=en|volume=32|issue=3|pages=2067–2079|doi=10.1137/17M1134482|issn=0895-4801|arxiv=1706.03951}}&amp;lt;/ref&amp;gt;).&lt;br /&gt;
&lt;br /&gt;
More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic if it is the degree sequence of some &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform hypergraph. In particular, a &amp;lt;math&amp;gt;2&amp;lt;/math&amp;gt;-graphic sequence is graphic. Deciding if a given sequence is &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic is doable in polynomial time for &amp;lt;math&amp;gt;k=2&amp;lt;/math&amp;gt; via the Erdős–Gallai theorem but is NP-complete for all &amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt; (Deza et al., 2018 ).&lt;br /&gt;
&lt;br /&gt;
一般来说，超图的度序列是其顶点度的非递增序列。一个序列是&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-graphic，如果它是一些&amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt;-uniform超图的度序列。特别是 &amp;lt; math &amp;gt; 2 &amp;lt;/math &amp;gt;-graphic 序列是图形。决定一个给定的序列是否是 &amp;lt; math &amp;gt; k &amp;lt;/math &amp;gt;-graphic 可以在 &amp;lt; math &amp;gt; k = 2 &amp;lt;/math &amp;gt;的多项式时间内通过Erdős–Gallai定理实现，但是当&amp;lt;math&amp;gt;k\ge 3&amp;lt;/math&amp;gt;时该问题转化为NP完全问题(Deza et al. ，2018)。&lt;br /&gt;
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==Special values==&lt;br /&gt;
特殊值&lt;br /&gt;
[[File:Depth-first-tree.png|thumb|&lt;br /&gt;
图3：An undirected graph with leaf nodes 4, 5, 6, 7, 10, 11, and 12 一个具有叶节点的无定向图]]&lt;br /&gt;
&lt;br /&gt;
*A vertex with degree 0 is called an [[isolated vertex]].&lt;br /&gt;
度值为0的顶点成为孤立顶点&lt;br /&gt;
*A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of [[tree (graph theory)|tree]]s in graph theory and especially [[tree (data structure)|tree]]s as [[data structure]]s.&lt;br /&gt;
度值为1的顶点称为叶顶点或尾顶点，该顶点的入射边称为悬挂边。在右侧的图中，{3,5}就是一个悬挂边。在图论中，该术语主要在研究树时使用，特别是具有树形结构的数据。&lt;br /&gt;
* A vertex with degree ''n''&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1  in a graph on ''n'' vertices is called a [[dominating vertex]].&lt;br /&gt;
在有n个顶点的图中，度值为n-1的顶点叫作主导顶点&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Global properties==&lt;br /&gt;
全局属性&lt;br /&gt;
*If each vertex of the graph has the same degree&amp;amp;nbsp;''k'' the graph is called a [[regular graph|''k''-regular graph]] and the graph itself is said to have degree&amp;amp;nbsp;''k''. Similarly, a [[bipartite graph]] in which every two vertices on the same side of the bipartition as each other have the same degree is called a [[biregular graph]].&lt;br /&gt;
如果一个图中的所有顶点的度值都为k，该图被称为k-正则图，该图的度值也为k。相似的，同侧每两个顶点都具有相同度值的二分图叫作二分正则图。&lt;br /&gt;
*An undirected, connected graph has an [[Eulerian path]] if and only if it has either 0 or 2 vertices of odd degree. If it has 0 vertices of odd degree, the Eulerian path is an Eulerian circuit.&lt;br /&gt;
当且仅当具有0或2个奇数度的顶点时，无向连通图才具有欧拉路径。 如果它具有0个奇数度的顶点，则欧拉路径为欧拉回路。&lt;br /&gt;
*A directed graph is a [[pseudoforest]] if and only if every vertex has outdegree at most&amp;amp;nbsp;1. A [[functional graph]] is a special case of a pseudoforest in which every vertex has outdegree exactly&amp;amp;nbsp;1.&lt;br /&gt;
有向图是当且仅当每个顶点的度值最大为1时才是伪森林。功能图是伪森林的特例，其中每个顶点的度数都恰好为1。&lt;br /&gt;
*By [[Brooks' theorem]], any graph other than a clique or an odd cycle has [[chromatic number]] at most&amp;amp;nbsp;Δ, and by [[Vizing's theorem]] any graph has [[chromatic index]] at most Δ&amp;amp;nbsp;+&amp;amp;nbsp;1.&lt;br /&gt;
根据布鲁克斯定理，除团簇或奇数循环外，任何图的色度数最大为Δ，而根据维辛定理，任何图的色度指数最大为Δ+ 1。&lt;br /&gt;
*A [[Degeneracy (graph theory)|''k''-degenerate graph]] is a graph in which each subgraph has a vertex of degree at most ''k''.&lt;br /&gt;
k简并图是其中每个子图最多具有度值为k的顶点的图。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==See also==&lt;br /&gt;
&lt;br /&gt;
*[[Indegree]], [[outdegree]] for [[digraph (mathematics)|digraph]]s&lt;br /&gt;
有向图的入度和出度&lt;br /&gt;
*[[Degree distribution]]&lt;br /&gt;
度分布&lt;br /&gt;
*[[bipartite graph|degree sequence]] for bipartite graphs&lt;br /&gt;
二分图的度序列&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Notes==&lt;br /&gt;
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{{reflist}}&lt;br /&gt;
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==References==&lt;br /&gt;
&lt;br /&gt;
*{{Citation | last1=Diestel | first1=Reinhard | title=Graph Theory | url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-3-540-26183-4 | year=2005 }}.&lt;br /&gt;
&lt;br /&gt;
*{{citation|first1=P.|last1=Erdős|author1-link=Paul Erdős|first2=T.|last2=Gallai|author2-link=Tibor Gallai|title=Gráfok előírt fokszámú pontokkal|language=Hungarian|journal=Matematikai Lapok|volume=11|year=1960|pages=264&amp;amp;ndash;274|url=http://www.renyi.hu/~p_erdos/1961-05.pdf}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
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&lt;br /&gt;
| last= Havel&lt;br /&gt;
&lt;br /&gt;
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| first= Václav&lt;br /&gt;
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| authorlink =V. J. Havel&lt;br /&gt;
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| journal = Časopis pro pěstování matematiky&lt;br /&gt;
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| url = http://eudml.org/doc/19050&lt;br /&gt;
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Http://eudml.org/doc/19050&lt;br /&gt;
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最后 = Hakimi | first = S.l. | authorlink = S.l. Hakimi&lt;br /&gt;
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0148049先生&lt;br /&gt;
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 | title = On realizability of a set of integers as degrees of the vertices of a linear graph. I&lt;br /&gt;
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| title = 一组整数作为线性图顶点度的可实现性。一&lt;br /&gt;
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1962}}.&lt;br /&gt;
&lt;br /&gt;
*{{citation&lt;br /&gt;
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&lt;br /&gt;
 | last1 = Sierksma | first1 = Gerard&lt;br /&gt;
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 | last2 = Hoogeveen | first2 = Han&lt;br /&gt;
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10.1002/jgt. 3190150209&lt;br /&gt;
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 | issue = 2&lt;br /&gt;
&lt;br /&gt;
 | issue = 2&lt;br /&gt;
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2&lt;br /&gt;
&lt;br /&gt;
 | journal = [[Journal of Graph Theory]]&lt;br /&gt;
&lt;br /&gt;
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1106533先生&lt;br /&gt;
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 | title = Seven criteria for integer sequences being graphic&lt;br /&gt;
&lt;br /&gt;
 | title = Seven criteria for integer sequences being graphic&lt;br /&gt;
&lt;br /&gt;
| title = 整数序列为图形的七个条件&lt;br /&gt;
&lt;br /&gt;
 | volume = 15&lt;br /&gt;
&lt;br /&gt;
 | volume = 15&lt;br /&gt;
&lt;br /&gt;
15&lt;br /&gt;
&lt;br /&gt;
 | year = 1991}}.&lt;br /&gt;
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 | year = 1991}}.&lt;br /&gt;
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1991}}.&lt;br /&gt;
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[[Category:Graph theory]]&lt;br /&gt;
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Category:Graph theory&lt;br /&gt;
&lt;br /&gt;
范畴: 图论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Degree (graph theory)]]. Its edit history can be viewed at [[度/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
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		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{Network Science}}&lt;br /&gt;
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A '''biological network''' is any [[network science|network]] that applies to [[biological system]]s. A network is any system with sub-units that are linked into a whole, such as species units linked into a whole [[food web]]. Biological networks provide a [[graph theory|mathematical representation]] of connections found in [[ecological network|ecological]], [[phylogenetic network|evolutionary]], and [[physiological]] studies, such as [[neural networks]].&amp;lt;ref name=&amp;quot;Proulx05&amp;quot;&amp;gt;{{cite journal | last1=Proulx | first1=S. R. | last2=Promislow | first2=D. E. L. | last3=Phillips | first3=P. C. | title=Network thinking in ecology and evolution | journal=Trends in Ecology and Evolution | volume=20 | issue=6 | pages=345–353 | year=2005 | doi=10.1016/j.tree.2005.04.004 | url=http://eeb19.biosci.arizona.edu/Faculty/Dornhaus/courses/materials/papers/Proulx%20Promislow%20Phillips%20networks%20ecol%20evol.pdf | pmid=16701391 | url-status=dead | archiveurl=https://web.archive.org/web/20110815122330/http://eeb19.biosci.arizona.edu/Faculty/Dornhaus/courses/materials/papers/Proulx%20Promislow%20Phillips%20networks%20ecol%20evol.pdf | archivedate=2011-08-15 }}&amp;lt;/ref&amp;gt; The analysis of biological networks with respect to human diseases has led to the field of [[network medicine]].&amp;lt;ref&amp;gt;{{cite journal | last1 = Barabási | first1 = A. L. | last2 = Gulbahce | first2 = N. | last3 = Loscalzo | first3 = J. | year = 2011 | title = Network medicine: a network-based approach to human disease | url = | journal = Nature Reviews Genetics | volume = 12 | issue = 1| pages = 56–68 | doi=10.1038/nrg2918 | pmid=21164525 | pmc=3140052}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite journal|last=Habibi|first=Iman|last2=Emamian|first2=Effat S.|last3=Abdi|first3=Ali|date=2014-10-07|title=Advanced Fault Diagnosis Methods in Molecular Networks|journal=PLOS ONE|volume=9|issue=10|pages=e108830|doi=10.1371/journal.pone.0108830|issn=1932-6203|pmc=4188586|pmid=25290670|bibcode=2014PLoSO...9j8830H}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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A biological network is any network that applies to biological systems. A network is any system with sub-units that are linked into a whole, such as species units linked into a whole food web. Biological networks provide a mathematical representation of connections found in ecological, evolutionary, and physiological studies, such as neural networks. The analysis of biological networks with respect to human diseases has led to the field of network medicine.&lt;br /&gt;
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任何应用于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生物系统 Biological Systems&amp;lt;/font&amp;gt;'''的网络都可称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 生物网络 Biological Network&amp;lt;/font&amp;gt;'''。网络是指内部子单元可连接成一个整体的系统，如大量单个物种连接成一个整体的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;食物链网 Food Web&amp;lt;/font&amp;gt;'''。生物网络为在生态学、进化学和生理学研究中发现的联系提供了数学形式的表达，例如'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 神经网络 Neural Networks&amp;lt;/font&amp;gt;'''。生物网络分析在人类疾病方面的应用，开创了网络医学领域。&lt;br /&gt;
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==Network biology and bioinformatics==&lt;br /&gt;
网络生物学和生物信息学&lt;br /&gt;
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Complex biological systems may be represented and analyzed as computable networks. For example, ecosystems can be modeled as networks of interacting species or a protein can be modeled as a network of amino acids. Breaking a protein down farther, [[amino acids]] can be represented as a network of connected [[atom]]s, such as [[carbon]], [[nitrogen]], and [[oxygen]]. [[vertex (graph theory)|Nodes]] and [[vertex (graph theory)|edges]] are the basic components of a network. Nodes represent units in the network, while edges represent the interactions between the units. Nodes can represent a wide array of biological units, from individual organisms to individual neurons in the brain. Two important properties of a network are [[vertex (graph theory)|degree]] and [[betweenness centrality]]. Degree (or connectivity, a distinct usage from that used in [[Connectivity (graph theory)|graph theory]]) is the number of edges that connect a node, while betweenness is a measure of how central a node is in a network.&amp;lt;ref name=&amp;quot;Proulx&amp;quot;&amp;gt;{{cite journal |author= Proulx, S.R. |title= Network thinking in ecology and evolution |journal= Trends in Ecology and Evolution |year= 2005  |volume= 20 |issue= 6 |pages= 345–353 |doi=10.1016/j.tree.2005.04.004 |pmid=16701391|display-authors=etal}}&amp;lt;/ref&amp;gt; Nodes with high betweenness essentially serve as bridges between different portions of  the network (i.e. interactions must pass through this node to reach other portions of the network). In social networks, nodes with high degree or high betweenness may play important roles in the overall composition of a network.&lt;br /&gt;
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Complex biological systems may be represented and analyzed as computable networks. For example, ecosystems can be modeled as networks of interacting species or a protein can be modeled as a network of amino acids. Breaking a protein down farther, amino acids can be represented as a network of connected atoms, such as carbon, nitrogen, and oxygen. Nodes and edges are the basic components of a network. Nodes represent units in the network, while edges represent the interactions between the units. Nodes can represent a wide array of biological units, from individual organisms to individual neurons in the brain. Two important properties of a network are degree and betweenness centrality. Degree (or connectivity, a distinct usage from that used in graph theory) is the number of edges that connect a node, while betweenness is a measure of how central a node is in a network. Nodes with high betweenness essentially serve as bridges between different portions of  the network (i.e. interactions must pass through this node to reach other portions of the network). In social networks, nodes with high degree or high betweenness may play important roles in the overall composition of a network.&lt;br /&gt;
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复杂的生物系统能用可计算的网络来表示和分析。例如，生态系统可以被模拟为相互作用的物种网络，一个蛋白质可以被模拟为氨基酸网络。进一步分解蛋白质，氨基酸可以表示为一个由相互连接的原子构成的网络，如碳、氮和氧。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 顶点 Nodes&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;边 Edges&amp;lt;/font&amp;gt;'''是网络的基本组成部分。顶点表示网络中的单元，边表示单元之间的相互作用。顶点可以代表一系列广泛的生物单元，从单个的生物体到大脑中单个的神经元。网络的两个重要性质是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度 Degree&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 介数中心性 Betweenness Centrality&amp;lt;/font&amp;gt;'''。度(或连通性，一个不同于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''的用法)是连接一个顶点的边的数量，而介数中心性是衡量一个顶点在网络中有多么靠近中心位置。具有高介数的顶点本质上充当网络不同部分之间的桥梁(即网络其他部分的交互，必须通过这个节点)。在社会网络中，具有较高度值和较高介数的节点可能在网络的整体组成中发挥重要作用。&lt;br /&gt;
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As early as the 1980s, researchers started viewing [[DNA]] or [[genome]]s as the dynamic storage of a language system with precise computable finite [[state (computer science)|states]] represented as a [[finite state machine]].&amp;lt;ref&amp;gt;{{cite book  |author= Searls, D. |title= Artificial intelligence and molecular biology |publisher= MIT Press |year= 1993 |location= Cambridge, MA}}&amp;lt;/ref&amp;gt; Recent [[complex system]]s research has also suggested some far-reaching commonality in the organization of information in problems from biology, [[computer science]], and [[physics]], such as the [[Bose–Einstein condensate]] (a special state of matter).&amp;lt;ref&amp;gt;{{cite journal |author1=Bianconi, G.  |author2=Barabasi A.  |lastauthoramp=yes |title= Bose-Einstein condensation in complex networks |journal= Phys. Rev. Lett. |year= 2001  |volume= 86 |issue= 24 |pages= 5632–5635 |doi=10.1103/physrevlett.86.5632|pmid=11415319  |arxiv=cond-mat/0011224|bibcode=2001PhRvL..86.5632B|hdl=2047/d20000696  }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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As early as the 1980s, researchers started viewing DNA or genomes as the dynamic storage of a language system with precise computable finite states represented as a finite state machine. Recent complex systems research has also suggested some far-reaching commonality in the organization of information in problems from biology, computer science, and physics, such as the Bose–Einstein condensate (a special state of matter).&lt;br /&gt;
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早在20世纪80年代，研究人员就开始将 DNA 或基因组视为一个可存储动态信息的语言系统，因其具有可精确计算的有限状态，所以可被看做一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有限状态机 Finite State Machine&amp;lt;/font&amp;gt;'''。最近的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;复杂系统 Complex System&amp;lt;/font&amp;gt;'''研究还表明，一些在生物学、计算机科学和物理学等领域的问题，从信息组织的角度来看存在一些意义深远的共性，例如玻色-爱因斯坦凝聚态(一种特殊的物质状态)。&lt;br /&gt;
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Bioinformatics has increasingly shifted its focus from individual [[gene]]s, proteins, and [[search algorithm]]s to large-scale networks often denoted as [[omics|-omes]] such as [[biome]], [[interactome]], [[genome]] and [[proteome]]. Such theoretical studies have revealed that biological networks share many features with other networks such as the [[Internet]] or [[social network]]s, e.g. their [[network topology]].&lt;br /&gt;
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Bioinformatics has increasingly shifted its focus from individual genes, proteins, and search algorithms to large-scale networks often denoted as -omes such as biome, interactome, genome and proteome. Such theoretical studies have revealed that biological networks share many features with other networks such as the Internet or social networks, e.g. their network topology.&lt;br /&gt;
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生物信息学已逐渐将关注点从单个基因、蛋白质和搜索算法逐渐转移到像'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生物组 Biome&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;交互组 Interactome&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因组 Genome&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;蛋白组 Proteome&amp;lt;/font&amp;gt;'''这样的大规模网络。这些理论研究揭示了生物网络与其他网络，如互联网或社交网络，有许多共同特征，例如:他们的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络拓扑结构 Network Topology&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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==Networks in biology==&lt;br /&gt;
生物学中的网络&lt;br /&gt;
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===Protein–protein interaction networks===&lt;br /&gt;
蛋白质-蛋白质互作网络&lt;br /&gt;
{{main article|interactome}}&lt;br /&gt;
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Many [[protein–protein interaction]]s (PPIs) in a cell form ''protein interaction networks'' (PINs) where proteins are ''nodes'' and their interactions are ''edges''.&amp;lt;ref&amp;gt;{{Cite journal|last=Habibi|first=Iman|last2=Emamian|first2=Effat S.|last3=Abdi|first3=Ali|date=2014-01-01|title=Quantitative analysis of intracellular communication and signaling errors in signaling networks|journal=BMC Systems Biology|volume=8|pages=89|doi=10.1186/s12918-014-0089-z|issn=1752-0509|pmc=4255782|pmid=25115405}}&amp;lt;/ref&amp;gt; PINs are the most intensely analyzed networks in biology. There are dozens of PPI detection methods to identify such interactions. The [[Y2H|yeast two-hybrid system]] is a commonly used experimental technique for the study of binary interactions.&amp;lt;ref&amp;gt;{{cite journal |author= Mashaghi, A. |title= Investigation of a protein complex network |journal= European Physical Journal |year= 2004  |volume= 41 |issue= 1 |pages= 113–121 |doi=10.1140/epjb/e2004-00301-0|display-authors=etal|arxiv= cond-mat/0304207|bibcode= 2004EPJB...41..113M}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Many protein–protein interactions (PPIs) in a cell form protein interaction networks (PINs) where proteins are nodes and their interactions are edges. PINs are the most intensely analyzed networks in biology. There are dozens of PPI detection methods to identify such interactions. The yeast two-hybrid system is a commonly used experimental technique for the study of binary interactions.&lt;br /&gt;
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在细胞中，大量'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;蛋白质之间相互作用 Protein-Protein Interactions(PPIs)&amp;lt;/font&amp;gt;'''形成'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;蛋白质相互作用网络 Protein Interaction Networks(PINs)&amp;lt;/font&amp;gt;'''，其中蛋白质是顶点，它们的相互作用是连边。蛋白质互作网络是生物学中分析最深入的网络，现有几十种基于PPIs的检测方法来识别蛋白质间的相互作用。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;酵母双杂交系统 Yeast Two-Hybrid System&amp;lt;/font&amp;gt;'''是一种研究二元相互作用的常用实验技术。&lt;br /&gt;
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Recent studies have indicated conservation of molecular networks through deep evolutionary time.&amp;lt;ref&amp;gt;{{cite journal |author= Sharan, R. |title= Conserved patterns of protein interaction in multiple species |journal= Proceedings of the National Academy of Sciences of the United States of America |year= 2005  |volume= 102 |issue= 6 |pages= 1974–1979 |doi=10.1073/pnas.0409522102|pmid= 15687504 |display-authors=etal|pmc= 548573|bibcode= 2005PNAS..102.1974S}}&amp;lt;/ref&amp;gt; Moreover, it has been discovered that proteins with high degrees of connectedness are more likely to be essential for survival than proteins with lesser degrees.&amp;lt;ref&amp;gt;{{cite journal |author= Jeong, H. |title= Lethality and centrality in protein networks |journal= Nature |year= 2001  |volume= 411 |issue= 6833 |pages= 41–42 |doi=10.1038/35075138 |pmid=11333967|display-authors=etal|bibcode= 2001Natur.411...41J |arxiv= cond-mat/0105306 }}&amp;lt;/ref&amp;gt; This suggests that the overall composition of the network (not simply interactions between protein pairs) is important for the overall functioning of an organism.&lt;br /&gt;
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Recent studies have indicated conservation of molecular networks through deep evolutionary time. Moreover, it has been discovered that proteins with high degrees of connectedness are more likely to be essential for survival than proteins with lesser degrees. This suggests that the overall composition of the network (not simply interactions between protein pairs) is important for the overall functioning of an organism.&lt;br /&gt;
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近年来的研究表明，分子网络在深层进化过程中是保守的（在进化过程中的改变较少）。此外，已经发现相较于低度值的蛋白质，具有高度值的蛋白质对物种的生存更加重要。这表明网络的各个部分的相互配合(不仅仅是蛋白质之间简单的相互作用)对于有机体的整体功能具有重要意义。&lt;br /&gt;
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===Gene regulatory networks (DNA–protein interaction networks)===&lt;br /&gt;
基因调控网络（DNA-蛋白质交互网络）&lt;br /&gt;
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{{main article| Gene regulatory network}}&lt;br /&gt;
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The activity of genes is regulated by [[transcription factors]], proteins that typically bind to [[DNA]]. Most transcription factors bind to multiple binding sites in a [[genome]]. As a result, all cells have complex [[gene regulatory networks]]. For instance, the [[human genome]] encodes on the order of 1,400 DNA-binding transcription factors that regulate the expression of more than 20,000 human genes.&amp;lt;ref&amp;gt;{{cite journal |author= Vaquerizas, J.-M. |title= A census of human transcription factors: function, expression and evolution |journal= Nature Reviews Genetics |year= 2009  |volume= 10 |issue= 4 |pages= 252–263 |pmid= 19274049 |doi=10.1038/nrg2538|display-authors=etal}}&amp;lt;/ref&amp;gt; Technologies to study gene regulatory networks include [[ChIP-chip]], [[ChIP-seq]], [[CliP-seq]], and others.&lt;br /&gt;
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The activity of genes is regulated by transcription factors, proteins that typically bind to DNA. Most transcription factors bind to multiple binding sites in a genome. As a result, all cells have complex gene regulatory networks. For instance, the human genome encodes on the order of 1,400 DNA-binding transcription factors that regulate the expression of more than 20,000 human genes. Technologies to study gene regulatory networks include ChIP-chip, ChIP-seq, CliP-seq, and others.&lt;br /&gt;
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与DNA结合的蛋白质中，最典型的是负责调控基因表达的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;转录因子 Transcription Factors&amp;lt;/font&amp;gt;'''。大多数转录因子可与基因组中的多个位点结合。因此，所有的细胞都有复杂的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因调控网络 Gene Regulatory Networks&amp;lt;/font&amp;gt;'''。例如，人类基因组中有1400个左右的基因编码可与DNA结合的转录因子，它们参与到调节超过20000个人类基因的表达过程。研究基因调控网络的技术包括 ChIP-chip、 ChIP-seq、 CliP-seq 等。&lt;br /&gt;
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===Gene co-expression networks (transcript–transcript association networks)===&lt;br /&gt;
基因共表达网络（转录-转录关联网络）&lt;br /&gt;
{{main article| Gene co-expression networks}}&lt;br /&gt;
  --～～～Gene co-expression networks  释义注意统一  &lt;br /&gt;
Gene co-expression networks can be interpreted as association networks between variables that measure transcript abundances. These networks have been used to provide a systems biologic analysis of DNA microarray data, RNA-seq data, miRNA data etc.&lt;br /&gt;
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Gene co-expression networks can be interpreted as association networks between variables that measure transcript abundances. These networks have been used to provide a systems biologic analysis of DNA microarray data, RNA-seq data, miRNA data etc.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因共表达网络 Gene Co-expression Networks &amp;lt;/font&amp;gt;'''可以看作是衡量转录丰度的变量之间的关联网络。这些网络已经被用于对DNA微阵列数据、 RNA-seq数据、 miRNA等数据进行系统生物学分析。&lt;br /&gt;
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[[Weighted correlation network analysis|weighted gene co-expression network analysis]] is widely used to identify co-expression modules and intramodular hub genes. Co-expression modules may correspond to cell types or pathways. Highly connected intramodular hubs can be interpreted as representatives of their respective module.&lt;br /&gt;
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weighted gene co-expression network analysis is widely used to identify co-expression modules and intramodular hub genes. Co-expression modules may correspond to cell types or pathways. Highly connected intramodular hubs can be interpreted as representatives of their respective module.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因共表达加权网络分析 Weighted Gene Co-expression Network Analysis &amp;lt;/font&amp;gt;'''被广泛应用于鉴定共表达模块（具有相似表达模式的基因很可能是紧密共调控的，功能紧密相关的或同一条信号通路或过程的成员，有其特定的生理意义，用网络的方式表示时，可划分到一个模块里）和模块内的核心基因。共表达模块可能与外部特征相关的信息关联在一起，如细胞类型或病症通路。具有大量连接的模块内节点可以代表模块内的其他基因，显示出这些基因共有的特点。&lt;br /&gt;
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===Metabolic networks===&lt;br /&gt;
代谢网络&lt;br /&gt;
{{main article|metabolic network}}&lt;br /&gt;
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The chemical compounds of a living cell are connected by biochemical reactions which convert one compound into another. The reactions are catalyzed by [[enzymes]]. Thus, all compounds in a cell are parts of an intricate biochemical network of reactions which is called [[metabolic network]]. It is possible to use network analyses to infer how selection acts on metabolic pathways.&amp;lt;ref name=&amp;quot;Proulx&amp;quot; /&amp;gt;&lt;br /&gt;
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The chemical compounds of a living cell are connected by biochemical reactions which convert one compound into another. The reactions are catalyzed by enzymes. Thus, all compounds in a cell are parts of an intricate biochemical network of reactions which is called metabolic network. It is possible to use network analyses to infer how selection acts on metabolic pathways.&lt;br /&gt;
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活细胞中的化合物相互之间进行生物化学反应，从而实现化合物之间的转变。这些反应是由酶催化的。因此，一个细胞中的所有化合物都是一个复杂的生化反应网络的一部分，该网络被称为代谢网络。可使用网络分析来推断筛选是如何影响代谢通路的。&lt;br /&gt;
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===Signaling networks===&lt;br /&gt;
信号网络&lt;br /&gt;
{{main article|Cell signaling}}&lt;br /&gt;
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Signals are transduced within cells or in between cells and thus form complex signaling networks. For instance, in the [[MAPK/ERK pathway]] is transduced from the cell surface to the cell nucleus by a series of protein–protein interactions, phosphorylation reactions, and other events. Signaling networks typically integrate [[interactome|protein–protein interaction networks]], [[gene regulatory networks]], and [[metabolic network]]s.&lt;br /&gt;
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Signals are transduced within cells or in between cells and thus form complex signaling networks. For instance, in the MAPK/ERK pathway is transduced from the cell surface to the cell nucleus by a series of protein–protein interactions, phosphorylation reactions, and other events. Signaling networks typically integrate protein–protein interaction networks, gene regulatory networks, and metabolic networks.&lt;br /&gt;
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信号在细胞内或细胞之间传递，从而形成复杂的信号网络。例如，MAPK/ERK 通路通过一系列蛋白质-蛋白质相互作用、磷酸化反应和其他事件将信号从细胞表面传递到细胞核内。信号网络通常包含了蛋白质-蛋白质相互作用网络、基因调控网络和代谢网络。&lt;br /&gt;
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===Neuronal networks===&lt;br /&gt;
神经网络&lt;br /&gt;
{{main article| Biological neural network}}&lt;br /&gt;
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The complex interactions in the [[brain]] make it a perfect candidate to apply network theory. [[Neuron]]s in the brain are deeply connected with one another and this results in complex networks being present in the structural and functional aspects of the brain.&amp;lt;ref&amp;gt;{{cite journal |author1=Bullmore, E.  |author2=O. Sporns  |lastauthoramp=yes |title= Complex brain networks: graph theoretical analysis of structural and functional systems |journal= Nature Reviews Neuroscience |year= 2009  |volume= 10 |issue= 3 |pages= 186–198 |doi=10.1038/nrn2575 |pmid=19190637}}&amp;lt;/ref&amp;gt; For instance, [[small-world network]] properties have been demonstrated in connections between cortical areas of the primate brain&amp;lt;ref&amp;gt;{{cite journal |author= Stephan, K.E. |title= Computational analysis of functional connectivity between areas of primate cerebral cortex |journal= [[Philosophical Transactions of the Royal Society B]]|year= 2000  |volume= 355 |issue= 1393 |pages= 111–126 |doi=10.1098/rstb.2000.0552|pmid= 10703047 |display-authors=etal|pmc= 1692715}}&amp;lt;/ref&amp;gt; or during swallowing in humans.&amp;lt;ref&amp;gt;{{Cite journal|date=2016-12-01|title=Functional connectivity patterns of normal human swallowing: difference among various viscosity swallows in normal and chin-tuck head positions|journal=Brain Research|volume=1652|pages=158–169|doi=10.1016/j.brainres.2016.09.041|pmid=27693396|pmc=5102805|issn=0006-8993|last1=Jestrović|first1=Iva|last2=Coyle|first2=James L|last3=Perera|first3=Subashan|last4=Sejdić|first4=Ervin}}&amp;lt;/ref&amp;gt; This suggests that cortical areas of the brain are not directly interacting with each other, but most areas can be reached from all others through only a few interactions.&lt;br /&gt;
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The complex interactions in the brain make it a perfect candidate to apply network theory. Neurons in the brain are deeply connected with one another and this results in complex networks being present in the structural and functional aspects of the brain. For instance, small-world network properties have been demonstrated in connections between cortical areas of the primate brain or during swallowing in humans. This suggests that cortical areas of the brain are not directly interacting with each other, but most areas can be reached from all others through only a few interactions.&lt;br /&gt;
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大脑中神经元之间复杂的相互作用使其成为应用网络理论的绝佳场景。大脑中的神经元彼此之间有着紧密的联系，其相互交织形成的复杂网络是人脑结构和功能的基础。例如，灵长类动物大脑皮层区域之间的连接或者人类吞咽时神经网络的行为已经证明了，这二者都具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;小世界网络属性Small-World Network &amp;lt;/font&amp;gt;'''。这表明大脑皮层各区域之间并不直接进行相互作用，而是大部分区域通过少量的节点区域进行相互沟通。&lt;br /&gt;
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===Food webs===&lt;br /&gt;
食物链网络&lt;br /&gt;
{{main article|Food web}}&lt;br /&gt;
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All organisms are connected to each other through feeding interactions. That is, if a species eats or is eaten by another species, they are connected in an intricate [[food web]] of predator and prey interactions. The stability of these interactions has been a long-standing question in ecology.&amp;lt;ref&amp;gt;{{cite journal |author= MacArthur, R.H. |author-link= Robert MacArthur|title= Fluctuations in animal populations and a measure of community stability |journal= Ecology |year= 1955  |volume= 36 |issue= 3 |pages= 533–536 |doi=10.2307/1929601|jstor= 1929601}}&amp;lt;/ref&amp;gt; That is to say, if certain individuals are removed, what happens to the network (i.e. does it collapse or adapt)? Network analysis can be used to explore food web stability and determine if certain network properties result in more stable networks. Moreover, network analysis can be used to determine how selective removals of species will influence the food web as a whole.&amp;lt;ref&amp;gt;{{cite journal |author= Dunne, J.A. |title= Network structure and biodiversity loss in food webs: robustness increases with connectance |journal= Ecology Letters |year= 2002  |volume= 5 |issue= 4 |pages= 558–567 |doi=10.1046/j.1461-0248.2002.00354.x|display-authors=etal}}&amp;lt;/ref&amp;gt; This is especially important considering the potential species loss due to global climate change.&lt;br /&gt;
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All organisms are connected to each other through feeding interactions. That is, if a species eats or is eaten by another species, they are connected in an intricate food web of predator and prey interactions. The stability of these interactions has been a long-standing question in ecology. That is to say, if certain individuals are removed, what happens to the network (i.e. does it collapse or adapt)? Network analysis can be used to explore food web stability and determine if certain network properties result in more stable networks. Moreover, network analysis can be used to determine how selective removals of species will influence the food web as a whole. This is especially important considering the potential species loss due to global climate change.&lt;br /&gt;
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所有的生物都具有捕食别的物种或被别的物种捕食的情况。也就是说，如果一个物种吃了另一个物种或被另一个物种吃了，它们就在一个复杂的捕食者和被捕食者相互作用的食物网中被联系起来。这些相互作用的稳定性长期困扰着生态学家们。具体来说，如果某些个体被移除，那么网络会发生什么？(即它是崩溃还是重新适应) ？网络分析可以用来探索食物网的稳定性，并确定某些网络特性是否会导致更稳定的网络。此外，网络分析可以用来确定物种的选择性迁移将如何影响整个食物链网。考虑到全球气候变化可能导致大量物种的消失，使用网络分析来研究食物链网的特性是尤其重要的。&lt;br /&gt;
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===Between-species interaction networks===&lt;br /&gt;
物种间交互网络&lt;br /&gt;
{{main article|social relation}}&lt;br /&gt;
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In biology, pairwise interactions have historically been the focus of intense study. With the recent advances in [[network science]], it has become possible to scale up pairwise interactions to include individuals of many species involved in many sets of interactions to understand the structure and function of larger [[ecological network]]s.&amp;lt;ref&amp;gt;{{cite journal |author= Bascompte, J. |title= Disentangling the web of life |journal= Science |year= 2009 |volume=325 |issue= 5939 |pages= 416–419 |doi=10.1126/science.1170749 |pmid=19628856|bibcode=2009Sci...325..416B }}&amp;lt;/ref&amp;gt; The use of [[Social network analysis|network analysis]] can allow for both the discovery and understanding how these complex interactions link together within the system’s network, a property which has previously been overlooked. This powerful tool allows for the study of various types of interactions (from [[competitive]] to [[cooperative]]) using the same general framework.&amp;lt;ref&amp;gt;{{cite journal |author= Krause, J. |title= Animal social networks: an introduction |journal= Behav. Ecol. Sociobiol.  |year= 2009 |volume= 63 |issue= 7 |pages= 967–973 |doi=10.1007/s00265-009-0747-0|display-authors=etal}}&amp;lt;/ref&amp;gt; For example, plant-[[pollinator]] interactions are mutually beneficial and often involve many different species of pollinators as well as many different species of plants. These interactions are critical to plant reproduction and thus the accumulation of resources at the base of the [[food chain]] for primary consumers, yet these interaction networks are threatened by [[Human impact on the environment|anthropogenic]] change. The use of network analysis can illuminate how [[pollination network]]s work and may in turn inform conservation efforts.&amp;lt;ref name=&amp;quot;Memmott, J., et al. 2004 2605-261&amp;quot;&amp;gt;{{cite journal |author= Memmott, J. |title= Tolerance of pollination networks to species extinctions |journal= [[Philosophical Transactions of the Royal Society B]] |year= 2004 |volume= 271 |issue= 1557 |pages= 2605–261 |doi=10.1098/rspb.2004.2909|pmid= 15615687 |pmc= 1691904 |display-authors=etal}}&amp;lt;/ref&amp;gt; Within pollination networks, nestedness (i.e., specialists interact with a subset of species that generalists interact with), redundancy (i.e., most plants are pollinated by many pollinators), and [[modularity (networks)|modularity]] play a large role in network stability.&amp;lt;ref name=&amp;quot;Memmott, J., et al. 2004 2605-261&amp;quot;/&amp;gt;&amp;lt;ref name=&amp;quot;Olesen, J., et al. 2007 19891–19896&amp;quot;&amp;gt;{{cite journal |author= Olesen, J. |title= The modularity of pollination networks |journal= PNAS |year= 2007 |volume= 104 |issue= 50 |pages= 19891–19896 |doi=10.1073/pnas.0706375104|pmid= 18056808 |pmc= 2148393 |display-authors=etal|bibcode= 2007PNAS..10419891O }}&amp;lt;/ref&amp;gt; These network properties may actually work to slow the spread of disturbance effects through the system and potentially buffer the pollination network from anthropogenic changes somewhat.&amp;lt;ref name=&amp;quot;Olesen, J., et al. 2007 19891–19896&amp;quot;/&amp;gt; More generally, the structure of species interactions within an ecological network can tell us something about the diversity, richness, and robustness of the network.&amp;lt;ref&amp;gt;{{cite journal |author= Campbell, V. |title= Experimental design and the outcome and interpretation of diversity-stability relations |journal= Oikos |year= 2011 |volume= 120 |issue= 3 |pages= 399–408 |doi=10.1111/j.1600-0706.2010.18768.x|display-authors=etal}}&amp;lt;/ref&amp;gt; Researchers can even compare current constructions of species interactions networks with historical reconstructions of ancient networks to determine how networks have changed over time.&amp;lt;ref&amp;gt;{{cite journal |author= Lotze, H. |title= Historical changes in marine resources, food-web structure and ecosystem functioning in the Adriatic Sea, Mediterranean |journal= Ecosystems |year= 2011 |volume= 14 |issue= 2 |pages= 198–222 |doi=10.1007/s10021-010-9404-8|display-authors=etal}}&amp;lt;/ref&amp;gt; Recent research into these complex species interactions networks is highly concerned with understanding what factors (e.g., diversity) lead to network stability.&amp;lt;ref&amp;gt;{{cite journal |author= Romanuk, T. |title= Maintenance of positive diversity-stability relations along a gradient of environmental stress |journal= PLoS ONE |year= 2010 |volume= 5 |issue= 4 |pages= e10378 |doi=10.1371/journal.pone.0010378|display-authors=etal |pmid=20436913 |pmc=2860506|bibcode=2010PLoSO...510378R }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In biology, pairwise interactions have historically been the focus of intense study. With the recent advances in network science, it has become possible to scale up pairwise interactions to include individuals of many species involved in many sets of interactions to understand the structure and function of larger ecological networks. The use of network analysis can allow for both the discovery and understanding how these complex interactions link together within the system’s network, a property which has previously been overlooked. This powerful tool allows for the study of various types of interactions (from competitive to cooperative) using the same general framework. For example, plant-pollinator interactions are mutually beneficial and often involve many different species of pollinators as well as many different species of plants. These interactions are critical to plant reproduction and thus the accumulation of resources at the base of the food chain for primary consumers, yet these interaction networks are threatened by anthropogenic change. The use of network analysis can illuminate how pollination networks work and may in turn inform conservation efforts. Within pollination networks, nestedness (i.e., specialists interact with a subset of species that generalists interact with), redundancy (i.e., most plants are pollinated by many pollinators), and modularity play a large role in network stability. These network properties may actually work to slow the spread of disturbance effects through the system and potentially buffer the pollination network from anthropogenic changes somewhat. Researchers can even compare current constructions of species interactions networks with historical reconstructions of ancient networks to determine how networks have changed over time. Recent research into these complex species interactions networks is highly concerned with understanding what factors (e.g., diversity) lead to network stability.&lt;br /&gt;
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在生物学中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;成对相互作用 Pairwise Interactions &amp;lt;/font&amp;gt;'''历来是研究的重点。随着近几年网络科学的发展，成对相互作用内涵很有可能被丰富，外延甚至能涵盖有许多物种个体参与的多组相互作用，进而帮助我们理解更大'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生态网络 Ecological Networks &amp;lt;/font&amp;gt;'''的结构和功能。使用网络分析可以发现和理解这些复杂的交互是如何连接成网络的，该形成过程之前经常被忽略。这个强大的工具使得同一个普世框架可以研究各种类型的交互(从竞争到合作)。例如，植物与传粉者之间的相互作用是互惠互利的，通常会涉及到许多不同种类的传粉者以及许多不同种类的植物。这些相互作用对植物的生殖和食物链底层初级消费者的资源积累至关重要，然而这些相互作用网络正受到人造变化的威胁。网络分析的使用可以说明授粉网络是如何工作的，反过来也可以为相关植物保护工作提供必要信息。在授粉网络中，内嵌性(某一领域的植物被少数几个昆虫授粉，剩下的各种昆虫只对少数几种植物授粉)、冗余性(即大多数植物是由许多授粉者授粉的)和模块性（一些物种相互之间连接的很紧密，与剩余的物种连接不紧密，由此形成一个个团簇，也可称之为“模块”）在网络稳定性中扮演着重要角色。这些网络属性实际上可以减缓干扰效应在系统上的传播，并可能缓冲授粉网络受到的人造变化的影响。研究人员甚至可以将物种相互作用网络的现有结构与该网络过去的结构进行比较，以确定网络在时间尺度上是如何演化的。最近对这些复杂物种间相互作用网络的研究聚焦于理解是什么因素(如多样性)造就了网络的稳定性。&lt;br /&gt;
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===Within-species interaction networks===&lt;br /&gt;
物种内交互网络&lt;br /&gt;
{{main article|social relation}}&lt;br /&gt;
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Network analysis provides the ability to quantify associations between individuals, which makes it possible to infer details about the network as a whole at the species and/or population level.&amp;lt;ref&amp;gt;{{cite journal |author= Croft, D.P |title= Social networks in the guppy (Poecilia reticulate) |journal= [[Philosophical Transactions of the Royal Society B]] |year= 2004 |volume= 271 |issue= Suppl |pages= S516–S519 |doi=10.1098/rsbl.2004.0206|pmid= 15801620 |pmc= 1810091 |display-authors=etal}}&amp;lt;/ref&amp;gt; One of the most attractive features of the network paradigm would be that it provides a single conceptual framework in which the social organisation of animals at all levels (individual, dyad, group, population) and for all types of interaction (aggressive, cooperative, sexual etc.) can be studied.&amp;lt;ref&amp;gt;{{Cite journal|last=Krause, Lusseau, James|first=Jens, David, Richard|date=1 May 2009|title=Animal social networks: an introduction|url=https://www.researchgate.net/publication/226996156_Animal_social_networks_An_introduction|journal=Behavioral Ecology and Sociobiology|volume=63|pages=967-973|doi=10.1007/s00265-009-0747-0|via=}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Network analysis provides the ability to quantify associations between individuals, which makes it possible to infer details about the network as a whole at the species and/or population level. One of the most attractive features of the network paradigm would be that it provides a single conceptual framework in which the social organisation of animals at all levels (individual, dyad, group, population) and for all types of interaction (aggressive, cooperative, sexual etc.) can be studied.&lt;br /&gt;
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网络分析可以量化个体之间的关联，这使得在物种和/或种群水平推断整个网络的细节成为可能。网络范式最吸引人的特征之一是，它提供了一个统一的概念框架，使得社会性动物在所有层次上(单，双，组，群)--[[用户:Ryan|Ryan]]（[[用户讨论:Ryan|讨论]]）和所有类型的互动中(攻击，合作，性等等)都可以被研究。&lt;br /&gt;
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Researchers interested in [[ethology]] across a multitude of taxa, from insects to primates, are starting to incorporate network analysis into their research. Researchers interested in social insects (e.g., ants and bees) have used network analyses to better understand division of labor, task allocation, and foraging optimization within colonies;&amp;lt;ref&amp;gt;{{cite journal |author= Dornhaus, A. |title= Benefits of recruitment in honey bees: Effects of ecology and colony size in an individual-based model |journal= Behavioral Ecology |year= 2006 |volume=17 |issue= 3 |pages= 336–344 |doi=10.1093/beheco/arj036|display-authors=etal|doi-access= free }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal |author= Linksvayer, T. |title= Developmental evolution in social insects: Regulatory networks from genes to societies |journal= Journal of Experimental Zoology Part B: Molecular and Developmental Evolution |year= 2012 |volume=318 |issue= 3 |pages= 159–169 |doi=10.1002/jez.b.22001|pmid= 22544713 |display-authors=etal}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal |author= Mullen, R. |title= A review of ant algorithms |journal= Expert Systems with Applications |year= 2009 |volume=36 |issue= 6 |pages= 9608–9617 |doi=10.1016/j.eswa.2009.01.020|display-authors=etal}}&amp;lt;/ref&amp;gt; Other researchers are interested in how certain network properties at the group and/or population level can explain individual level behaviors. Studies have demonstrated how animal social network structure can be influenced by factors ranging from characteristics of the environment to characteristics of the individual, such as developmental experience and personality. At the level of the individual, the patterning of social connections can be an important determinant of [[Fitness (biology)|fitness]], predicting both survival and reproductive success. At the population level, network structure can influence the patterning of ecological and evolutionary processes, such as [[frequency-dependent selection]] and disease and information transmission.&amp;lt;ref&amp;gt;{{Cite journal|last=Croft, Darden, Wey|first=Darren P., Safi K., Tina W.|date=2016|title=Current directions in animal social networks|url=https://www.researchgate.net/publication/310819524_Current_directions_in_animal_social_networks|journal=Current Opinion in Behavioral Sciences|volume=12|pages=52-58|doi=10.1016/j.cobeha.2016.09.001|via=|hdl=10871/23348|hdl-access=free}}&amp;lt;/ref&amp;gt; For instance, a study on [[Wire-tailed manakin|wire-tailed manakins]] (a small passerine bird)  found that a male’s [[degree (graph theory)|degree]] in the network largely predicted the ability of the male to rise in the social hierarchy (i.e. eventually obtain a territory and matings).&amp;lt;ref&amp;gt;{{cite journal |author= Ryder, T.B. |title= Social networks in the lek-mating wire-tailed manakin (''Pipra filicauda'') |journal= [[Philosophical Transactions of the Royal Society B]]  |year= 2008 |volume= 275 |issue= 1641 |pages= 1367–1374 |doi=10.1098/rspb.2008.0205 |pmid=18381257 |pmc=2602714|display-authors=etal}}&amp;lt;/ref&amp;gt; In [[bottlenose dolphin]] groups, an individual’s degree and [[betweenness centrality]] values may predict whether or not that individual will exhibit certain behaviors, like the use of side flopping and upside-down lobtailing to lead group traveling efforts; individuals with high betweenness values are more connected and can obtain more information, and thus are better suited to lead group travel and therefore tend to exhibit these signaling behaviors more than other group members.&amp;lt;ref&amp;gt;{{cite journal |author= Lusseau, D. |title= Evidence for social role in a dolphin social network |journal= Evolutionary Ecology |year= 2007 |volume=21 |issue= 3 |pages= 357–366 |doi=10.1007/s10682-006-9105-0|arxiv= q-bio/0607048 }}&amp;lt;/ref&amp;gt; &lt;br /&gt;
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Researchers interested in ethology across a multitude of taxa, from insects to primates, are starting to incorporate network analysis into their research. Researchers interested in social insects (e.g., ants and bees) have used network analyses to better understand division of labor, task allocation, and foraging optimization within colonies; Other researchers are interested in how certain network properties at the group and/or population level can explain individual level behaviors. Studies have demonstrated how animal social network structure can be influenced by factors ranging from characteristics of the environment to characteristics of the individual, such as developmental experience and personality. At the level of the individual, the patterning of social connections can be an important determinant of fitness, predicting both survival and reproductive success. At the population level, network structure can influence the patterning of ecological and evolutionary processes, such as frequency-dependent selection and disease and information transmission. For instance, a study on wire-tailed manakins (a small passerine bird)  found that a male’s degree in the network largely predicted the ability of the male to rise in the social hierarchy (i.e. eventually obtain a territory and matings). In bottlenose dolphin groups, an individual’s degree and betweenness centrality values may predict whether or not that individual will exhibit certain behaviors, like the use of side flopping and upside-down lobtailing to lead group traveling efforts; individuals with high betweenness values are more connected and can obtain more information, and thus are better suited to lead group travel and therefore tend to exhibit these signaling behaviors more than other group members. &lt;br /&gt;
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从昆虫到灵长类动物，对动物行为学感兴趣的研究人员，开始将网络分析纳入到他们的研究中。对社会性昆虫(如蚂蚁和蜜蜂)感兴趣的研究人员利用网络分析来更好地理解，群体内昆虫的分工、任务分配和觅食优化; 其他研究人员感兴趣的点则在群体和/或种群水平的某些网络特性如何影响个体的行为。研究表明，环境特征和个体特征，如经验发展和人格等因素，可影响动物的社会网络结构。在个体层面上，社会联系的模式是个体适应性的一个关键决定因素，可能预测着个体生存和繁殖的成功概率。在种群水平上，网络结构可以影响生态和进化过程的模式，如频率依赖性选择和疾病及信息的传递。例如，一项针对线尾 manakins (一种小型雀形目鸟)的研究发现，在网络中雄性个体的度值在很大程度上预测了雄性在社会等级中进阶的能力(即最终可获得领地和交配机会)。在宽吻海豚群体中，个体的度值和介数中心性可以预测个体是否会表现出某些行为，比如使用侧翻和头潜入水中尾巴举向空中来表示自己愿意领导团队的迁移; 介数值较高的个体有着更多的连接，可以获得更多信息，因此更适合领导团队迁行，因此比其他团队成员更容易表现出传递这些信号的行为。&lt;br /&gt;
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[[Social network analysis]] can also be used to describe the social organization within a species more generally, which frequently reveals important proximate mechanisms promoting the use of certain behavioral strategies. These descriptions are frequently linked to ecological properties (e.g., resource distribution). For example, network analyses revealed subtle differences in the group dynamics of two related equid [[fission-fusion society|fission-fusion]] species, [[Grevy’s zebra]] and [[onager]]s, living in variable environments; Grevy’s zebras show distinct preferences in their association choices when they fission into smaller groups, whereas onagers do not.&amp;lt;ref&amp;gt;{{cite journal |author= Sundaresan, S. |title= Network metrics reveal differences in social organization between two fission-fusion species, Grevy's zebra and onager |journal= Oecologia |year= 2007 |volume= 151 |issue= 1 |pages= 140–149 |doi=10.1007/s00442-006-0553-6|pmid= 16964497 |display-authors=etal|bibcode= 2007Oecol.151..140S }}&amp;lt;/ref&amp;gt; Similarly, researchers interested in primates have also utilized network analyses to compare social organizations across the diverse [[primate]] order, suggesting that using network measures (such as [[centrality]], [[assortativity]], [[modularity (networks)|modularity]], and betweenness) may be useful in terms of explaining the types of social behaviors we see within certain groups and not others.&amp;lt;ref&amp;gt;{{cite journal |author1=Kasper, C. |author2=Voelkl, B. |title= A social network analysis of primate groups |journal= Primates |year= 2009 |volume= 50 |issue=4 |pages= 343–356 |doi=10.1007/s10329-009-0153-2|pmid=19533270 }}&amp;lt;/ref&amp;gt; &lt;br /&gt;
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Social network analysis can also be used to describe the social organization within a species more generally, which frequently reveals important proximate mechanisms promoting the use of certain behavioral strategies. These descriptions are frequently linked to ecological properties (e.g., resource distribution). For example, network analyses revealed subtle differences in the group dynamics of two related equid fission-fusion species, Grevy’s zebra and onagers, living in variable environments; Grevy’s zebras show distinct preferences in their association choices when they fission into smaller groups, whereas onagers do not. Similarly, researchers interested in primates have also utilized network analyses to compare social organizations across the diverse primate order, suggesting that using network measures (such as centrality, assortativity, modularity, and betweenness) may be useful in terms of explaining the types of social behaviors we see within certain groups and not others. &lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会网络分析 Social Network Analysis &amp;lt;/font&amp;gt;'''也可以用来更广泛地描述一个物种内的社会组织，它经常揭示出那些促进某些行为策略使用的重要邻近机制。这些描述经常与生态属性(例如，资源分配)联系在一起。例如，网络分析揭示了生活在变化环境中的两个相关的等裂变融合物种——格雷维斑马和骑驴——的群体动力学的细微差异（裂变融合社会是这样一个社会，其规模和组成随着时间的流逝以及动物在整个环境中的移动而变化；动物会合并成一团（融合）-例如在一个地方睡觉-或分裂（裂变）-例如白天在小组中觅食）; 格雷维斑马在分裂为较小的群体时，在关联选择上表现出明显的偏好，而骑驴则不然。同样，对灵长类动物感兴趣的研究人员也利用网络分析来比较不同灵长类动物的社会组织，这表明以网络的视角进行测量(如集中性、协调性、模块性和中介性)可能有助于解释我们在某些特定群体中看到的社会行为类型。&lt;br /&gt;
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Finally, social network analysis can also reveal important fluctuations in animal behaviors across changing environments. For example, network analyses in female [[chacma baboon]]s (''Papio hamadryas ursinus'') revealed important dynamic changes across seasons which were previously unknown; instead of creating stable, long-lasting social bonds with friends, baboons were found to exhibit more variable relationships which were dependent on short-term contingencies related to group level dynamics as well as environmental variability.&amp;lt;ref&amp;gt;{{cite journal |author= Henzi, S. |title= Cyclicity in the structure of female baboon social networks |journal= Behavioral Ecology and Sociobiology |year= 2009 |volume= 63 |issue= 7 |pages= 1015–1021 |doi=10.1007/s00265-009-0720-y|display-authors=etal}}&amp;lt;/ref&amp;gt; Changes in an individual's social network environment can also influence characteristics such as 'personality': for example, social spiders that huddle with bolder neighbours tend to increase also in boldness.&amp;lt;ref&amp;gt;{{cite journal |author= Hunt, ER. |title= Social interactions shape individual and collective personality in social spiders |journal= Proceedings of the Royal Society B |year= 2018 |volume= 285 |issue= 1886 |pages= 20181366 |doi=10.1098/rspb.2018.1366|pmid= 30185649 |pmc= 6158534 |display-authors=etal}}&amp;lt;/ref&amp;gt; This is a very small set of broad examples of how researchers can use network analysis to study animal behavior. Research in this area is currently expanding very rapidly, especially since the broader development of animal borne tags and [[computer vision]] that can be used to automate the collection of social associations.&amp;lt;ref&amp;gt;{{Cite journal|last=Krause J, Krause S, Arlinghaus R, Psorakis I, Roberts S, Rutz C|first=|date=2013|title=Reality mining of animal social systems.|url=https://risweb.st-andrews.ac.uk/portal/en/researchoutput/reality-mining-of-animal-social-systems(570e21d0-527e-4275-8926-c4b479c55ee4).html|journal=Trends in Ecology and Evolution|volume=28|pages=541-551|doi=10.1016/j.tree.2013.06.002|via=}}&amp;lt;/ref&amp;gt; Social network analysis is a valuable tool for studying animal behavior across all animal species, and has the potential to uncover new information about animal behavior and social ecology that was previously poorly understood.&lt;br /&gt;
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Finally, social network analysis can also reveal important fluctuations in animal behaviors across changing environments. For example, network analyses in female chacma baboons (Papio hamadryas ursinus) revealed important dynamic changes across seasons which were previously unknown; instead of creating stable, long-lasting social bonds with friends, baboons were found to exhibit more variable relationships which were dependent on short-term contingencies related to group level dynamics as well as environmental variability. Changes in an individual's social network environment can also influence characteristics such as 'personality': for example, social spiders that huddle with bolder neighbours tend to increase also in boldness. This is a very small set of broad examples of how researchers can use network analysis to study animal behavior. Research in this area is currently expanding very rapidly, especially since the broader development of animal borne tags and computer vision that can be used to automate the collection of social associations. Social network analysis is a valuable tool for studying animal behavior across all animal species, and has the potential to uncover new information about animal behavior and social ecology that was previously poorly understood.&lt;br /&gt;
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最后，社会网络分析还可以揭示动物行为在不断变化的环境中的关键波动。例如，对雌性沙卡马狒狒(Papio hamadryas ursinus)的网络分析揭示了以前未知的跨季节的重要动态变化; 结果发现，狒狒之间没有建立稳定、持久的社会联系，相反，它们之间的关系是多变的。这些关系依赖于在动态的群体层面产生的短期偶然事件以及环境中的变化。个体社交网络环境的变化也会影响个体的性格特征，例如，与胆大的邻居挤在一起的蜘蛛也会更加大胆。这只是研究人员利用网络分析来研究动物行为的一小部分。这一领域的研究目前正在迅速扩展，特别是随着动物身体标记技术和计算机视觉技术的广泛发展，这些技术可以用来自动收集个体的社会关系。社会网络分析是研究所有物种的行为的一个有价值的工具，并且有可能发现以前鲜为人知的关于动物行为和社会生态的新信息。&lt;br /&gt;
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==See also==&lt;br /&gt;
&lt;br /&gt;
* [[List of omics topics in biology]]&lt;br /&gt;
生物学中的组学&lt;br /&gt;
* [[Biological network inference]]&lt;br /&gt;
生物网络推理&lt;br /&gt;
* [[Applied Statistics]]&lt;br /&gt;
应用统计学&lt;br /&gt;
* [[Biostatistics]]&lt;br /&gt;
生物统计学&lt;br /&gt;
* [[Computational Biology]]&lt;br /&gt;
计算生物学&lt;br /&gt;
* [[Systems biology]]&lt;br /&gt;
系统生物学&lt;br /&gt;
* [[Weighted correlation network analysis]] &lt;br /&gt;
权重相关性网络分析&lt;br /&gt;
* [[Interactome]]&lt;br /&gt;
交互组&lt;br /&gt;
* [[Network medicine]]&lt;br /&gt;
网络医学&lt;br /&gt;
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&lt;br /&gt;
== External links ==&lt;br /&gt;
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* [http://www.nettab.org Network Tools and Applications in Biology] (NETTAB) workshops.&lt;br /&gt;
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==References==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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==Books==&lt;br /&gt;
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*E. Estrada,  &amp;quot;The Structure of Complex Networks: Theory and Applications&amp;quot;, Oxford University Press, 2011, {{ISBN|978-0-199-59175-6}}&lt;br /&gt;
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*J. Krause, R. James, D. Franks, D. Croft, &amp;quot;Animal Social Networks&amp;quot;, Oxford University Press, 2015, {{ISBN|978-0199679041}}&lt;br /&gt;
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==External links==&lt;br /&gt;
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* [https://web.archive.org/web/20110920220214/http://www.networkbio.org/ Networkbio.org], The site of the series of Integrative Network Biology (INB) meetings. For the 2012 event also see www.networkbio.org&lt;br /&gt;
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* [http://networkbiology.org Networkbiology.org], NetworkBiology wiki site.&lt;br /&gt;
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* [http://www.lindinglab.org LindingLab.org], Technical University of Denmark (DTU) studies Network Biology and Cellular Information Processing, and is also organizing the Denmark branch of the annual &amp;quot;[http://www.lindinglab.org/events ''Integrative Network Biology and Cancer'']&amp;quot; symposium series.&lt;br /&gt;
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* [http://nrnb.org NRNB.org], The National Resource for Network Biology. A US National Institute of Health (NIH) Biomedical Technology Research Center dedicated to the study of biological networks.&lt;br /&gt;
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*[http://networkrepository.com/asn.php Network Repository] The first interactive data and network data repository with real-time visual analytics.&lt;br /&gt;
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*[https://bansallab.github.io/asnr/ Animal Social Network Repository(ASNR)] The first multi-taxonomic repository that collates 790 social networks from more than 45 species, including those of mammals, reptiles, fish, birds, and insects &lt;br /&gt;
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*[http://konect.uni-koblenz.de/networks/ KONECT – Koblenz network collection.]&lt;br /&gt;
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[[Category:Biological techniques and tools]]&lt;br /&gt;
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Category:Biological techniques and tools&lt;br /&gt;
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类别: 生物技术和工具&lt;br /&gt;
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[[Category:Bioinformatics]]&lt;br /&gt;
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Category:Bioinformatics&lt;br /&gt;
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类别: 生物信息学&lt;br /&gt;
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[[Category:Systems biology]]&lt;br /&gt;
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Category:Systems biology&lt;br /&gt;
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分类: 系统生物学&lt;br /&gt;
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[[Category:Networks]]&lt;br /&gt;
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Category:Networks&lt;br /&gt;
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类别: 网络&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Biological network]]. Its edit history can be viewed at [[生物网络/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E7%94%9F%E7%89%A9%E7%BD%91%E7%BB%9C&amp;diff=12556</id>
		<title>生物网络</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E7%94%9F%E7%89%A9%E7%BD%91%E7%BB%9C&amp;diff=12556"/>
		<updated>2020-08-15T04:07:36Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{Network Science}}&lt;br /&gt;
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{{refimprove|date=October 2011}}&lt;br /&gt;
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A '''biological network''' is any [[network science|network]] that applies to [[biological system]]s. A network is any system with sub-units that are linked into a whole, such as species units linked into a whole [[food web]]. Biological networks provide a [[graph theory|mathematical representation]] of connections found in [[ecological network|ecological]], [[phylogenetic network|evolutionary]], and [[physiological]] studies, such as [[neural networks]].&amp;lt;ref name=&amp;quot;Proulx05&amp;quot;&amp;gt;{{cite journal | last1=Proulx | first1=S. R. | last2=Promislow | first2=D. E. L. | last3=Phillips | first3=P. C. | title=Network thinking in ecology and evolution | journal=Trends in Ecology and Evolution | volume=20 | issue=6 | pages=345–353 | year=2005 | doi=10.1016/j.tree.2005.04.004 | url=http://eeb19.biosci.arizona.edu/Faculty/Dornhaus/courses/materials/papers/Proulx%20Promislow%20Phillips%20networks%20ecol%20evol.pdf | pmid=16701391 | url-status=dead | archiveurl=https://web.archive.org/web/20110815122330/http://eeb19.biosci.arizona.edu/Faculty/Dornhaus/courses/materials/papers/Proulx%20Promislow%20Phillips%20networks%20ecol%20evol.pdf | archivedate=2011-08-15 }}&amp;lt;/ref&amp;gt; The analysis of biological networks with respect to human diseases has led to the field of [[network medicine]].&amp;lt;ref&amp;gt;{{cite journal | last1 = Barabási | first1 = A. L. | last2 = Gulbahce | first2 = N. | last3 = Loscalzo | first3 = J. | year = 2011 | title = Network medicine: a network-based approach to human disease | url = | journal = Nature Reviews Genetics | volume = 12 | issue = 1| pages = 56–68 | doi=10.1038/nrg2918 | pmid=21164525 | pmc=3140052}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite journal|last=Habibi|first=Iman|last2=Emamian|first2=Effat S.|last3=Abdi|first3=Ali|date=2014-10-07|title=Advanced Fault Diagnosis Methods in Molecular Networks|journal=PLOS ONE|volume=9|issue=10|pages=e108830|doi=10.1371/journal.pone.0108830|issn=1932-6203|pmc=4188586|pmid=25290670|bibcode=2014PLoSO...9j8830H}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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A biological network is any network that applies to biological systems. A network is any system with sub-units that are linked into a whole, such as species units linked into a whole food web. Biological networks provide a mathematical representation of connections found in ecological, evolutionary, and physiological studies, such as neural networks. The analysis of biological networks with respect to human diseases has led to the field of network medicine.&lt;br /&gt;
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任何应用于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生物系统 Biological Systems&amp;lt;/font&amp;gt;'''的网络都可称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 生物网络 Biological Network&amp;lt;/font&amp;gt;'''。网络是指内部子单元可连接成一个整体的系统，如大量单个物种连接成一个整体的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;食物链网 Food Web&amp;lt;/font&amp;gt;'''。生物网络为在生态学、进化学和生理学研究中发现的联系提供了数学形式的表达，例如'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 神经网络 Neural Networks&amp;lt;/font&amp;gt;'''。生物网络分析在人类疾病方面的应用，开创了网络医学领域。&lt;br /&gt;
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==Network biology and bioinformatics==&lt;br /&gt;
网络生物学和生物信息学&lt;br /&gt;
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Complex biological systems may be represented and analyzed as computable networks. For example, ecosystems can be modeled as networks of interacting species or a protein can be modeled as a network of amino acids. Breaking a protein down farther, [[amino acids]] can be represented as a network of connected [[atom]]s, such as [[carbon]], [[nitrogen]], and [[oxygen]]. [[vertex (graph theory)|Nodes]] and [[vertex (graph theory)|edges]] are the basic components of a network. Nodes represent units in the network, while edges represent the interactions between the units. Nodes can represent a wide array of biological units, from individual organisms to individual neurons in the brain. Two important properties of a network are [[vertex (graph theory)|degree]] and [[betweenness centrality]]. Degree (or connectivity, a distinct usage from that used in [[Connectivity (graph theory)|graph theory]]) is the number of edges that connect a node, while betweenness is a measure of how central a node is in a network.&amp;lt;ref name=&amp;quot;Proulx&amp;quot;&amp;gt;{{cite journal |author= Proulx, S.R. |title= Network thinking in ecology and evolution |journal= Trends in Ecology and Evolution |year= 2005  |volume= 20 |issue= 6 |pages= 345–353 |doi=10.1016/j.tree.2005.04.004 |pmid=16701391|display-authors=etal}}&amp;lt;/ref&amp;gt; Nodes with high betweenness essentially serve as bridges between different portions of  the network (i.e. interactions must pass through this node to reach other portions of the network). In social networks, nodes with high degree or high betweenness may play important roles in the overall composition of a network.&lt;br /&gt;
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Complex biological systems may be represented and analyzed as computable networks. For example, ecosystems can be modeled as networks of interacting species or a protein can be modeled as a network of amino acids. Breaking a protein down farther, amino acids can be represented as a network of connected atoms, such as carbon, nitrogen, and oxygen. Nodes and edges are the basic components of a network. Nodes represent units in the network, while edges represent the interactions between the units. Nodes can represent a wide array of biological units, from individual organisms to individual neurons in the brain. Two important properties of a network are degree and betweenness centrality. Degree (or connectivity, a distinct usage from that used in graph theory) is the number of edges that connect a node, while betweenness is a measure of how central a node is in a network. Nodes with high betweenness essentially serve as bridges between different portions of  the network (i.e. interactions must pass through this node to reach other portions of the network). In social networks, nodes with high degree or high betweenness may play important roles in the overall composition of a network.&lt;br /&gt;
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复杂的生物系统能用可计算的网络来表示和分析。例如，生态系统可以被模拟为相互作用的物种网络，一个蛋白质可以被模拟为氨基酸网络。进一步分解蛋白质，氨基酸可以表示为一个由相互连接的原子构成的网络，如碳、氮和氧。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 节点 Nodes&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;边 Edges&amp;lt;/font&amp;gt;'''是网络的基本组成部分。节点表示网络中的单元，边表示单元之间的相互作用。节点可以代表一系列广泛的生物单元，从单个的生物体到大脑中单个的神经元。网络的两个重要性质是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度 Degree&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 介数中心性 Betweenness Centrality&amp;lt;/font&amp;gt;'''。度(或连通性，一个不同于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''的用法)是连接一个节点的边的数量，而介数中心性是衡量一个节点在网络中有多么靠近中心位置。具有高介数的节点本质上充当网络不同部分之间的桥梁(即网络其他部分的交互，必须通过这个节点)。在社会网络中，具有较高度值和较高介数的节点可能在网络的整体组成中发挥重要作用。&lt;br /&gt;
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As early as the 1980s, researchers started viewing [[DNA]] or [[genome]]s as the dynamic storage of a language system with precise computable finite [[state (computer science)|states]] represented as a [[finite state machine]].&amp;lt;ref&amp;gt;{{cite book  |author= Searls, D. |title= Artificial intelligence and molecular biology |publisher= MIT Press |year= 1993 |location= Cambridge, MA}}&amp;lt;/ref&amp;gt; Recent [[complex system]]s research has also suggested some far-reaching commonality in the organization of information in problems from biology, [[computer science]], and [[physics]], such as the [[Bose–Einstein condensate]] (a special state of matter).&amp;lt;ref&amp;gt;{{cite journal |author1=Bianconi, G.  |author2=Barabasi A.  |lastauthoramp=yes |title= Bose-Einstein condensation in complex networks |journal= Phys. Rev. Lett. |year= 2001  |volume= 86 |issue= 24 |pages= 5632–5635 |doi=10.1103/physrevlett.86.5632|pmid=11415319  |arxiv=cond-mat/0011224|bibcode=2001PhRvL..86.5632B|hdl=2047/d20000696  }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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As early as the 1980s, researchers started viewing DNA or genomes as the dynamic storage of a language system with precise computable finite states represented as a finite state machine. Recent complex systems research has also suggested some far-reaching commonality in the organization of information in problems from biology, computer science, and physics, such as the Bose–Einstein condensate (a special state of matter).&lt;br /&gt;
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早在20世纪80年代，研究人员就开始将 DNA 或基因组视为一个可存储动态信息的语言系统，因其具有可精确计算的有限状态，所以可被看做一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有限状态机 Finite State Machine&amp;lt;/font&amp;gt;'''。最近的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;复杂系统 Complex System&amp;lt;/font&amp;gt;'''研究还表明，一些在生物学、计算机科学和物理学等领域的问题，从信息组织的角度来看存在一些意义深远的共性，例如玻色-爱因斯坦凝聚态(一种特殊的物质状态)。&lt;br /&gt;
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Bioinformatics has increasingly shifted its focus from individual [[gene]]s, proteins, and [[search algorithm]]s to large-scale networks often denoted as [[omics|-omes]] such as [[biome]], [[interactome]], [[genome]] and [[proteome]]. Such theoretical studies have revealed that biological networks share many features with other networks such as the [[Internet]] or [[social network]]s, e.g. their [[network topology]].&lt;br /&gt;
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Bioinformatics has increasingly shifted its focus from individual genes, proteins, and search algorithms to large-scale networks often denoted as -omes such as biome, interactome, genome and proteome. Such theoretical studies have revealed that biological networks share many features with other networks such as the Internet or social networks, e.g. their network topology.&lt;br /&gt;
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生物信息学已逐渐将关注点从单个基因、蛋白质和搜索算法逐渐转移到像'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生物组 Biome&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;交互组 Interactome&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因组 Genome&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;蛋白组 Proteome&amp;lt;/font&amp;gt;'''这样的大规模网络。这些理论研究揭示了生物网络与其他网络，如互联网或社交网络，有许多共同特征，例如:他们的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络拓扑结构 Network Topology&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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==Networks in biology==&lt;br /&gt;
生物学中的网络&lt;br /&gt;
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===Protein–protein interaction networks===&lt;br /&gt;
蛋白质-蛋白质互作网络&lt;br /&gt;
{{main article|interactome}}&lt;br /&gt;
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Many [[protein–protein interaction]]s (PPIs) in a cell form ''protein interaction networks'' (PINs) where proteins are ''nodes'' and their interactions are ''edges''.&amp;lt;ref&amp;gt;{{Cite journal|last=Habibi|first=Iman|last2=Emamian|first2=Effat S.|last3=Abdi|first3=Ali|date=2014-01-01|title=Quantitative analysis of intracellular communication and signaling errors in signaling networks|journal=BMC Systems Biology|volume=8|pages=89|doi=10.1186/s12918-014-0089-z|issn=1752-0509|pmc=4255782|pmid=25115405}}&amp;lt;/ref&amp;gt; PINs are the most intensely analyzed networks in biology. There are dozens of PPI detection methods to identify such interactions. The [[Y2H|yeast two-hybrid system]] is a commonly used experimental technique for the study of binary interactions.&amp;lt;ref&amp;gt;{{cite journal |author= Mashaghi, A. |title= Investigation of a protein complex network |journal= European Physical Journal |year= 2004  |volume= 41 |issue= 1 |pages= 113–121 |doi=10.1140/epjb/e2004-00301-0|display-authors=etal|arxiv= cond-mat/0304207|bibcode= 2004EPJB...41..113M}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Many protein–protein interactions (PPIs) in a cell form protein interaction networks (PINs) where proteins are nodes and their interactions are edges. PINs are the most intensely analyzed networks in biology. There are dozens of PPI detection methods to identify such interactions. The yeast two-hybrid system is a commonly used experimental technique for the study of binary interactions.&lt;br /&gt;
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在细胞中，大量'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;蛋白质之间相互作用 Protein-Protein Interactions(PPIs)&amp;lt;/font&amp;gt;'''形成'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;蛋白质相互作用网络 Protein Interaction Networks(PINs)&amp;lt;/font&amp;gt;'''，其中蛋白质是节点，它们的相互作用是连边。蛋白质互作网络是生物学中分析最深入的网络，现有几十种基于PPIs的检测方法来识别蛋白质间的相互作用。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;酵母双杂交系统 Yeast Two-Hybrid System&amp;lt;/font&amp;gt;'''是一种研究二元相互作用的常用实验技术。&lt;br /&gt;
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Recent studies have indicated conservation of molecular networks through deep evolutionary time.&amp;lt;ref&amp;gt;{{cite journal |author= Sharan, R. |title= Conserved patterns of protein interaction in multiple species |journal= Proceedings of the National Academy of Sciences of the United States of America |year= 2005  |volume= 102 |issue= 6 |pages= 1974–1979 |doi=10.1073/pnas.0409522102|pmid= 15687504 |display-authors=etal|pmc= 548573|bibcode= 2005PNAS..102.1974S}}&amp;lt;/ref&amp;gt; Moreover, it has been discovered that proteins with high degrees of connectedness are more likely to be essential for survival than proteins with lesser degrees.&amp;lt;ref&amp;gt;{{cite journal |author= Jeong, H. |title= Lethality and centrality in protein networks |journal= Nature |year= 2001  |volume= 411 |issue= 6833 |pages= 41–42 |doi=10.1038/35075138 |pmid=11333967|display-authors=etal|bibcode= 2001Natur.411...41J |arxiv= cond-mat/0105306 }}&amp;lt;/ref&amp;gt; This suggests that the overall composition of the network (not simply interactions between protein pairs) is important for the overall functioning of an organism.&lt;br /&gt;
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Recent studies have indicated conservation of molecular networks through deep evolutionary time. Moreover, it has been discovered that proteins with high degrees of connectedness are more likely to be essential for survival than proteins with lesser degrees. This suggests that the overall composition of the network (not simply interactions between protein pairs) is important for the overall functioning of an organism.&lt;br /&gt;
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近年来的研究表明，分子网络在深层进化过程中是保守的（在进化过程中的改变较少）。此外，已经发现相较于低度值的蛋白质，具有高度值的蛋白质对物种的生存更加重要。这表明网络的各个部分的相互配合(不仅仅是蛋白质之间简单的相互作用)对于有机体的整体功能具有重要意义。&lt;br /&gt;
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===Gene regulatory networks (DNA–protein interaction networks)===&lt;br /&gt;
基因调控网络（DNA-蛋白质交互网络）&lt;br /&gt;
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{{main article| Gene regulatory network}}&lt;br /&gt;
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The activity of genes is regulated by [[transcription factors]], proteins that typically bind to [[DNA]]. Most transcription factors bind to multiple binding sites in a [[genome]]. As a result, all cells have complex [[gene regulatory networks]]. For instance, the [[human genome]] encodes on the order of 1,400 DNA-binding transcription factors that regulate the expression of more than 20,000 human genes.&amp;lt;ref&amp;gt;{{cite journal |author= Vaquerizas, J.-M. |title= A census of human transcription factors: function, expression and evolution |journal= Nature Reviews Genetics |year= 2009  |volume= 10 |issue= 4 |pages= 252–263 |pmid= 19274049 |doi=10.1038/nrg2538|display-authors=etal}}&amp;lt;/ref&amp;gt; Technologies to study gene regulatory networks include [[ChIP-chip]], [[ChIP-seq]], [[CliP-seq]], and others.&lt;br /&gt;
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The activity of genes is regulated by transcription factors, proteins that typically bind to DNA. Most transcription factors bind to multiple binding sites in a genome. As a result, all cells have complex gene regulatory networks. For instance, the human genome encodes on the order of 1,400 DNA-binding transcription factors that regulate the expression of more than 20,000 human genes. Technologies to study gene regulatory networks include ChIP-chip, ChIP-seq, CliP-seq, and others.&lt;br /&gt;
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与DNA结合的蛋白质中，最典型的是负责调控基因表达的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;转录因子 Transcription Factors&amp;lt;/font&amp;gt;'''。大多数转录因子可与基因组中的多个位点结合。因此，所有的细胞都有复杂的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因调控网络 Gene Regulatory Networks&amp;lt;/font&amp;gt;'''。例如，人类基因组中有1400个左右的基因编码可与DNA结合的转录因子，它们参与到调节超过20000个人类基因的表达过程。研究基因调控网络的技术包括 ChIP-chip、 ChIP-seq、 CliP-seq 等。&lt;br /&gt;
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===Gene co-expression networks (transcript–transcript association networks)===&lt;br /&gt;
基因共表达网络（转录-转录关联网络）&lt;br /&gt;
{{main article| Gene co-expression networks}}&lt;br /&gt;
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Gene co-expression networks can be interpreted as association networks between variables that measure transcript abundances. These networks have been used to provide a systems biologic analysis of DNA microarray data, RNA-seq data, miRNA data etc.&lt;br /&gt;
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Gene co-expression networks can be interpreted as association networks between variables that measure transcript abundances. These networks have been used to provide a systems biologic analysis of DNA microarray data, RNA-seq data, miRNA data etc.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因共同表达网络 Gene Co-expression Networks &amp;lt;/font&amp;gt;'''可以看作是衡量转录丰度的变量之间的关联网络。这些网络已经被用于对DNA微阵列数据、 RNA-seq数据、 miRNA等数据进行系统生物学分析。&lt;br /&gt;
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[[Weighted correlation network analysis|weighted gene co-expression network analysis]] is widely used to identify co-expression modules and intramodular hub genes. Co-expression modules may correspond to cell types or pathways. Highly connected intramodular hubs can be interpreted as representatives of their respective module.&lt;br /&gt;
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weighted gene co-expression network analysis is widely used to identify co-expression modules and intramodular hub genes. Co-expression modules may correspond to cell types or pathways. Highly connected intramodular hubs can be interpreted as representatives of their respective module.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因共表达加权网络分析 Weighted Gene Co-expression Network Analysis &amp;lt;/font&amp;gt;'''被广泛应用于鉴定共表达模块（具有相似表达模式的基因很可能是紧密共调控的，功能紧密相关的或同一条信号通路或过程的成员，有其特定的生理意义，用网络的方式表示时，可划分到一个模块里）和模块内的核心基因。共表达模块可能与外部特征相关的信息关联在一起，如细胞类型或病症通路。具有大量连接的模块内节点可以代表模块内的其他基因，显示出这些基因共有的特点。&lt;br /&gt;
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===Metabolic networks===&lt;br /&gt;
代谢网络&lt;br /&gt;
{{main article|metabolic network}}&lt;br /&gt;
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The chemical compounds of a living cell are connected by biochemical reactions which convert one compound into another. The reactions are catalyzed by [[enzymes]]. Thus, all compounds in a cell are parts of an intricate biochemical network of reactions which is called [[metabolic network]]. It is possible to use network analyses to infer how selection acts on metabolic pathways.&amp;lt;ref name=&amp;quot;Proulx&amp;quot; /&amp;gt;&lt;br /&gt;
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The chemical compounds of a living cell are connected by biochemical reactions which convert one compound into another. The reactions are catalyzed by enzymes. Thus, all compounds in a cell are parts of an intricate biochemical network of reactions which is called metabolic network. It is possible to use network analyses to infer how selection acts on metabolic pathways.&lt;br /&gt;
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活细胞中的化合物相互之间进行生物化学反应，从而实现化合物之间的转变。这些反应是由酶催化的。因此，一个细胞中的所有化合物都是一个复杂的生化反应网络的一部分，该网络被称为代谢网络。可使用网络分析来推断筛选是如何影响代谢通路的。&lt;br /&gt;
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===Signaling networks===&lt;br /&gt;
信号网络&lt;br /&gt;
{{main article|Cell signaling}}&lt;br /&gt;
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Signals are transduced within cells or in between cells and thus form complex signaling networks. For instance, in the [[MAPK/ERK pathway]] is transduced from the cell surface to the cell nucleus by a series of protein–protein interactions, phosphorylation reactions, and other events. Signaling networks typically integrate [[interactome|protein–protein interaction networks]], [[gene regulatory networks]], and [[metabolic network]]s.&lt;br /&gt;
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Signals are transduced within cells or in between cells and thus form complex signaling networks. For instance, in the MAPK/ERK pathway is transduced from the cell surface to the cell nucleus by a series of protein–protein interactions, phosphorylation reactions, and other events. Signaling networks typically integrate protein–protein interaction networks, gene regulatory networks, and metabolic networks.&lt;br /&gt;
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信号在细胞内或细胞之间传递，从而形成复杂的信号网络。例如，MAPK/ERK 通路通过一系列蛋白质-蛋白质相互作用、磷酸化反应和其他事件将信号从细胞表面传递到细胞核内。信号网络通常包含了蛋白质-蛋白质相互作用网络、基因调控网络和代谢网络。&lt;br /&gt;
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===Neuronal networks===&lt;br /&gt;
神经网络&lt;br /&gt;
{{main article| Biological neural network}}&lt;br /&gt;
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The complex interactions in the [[brain]] make it a perfect candidate to apply network theory. [[Neuron]]s in the brain are deeply connected with one another and this results in complex networks being present in the structural and functional aspects of the brain.&amp;lt;ref&amp;gt;{{cite journal |author1=Bullmore, E.  |author2=O. Sporns  |lastauthoramp=yes |title= Complex brain networks: graph theoretical analysis of structural and functional systems |journal= Nature Reviews Neuroscience |year= 2009  |volume= 10 |issue= 3 |pages= 186–198 |doi=10.1038/nrn2575 |pmid=19190637}}&amp;lt;/ref&amp;gt; For instance, [[small-world network]] properties have been demonstrated in connections between cortical areas of the primate brain&amp;lt;ref&amp;gt;{{cite journal |author= Stephan, K.E. |title= Computational analysis of functional connectivity between areas of primate cerebral cortex |journal= [[Philosophical Transactions of the Royal Society B]]|year= 2000  |volume= 355 |issue= 1393 |pages= 111–126 |doi=10.1098/rstb.2000.0552|pmid= 10703047 |display-authors=etal|pmc= 1692715}}&amp;lt;/ref&amp;gt; or during swallowing in humans.&amp;lt;ref&amp;gt;{{Cite journal|date=2016-12-01|title=Functional connectivity patterns of normal human swallowing: difference among various viscosity swallows in normal and chin-tuck head positions|journal=Brain Research|volume=1652|pages=158–169|doi=10.1016/j.brainres.2016.09.041|pmid=27693396|pmc=5102805|issn=0006-8993|last1=Jestrović|first1=Iva|last2=Coyle|first2=James L|last3=Perera|first3=Subashan|last4=Sejdić|first4=Ervin}}&amp;lt;/ref&amp;gt; This suggests that cortical areas of the brain are not directly interacting with each other, but most areas can be reached from all others through only a few interactions.&lt;br /&gt;
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The complex interactions in the brain make it a perfect candidate to apply network theory. Neurons in the brain are deeply connected with one another and this results in complex networks being present in the structural and functional aspects of the brain. For instance, small-world network properties have been demonstrated in connections between cortical areas of the primate brain or during swallowing in humans. This suggests that cortical areas of the brain are not directly interacting with each other, but most areas can be reached from all others through only a few interactions.&lt;br /&gt;
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大脑中神经元之间复杂的相互作用使其成为应用网络理论的绝佳场景。大脑中的神经元彼此之间有着紧密的联系，其相互交织形成的复杂网络是人脑结构和功能的基础。例如，灵长类动物大脑皮层区域之间的连接或者人类吞咽时神经网络的行为已经证明了，这二者都具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;小世界网络属性Small-World Network &amp;lt;/font&amp;gt;'''。这表明大脑皮层各区域之间并不直接进行相互作用，而是大部分区域通过少量的节点区域进行相互沟通。&lt;br /&gt;
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===Food webs===&lt;br /&gt;
食物链网络&lt;br /&gt;
{{main article|Food web}}&lt;br /&gt;
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All organisms are connected to each other through feeding interactions. That is, if a species eats or is eaten by another species, they are connected in an intricate [[food web]] of predator and prey interactions. The stability of these interactions has been a long-standing question in ecology.&amp;lt;ref&amp;gt;{{cite journal |author= MacArthur, R.H. |author-link= Robert MacArthur|title= Fluctuations in animal populations and a measure of community stability |journal= Ecology |year= 1955  |volume= 36 |issue= 3 |pages= 533–536 |doi=10.2307/1929601|jstor= 1929601}}&amp;lt;/ref&amp;gt; That is to say, if certain individuals are removed, what happens to the network (i.e. does it collapse or adapt)? Network analysis can be used to explore food web stability and determine if certain network properties result in more stable networks. Moreover, network analysis can be used to determine how selective removals of species will influence the food web as a whole.&amp;lt;ref&amp;gt;{{cite journal |author= Dunne, J.A. |title= Network structure and biodiversity loss in food webs: robustness increases with connectance |journal= Ecology Letters |year= 2002  |volume= 5 |issue= 4 |pages= 558–567 |doi=10.1046/j.1461-0248.2002.00354.x|display-authors=etal}}&amp;lt;/ref&amp;gt; This is especially important considering the potential species loss due to global climate change.&lt;br /&gt;
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All organisms are connected to each other through feeding interactions. That is, if a species eats or is eaten by another species, they are connected in an intricate food web of predator and prey interactions. The stability of these interactions has been a long-standing question in ecology. That is to say, if certain individuals are removed, what happens to the network (i.e. does it collapse or adapt)? Network analysis can be used to explore food web stability and determine if certain network properties result in more stable networks. Moreover, network analysis can be used to determine how selective removals of species will influence the food web as a whole. This is especially important considering the potential species loss due to global climate change.&lt;br /&gt;
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所有的生物都具有捕食别的物种或被别的物种捕食的情况。也就是说，如果一个物种吃了另一个物种或被另一个物种吃了，它们就在一个复杂的捕食者和被捕食者相互作用的食物网中被联系起来。这些相互作用的稳定性长期困扰着生态学家们。具体来说，如果某些个体被移除，那么网络会发生什么？(即它是崩溃还是重新适应) ？网络分析可以用来探索食物网的稳定性，并确定某些网络特性是否会导致更稳定的网络。此外，网络分析可以用来确定物种的选择性迁移将如何影响整个食物链网。考虑到全球气候变化可能导致大量物种的消失，使用网络分析来研究食物链网的特性是尤其重要的。&lt;br /&gt;
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===Between-species interaction networks===&lt;br /&gt;
物种间交互网络&lt;br /&gt;
{{main article|social relation}}&lt;br /&gt;
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In biology, pairwise interactions have historically been the focus of intense study. With the recent advances in [[network science]], it has become possible to scale up pairwise interactions to include individuals of many species involved in many sets of interactions to understand the structure and function of larger [[ecological network]]s.&amp;lt;ref&amp;gt;{{cite journal |author= Bascompte, J. |title= Disentangling the web of life |journal= Science |year= 2009 |volume=325 |issue= 5939 |pages= 416–419 |doi=10.1126/science.1170749 |pmid=19628856|bibcode=2009Sci...325..416B }}&amp;lt;/ref&amp;gt; The use of [[Social network analysis|network analysis]] can allow for both the discovery and understanding how these complex interactions link together within the system’s network, a property which has previously been overlooked. This powerful tool allows for the study of various types of interactions (from [[competitive]] to [[cooperative]]) using the same general framework.&amp;lt;ref&amp;gt;{{cite journal |author= Krause, J. |title= Animal social networks: an introduction |journal= Behav. Ecol. Sociobiol.  |year= 2009 |volume= 63 |issue= 7 |pages= 967–973 |doi=10.1007/s00265-009-0747-0|display-authors=etal}}&amp;lt;/ref&amp;gt; For example, plant-[[pollinator]] interactions are mutually beneficial and often involve many different species of pollinators as well as many different species of plants. These interactions are critical to plant reproduction and thus the accumulation of resources at the base of the [[food chain]] for primary consumers, yet these interaction networks are threatened by [[Human impact on the environment|anthropogenic]] change. The use of network analysis can illuminate how [[pollination network]]s work and may in turn inform conservation efforts.&amp;lt;ref name=&amp;quot;Memmott, J., et al. 2004 2605-261&amp;quot;&amp;gt;{{cite journal |author= Memmott, J. |title= Tolerance of pollination networks to species extinctions |journal= [[Philosophical Transactions of the Royal Society B]] |year= 2004 |volume= 271 |issue= 1557 |pages= 2605–261 |doi=10.1098/rspb.2004.2909|pmid= 15615687 |pmc= 1691904 |display-authors=etal}}&amp;lt;/ref&amp;gt; Within pollination networks, nestedness (i.e., specialists interact with a subset of species that generalists interact with), redundancy (i.e., most plants are pollinated by many pollinators), and [[modularity (networks)|modularity]] play a large role in network stability.&amp;lt;ref name=&amp;quot;Memmott, J., et al. 2004 2605-261&amp;quot;/&amp;gt;&amp;lt;ref name=&amp;quot;Olesen, J., et al. 2007 19891–19896&amp;quot;&amp;gt;{{cite journal |author= Olesen, J. |title= The modularity of pollination networks |journal= PNAS |year= 2007 |volume= 104 |issue= 50 |pages= 19891–19896 |doi=10.1073/pnas.0706375104|pmid= 18056808 |pmc= 2148393 |display-authors=etal|bibcode= 2007PNAS..10419891O }}&amp;lt;/ref&amp;gt; These network properties may actually work to slow the spread of disturbance effects through the system and potentially buffer the pollination network from anthropogenic changes somewhat.&amp;lt;ref name=&amp;quot;Olesen, J., et al. 2007 19891–19896&amp;quot;/&amp;gt; More generally, the structure of species interactions within an ecological network can tell us something about the diversity, richness, and robustness of the network.&amp;lt;ref&amp;gt;{{cite journal |author= Campbell, V. |title= Experimental design and the outcome and interpretation of diversity-stability relations |journal= Oikos |year= 2011 |volume= 120 |issue= 3 |pages= 399–408 |doi=10.1111/j.1600-0706.2010.18768.x|display-authors=etal}}&amp;lt;/ref&amp;gt; Researchers can even compare current constructions of species interactions networks with historical reconstructions of ancient networks to determine how networks have changed over time.&amp;lt;ref&amp;gt;{{cite journal |author= Lotze, H. |title= Historical changes in marine resources, food-web structure and ecosystem functioning in the Adriatic Sea, Mediterranean |journal= Ecosystems |year= 2011 |volume= 14 |issue= 2 |pages= 198–222 |doi=10.1007/s10021-010-9404-8|display-authors=etal}}&amp;lt;/ref&amp;gt; Recent research into these complex species interactions networks is highly concerned with understanding what factors (e.g., diversity) lead to network stability.&amp;lt;ref&amp;gt;{{cite journal |author= Romanuk, T. |title= Maintenance of positive diversity-stability relations along a gradient of environmental stress |journal= PLoS ONE |year= 2010 |volume= 5 |issue= 4 |pages= e10378 |doi=10.1371/journal.pone.0010378|display-authors=etal |pmid=20436913 |pmc=2860506|bibcode=2010PLoSO...510378R }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In biology, pairwise interactions have historically been the focus of intense study. With the recent advances in network science, it has become possible to scale up pairwise interactions to include individuals of many species involved in many sets of interactions to understand the structure and function of larger ecological networks. The use of network analysis can allow for both the discovery and understanding how these complex interactions link together within the system’s network, a property which has previously been overlooked. This powerful tool allows for the study of various types of interactions (from competitive to cooperative) using the same general framework. For example, plant-pollinator interactions are mutually beneficial and often involve many different species of pollinators as well as many different species of plants. These interactions are critical to plant reproduction and thus the accumulation of resources at the base of the food chain for primary consumers, yet these interaction networks are threatened by anthropogenic change. The use of network analysis can illuminate how pollination networks work and may in turn inform conservation efforts. Within pollination networks, nestedness (i.e., specialists interact with a subset of species that generalists interact with), redundancy (i.e., most plants are pollinated by many pollinators), and modularity play a large role in network stability. These network properties may actually work to slow the spread of disturbance effects through the system and potentially buffer the pollination network from anthropogenic changes somewhat. Researchers can even compare current constructions of species interactions networks with historical reconstructions of ancient networks to determine how networks have changed over time. Recent research into these complex species interactions networks is highly concerned with understanding what factors (e.g., diversity) lead to network stability.&lt;br /&gt;
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在生物学中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;成对相互作用 Pairwise Interactions &amp;lt;/font&amp;gt;'''历来是研究的重点。随着近几年网络科学的发展，成对相互作用内涵很有可能被丰富，外延甚至能涵盖有许多物种个体参与的多组相互作用，进而帮助我们理解更大'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生态网络 Ecological Networks &amp;lt;/font&amp;gt;'''的结构和功能。使用网络分析可以发现和理解这些复杂的交互是如何连接成网络的，该形成过程之前经常被忽略。这个强大的工具使得同一个普世框架可以研究各种类型的交互(从竞争到合作)。例如，植物与传粉者之间的相互作用是互惠互利的，通常会涉及到许多不同种类的传粉者以及许多不同种类的植物。这些相互作用对植物的生殖和食物链底层初级消费者的资源积累至关重要，然而这些相互作用网络正受到人造变化的威胁。网络分析的使用可以说明授粉网络是如何工作的，反过来也可以为相关植物保护工作提供必要信息。在授粉网络中，内嵌性(某一领域的植物被少数几个昆虫授粉，剩下的各种昆虫只对少数几种植物授粉)、冗余性(即大多数植物是由许多授粉者授粉的)和模块性（一些物种相互之间连接的很紧密，与剩余的物种连接不紧密，由此形成一个个团簇，也可称之为“模块”）在网络稳定性中扮演着重要角色。这些网络属性实际上可以减缓干扰效应在系统上的传播，并可能缓冲授粉网络受到的人造变化的影响。研究人员甚至可以将物种相互作用网络的现有结构与该网络过去的结构进行比较，以确定网络在时间尺度上是如何演化的。最近对这些复杂物种间相互作用网络的研究聚焦于理解是什么因素(如多样性)造就了网络的稳定性。&lt;br /&gt;
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===Within-species interaction networks===&lt;br /&gt;
物种内交互网络&lt;br /&gt;
{{main article|social relation}}&lt;br /&gt;
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Network analysis provides the ability to quantify associations between individuals, which makes it possible to infer details about the network as a whole at the species and/or population level.&amp;lt;ref&amp;gt;{{cite journal |author= Croft, D.P |title= Social networks in the guppy (Poecilia reticulate) |journal= [[Philosophical Transactions of the Royal Society B]] |year= 2004 |volume= 271 |issue= Suppl |pages= S516–S519 |doi=10.1098/rsbl.2004.0206|pmid= 15801620 |pmc= 1810091 |display-authors=etal}}&amp;lt;/ref&amp;gt; One of the most attractive features of the network paradigm would be that it provides a single conceptual framework in which the social organisation of animals at all levels (individual, dyad, group, population) and for all types of interaction (aggressive, cooperative, sexual etc.) can be studied.&amp;lt;ref&amp;gt;{{Cite journal|last=Krause, Lusseau, James|first=Jens, David, Richard|date=1 May 2009|title=Animal social networks: an introduction|url=https://www.researchgate.net/publication/226996156_Animal_social_networks_An_introduction|journal=Behavioral Ecology and Sociobiology|volume=63|pages=967-973|doi=10.1007/s00265-009-0747-0|via=}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Network analysis provides the ability to quantify associations between individuals, which makes it possible to infer details about the network as a whole at the species and/or population level. One of the most attractive features of the network paradigm would be that it provides a single conceptual framework in which the social organisation of animals at all levels (individual, dyad, group, population) and for all types of interaction (aggressive, cooperative, sexual etc.) can be studied.&lt;br /&gt;
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网络分析可以量化个体之间的关联，这使得在物种和/或种群水平推断整个网络的细节成为可能。网络范式最吸引人的特征之一是，它提供了一个统一的概念框架，使得社会性动物在所有层次上(单，双，组，群)--[[用户:Ryan|Ryan]]（[[用户讨论:Ryan|讨论]]）和所有类型的互动中(攻击，合作，性等等)都可以被研究。&lt;br /&gt;
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Researchers interested in [[ethology]] across a multitude of taxa, from insects to primates, are starting to incorporate network analysis into their research. Researchers interested in social insects (e.g., ants and bees) have used network analyses to better understand division of labor, task allocation, and foraging optimization within colonies;&amp;lt;ref&amp;gt;{{cite journal |author= Dornhaus, A. |title= Benefits of recruitment in honey bees: Effects of ecology and colony size in an individual-based model |journal= Behavioral Ecology |year= 2006 |volume=17 |issue= 3 |pages= 336–344 |doi=10.1093/beheco/arj036|display-authors=etal|doi-access= free }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal |author= Linksvayer, T. |title= Developmental evolution in social insects: Regulatory networks from genes to societies |journal= Journal of Experimental Zoology Part B: Molecular and Developmental Evolution |year= 2012 |volume=318 |issue= 3 |pages= 159–169 |doi=10.1002/jez.b.22001|pmid= 22544713 |display-authors=etal}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal |author= Mullen, R. |title= A review of ant algorithms |journal= Expert Systems with Applications |year= 2009 |volume=36 |issue= 6 |pages= 9608–9617 |doi=10.1016/j.eswa.2009.01.020|display-authors=etal}}&amp;lt;/ref&amp;gt; Other researchers are interested in how certain network properties at the group and/or population level can explain individual level behaviors. Studies have demonstrated how animal social network structure can be influenced by factors ranging from characteristics of the environment to characteristics of the individual, such as developmental experience and personality. At the level of the individual, the patterning of social connections can be an important determinant of [[Fitness (biology)|fitness]], predicting both survival and reproductive success. At the population level, network structure can influence the patterning of ecological and evolutionary processes, such as [[frequency-dependent selection]] and disease and information transmission.&amp;lt;ref&amp;gt;{{Cite journal|last=Croft, Darden, Wey|first=Darren P., Safi K., Tina W.|date=2016|title=Current directions in animal social networks|url=https://www.researchgate.net/publication/310819524_Current_directions_in_animal_social_networks|journal=Current Opinion in Behavioral Sciences|volume=12|pages=52-58|doi=10.1016/j.cobeha.2016.09.001|via=|hdl=10871/23348|hdl-access=free}}&amp;lt;/ref&amp;gt; For instance, a study on [[Wire-tailed manakin|wire-tailed manakins]] (a small passerine bird)  found that a male’s [[degree (graph theory)|degree]] in the network largely predicted the ability of the male to rise in the social hierarchy (i.e. eventually obtain a territory and matings).&amp;lt;ref&amp;gt;{{cite journal |author= Ryder, T.B. |title= Social networks in the lek-mating wire-tailed manakin (''Pipra filicauda'') |journal= [[Philosophical Transactions of the Royal Society B]]  |year= 2008 |volume= 275 |issue= 1641 |pages= 1367–1374 |doi=10.1098/rspb.2008.0205 |pmid=18381257 |pmc=2602714|display-authors=etal}}&amp;lt;/ref&amp;gt; In [[bottlenose dolphin]] groups, an individual’s degree and [[betweenness centrality]] values may predict whether or not that individual will exhibit certain behaviors, like the use of side flopping and upside-down lobtailing to lead group traveling efforts; individuals with high betweenness values are more connected and can obtain more information, and thus are better suited to lead group travel and therefore tend to exhibit these signaling behaviors more than other group members.&amp;lt;ref&amp;gt;{{cite journal |author= Lusseau, D. |title= Evidence for social role in a dolphin social network |journal= Evolutionary Ecology |year= 2007 |volume=21 |issue= 3 |pages= 357–366 |doi=10.1007/s10682-006-9105-0|arxiv= q-bio/0607048 }}&amp;lt;/ref&amp;gt; &lt;br /&gt;
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Researchers interested in ethology across a multitude of taxa, from insects to primates, are starting to incorporate network analysis into their research. Researchers interested in social insects (e.g., ants and bees) have used network analyses to better understand division of labor, task allocation, and foraging optimization within colonies; Other researchers are interested in how certain network properties at the group and/or population level can explain individual level behaviors. Studies have demonstrated how animal social network structure can be influenced by factors ranging from characteristics of the environment to characteristics of the individual, such as developmental experience and personality. At the level of the individual, the patterning of social connections can be an important determinant of fitness, predicting both survival and reproductive success. At the population level, network structure can influence the patterning of ecological and evolutionary processes, such as frequency-dependent selection and disease and information transmission. For instance, a study on wire-tailed manakins (a small passerine bird)  found that a male’s degree in the network largely predicted the ability of the male to rise in the social hierarchy (i.e. eventually obtain a territory and matings). In bottlenose dolphin groups, an individual’s degree and betweenness centrality values may predict whether or not that individual will exhibit certain behaviors, like the use of side flopping and upside-down lobtailing to lead group traveling efforts; individuals with high betweenness values are more connected and can obtain more information, and thus are better suited to lead group travel and therefore tend to exhibit these signaling behaviors more than other group members. &lt;br /&gt;
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从昆虫到灵长类动物，对动物行为学感兴趣的研究人员，开始将网络分析纳入到他们的研究中。对社会性昆虫(如蚂蚁和蜜蜂)感兴趣的研究人员利用网络分析来更好地理解，群体内昆虫的分工、任务分配和觅食优化; 其他研究人员感兴趣的点则在群体和/或种群水平的某些网络特性如何影响个体的行为。研究表明，环境特征和个体特征，如经验发展和人格等因素，可影响动物的社会网络结构。在个体层面上，社会联系的模式是个体适应性的一个关键决定因素，可能预测着个体生存和繁殖的成功概率。在种群水平上，网络结构可以影响生态和进化过程的模式，如频率依赖性选择和疾病及信息的传递。例如，一项针对线尾 manakins (一种小型雀形目鸟)的研究发现，在网络中雄性个体的度值在很大程度上预测了雄性在社会等级中进阶的能力(即最终可获得领地和交配机会)。在宽吻海豚群体中，个体的度值和介数中心性可以预测个体是否会表现出某些行为，比如使用侧翻和头潜入水中尾巴举向空中来表示自己愿意领导团队的迁移; 介数值较高的个体有着更多的连接，可以获得更多信息，因此更适合领导团队迁行，因此比其他团队成员更容易表现出传递这些信号的行为。&lt;br /&gt;
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[[Social network analysis]] can also be used to describe the social organization within a species more generally, which frequently reveals important proximate mechanisms promoting the use of certain behavioral strategies. These descriptions are frequently linked to ecological properties (e.g., resource distribution). For example, network analyses revealed subtle differences in the group dynamics of two related equid [[fission-fusion society|fission-fusion]] species, [[Grevy’s zebra]] and [[onager]]s, living in variable environments; Grevy’s zebras show distinct preferences in their association choices when they fission into smaller groups, whereas onagers do not.&amp;lt;ref&amp;gt;{{cite journal |author= Sundaresan, S. |title= Network metrics reveal differences in social organization between two fission-fusion species, Grevy's zebra and onager |journal= Oecologia |year= 2007 |volume= 151 |issue= 1 |pages= 140–149 |doi=10.1007/s00442-006-0553-6|pmid= 16964497 |display-authors=etal|bibcode= 2007Oecol.151..140S }}&amp;lt;/ref&amp;gt; Similarly, researchers interested in primates have also utilized network analyses to compare social organizations across the diverse [[primate]] order, suggesting that using network measures (such as [[centrality]], [[assortativity]], [[modularity (networks)|modularity]], and betweenness) may be useful in terms of explaining the types of social behaviors we see within certain groups and not others.&amp;lt;ref&amp;gt;{{cite journal |author1=Kasper, C. |author2=Voelkl, B. |title= A social network analysis of primate groups |journal= Primates |year= 2009 |volume= 50 |issue=4 |pages= 343–356 |doi=10.1007/s10329-009-0153-2|pmid=19533270 }}&amp;lt;/ref&amp;gt; &lt;br /&gt;
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Social network analysis can also be used to describe the social organization within a species more generally, which frequently reveals important proximate mechanisms promoting the use of certain behavioral strategies. These descriptions are frequently linked to ecological properties (e.g., resource distribution). For example, network analyses revealed subtle differences in the group dynamics of two related equid fission-fusion species, Grevy’s zebra and onagers, living in variable environments; Grevy’s zebras show distinct preferences in their association choices when they fission into smaller groups, whereas onagers do not. Similarly, researchers interested in primates have also utilized network analyses to compare social organizations across the diverse primate order, suggesting that using network measures (such as centrality, assortativity, modularity, and betweenness) may be useful in terms of explaining the types of social behaviors we see within certain groups and not others. &lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会网络分析 Social Network Analysis &amp;lt;/font&amp;gt;'''也可以用来更广泛地描述一个物种内的社会组织，它经常揭示出那些促进某些行为策略使用的重要邻近机制。这些描述经常与生态属性(例如，资源分配)联系在一起。例如，网络分析揭示了生活在变化环境中的两个相关的等裂变融合物种——格雷维斑马和骑驴——的群体动力学的细微差异（裂变融合社会是这样一个社会，其规模和组成随着时间的流逝以及动物在整个环境中的移动而变化；动物会合并成一团（融合）-例如在一个地方睡觉-或分裂（裂变）-例如白天在小组中觅食）; 格雷维斑马在分裂为较小的群体时，在关联选择上表现出明显的偏好，而骑驴则不然。同样，对灵长类动物感兴趣的研究人员也利用网络分析来比较不同灵长类动物的社会组织，这表明以网络的视角进行测量(如集中性、协调性、模块性和中介性)可能有助于解释我们在某些特定群体中看到的社会行为类型。&lt;br /&gt;
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Finally, social network analysis can also reveal important fluctuations in animal behaviors across changing environments. For example, network analyses in female [[chacma baboon]]s (''Papio hamadryas ursinus'') revealed important dynamic changes across seasons which were previously unknown; instead of creating stable, long-lasting social bonds with friends, baboons were found to exhibit more variable relationships which were dependent on short-term contingencies related to group level dynamics as well as environmental variability.&amp;lt;ref&amp;gt;{{cite journal |author= Henzi, S. |title= Cyclicity in the structure of female baboon social networks |journal= Behavioral Ecology and Sociobiology |year= 2009 |volume= 63 |issue= 7 |pages= 1015–1021 |doi=10.1007/s00265-009-0720-y|display-authors=etal}}&amp;lt;/ref&amp;gt; Changes in an individual's social network environment can also influence characteristics such as 'personality': for example, social spiders that huddle with bolder neighbours tend to increase also in boldness.&amp;lt;ref&amp;gt;{{cite journal |author= Hunt, ER. |title= Social interactions shape individual and collective personality in social spiders |journal= Proceedings of the Royal Society B |year= 2018 |volume= 285 |issue= 1886 |pages= 20181366 |doi=10.1098/rspb.2018.1366|pmid= 30185649 |pmc= 6158534 |display-authors=etal}}&amp;lt;/ref&amp;gt; This is a very small set of broad examples of how researchers can use network analysis to study animal behavior. Research in this area is currently expanding very rapidly, especially since the broader development of animal borne tags and [[computer vision]] that can be used to automate the collection of social associations.&amp;lt;ref&amp;gt;{{Cite journal|last=Krause J, Krause S, Arlinghaus R, Psorakis I, Roberts S, Rutz C|first=|date=2013|title=Reality mining of animal social systems.|url=https://risweb.st-andrews.ac.uk/portal/en/researchoutput/reality-mining-of-animal-social-systems(570e21d0-527e-4275-8926-c4b479c55ee4).html|journal=Trends in Ecology and Evolution|volume=28|pages=541-551|doi=10.1016/j.tree.2013.06.002|via=}}&amp;lt;/ref&amp;gt; Social network analysis is a valuable tool for studying animal behavior across all animal species, and has the potential to uncover new information about animal behavior and social ecology that was previously poorly understood.&lt;br /&gt;
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Finally, social network analysis can also reveal important fluctuations in animal behaviors across changing environments. For example, network analyses in female chacma baboons (Papio hamadryas ursinus) revealed important dynamic changes across seasons which were previously unknown; instead of creating stable, long-lasting social bonds with friends, baboons were found to exhibit more variable relationships which were dependent on short-term contingencies related to group level dynamics as well as environmental variability. Changes in an individual's social network environment can also influence characteristics such as 'personality': for example, social spiders that huddle with bolder neighbours tend to increase also in boldness. This is a very small set of broad examples of how researchers can use network analysis to study animal behavior. Research in this area is currently expanding very rapidly, especially since the broader development of animal borne tags and computer vision that can be used to automate the collection of social associations. Social network analysis is a valuable tool for studying animal behavior across all animal species, and has the potential to uncover new information about animal behavior and social ecology that was previously poorly understood.&lt;br /&gt;
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最后，社会网络分析还可以揭示动物行为在不断变化的环境中的关键波动。例如，对雌性沙卡马狒狒(Papio hamadryas ursinus)的网络分析揭示了以前未知的跨季节的重要动态变化; 结果发现，狒狒之间没有建立稳定、持久的社会联系，相反，它们之间的关系是多变的。这些关系依赖于在动态的群体层面产生的短期偶然事件以及环境中的变化。个体社交网络环境的变化也会影响个体的性格特征，例如，与胆大的邻居挤在一起的蜘蛛也会更加大胆。这只是研究人员利用网络分析来研究动物行为的一小部分。这一领域的研究目前正在迅速扩展，特别是随着动物身体标记技术和计算机视觉技术的广泛发展，这些技术可以用来自动收集个体的社会关系。社会网络分析是研究所有物种的行为的一个有价值的工具，并且有可能发现以前鲜为人知的关于动物行为和社会生态的新信息。&lt;br /&gt;
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==See also==&lt;br /&gt;
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* [[List of omics topics in biology]]&lt;br /&gt;
生物学中的组学&lt;br /&gt;
* [[Biological network inference]]&lt;br /&gt;
生物网络推理&lt;br /&gt;
* [[Applied Statistics]]&lt;br /&gt;
应用统计学&lt;br /&gt;
* [[Biostatistics]]&lt;br /&gt;
生物统计学&lt;br /&gt;
* [[Computational Biology]]&lt;br /&gt;
计算生物学&lt;br /&gt;
* [[Systems biology]]&lt;br /&gt;
系统生物学&lt;br /&gt;
* [[Weighted correlation network analysis]] &lt;br /&gt;
权重相关性网络分析&lt;br /&gt;
* [[Interactome]]&lt;br /&gt;
交互组&lt;br /&gt;
* [[Network medicine]]&lt;br /&gt;
网络医学&lt;br /&gt;
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== External links ==&lt;br /&gt;
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* [http://www.nettab.org Network Tools and Applications in Biology] (NETTAB) workshops.&lt;br /&gt;
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==References==&lt;br /&gt;
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{{Reflist}}&lt;br /&gt;
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==Books==&lt;br /&gt;
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*E. Estrada,  &amp;quot;The Structure of Complex Networks: Theory and Applications&amp;quot;, Oxford University Press, 2011, {{ISBN|978-0-199-59175-6}}&lt;br /&gt;
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*J. Krause, R. James, D. Franks, D. Croft, &amp;quot;Animal Social Networks&amp;quot;, Oxford University Press, 2015, {{ISBN|978-0199679041}}&lt;br /&gt;
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==External links==&lt;br /&gt;
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* [https://web.archive.org/web/20110920220214/http://www.networkbio.org/ Networkbio.org], The site of the series of Integrative Network Biology (INB) meetings. For the 2012 event also see www.networkbio.org&lt;br /&gt;
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* [http://networkbiology.org Networkbiology.org], NetworkBiology wiki site.&lt;br /&gt;
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* [http://www.lindinglab.org LindingLab.org], Technical University of Denmark (DTU) studies Network Biology and Cellular Information Processing, and is also organizing the Denmark branch of the annual &amp;quot;[http://www.lindinglab.org/events ''Integrative Network Biology and Cancer'']&amp;quot; symposium series.&lt;br /&gt;
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* [http://nrnb.org NRNB.org], The National Resource for Network Biology. A US National Institute of Health (NIH) Biomedical Technology Research Center dedicated to the study of biological networks.&lt;br /&gt;
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*[http://networkrepository.com/asn.php Network Repository] The first interactive data and network data repository with real-time visual analytics.&lt;br /&gt;
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*[https://bansallab.github.io/asnr/ Animal Social Network Repository(ASNR)] The first multi-taxonomic repository that collates 790 social networks from more than 45 species, including those of mammals, reptiles, fish, birds, and insects &lt;br /&gt;
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*[http://konect.uni-koblenz.de/networks/ KONECT – Koblenz network collection.]&lt;br /&gt;
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[[Category:Biological techniques and tools]]&lt;br /&gt;
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Category:Biological techniques and tools&lt;br /&gt;
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类别: 生物技术和工具&lt;br /&gt;
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[[Category:Bioinformatics]]&lt;br /&gt;
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Category:Bioinformatics&lt;br /&gt;
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类别: 生物信息学&lt;br /&gt;
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[[Category:Systems biology]]&lt;br /&gt;
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Category:Systems biology&lt;br /&gt;
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分类: 系统生物学&lt;br /&gt;
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[[Category:Networks]]&lt;br /&gt;
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Category:Networks&lt;br /&gt;
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类别: 网络&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Biological network]]. Its edit history can be viewed at [[生物网络/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E7%94%9F%E7%89%A9%E7%BD%91%E7%BB%9C&amp;diff=12555</id>
		<title>生物网络</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E7%94%9F%E7%89%A9%E7%BD%91%E7%BB%9C&amp;diff=12555"/>
		<updated>2020-08-15T04:02:59Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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{{Network Science}}&lt;br /&gt;
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{{refimprove|date=October 2011}}&lt;br /&gt;
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A '''biological network''' is any [[network science|network]] that applies to [[biological system]]s. A network is any system with sub-units that are linked into a whole, such as species units linked into a whole [[food web]]. Biological networks provide a [[graph theory|mathematical representation]] of connections found in [[ecological network|ecological]], [[phylogenetic network|evolutionary]], and [[physiological]] studies, such as [[neural networks]].&amp;lt;ref name=&amp;quot;Proulx05&amp;quot;&amp;gt;{{cite journal | last1=Proulx | first1=S. R. | last2=Promislow | first2=D. E. L. | last3=Phillips | first3=P. C. | title=Network thinking in ecology and evolution | journal=Trends in Ecology and Evolution | volume=20 | issue=6 | pages=345–353 | year=2005 | doi=10.1016/j.tree.2005.04.004 | url=http://eeb19.biosci.arizona.edu/Faculty/Dornhaus/courses/materials/papers/Proulx%20Promislow%20Phillips%20networks%20ecol%20evol.pdf | pmid=16701391 | url-status=dead | archiveurl=https://web.archive.org/web/20110815122330/http://eeb19.biosci.arizona.edu/Faculty/Dornhaus/courses/materials/papers/Proulx%20Promislow%20Phillips%20networks%20ecol%20evol.pdf | archivedate=2011-08-15 }}&amp;lt;/ref&amp;gt; The analysis of biological networks with respect to human diseases has led to the field of [[network medicine]].&amp;lt;ref&amp;gt;{{cite journal | last1 = Barabási | first1 = A. L. | last2 = Gulbahce | first2 = N. | last3 = Loscalzo | first3 = J. | year = 2011 | title = Network medicine: a network-based approach to human disease | url = | journal = Nature Reviews Genetics | volume = 12 | issue = 1| pages = 56–68 | doi=10.1038/nrg2918 | pmid=21164525 | pmc=3140052}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{Cite journal|last=Habibi|first=Iman|last2=Emamian|first2=Effat S.|last3=Abdi|first3=Ali|date=2014-10-07|title=Advanced Fault Diagnosis Methods in Molecular Networks|journal=PLOS ONE|volume=9|issue=10|pages=e108830|doi=10.1371/journal.pone.0108830|issn=1932-6203|pmc=4188586|pmid=25290670|bibcode=2014PLoSO...9j8830H}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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A biological network is any network that applies to biological systems. A network is any system with sub-units that are linked into a whole, such as species units linked into a whole food web. Biological networks provide a mathematical representation of connections found in ecological, evolutionary, and physiological studies, such as neural networks. The analysis of biological networks with respect to human diseases has led to the field of network medicine.&lt;br /&gt;
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任何应用于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生物系统 Biological Systems&amp;lt;/font&amp;gt;'''的网络都可称为'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 生物网络 Biological Network&amp;lt;/font&amp;gt;'''。网络是指内部子单元可连接成一个整体的系统，如大量单个物种连接成一个整体的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;食物链网 Food Web&amp;lt;/font&amp;gt;'''。生物网络为在生态学、进化学和生理学研究中发现的联系提供了数学形式的表达，例如'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 神经网络 Neural Networks&amp;lt;/font&amp;gt;'''。生物网络分析在人类疾病方面的应用，开创了网络医学领域。&lt;br /&gt;
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==Network biology and bioinformatics==&lt;br /&gt;
网络生物学和生物信息学&lt;br /&gt;
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Complex biological systems may be represented and analyzed as computable networks. For example, ecosystems can be modeled as networks of interacting species or a protein can be modeled as a network of amino acids. Breaking a protein down farther, [[amino acids]] can be represented as a network of connected [[atom]]s, such as [[carbon]], [[nitrogen]], and [[oxygen]]. [[vertex (graph theory)|Nodes]] and [[vertex (graph theory)|edges]] are the basic components of a network. Nodes represent units in the network, while edges represent the interactions between the units. Nodes can represent a wide array of biological units, from individual organisms to individual neurons in the brain. Two important properties of a network are [[vertex (graph theory)|degree]] and [[betweenness centrality]]. Degree (or connectivity, a distinct usage from that used in [[Connectivity (graph theory)|graph theory]]) is the number of edges that connect a node, while betweenness is a measure of how central a node is in a network.&amp;lt;ref name=&amp;quot;Proulx&amp;quot;&amp;gt;{{cite journal |author= Proulx, S.R. |title= Network thinking in ecology and evolution |journal= Trends in Ecology and Evolution |year= 2005  |volume= 20 |issue= 6 |pages= 345–353 |doi=10.1016/j.tree.2005.04.004 |pmid=16701391|display-authors=etal}}&amp;lt;/ref&amp;gt; Nodes with high betweenness essentially serve as bridges between different portions of  the network (i.e. interactions must pass through this node to reach other portions of the network). In social networks, nodes with high degree or high betweenness may play important roles in the overall composition of a network.&lt;br /&gt;
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Complex biological systems may be represented and analyzed as computable networks. For example, ecosystems can be modeled as networks of interacting species or a protein can be modeled as a network of amino acids. Breaking a protein down farther, amino acids can be represented as a network of connected atoms, such as carbon, nitrogen, and oxygen. Nodes and edges are the basic components of a network. Nodes represent units in the network, while edges represent the interactions between the units. Nodes can represent a wide array of biological units, from individual organisms to individual neurons in the brain. Two important properties of a network are degree and betweenness centrality. Degree (or connectivity, a distinct usage from that used in graph theory) is the number of edges that connect a node, while betweenness is a measure of how central a node is in a network. Nodes with high betweenness essentially serve as bridges between different portions of  the network (i.e. interactions must pass through this node to reach other portions of the network). In social networks, nodes with high degree or high betweenness may play important roles in the overall composition of a network.&lt;br /&gt;
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复杂的生物系统能用可计算的网络来表示和分析。例如，生态系统可以被模拟为相互作用的物种网络，一个蛋白质可以被模拟为氨基酸网络。进一步分解蛋白质，氨基酸可以表示为一个由相互连接的原子构成的网络，如碳、氮和氧。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 节点 Nodes&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;边 Edges&amp;lt;/font&amp;gt;'''是网络的基本组成部分。节点表示网络中的单元，边表示单元之间的相互作用。节点可以代表一系列广泛的生物单元，从单个的生物体到大脑中单个的神经元。网络的两个重要性质是'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;度 Degree&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 介数中心性 Betweenness Centrality&amp;lt;/font&amp;gt;'''。度(或连通性，一个不同于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;图论 Graph Theory&amp;lt;/font&amp;gt;'''的用法)是连接一个节点的边的数量，而介数中心性是衡量一个节点在网络中有多么靠近中心位置。具有高介数的节点本质上充当网络不同部分之间的桥梁(即网络其他部分的交互，必须通过这个节点)。在社会网络中，具有较高度值和较高介数的节点可能在网络的整体组成中发挥重要作用。&lt;br /&gt;
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As early as the 1980s, researchers started viewing [[DNA]] or [[genome]]s as the dynamic storage of a language system with precise computable finite [[state (computer science)|states]] represented as a [[finite state machine]].&amp;lt;ref&amp;gt;{{cite book  |author= Searls, D. |title= Artificial intelligence and molecular biology |publisher= MIT Press |year= 1993 |location= Cambridge, MA}}&amp;lt;/ref&amp;gt; Recent [[complex system]]s research has also suggested some far-reaching commonality in the organization of information in problems from biology, [[computer science]], and [[physics]], such as the [[Bose–Einstein condensate]] (a special state of matter).&amp;lt;ref&amp;gt;{{cite journal |author1=Bianconi, G.  |author2=Barabasi A.  |lastauthoramp=yes |title= Bose-Einstein condensation in complex networks |journal= Phys. Rev. Lett. |year= 2001  |volume= 86 |issue= 24 |pages= 5632–5635 |doi=10.1103/physrevlett.86.5632|pmid=11415319  |arxiv=cond-mat/0011224|bibcode=2001PhRvL..86.5632B|hdl=2047/d20000696  }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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As early as the 1980s, researchers started viewing DNA or genomes as the dynamic storage of a language system with precise computable finite states represented as a finite state machine. Recent complex systems research has also suggested some far-reaching commonality in the organization of information in problems from biology, computer science, and physics, such as the Bose–Einstein condensate (a special state of matter).&lt;br /&gt;
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早在20世纪80年代，研究人员就开始将 DNA 或基因组视为一个可存储动态信息的语言系统，因其具有可精确计算的有限状态，所以可被看做一个'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;有限状态机 Finite State Machine&amp;lt;/font&amp;gt;'''。最近的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;复杂系统 Complex System&amp;lt;/font&amp;gt;'''研究还表明，一些在生物学、计算机科学和物理学等领域的问题，从信息组织的角度来看存在一些意义深远的共性，例如玻色-爱因斯坦凝聚态(一种特殊的物质状态)。&lt;br /&gt;
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Bioinformatics has increasingly shifted its focus from individual [[gene]]s, proteins, and [[search algorithm]]s to large-scale networks often denoted as [[omics|-omes]] such as [[biome]], [[interactome]], [[genome]] and [[proteome]]. Such theoretical studies have revealed that biological networks share many features with other networks such as the [[Internet]] or [[social network]]s, e.g. their [[network topology]].&lt;br /&gt;
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Bioinformatics has increasingly shifted its focus from individual genes, proteins, and search algorithms to large-scale networks often denoted as -omes such as biome, interactome, genome and proteome. Such theoretical studies have revealed that biological networks share many features with other networks such as the Internet or social networks, e.g. their network topology.&lt;br /&gt;
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生物信息学已逐渐将关注点从单个基因、蛋白质和搜索算法逐渐转移到像'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生物组 Biome&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;交互组 Interactome&amp;lt;/font&amp;gt;'''、'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因组 Genome&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;蛋白组 Proteome&amp;lt;/font&amp;gt;'''这样的大规模网络。这些理论研究揭示了生物网络与其他网络，如互联网或社交网络，有许多共同特征，例如:他们的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;网络拓扑结构 Network Topology&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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==Networks in biology==&lt;br /&gt;
生物学中的网络&lt;br /&gt;
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===Protein–protein interaction networks===&lt;br /&gt;
蛋白质-蛋白质互作网络&lt;br /&gt;
{{main article|interactome}}&lt;br /&gt;
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Many [[protein–protein interaction]]s (PPIs) in a cell form ''protein interaction networks'' (PINs) where proteins are ''nodes'' and their interactions are ''edges''.&amp;lt;ref&amp;gt;{{Cite journal|last=Habibi|first=Iman|last2=Emamian|first2=Effat S.|last3=Abdi|first3=Ali|date=2014-01-01|title=Quantitative analysis of intracellular communication and signaling errors in signaling networks|journal=BMC Systems Biology|volume=8|pages=89|doi=10.1186/s12918-014-0089-z|issn=1752-0509|pmc=4255782|pmid=25115405}}&amp;lt;/ref&amp;gt; PINs are the most intensely analyzed networks in biology. There are dozens of PPI detection methods to identify such interactions. The [[Y2H|yeast two-hybrid system]] is a commonly used experimental technique for the study of binary interactions.&amp;lt;ref&amp;gt;{{cite journal |author= Mashaghi, A. |title= Investigation of a protein complex network |journal= European Physical Journal |year= 2004  |volume= 41 |issue= 1 |pages= 113–121 |doi=10.1140/epjb/e2004-00301-0|display-authors=etal|arxiv= cond-mat/0304207|bibcode= 2004EPJB...41..113M}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Many protein–protein interactions (PPIs) in a cell form protein interaction networks (PINs) where proteins are nodes and their interactions are edges. PINs are the most intensely analyzed networks in biology. There are dozens of PPI detection methods to identify such interactions. The yeast two-hybrid system is a commonly used experimental technique for the study of binary interactions.&lt;br /&gt;
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在细胞中，大量'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;蛋白质之间相互作用 Protein-Protein Interactions(PPIs)&amp;lt;/font&amp;gt;'''形成'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;蛋白质相互作用网络 Protein Interaction Networks(PINs)&amp;lt;/font&amp;gt;'''，其中蛋白质是节点，它们的相互作用是连边。蛋白质互作网络是生物学中分析最深入的网络，现有几十种基于PPIs的检测方法来识别蛋白质间的相互作用。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;酵母双杂交系统 Yeast Two-Hybrid System&amp;lt;/font&amp;gt;'''是一种研究二元相互作用的常用实验技术。&lt;br /&gt;
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Recent studies have indicated conservation of molecular networks through deep evolutionary time.&amp;lt;ref&amp;gt;{{cite journal |author= Sharan, R. |title= Conserved patterns of protein interaction in multiple species |journal= Proceedings of the National Academy of Sciences of the United States of America |year= 2005  |volume= 102 |issue= 6 |pages= 1974–1979 |doi=10.1073/pnas.0409522102|pmid= 15687504 |display-authors=etal|pmc= 548573|bibcode= 2005PNAS..102.1974S}}&amp;lt;/ref&amp;gt; Moreover, it has been discovered that proteins with high degrees of connectedness are more likely to be essential for survival than proteins with lesser degrees.&amp;lt;ref&amp;gt;{{cite journal |author= Jeong, H. |title= Lethality and centrality in protein networks |journal= Nature |year= 2001  |volume= 411 |issue= 6833 |pages= 41–42 |doi=10.1038/35075138 |pmid=11333967|display-authors=etal|bibcode= 2001Natur.411...41J |arxiv= cond-mat/0105306 }}&amp;lt;/ref&amp;gt; This suggests that the overall composition of the network (not simply interactions between protein pairs) is important for the overall functioning of an organism.&lt;br /&gt;
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Recent studies have indicated conservation of molecular networks through deep evolutionary time. Moreover, it has been discovered that proteins with high degrees of connectedness are more likely to be essential for survival than proteins with lesser degrees. This suggests that the overall composition of the network (not simply interactions between protein pairs) is important for the overall functioning of an organism.&lt;br /&gt;
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近年来的研究表明，分子网络在深层进化过程中是保守的（在进化过程中的改变较少）。此外，已经发现相较于低度值的蛋白质，具有高度值的蛋白质对物种的生存更加重要。这表明网络的各个部分的相互配合(不仅仅是蛋白质之间简单的相互作用)对于有机体的整体功能具有重要意义。&lt;br /&gt;
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===Gene regulatory networks (DNA–protein interaction networks)===&lt;br /&gt;
基因调控网络（DNA-蛋白质交互网络）&lt;br /&gt;
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{{main article| Gene regulatory network}}&lt;br /&gt;
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The activity of genes is regulated by [[transcription factors]], proteins that typically bind to [[DNA]]. Most transcription factors bind to multiple binding sites in a [[genome]]. As a result, all cells have complex [[gene regulatory networks]]. For instance, the [[human genome]] encodes on the order of 1,400 DNA-binding transcription factors that regulate the expression of more than 20,000 human genes.&amp;lt;ref&amp;gt;{{cite journal |author= Vaquerizas, J.-M. |title= A census of human transcription factors: function, expression and evolution |journal= Nature Reviews Genetics |year= 2009  |volume= 10 |issue= 4 |pages= 252–263 |pmid= 19274049 |doi=10.1038/nrg2538|display-authors=etal}}&amp;lt;/ref&amp;gt; Technologies to study gene regulatory networks include [[ChIP-chip]], [[ChIP-seq]], [[CliP-seq]], and others.&lt;br /&gt;
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The activity of genes is regulated by transcription factors, proteins that typically bind to DNA. Most transcription factors bind to multiple binding sites in a genome. As a result, all cells have complex gene regulatory networks. For instance, the human genome encodes on the order of 1,400 DNA-binding transcription factors that regulate the expression of more than 20,000 human genes. Technologies to study gene regulatory networks include ChIP-chip, ChIP-seq, CliP-seq, and others.&lt;br /&gt;
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与DNA结合的蛋白质中，最典型的是负责调控基因表达的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;转录因子 Transcription Factors&amp;lt;/font&amp;gt;'''。大多数转录因子可与基因组中的多个位点结合。因此，所有的细胞都有复杂的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因调控网络 Gene Regulatory Networks&amp;lt;/font&amp;gt;'''。例如，人类基因组中有1400个左右的基因编码可与DNA结合的转录因子，它们参与到调节超过20000个人类基因的表达过程。研究基因调控网络的技术包括 ChIP-chip、 ChIP-seq、 CliP-seq 等。&lt;br /&gt;
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===Gene co-expression networks (transcript–transcript association networks)===&lt;br /&gt;
基因共表达网络（转录-转录关联网络）&lt;br /&gt;
{{main article| Gene co-expression networks}}&lt;br /&gt;
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Gene co-expression networks can be interpreted as association networks between variables that measure transcript abundances. These networks have been used to provide a systems biologic analysis of DNA microarray data, RNA-seq data, miRNA data etc.&lt;br /&gt;
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Gene co-expression networks can be interpreted as association networks between variables that measure transcript abundances. These networks have been used to provide a systems biologic analysis of DNA microarray data, RNA-seq data, miRNA data etc.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因共同表达网络 Gene Co-expression Networks &amp;lt;/font&amp;gt;'''可以看作是衡量转录丰度的变量之间的关联网络。这些网络已经被用于对DNA微阵列数据、 RNA-seq数据、 miRNA等数据进行系统生物学分析。&lt;br /&gt;
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[[Weighted correlation network analysis|weighted gene co-expression network analysis]] is widely used to identify co-expression modules and intramodular hub genes. Co-expression modules may correspond to cell types or pathways. Highly connected intramodular hubs can be interpreted as representatives of their respective module.&lt;br /&gt;
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weighted gene co-expression network analysis is widely used to identify co-expression modules and intramodular hub genes. Co-expression modules may correspond to cell types or pathways. Highly connected intramodular hubs can be interpreted as representatives of their respective module.&lt;br /&gt;
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'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;基因共表达加权网络分析 Weighted Gene Co-expression Network Analysis &amp;lt;/font&amp;gt;'''被广泛应用于鉴定共表达模块（具有相似表达模式的基因很可能是紧密共调控的，功能紧密相关的或同一条信号通路或过程的成员，有其特定的生理意义，用网络的方式表示时，可划分到一个模块里）和模块内的核心基因。共表达模块可能与外部特征相关的信息关联在一起，如细胞类型或病症通路。具有大量连接的模块内节点可以代表模块内的其他基因，显示出这些基因共有的特点。&lt;br /&gt;
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===Metabolic networks===&lt;br /&gt;
代谢网络&lt;br /&gt;
{{main article|metabolic network}}&lt;br /&gt;
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The chemical compounds of a living cell are connected by biochemical reactions which convert one compound into another. The reactions are catalyzed by [[enzymes]]. Thus, all compounds in a cell are parts of an intricate biochemical network of reactions which is called [[metabolic network]]. It is possible to use network analyses to infer how selection acts on metabolic pathways.&amp;lt;ref name=&amp;quot;Proulx&amp;quot; /&amp;gt;&lt;br /&gt;
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The chemical compounds of a living cell are connected by biochemical reactions which convert one compound into another. The reactions are catalyzed by enzymes. Thus, all compounds in a cell are parts of an intricate biochemical network of reactions which is called metabolic network. It is possible to use network analyses to infer how selection acts on metabolic pathways.&lt;br /&gt;
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活细胞中的化合物相互之间进行生物化学反应，从而实现化合物之间的转变。这些反应是由酶催化的。因此，一个细胞中的所有化合物都是一个复杂的生化反应网络的一部分，该网络被称为代谢网络。可使用网络分析来推断筛选是如何影响代谢通路的。&lt;br /&gt;
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===Signaling networks===&lt;br /&gt;
信号网络&lt;br /&gt;
{{main article|Cell signaling}}&lt;br /&gt;
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Signals are transduced within cells or in between cells and thus form complex signaling networks. For instance, in the [[MAPK/ERK pathway]] is transduced from the cell surface to the cell nucleus by a series of protein–protein interactions, phosphorylation reactions, and other events. Signaling networks typically integrate [[interactome|protein–protein interaction networks]], [[gene regulatory networks]], and [[metabolic network]]s.&lt;br /&gt;
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Signals are transduced within cells or in between cells and thus form complex signaling networks. For instance, in the MAPK/ERK pathway is transduced from the cell surface to the cell nucleus by a series of protein–protein interactions, phosphorylation reactions, and other events. Signaling networks typically integrate protein–protein interaction networks, gene regulatory networks, and metabolic networks.&lt;br /&gt;
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信号在细胞内或细胞之间传递，从而形成复杂的信号网络。例如，MAPK/ERK 通路通过一系列蛋白质-蛋白质相互作用、磷酸化反应和其他事件将信号从细胞表面传递到细胞核内。信号网络通常包含了蛋白质-蛋白质相互作用网络、基因调控网络和代谢网络。&lt;br /&gt;
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===Neuronal networks===&lt;br /&gt;
神经网络&lt;br /&gt;
{{main article| Biological neural network}}&lt;br /&gt;
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The complex interactions in the [[brain]] make it a perfect candidate to apply network theory. [[Neuron]]s in the brain are deeply connected with one another and this results in complex networks being present in the structural and functional aspects of the brain.&amp;lt;ref&amp;gt;{{cite journal |author1=Bullmore, E.  |author2=O. Sporns  |lastauthoramp=yes |title= Complex brain networks: graph theoretical analysis of structural and functional systems |journal= Nature Reviews Neuroscience |year= 2009  |volume= 10 |issue= 3 |pages= 186–198 |doi=10.1038/nrn2575 |pmid=19190637}}&amp;lt;/ref&amp;gt; For instance, [[small-world network]] properties have been demonstrated in connections between cortical areas of the primate brain&amp;lt;ref&amp;gt;{{cite journal |author= Stephan, K.E. |title= Computational analysis of functional connectivity between areas of primate cerebral cortex |journal= [[Philosophical Transactions of the Royal Society B]]|year= 2000  |volume= 355 |issue= 1393 |pages= 111–126 |doi=10.1098/rstb.2000.0552|pmid= 10703047 |display-authors=etal|pmc= 1692715}}&amp;lt;/ref&amp;gt; or during swallowing in humans.&amp;lt;ref&amp;gt;{{Cite journal|date=2016-12-01|title=Functional connectivity patterns of normal human swallowing: difference among various viscosity swallows in normal and chin-tuck head positions|journal=Brain Research|volume=1652|pages=158–169|doi=10.1016/j.brainres.2016.09.041|pmid=27693396|pmc=5102805|issn=0006-8993|last1=Jestrović|first1=Iva|last2=Coyle|first2=James L|last3=Perera|first3=Subashan|last4=Sejdić|first4=Ervin}}&amp;lt;/ref&amp;gt; This suggests that cortical areas of the brain are not directly interacting with each other, but most areas can be reached from all others through only a few interactions.&lt;br /&gt;
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The complex interactions in the brain make it a perfect candidate to apply network theory. Neurons in the brain are deeply connected with one another and this results in complex networks being present in the structural and functional aspects of the brain. For instance, small-world network properties have been demonstrated in connections between cortical areas of the primate brain or during swallowing in humans. This suggests that cortical areas of the brain are not directly interacting with each other, but most areas can be reached from all others through only a few interactions.&lt;br /&gt;
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大脑中神经元之间复杂的相互作用使其成为应用网络理论的绝佳场景。大脑中的神经元彼此之间有着紧密的联系，其相互交织形成的复杂网络是人脑结构和功能的基础。例如，灵长类动物大脑皮层区域之间的连接或者人类吞咽时神经网络的行为已经证明了，这二者都具有'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;小世界网络属性Small-World Network &amp;lt;/font&amp;gt;'''。这表明大脑皮层各区域之间并不直接进行相互作用，而是大部分区域可以通过少量的节点区域进行相互沟通。&lt;br /&gt;
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===Food webs===&lt;br /&gt;
食物链网络&lt;br /&gt;
{{main article|Food web}}&lt;br /&gt;
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All organisms are connected to each other through feeding interactions. That is, if a species eats or is eaten by another species, they are connected in an intricate [[food web]] of predator and prey interactions. The stability of these interactions has been a long-standing question in ecology.&amp;lt;ref&amp;gt;{{cite journal |author= MacArthur, R.H. |author-link= Robert MacArthur|title= Fluctuations in animal populations and a measure of community stability |journal= Ecology |year= 1955  |volume= 36 |issue= 3 |pages= 533–536 |doi=10.2307/1929601|jstor= 1929601}}&amp;lt;/ref&amp;gt; That is to say, if certain individuals are removed, what happens to the network (i.e. does it collapse or adapt)? Network analysis can be used to explore food web stability and determine if certain network properties result in more stable networks. Moreover, network analysis can be used to determine how selective removals of species will influence the food web as a whole.&amp;lt;ref&amp;gt;{{cite journal |author= Dunne, J.A. |title= Network structure and biodiversity loss in food webs: robustness increases with connectance |journal= Ecology Letters |year= 2002  |volume= 5 |issue= 4 |pages= 558–567 |doi=10.1046/j.1461-0248.2002.00354.x|display-authors=etal}}&amp;lt;/ref&amp;gt; This is especially important considering the potential species loss due to global climate change.&lt;br /&gt;
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All organisms are connected to each other through feeding interactions. That is, if a species eats or is eaten by another species, they are connected in an intricate food web of predator and prey interactions. The stability of these interactions has been a long-standing question in ecology. That is to say, if certain individuals are removed, what happens to the network (i.e. does it collapse or adapt)? Network analysis can be used to explore food web stability and determine if certain network properties result in more stable networks. Moreover, network analysis can be used to determine how selective removals of species will influence the food web as a whole. This is especially important considering the potential species loss due to global climate change.&lt;br /&gt;
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所有的生物都具有捕食别的物种或被别的物种捕食的情况。也就是说，如果一个物种吃了另一个物种或被另一个物种吃了，它们就在一个复杂的捕食者和被捕食者相互作用的食物网中被联系起来。这些相互作用的稳定性长期困扰着生态学家们。具体来说，如果某些个体被移除，那么网络会发生什么？(即它是崩溃还是重新适应) ？网络分析可以用来探索食物网的稳定性，并确定某些网络特性是否会导致更稳定的网络。此外，网络分析可以用来确定物种的选择性迁移将如何影响整个食物链网。考虑到全球气候变化可能导致大量物种的消失，使用网络分析来研究食物链网的特性是尤其重要的。&lt;br /&gt;
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===Between-species interaction networks===&lt;br /&gt;
物种间交互网络&lt;br /&gt;
{{main article|social relation}}&lt;br /&gt;
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In biology, pairwise interactions have historically been the focus of intense study. With the recent advances in [[network science]], it has become possible to scale up pairwise interactions to include individuals of many species involved in many sets of interactions to understand the structure and function of larger [[ecological network]]s.&amp;lt;ref&amp;gt;{{cite journal |author= Bascompte, J. |title= Disentangling the web of life |journal= Science |year= 2009 |volume=325 |issue= 5939 |pages= 416–419 |doi=10.1126/science.1170749 |pmid=19628856|bibcode=2009Sci...325..416B }}&amp;lt;/ref&amp;gt; The use of [[Social network analysis|network analysis]] can allow for both the discovery and understanding how these complex interactions link together within the system’s network, a property which has previously been overlooked. This powerful tool allows for the study of various types of interactions (from [[competitive]] to [[cooperative]]) using the same general framework.&amp;lt;ref&amp;gt;{{cite journal |author= Krause, J. |title= Animal social networks: an introduction |journal= Behav. Ecol. Sociobiol.  |year= 2009 |volume= 63 |issue= 7 |pages= 967–973 |doi=10.1007/s00265-009-0747-0|display-authors=etal}}&amp;lt;/ref&amp;gt; For example, plant-[[pollinator]] interactions are mutually beneficial and often involve many different species of pollinators as well as many different species of plants. These interactions are critical to plant reproduction and thus the accumulation of resources at the base of the [[food chain]] for primary consumers, yet these interaction networks are threatened by [[Human impact on the environment|anthropogenic]] change. The use of network analysis can illuminate how [[pollination network]]s work and may in turn inform conservation efforts.&amp;lt;ref name=&amp;quot;Memmott, J., et al. 2004 2605-261&amp;quot;&amp;gt;{{cite journal |author= Memmott, J. |title= Tolerance of pollination networks to species extinctions |journal= [[Philosophical Transactions of the Royal Society B]] |year= 2004 |volume= 271 |issue= 1557 |pages= 2605–261 |doi=10.1098/rspb.2004.2909|pmid= 15615687 |pmc= 1691904 |display-authors=etal}}&amp;lt;/ref&amp;gt; Within pollination networks, nestedness (i.e., specialists interact with a subset of species that generalists interact with), redundancy (i.e., most plants are pollinated by many pollinators), and [[modularity (networks)|modularity]] play a large role in network stability.&amp;lt;ref name=&amp;quot;Memmott, J., et al. 2004 2605-261&amp;quot;/&amp;gt;&amp;lt;ref name=&amp;quot;Olesen, J., et al. 2007 19891–19896&amp;quot;&amp;gt;{{cite journal |author= Olesen, J. |title= The modularity of pollination networks |journal= PNAS |year= 2007 |volume= 104 |issue= 50 |pages= 19891–19896 |doi=10.1073/pnas.0706375104|pmid= 18056808 |pmc= 2148393 |display-authors=etal|bibcode= 2007PNAS..10419891O }}&amp;lt;/ref&amp;gt; These network properties may actually work to slow the spread of disturbance effects through the system and potentially buffer the pollination network from anthropogenic changes somewhat.&amp;lt;ref name=&amp;quot;Olesen, J., et al. 2007 19891–19896&amp;quot;/&amp;gt; More generally, the structure of species interactions within an ecological network can tell us something about the diversity, richness, and robustness of the network.&amp;lt;ref&amp;gt;{{cite journal |author= Campbell, V. |title= Experimental design and the outcome and interpretation of diversity-stability relations |journal= Oikos |year= 2011 |volume= 120 |issue= 3 |pages= 399–408 |doi=10.1111/j.1600-0706.2010.18768.x|display-authors=etal}}&amp;lt;/ref&amp;gt; Researchers can even compare current constructions of species interactions networks with historical reconstructions of ancient networks to determine how networks have changed over time.&amp;lt;ref&amp;gt;{{cite journal |author= Lotze, H. |title= Historical changes in marine resources, food-web structure and ecosystem functioning in the Adriatic Sea, Mediterranean |journal= Ecosystems |year= 2011 |volume= 14 |issue= 2 |pages= 198–222 |doi=10.1007/s10021-010-9404-8|display-authors=etal}}&amp;lt;/ref&amp;gt; Recent research into these complex species interactions networks is highly concerned with understanding what factors (e.g., diversity) lead to network stability.&amp;lt;ref&amp;gt;{{cite journal |author= Romanuk, T. |title= Maintenance of positive diversity-stability relations along a gradient of environmental stress |journal= PLoS ONE |year= 2010 |volume= 5 |issue= 4 |pages= e10378 |doi=10.1371/journal.pone.0010378|display-authors=etal |pmid=20436913 |pmc=2860506|bibcode=2010PLoSO...510378R }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In biology, pairwise interactions have historically been the focus of intense study. With the recent advances in network science, it has become possible to scale up pairwise interactions to include individuals of many species involved in many sets of interactions to understand the structure and function of larger ecological networks. The use of network analysis can allow for both the discovery and understanding how these complex interactions link together within the system’s network, a property which has previously been overlooked. This powerful tool allows for the study of various types of interactions (from competitive to cooperative) using the same general framework. For example, plant-pollinator interactions are mutually beneficial and often involve many different species of pollinators as well as many different species of plants. These interactions are critical to plant reproduction and thus the accumulation of resources at the base of the food chain for primary consumers, yet these interaction networks are threatened by anthropogenic change. The use of network analysis can illuminate how pollination networks work and may in turn inform conservation efforts. Within pollination networks, nestedness (i.e., specialists interact with a subset of species that generalists interact with), redundancy (i.e., most plants are pollinated by many pollinators), and modularity play a large role in network stability. These network properties may actually work to slow the spread of disturbance effects through the system and potentially buffer the pollination network from anthropogenic changes somewhat. Researchers can even compare current constructions of species interactions networks with historical reconstructions of ancient networks to determine how networks have changed over time. Recent research into these complex species interactions networks is highly concerned with understanding what factors (e.g., diversity) lead to network stability.&lt;br /&gt;
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在生物学中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;成对相互作用 Pairwise Interactions &amp;lt;/font&amp;gt;'''历来是研究的重点。随着近几年网络科学的发展，成对相互作用内涵很有可能被丰富，外延甚至能涵盖有许多物种个体参与的多组相互作用，进而帮助我们理解更大'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;生态网络 Ecological Networks &amp;lt;/font&amp;gt;'''的结构和功能。使用网络分析可以发现和理解这些复杂的交互是如何连接成网络的，该形成过程之前经常被忽略。这个强大的工具使得同一个普世框架可以研究各种类型的交互(从竞争到合作)。例如，植物与传粉者之间的相互作用是互惠互利的，通常会涉及到许多不同种类的传粉者以及许多不同种类的植物。这些相互作用对植物的生殖和食物链底层初级消费者的资源积累至关重要，然而这些相互作用网络正受到人造变化的威胁。网络分析的使用可以说明授粉网络是如何工作的，反过来也可以为相关植物保护工作提供必要信息。在授粉网络中，内嵌性(某一领域的植物被少数几个昆虫授粉，剩下的各种昆虫只对少数几种植物授粉)、冗余性(即大多数植物是由许多授粉者授粉的)和模块性（一些物种相互之间连接的很紧密，与剩余的物种连接不紧密，由此形成一个个团簇，也可称之为“模块”）在网络稳定性中扮演着重要角色。这些网络属性实际上可以减缓干扰效应在系统上的传播，并可能缓冲授粉网络受到的人造变化的影响。研究人员甚至可以将物种相互作用网络的现有结构与该网络过去的结构进行比较，以确定网络在时间尺度上是如何演化的。最近对这些复杂物种间相互作用网络的研究聚焦于理解是什么因素(如多样性)造就了网络的稳定性。&lt;br /&gt;
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===Within-species interaction networks===&lt;br /&gt;
物种内交互网络&lt;br /&gt;
{{main article|social relation}}&lt;br /&gt;
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Network analysis provides the ability to quantify associations between individuals, which makes it possible to infer details about the network as a whole at the species and/or population level.&amp;lt;ref&amp;gt;{{cite journal |author= Croft, D.P |title= Social networks in the guppy (Poecilia reticulate) |journal= [[Philosophical Transactions of the Royal Society B]] |year= 2004 |volume= 271 |issue= Suppl |pages= S516–S519 |doi=10.1098/rsbl.2004.0206|pmid= 15801620 |pmc= 1810091 |display-authors=etal}}&amp;lt;/ref&amp;gt; One of the most attractive features of the network paradigm would be that it provides a single conceptual framework in which the social organisation of animals at all levels (individual, dyad, group, population) and for all types of interaction (aggressive, cooperative, sexual etc.) can be studied.&amp;lt;ref&amp;gt;{{Cite journal|last=Krause, Lusseau, James|first=Jens, David, Richard|date=1 May 2009|title=Animal social networks: an introduction|url=https://www.researchgate.net/publication/226996156_Animal_social_networks_An_introduction|journal=Behavioral Ecology and Sociobiology|volume=63|pages=967-973|doi=10.1007/s00265-009-0747-0|via=}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Network analysis provides the ability to quantify associations between individuals, which makes it possible to infer details about the network as a whole at the species and/or population level. One of the most attractive features of the network paradigm would be that it provides a single conceptual framework in which the social organisation of animals at all levels (individual, dyad, group, population) and for all types of interaction (aggressive, cooperative, sexual etc.) can be studied.&lt;br /&gt;
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网络分析可以量化个体之间的关联，这使得在物种和/或种群水平推断整个网络的细节成为可能。网络范式最吸引人的特征之一是，它提供了一个统一的概念框架，使得社会性动物在所有层次上(单，双，组，群)--[[用户:Ryan|Ryan]]（[[用户讨论:Ryan|讨论]]）和所有类型的互动中(攻击，合作，性等等)都可以被研究。&lt;br /&gt;
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Researchers interested in [[ethology]] across a multitude of taxa, from insects to primates, are starting to incorporate network analysis into their research. Researchers interested in social insects (e.g., ants and bees) have used network analyses to better understand division of labor, task allocation, and foraging optimization within colonies;&amp;lt;ref&amp;gt;{{cite journal |author= Dornhaus, A. |title= Benefits of recruitment in honey bees: Effects of ecology and colony size in an individual-based model |journal= Behavioral Ecology |year= 2006 |volume=17 |issue= 3 |pages= 336–344 |doi=10.1093/beheco/arj036|display-authors=etal|doi-access= free }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal |author= Linksvayer, T. |title= Developmental evolution in social insects: Regulatory networks from genes to societies |journal= Journal of Experimental Zoology Part B: Molecular and Developmental Evolution |year= 2012 |volume=318 |issue= 3 |pages= 159–169 |doi=10.1002/jez.b.22001|pmid= 22544713 |display-authors=etal}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal |author= Mullen, R. |title= A review of ant algorithms |journal= Expert Systems with Applications |year= 2009 |volume=36 |issue= 6 |pages= 9608–9617 |doi=10.1016/j.eswa.2009.01.020|display-authors=etal}}&amp;lt;/ref&amp;gt; Other researchers are interested in how certain network properties at the group and/or population level can explain individual level behaviors. Studies have demonstrated how animal social network structure can be influenced by factors ranging from characteristics of the environment to characteristics of the individual, such as developmental experience and personality. At the level of the individual, the patterning of social connections can be an important determinant of [[Fitness (biology)|fitness]], predicting both survival and reproductive success. At the population level, network structure can influence the patterning of ecological and evolutionary processes, such as [[frequency-dependent selection]] and disease and information transmission.&amp;lt;ref&amp;gt;{{Cite journal|last=Croft, Darden, Wey|first=Darren P., Safi K., Tina W.|date=2016|title=Current directions in animal social networks|url=https://www.researchgate.net/publication/310819524_Current_directions_in_animal_social_networks|journal=Current Opinion in Behavioral Sciences|volume=12|pages=52-58|doi=10.1016/j.cobeha.2016.09.001|via=|hdl=10871/23348|hdl-access=free}}&amp;lt;/ref&amp;gt; For instance, a study on [[Wire-tailed manakin|wire-tailed manakins]] (a small passerine bird)  found that a male’s [[degree (graph theory)|degree]] in the network largely predicted the ability of the male to rise in the social hierarchy (i.e. eventually obtain a territory and matings).&amp;lt;ref&amp;gt;{{cite journal |author= Ryder, T.B. |title= Social networks in the lek-mating wire-tailed manakin (''Pipra filicauda'') |journal= [[Philosophical Transactions of the Royal Society B]]  |year= 2008 |volume= 275 |issue= 1641 |pages= 1367–1374 |doi=10.1098/rspb.2008.0205 |pmid=18381257 |pmc=2602714|display-authors=etal}}&amp;lt;/ref&amp;gt; In [[bottlenose dolphin]] groups, an individual’s degree and [[betweenness centrality]] values may predict whether or not that individual will exhibit certain behaviors, like the use of side flopping and upside-down lobtailing to lead group traveling efforts; individuals with high betweenness values are more connected and can obtain more information, and thus are better suited to lead group travel and therefore tend to exhibit these signaling behaviors more than other group members.&amp;lt;ref&amp;gt;{{cite journal |author= Lusseau, D. |title= Evidence for social role in a dolphin social network |journal= Evolutionary Ecology |year= 2007 |volume=21 |issue= 3 |pages= 357–366 |doi=10.1007/s10682-006-9105-0|arxiv= q-bio/0607048 }}&amp;lt;/ref&amp;gt; &lt;br /&gt;
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Researchers interested in ethology across a multitude of taxa, from insects to primates, are starting to incorporate network analysis into their research. Researchers interested in social insects (e.g., ants and bees) have used network analyses to better understand division of labor, task allocation, and foraging optimization within colonies; Other researchers are interested in how certain network properties at the group and/or population level can explain individual level behaviors. Studies have demonstrated how animal social network structure can be influenced by factors ranging from characteristics of the environment to characteristics of the individual, such as developmental experience and personality. At the level of the individual, the patterning of social connections can be an important determinant of fitness, predicting both survival and reproductive success. At the population level, network structure can influence the patterning of ecological and evolutionary processes, such as frequency-dependent selection and disease and information transmission. For instance, a study on wire-tailed manakins (a small passerine bird)  found that a male’s degree in the network largely predicted the ability of the male to rise in the social hierarchy (i.e. eventually obtain a territory and matings). In bottlenose dolphin groups, an individual’s degree and betweenness centrality values may predict whether or not that individual will exhibit certain behaviors, like the use of side flopping and upside-down lobtailing to lead group traveling efforts; individuals with high betweenness values are more connected and can obtain more information, and thus are better suited to lead group travel and therefore tend to exhibit these signaling behaviors more than other group members. &lt;br /&gt;
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从昆虫到灵长类动物，对动物行为学感兴趣的研究人员，开始将网络分析纳入到他们的研究中。对社会性昆虫(如蚂蚁和蜜蜂)感兴趣的研究人员利用网络分析来更好地理解，群体内昆虫的分工、任务分配和觅食优化; 其他研究人员感兴趣的点则在群体和/或种群水平的某些网络特性如何影响个体的行为。研究表明，环境特征和个体特征，如经验发展和人格等因素，可影响动物的社会网络结构。在个体层面上，社会联系的模式是个体适应性的一个关键决定因素，可能预测着个体生存和繁殖的成功概率。在种群水平上，网络结构可以影响生态和进化过程的模式，如频率依赖性选择和疾病及信息的传递。例如，一项针对线尾 manakins (一种小型雀形目鸟)的研究发现，在网络中雄性个体的度值在很大程度上预测了雄性在社会等级中进阶的能力(即最终可获得领地和交配机会)。在宽吻海豚群体中，个体的度值和介数中心性可以预测个体是否会表现出某些行为，比如使用侧翻和头潜入水中尾巴举向空中来表示自己愿意领导团队的迁移; 介数值较高的个体有着更多的连接，可以获得更多信息，因此更适合领导团队迁行，因此比其他团队成员更容易表现出传递这些信号的行为。&lt;br /&gt;
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[[Social network analysis]] can also be used to describe the social organization within a species more generally, which frequently reveals important proximate mechanisms promoting the use of certain behavioral strategies. These descriptions are frequently linked to ecological properties (e.g., resource distribution). For example, network analyses revealed subtle differences in the group dynamics of two related equid [[fission-fusion society|fission-fusion]] species, [[Grevy’s zebra]] and [[onager]]s, living in variable environments; Grevy’s zebras show distinct preferences in their association choices when they fission into smaller groups, whereas onagers do not.&amp;lt;ref&amp;gt;{{cite journal |author= Sundaresan, S. |title= Network metrics reveal differences in social organization between two fission-fusion species, Grevy's zebra and onager |journal= Oecologia |year= 2007 |volume= 151 |issue= 1 |pages= 140–149 |doi=10.1007/s00442-006-0553-6|pmid= 16964497 |display-authors=etal|bibcode= 2007Oecol.151..140S }}&amp;lt;/ref&amp;gt; Similarly, researchers interested in primates have also utilized network analyses to compare social organizations across the diverse [[primate]] order, suggesting that using network measures (such as [[centrality]], [[assortativity]], [[modularity (networks)|modularity]], and betweenness) may be useful in terms of explaining the types of social behaviors we see within certain groups and not others.&amp;lt;ref&amp;gt;{{cite journal |author1=Kasper, C. |author2=Voelkl, B. |title= A social network analysis of primate groups |journal= Primates |year= 2009 |volume= 50 |issue=4 |pages= 343–356 |doi=10.1007/s10329-009-0153-2|pmid=19533270 }}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
Social network analysis can also be used to describe the social organization within a species more generally, which frequently reveals important proximate mechanisms promoting the use of certain behavioral strategies. These descriptions are frequently linked to ecological properties (e.g., resource distribution). For example, network analyses revealed subtle differences in the group dynamics of two related equid fission-fusion species, Grevy’s zebra and onagers, living in variable environments; Grevy’s zebras show distinct preferences in their association choices when they fission into smaller groups, whereas onagers do not. Similarly, researchers interested in primates have also utilized network analyses to compare social organizations across the diverse primate order, suggesting that using network measures (such as centrality, assortativity, modularity, and betweenness) may be useful in terms of explaining the types of social behaviors we see within certain groups and not others. &lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会网络分析 Social Network Analysis &amp;lt;/font&amp;gt;'''也可以用来更广泛地描述一个物种内的社会组织，它经常揭示出那些促进某些行为策略使用的重要邻近机制。这些描述经常与生态属性(例如，资源分配)联系在一起。例如，网络分析揭示了生活在变化环境中的两个相关的等裂变融合物种——格雷维斑马和骑驴——的群体动力学的细微差异（裂变融合社会是这样一个社会，其规模和组成随着时间的流逝以及动物在整个环境中的移动而变化；动物会合并成一团（融合）-例如在一个地方睡觉-或分裂（裂变）-例如白天在小组中觅食）; 格雷维斑马在分裂为较小的群体时，在关联选择上表现出明显的偏好，而骑驴则不然。同样，对灵长类动物感兴趣的研究人员也利用网络分析来比较不同灵长类动物的社会组织，这表明以网络的视角进行测量(如集中性、协调性、模块性和中介性)可能有助于解释我们在某些特定群体中看到的社会行为类型。&lt;br /&gt;
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Finally, social network analysis can also reveal important fluctuations in animal behaviors across changing environments. For example, network analyses in female [[chacma baboon]]s (''Papio hamadryas ursinus'') revealed important dynamic changes across seasons which were previously unknown; instead of creating stable, long-lasting social bonds with friends, baboons were found to exhibit more variable relationships which were dependent on short-term contingencies related to group level dynamics as well as environmental variability.&amp;lt;ref&amp;gt;{{cite journal |author= Henzi, S. |title= Cyclicity in the structure of female baboon social networks |journal= Behavioral Ecology and Sociobiology |year= 2009 |volume= 63 |issue= 7 |pages= 1015–1021 |doi=10.1007/s00265-009-0720-y|display-authors=etal}}&amp;lt;/ref&amp;gt; Changes in an individual's social network environment can also influence characteristics such as 'personality': for example, social spiders that huddle with bolder neighbours tend to increase also in boldness.&amp;lt;ref&amp;gt;{{cite journal |author= Hunt, ER. |title= Social interactions shape individual and collective personality in social spiders |journal= Proceedings of the Royal Society B |year= 2018 |volume= 285 |issue= 1886 |pages= 20181366 |doi=10.1098/rspb.2018.1366|pmid= 30185649 |pmc= 6158534 |display-authors=etal}}&amp;lt;/ref&amp;gt; This is a very small set of broad examples of how researchers can use network analysis to study animal behavior. Research in this area is currently expanding very rapidly, especially since the broader development of animal borne tags and [[computer vision]] that can be used to automate the collection of social associations.&amp;lt;ref&amp;gt;{{Cite journal|last=Krause J, Krause S, Arlinghaus R, Psorakis I, Roberts S, Rutz C|first=|date=2013|title=Reality mining of animal social systems.|url=https://risweb.st-andrews.ac.uk/portal/en/researchoutput/reality-mining-of-animal-social-systems(570e21d0-527e-4275-8926-c4b479c55ee4).html|journal=Trends in Ecology and Evolution|volume=28|pages=541-551|doi=10.1016/j.tree.2013.06.002|via=}}&amp;lt;/ref&amp;gt; Social network analysis is a valuable tool for studying animal behavior across all animal species, and has the potential to uncover new information about animal behavior and social ecology that was previously poorly understood.&lt;br /&gt;
&lt;br /&gt;
Finally, social network analysis can also reveal important fluctuations in animal behaviors across changing environments. For example, network analyses in female chacma baboons (Papio hamadryas ursinus) revealed important dynamic changes across seasons which were previously unknown; instead of creating stable, long-lasting social bonds with friends, baboons were found to exhibit more variable relationships which were dependent on short-term contingencies related to group level dynamics as well as environmental variability. Changes in an individual's social network environment can also influence characteristics such as 'personality': for example, social spiders that huddle with bolder neighbours tend to increase also in boldness. This is a very small set of broad examples of how researchers can use network analysis to study animal behavior. Research in this area is currently expanding very rapidly, especially since the broader development of animal borne tags and computer vision that can be used to automate the collection of social associations. Social network analysis is a valuable tool for studying animal behavior across all animal species, and has the potential to uncover new information about animal behavior and social ecology that was previously poorly understood.&lt;br /&gt;
&lt;br /&gt;
最后，社会网络分析还可以揭示动物行为在不断变化的环境中的关键波动。例如，对雌性沙卡马狒狒(Papio hamadryas ursinus)的网络分析揭示了以前未知的跨季节的重要动态变化; 结果发现，狒狒之间没有建立稳定、持久的社会联系，相反，它们之间的关系是多变的。这些关系依赖于在动态的群体层面产生的短期偶然事件以及环境中的变化。个体社交网络环境的变化也会影响个体的性格特征，例如，与胆大的邻居挤在一起的蜘蛛也会更加大胆。这只是研究人员利用网络分析来研究动物行为的一小部分。这一领域的研究目前正在迅速扩展，特别是随着动物身体标记技术和计算机视觉技术的广泛发展，这些技术可以用来自动收集个体的社会关系。社会网络分析是研究所有物种的行为的一个有价值的工具，并且有可能发现以前鲜为人知的关于动物行为和社会生态的新信息。&lt;br /&gt;
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&lt;br /&gt;
==See also==&lt;br /&gt;
&lt;br /&gt;
* [[List of omics topics in biology]]&lt;br /&gt;
生物学中的组学&lt;br /&gt;
* [[Biological network inference]]&lt;br /&gt;
生物网络推理&lt;br /&gt;
* [[Applied Statistics]]&lt;br /&gt;
应用统计学&lt;br /&gt;
* [[Biostatistics]]&lt;br /&gt;
生物统计学&lt;br /&gt;
* [[Computational Biology]]&lt;br /&gt;
计算生物学&lt;br /&gt;
* [[Systems biology]]&lt;br /&gt;
系统生物学&lt;br /&gt;
* [[Weighted correlation network analysis]] &lt;br /&gt;
权重相关性网络分析&lt;br /&gt;
* [[Interactome]]&lt;br /&gt;
交互组&lt;br /&gt;
* [[Network medicine]]&lt;br /&gt;
网络医学&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== External links ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
* [http://www.nettab.org Network Tools and Applications in Biology] (NETTAB) workshops.&lt;br /&gt;
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&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&lt;br /&gt;
{{Reflist}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Books==&lt;br /&gt;
&lt;br /&gt;
*E. Estrada,  &amp;quot;The Structure of Complex Networks: Theory and Applications&amp;quot;, Oxford University Press, 2011, {{ISBN|978-0-199-59175-6}}&lt;br /&gt;
&lt;br /&gt;
*J. Krause, R. James, D. Franks, D. Croft, &amp;quot;Animal Social Networks&amp;quot;, Oxford University Press, 2015, {{ISBN|978-0199679041}}&lt;br /&gt;
&lt;br /&gt;
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&lt;br /&gt;
==External links==&lt;br /&gt;
&lt;br /&gt;
* [https://web.archive.org/web/20110920220214/http://www.networkbio.org/ Networkbio.org], The site of the series of Integrative Network Biology (INB) meetings. For the 2012 event also see www.networkbio.org&lt;br /&gt;
&lt;br /&gt;
* [http://networkbiology.org Networkbiology.org], NetworkBiology wiki site.&lt;br /&gt;
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* [http://www.lindinglab.org LindingLab.org], Technical University of Denmark (DTU) studies Network Biology and Cellular Information Processing, and is also organizing the Denmark branch of the annual &amp;quot;[http://www.lindinglab.org/events ''Integrative Network Biology and Cancer'']&amp;quot; symposium series.&lt;br /&gt;
&lt;br /&gt;
* [http://nrnb.org NRNB.org], The National Resource for Network Biology. A US National Institute of Health (NIH) Biomedical Technology Research Center dedicated to the study of biological networks.&lt;br /&gt;
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*[http://networkrepository.com/asn.php Network Repository] The first interactive data and network data repository with real-time visual analytics.&lt;br /&gt;
&lt;br /&gt;
*[https://bansallab.github.io/asnr/ Animal Social Network Repository(ASNR)] The first multi-taxonomic repository that collates 790 social networks from more than 45 species, including those of mammals, reptiles, fish, birds, and insects &lt;br /&gt;
&lt;br /&gt;
*[http://konect.uni-koblenz.de/networks/ KONECT – Koblenz network collection.]&lt;br /&gt;
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&lt;br /&gt;
[[Category:Biological techniques and tools]]&lt;br /&gt;
&lt;br /&gt;
Category:Biological techniques and tools&lt;br /&gt;
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类别: 生物技术和工具&lt;br /&gt;
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[[Category:Bioinformatics]]&lt;br /&gt;
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Category:Bioinformatics&lt;br /&gt;
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类别: 生物信息学&lt;br /&gt;
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[[Category:Systems biology]]&lt;br /&gt;
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Category:Systems biology&lt;br /&gt;
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分类: 系统生物学&lt;br /&gt;
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[[Category:Networks]]&lt;br /&gt;
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Category:Networks&lt;br /&gt;
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类别: 网络&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Biological network]]. Its edit history can be viewed at [[生物网络/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E9%93%BE%E8%B7%AF%E5%88%86%E6%9E%90&amp;diff=12551</id>
		<title>链路分析</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E9%93%BE%E8%B7%AF%E5%88%86%E6%9E%90&amp;diff=12551"/>
		<updated>2020-08-15T02:12:48Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
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&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
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In [[network theory]], '''link analysis''' is a [[data-analysis]] technique used to evaluate relationships (connections) between nodes. Relationships may be identified among various types of nodes (objects), including [[organization]]s, [[people]] and [[Financial transaction|transactions]]. Link analysis has been used for investigation of criminal activity ([[fraud detection]], [[counterterrorism]], and [[Intelligence (information gathering)|intelligence]]), [[Computer security|computer security analysis]], [[search engine optimization]], [[market research]], [[medical research]], and art.&lt;br /&gt;
&lt;br /&gt;
In network theory, link analysis is a data-analysis technique used to evaluate relationships (connections) between nodes. Relationships may be identified among various types of nodes (objects), including organizations, people and transactions. Link analysis has been used for investigation of criminal activity (fraud detection, counterterrorism, and intelligence), computer security analysis, search engine optimization, market research, medical research, and art.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 网络理论 Network Theory&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 链路分析 Link Analysis&amp;lt;/font&amp;gt;'''是一种用于评估节点之间关系(连接)的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 数据分析 Data Analysis&amp;lt;/font&amp;gt;'''技术。该技术可以鉴别各种类型节点(对象)之间的关系，包括组织、人群和市场交易双方。链路分析已被应用于诸多领域，如打击犯罪活动(如欺诈侦查、反恐和情报)、计算机安全分析、搜索引擎优化、市场调查、医学研究和艺术。&lt;br /&gt;
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==Knowledge discovery==&lt;br /&gt;
知识发现&lt;br /&gt;
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[[Knowledge discovery]] is an [[Iteration|iterative]] and [[interactive]] process used to [[Identification (information)|identify]], analyze and visualize patterns in data.&amp;lt;ref&amp;gt;{{cite web|url=https://www.torproject.org/about/overview.html.en|title=Tor Project: Overview|first=The Tor Project|last=Inc.|publisher=}}&amp;lt;/ref&amp;gt; Network analysis, link analysis and [[social network analysis]] are all methods of knowledge discovery, each a corresponding subset of the prior method. Most knowledge discovery methods follow these steps (at the highest level):&amp;lt;ref&amp;gt;Ahonen, H., [http://www.cs.helsinki.fi/u/hahonen/features.txt Features of Knowledge Discovery Systems].&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Knowledge discovery is an iterative and interactive process used to identify, analyze and visualize patterns in data. Network analysis, link analysis and social network analysis are all methods of knowledge discovery, each a corresponding subset of the prior method. Most knowledge discovery methods follow these steps (at the highest level):&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 知识的发现 Knowledge Discovery&amp;lt;/font&amp;gt;'''，是指不断地识别、分析和可视化数据中的内在模式，这是一个持续迭代和交互的过程。网络分析、链路分析和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会网络分析 Social Network Analysis&amp;lt;/font&amp;gt;'''都是知识发现的方法，它们都是属于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 先验方法 Prior Method&amp;lt;/font&amp;gt;'''。大多数知识发现方法遵循以下几个步骤('''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;在最高级别&amp;lt;/font&amp;gt;''' at the highest level) :&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）遇到拿不准的专业名词翻译 可从cnki翻译助手检索一下该名词，下为链接http://dict.cnki.net/dict_result.aspx  选取引用量较高的释义进行翻译 譬如Knowledge Discovery 可以暂时定为知识发现  后续专家审校进行反馈再核实其准不准确&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
# [[Data processing]]&lt;br /&gt;
&lt;br /&gt;
 Data processing&lt;br /&gt;
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数据处理&lt;br /&gt;
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# [[Data transformation|Transformation]]&lt;br /&gt;
&lt;br /&gt;
 Transformation&lt;br /&gt;
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数据转变&lt;br /&gt;
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# [[Data analysis|Analysis]]&lt;br /&gt;
&lt;br /&gt;
 Analysis&lt;br /&gt;
&lt;br /&gt;
数据分析&lt;br /&gt;
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# [[Data visualization|Visualization]]&lt;br /&gt;
&lt;br /&gt;
 Visualization&lt;br /&gt;
&lt;br /&gt;
数据可视化&lt;br /&gt;
&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）这里很棒 补充出“数据”这个限定词&lt;br /&gt;
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Data gathering and processing requires access to data and has several inherent issues, including [[information overload]] and data errors. Once data is collected, it will need to be transformed into a format that can be effectively used by both human and computer analyzers. Manual or computer-generated visualizations tools may be mapped from the data, including network charts. Several algorithms exist to help with analysis of data – [[Dijkstra's algorithm|Dijkstra’s algorithm]], [[breadth-first search]], and [[depth-first search]].&lt;br /&gt;
&lt;br /&gt;
Data gathering and processing requires access to data and has several inherent issues, including information overload and data errors. Once data is collected, it will need to be transformed into a format that can be effectively used by both human and computer analyzers. Manual or computer-generated visualizations tools may be mapped from the data, including network charts. Several algorithms exist to help with analysis of data – Dijkstra’s algorithm, breadth-first search, and depth-first search.&lt;br /&gt;
&lt;br /&gt;
数据的收集和处理是首先进行的过程，但此过程存在一些固有的问题，包括'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;信息超载 Information Overload&amp;lt;/font&amp;gt;'''和数据错误等。在数据被收集后，它将转换成一种人和计算机分析程序都能有效使用的格式。之后基于数据，可使用计算机生成的或人工操作的可视化工具进行作图（如网络图）。目前有几种算法可以帮助人类进行数据分析-'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Dijkstra算法&amp;lt;/font&amp;gt;'''，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;广度优先搜索 Breadth-First Search&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 深度优先搜索 Depth-First Search&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
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Link analysis focuses on analysis of relationships among nodes through [[Visualization (computer graphics)|visualization methods]] ([[network charts]], association matrix). Here is an example of the relationships that may be mapped for crime investigations:&amp;lt;ref name=Krebs&amp;gt;Krebs, V. E. 2001, [http://vlado.fmf.uni-lj.si/pub/networks/doc/Seminar/Krebs.pdf Mapping networks of terrorist cells] {{webarchive|url=https://web.archive.org/web/20110720000539/http://vlado.fmf.uni-lj.si/pub/networks/doc/Seminar/Krebs.pdf |date=2011-07-20 }}, Connections 24, 43–52.&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
Link analysis focuses on analysis of relationships among nodes through visualization methods (network charts, association matrix). Here is an example of the relationships that may be mapped for crime investigations: &lt;br /&gt;
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链路分析主要通过可视化方法(网络图、关联矩阵)分析节点之间的关系。这里有一个基于罪犯和社会各部分关系绘制网图的例子:&lt;br /&gt;
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&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
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{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
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{ | class“ wikitable”&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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! Relationship/Network !! Data Sources&lt;br /&gt;
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! Relationship/Network !! Data Sources&lt;br /&gt;
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!关系 / 网络! ！数据来源&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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| 1. Trust || Prior contacts in family, neighborhood, school, military, club or organization. Public and court records. Data may only be available in suspect's native country.&lt;br /&gt;
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| 1. Trust || Prior contacts in family, neighborhood, school, military, club or organization. Public and court records. Data may only be available in suspect's native country.&lt;br /&gt;
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| 1.信任 | 嫌疑人在家庭、社区、学校、军队、俱乐部或组织中已有的联系。公开信息及法庭纪录。以及只能在嫌疑人本国获得的数据。&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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|-&lt;br /&gt;
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| 2. Task || Logs and records of phone calls, electronic mail, chat rooms, instant messages, Web site visits. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 2. Task || Logs and records of phone calls, electronic mail, chat rooms, instant messages, Web site visits. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 2.任务 | | 电话、电子邮件、聊天室、即时消息、网站访问的日志和记录。出入境纪录。'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;人类智能: 会议评论和公共活动的出席。&amp;lt;/font&amp;gt;'''Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 3. Money &amp;amp; Resources || Bank account and money transfer records. Pattern and location of credit card use. Prior court records. Human intelligence: observation of visits to alternate banking resources such as [[Hawala]].&lt;br /&gt;
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| 3. Money &amp;amp; Resources || Bank account and money transfer records. Pattern and location of credit card use. Prior court records. Human intelligence: observation of visits to alternate banking resources such as Hawala.&lt;br /&gt;
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| 3.资金和资源 | 银行账户和汇款记录。信用卡使用地点及使用习惯。以前的法庭记录。'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;人类智能: 访问其他银行资源的观察，如 Hawala。&amp;lt;/font&amp;gt;'''Human intelligence: observation of visits to alternate banking resources such as Hawala.&lt;br /&gt;
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| 4. Strategy &amp;amp; Goals || Web sites. Videos and encrypted disks delivered by courier. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 4. Strategy &amp;amp; Goals || Web sites. Videos and encrypted disks delivered by courier. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 4.策略与目标 | 网站。由快递公司递送的视频和加密光盘。出入境纪录。'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;人类智能: 会议评论和公共活动的出席。&amp;lt;/font&amp;gt;'''Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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Link analysis is used for 3 primary purposes:&amp;lt;ref name=&amp;quot;Link Analysis Workbench&amp;quot;&amp;gt;[https://docs.google.com/viewer?a=v&amp;amp;q=cache:R4_7k-udkxMJ:www.fas.org/irp/eprint/law.pdf+%22link+analysis%22&amp;amp;hl=en&amp;amp;gl=uk&amp;amp;pid=bl&amp;amp;srcid=ADGEESjUVRJFP2O2eMGzSuW4WtGgujUHVeuUYDQ9wqOkJUeVMHOwpagPa65ypDM5Fgma2AlwDyFBblZwXNMoYDCHwCQuTQvK-HnmqbW-z5A-MOwMKLy8nk6_uHLa0CiUAql-kAilAYcd&amp;amp;sig=AHIEtbRgGVNE4PMZ0a2vipzlDTd6OLX4fA Link Analysis Workbench], Air Force Research Laboratory Information Directorate, Rome Research Site, Rome, New York, September 2004.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Link analysis is used for 3 primary purposes:&lt;br /&gt;
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链接分析主要有3个作用:&lt;br /&gt;
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# Find matches in data for known patterns of interest;&lt;br /&gt;
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 Find matches in data for known patterns of interest;&lt;br /&gt;
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在数据中寻找有意义的已知模式;&lt;br /&gt;
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# Find anomalies where known patterns are violated;&lt;br /&gt;
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 Find anomalies where known patterns are violated;&lt;br /&gt;
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发现与已知模式不相符的异常数据;&lt;br /&gt;
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# Discover new patterns of interest (social network analysis, [[data mining]]).&lt;br /&gt;
&lt;br /&gt;
 Discover new patterns of interest (social network analysis, data mining).&lt;br /&gt;
&lt;br /&gt;
发现有意义的新模式(社会网络分析、数据挖掘)。&lt;br /&gt;
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==History==&lt;br /&gt;
&lt;br /&gt;
Klerks categorized link analysis tools into 3 generations.&amp;lt;ref&amp;gt;{{cite journal | last = Klerks | first = P. | year = 2001 | title = The network paradigm applied to criminal organizations: Theoretical nitpicking or a relevant doctrine for investigators? Recent developments in the Netherlands | citeseerx = 10.1.1.129.4720 | journal = Connections | volume = 24 | pages = 53–65 }}&amp;lt;/ref&amp;gt; The first generation was introduced in 1975 as the Anacpapa Chart of Harper and Harris.&amp;lt;ref&amp;gt;Harper and Harris, The Analysis of Criminal Intelligence, Human Factors and Ergonomics Society Annual Meeting Proceedings, 19(2), 1975, pp. 232-238.&amp;lt;/ref&amp;gt; This method requires that a domain expert review data files, identify associations by constructing an association matrix, create a link chart for visualization and finally analyze the network chart to identify patterns of interest. This method requires extensive domain knowledge and is extremely time-consuming when reviewing vast amounts of data.&lt;br /&gt;
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[[File:Association Matrix.png|thumb|&lt;br /&gt;
图1：Association Matrix 关联矩阵]]&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）点击编辑后会发现[[File: 开头关于图片的语法  该图注需要进行翻译且遵循格式【图1：英文原文＋翻译内容】&lt;br /&gt;
&lt;br /&gt;
Klerks categorized link analysis tools into 3 generations. The first generation was introduced in 1975 as the Anacpapa Chart of Harper and Harris. This method requires that a domain expert review data files, identify associations by constructing an association matrix, create a link chart for visualization and finally analyze the network chart to identify patterns of interest. This method requires extensive domain knowledge and is extremely time-consuming when reviewing vast amounts of data.&lt;br /&gt;
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Klerks把链接分析工具分为三代。第一代是由哈珀和哈里斯在1975年引入的，阿纳卡帕图。这种方法需要一个领域内的专家来审查数据文件，通过构造一个关联矩阵来识别关联，然后创建一个可视化的链路图，最后通过分析网络图来识别有意义的模式。这种方法需要广泛的领域知识，且因要人工审查大量数据，所以非常耗时。&lt;br /&gt;
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In addition to the association matrix, the activities matrix can be used to produce actionable information, which has practical value and use to law-enforcement. The activities matrix, as the term might imply, centers on the actions and activities of people with respect to locations. Whereas the association matrix focuses on the relationships between people, organizations, and/or properties. The distinction between these two types of matrices, while minor, is nonetheless significant in terms of the output of the analysis completed or rendered.&amp;lt;ref&amp;gt;{{cite web|url=http://www.globalsecurity.org/military/library/policy/army/fm/3-07-22/app-f.htm|title=FMI 3-07.22 Appendix F Intelligence Analysis Tools and Indicators|first=John|last=Pike|publisher=}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[https://rdl.train.army.mil/catalog/view/100.ATSC/41449AB4-E8E0-46C4-8443-E4276B6F0481-1274576841878/3-24/appb.htm Social Network Analysis and Other Analytical Tools] {{webarchive|url=https://web.archive.org/web/20140308233614/https://rdl.train.army.mil/catalog/view/100.ATSC/41449AB4-E8E0-46C4-8443-E4276B6F0481-1274576841878/3-24/appb.htm |date=2014-03-08 }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite web|url=http://www.nasa.gov/audience/foreducators/topnav/materials/listbytype/Aeronautics_Activity_Matrices.html|title=Aeronautics Educator Guide - Activity Matrices|first=Rebecca Whitaker|last=MSFC|date=10 July 2009|publisher=}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[https://rdl.train.army.mil/catalog/view/100.ATSC/0EF89CA1-2680-4782-B103-D2F5DC941188-1274309335668/7-98-1/chap2l6.htm Personality/Activity Matrix] {{webarchive|url=https://web.archive.org/web/20140308234135/https://rdl.train.army.mil/catalog/view/100.ATSC/0EF89CA1-2680-4782-B103-D2F5DC941188-1274309335668/7-98-1/chap2l6.htm |date=2014-03-08 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In addition to the association matrix, the activities matrix can be used to produce actionable information, which has practical value and use to law-enforcement. The activities matrix, as the term might imply, centers on the actions and activities of people with respect to locations. Whereas the association matrix focuses on the relationships between people, organizations, and/or properties. The distinction between these two types of matrices, while minor, is nonetheless significant in terms of the output of the analysis completed or rendered.&lt;br /&gt;
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除了关联矩阵外，活动矩阵也可用于生成对执法活动具有实用和使用价值的可操作的信息。正如这个术语可能暗示的那样，活动矩阵关注的是人们基于地点的行动和活动。而关联矩阵关注的是人，组织和/或属性之间的关系。这两类矩阵之间的区别虽然很小，但就已完成或经过分析的数据来看，区别还是很重要的。&lt;br /&gt;
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Second generation tools consist of automatic graphics-based analysis tools such as IBM i2 Analyst’s Notebook, Netmap, [[ClueMaker]] and Watson. These tools offer the ability to automate the construction and updates of the link chart once an association matrix is manually created, however, analysis of the resulting charts and graphs still requires an expert with extensive domain knowledge.&lt;br /&gt;
&lt;br /&gt;
Second generation tools consist of automatic graphics-based analysis tools such as IBM i2 Analyst’s Notebook, Netmap, ClueMaker and Watson. These tools offer the ability to automate the construction and updates of the link chart once an association matrix is manually created, however, analysis of the resulting charts and graphs still requires an expert with extensive domain knowledge.&lt;br /&gt;
&lt;br /&gt;
第二代工具包括基于图形的自动分析工具，如 IBM i 2 Analyst’ s Notebook、 Netmap、 ClueMaker 和 Watson。在手动创建关联矩阵的情况下，这些工具提供了自动构建和更新链接图表的能力。然而，对结果图表的分析仍然需要在该领域具有广泛知识的专家。&lt;br /&gt;
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The third generation of link-analysis tools like DataWalk allow the automatic visualization of linkages between elements in a data set, that can then serve as the canvas for further exploration or manual updates.&lt;br /&gt;
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The third generation of link-analysis tools like DataWalk allow the automatic visualization of linkages between elements in a data set, that can then serve as the canvas for further exploration or manual updates.&lt;br /&gt;
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像 DataWalk 这样的第三代链接分析工具能自动可视化数据集中元素之间的链接，其结果可允许进一步的人工探索与改进。&lt;br /&gt;
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==Applications==&lt;br /&gt;
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* [[ViCAP|FBI Violent Criminal Apprehension Program (ViCAP)]]&lt;br /&gt;
联邦调查局暴力刑事逮捕程序&lt;br /&gt;
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* Iowa State Sex Crimes Analysis System&lt;br /&gt;
爱荷华州性犯罪分析系统&lt;br /&gt;
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* Minnesota State Sex Crimes Analysis System (MIN/SCAP)&lt;br /&gt;
明尼苏达州性犯罪分析系统&lt;br /&gt;
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* Washington State Homicide Investigation Tracking System (HITS)&amp;lt;ref&amp;gt;{{cite web|url=http://www.atg.wa.gov/HITS.aspx |title=Archived copy |accessdate=2010-10-31 |url-status=dead |archiveurl=https://web.archive.org/web/20101021005202/http://atg.wa.gov/HITS.aspx |archivedate=2010-10-21 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
华盛顿州凶杀案调查追踪系统&lt;br /&gt;
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* New York State Homicide Investigation &amp;amp; Lead Tracking (HALT)&lt;br /&gt;
纽约州凶杀案调查和线索追踪系统&lt;br /&gt;
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* New Jersey Homicide Evaluation &amp;amp; Assessment Tracking (HEAT)&amp;lt;ref&amp;gt;{{cite web|url=http://www.state.nj.us/njsp/divorg/invest/invest.html |title=Archived copy |accessdate=2010-10-31 |url-status=dead |archiveurl=https://web.archive.org/web/20090325004722/http://www.state.nj.us/njsp/divorg/invest/invest.html |archivedate=2009-03-25 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
新泽西州凶杀案评估与测评跟踪系统&lt;br /&gt;
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* Pennsylvania State ATAC Program.&lt;br /&gt;
宾夕法尼亚州ATAC程序&lt;br /&gt;
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* Violent Crime Linkage Analysis System (ViCLAS)&amp;lt;ref&amp;gt;{{cite web|url=http://www.rcmp-grc.gc.ca/tops-opst/bs-sc/viclas-salvac-eng.htm |title=Archived copy |accessdate=2010-10-31 |url-status=dead |archiveurl=https://web.archive.org/web/20101202144141/http://www.rcmp-grc.gc.ca/tops-opst/bs-sc/viclas-salvac-eng.htm |archivedate=2010-12-02 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
暴力犯罪联系分析系统&lt;br /&gt;
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==Issues with link analysis==&lt;br /&gt;
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===Information overload===&lt;br /&gt;
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With the vast amounts of data and information that are stored electronically, users are confronted with multiple unrelated sources of information available for analysis. Data analysis techniques are required to make effective and efficient use of the data. Palshikar classifies data analysis techniques into two categories – ([[Statistics|statistical]] [[Models of computation|models]], [[Time series analysis|time-series analysis]], [[Clustering coefficient|clustering]] and [[Statistical classification|classification]], matching algorithms to detect anomalies) and [[Artificial intelligence|artificial intelligence (AI)]] techniques (data mining, [[expert systems]], [[pattern recognition]], [[Machine learning|machine learning techniques]], [[neural network]]s).&amp;lt;ref&amp;gt;Palshikar, G. K., [http://www.intelligententerprise.com//020528/509feat3_1.jhtml The Hidden Truth], Intelligent Enterprise, May 2002.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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With the vast amounts of data and information that are stored electronically, users are confronted with multiple unrelated sources of information available for analysis. Data analysis techniques are required to make effective and efficient use of the data. Palshikar classifies data analysis techniques into two categories – (statistical models, time-series analysis, clustering and classification, matching algorithms to detect anomalies) and artificial intelligence (AI) techniques (data mining, expert systems, pattern recognition, machine learning techniques, neural networks).&lt;br /&gt;
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由于大量数据和信息以电子形式存储，用户可能会面临拥有多种不相关的信息来源却不知如何分析的难题。数据分析技术的使用可以帮助有效和高效地利用数据。Palshikar 将数据分析技术分为两大类(统计模型、时间序列分析、聚类分类、异常检测匹配算法)和人工智能(AI)技术(数据挖掘、专家系统、模式识别、机器学习技术、神经网络)。&lt;br /&gt;
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Bolton &amp;amp; Hand define statistical data analysis as either supervised or unsupervised methods.&amp;lt;ref&amp;gt;Bolton, R. J. &amp;amp; Hand, D. J., Statistical Fraud Detection: A Review, Statistical Science, 2002, 17(3), pp. 235-255.&amp;lt;/ref&amp;gt; [[Supervised learning|Supervised learning methods]] require that rules are defined within the system to establish what is expected or unexpected behavior. [[Unsupervised learning|Unsupervised learning methods]] review data in comparison to the norm and detect statistical outliers. Supervised learning methods are limited in the scenarios that can be handled as this method requires that training rules are established based on previous patterns. Unsupervised learning methods can provide detection of broader issues, however, may result in a higher false-positive ratio if the behavioral norm is not well established or understood.&lt;br /&gt;
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Bolton &amp;amp; Hand define statistical data analysis as either supervised or unsupervised methods. Supervised learning methods require that rules are defined within the system to establish what is expected or unexpected behavior. Unsupervised learning methods review data in comparison to the norm and detect statistical outliers. Supervised learning methods are limited in the scenarios that can be handled as this method requires that training rules are established based on previous patterns. Unsupervised learning methods can provide detection of broader issues, however, may result in a higher false-positive ratio if the behavioral norm is not well established or understood.&lt;br /&gt;
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Bolton &amp;amp; Hand 将统计数据分析定义为有监督或无监督的方法。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;监督式学习方法 Supervised Learning Methods&amp;lt;/font&amp;gt;'''要求在系统中有明确的规则来指出什么是预期行为，什么是意外行为。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;非监督式学习方法 Unsupervised Learning Methods&amp;lt;/font&amp;gt;'''在审视数据时，通过将数据与正常值的比较，来发现统计异常值。监督式学习方法能处理的场景是有限的，因为这种方法需要基于以前的模式建立训练规则。非监督式学习方法可以对更广泛的问题采取进攻。但是，如果数据的行为规范没有很好的建立或被机器理解 --[[用户:Ryan|Ryan]]（[[用户讨论:Ryan|讨论]]）该句存疑，其结果可能会导致较高的假阳性率（本身不是正常值，但识别为正常值，说明算法预测了“正确”或“有”的判断，但却判断错误了）。&lt;br /&gt;
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Data itself has inherent issues including integrity (or lack of) and continuous changes. Data may contain “errors of omission and commission because of faulty collection or handling, and when entities are actively attempting to deceive and/or conceal their actions”.&amp;lt;ref name=&amp;quot;Link Analysis Workbench&amp;quot;/&amp;gt; Sparrow&amp;lt;ref&amp;gt;Sparrow M.K. 1991. Network Vulnerabilities and Strategic Intelligence in Law Enforcement’, [[International Journal of Intelligence and Counterintelligence]] Vol. 5 #3.&amp;lt;/ref&amp;gt; highlights incompleteness (inevitability of missing data or links), fuzzy boundaries (subjectivity in deciding what to include) and dynamic changes (recognition that data is ever-changing) as the three primary problems with data analysis.&amp;lt;ref name=Krebs/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Data itself has inherent issues including integrity (or lack of) and continuous changes. Data may contain “errors of omission and commission because of faulty collection or handling, and when entities are actively attempting to deceive and/or conceal their actions”. Sparrow highlights incompleteness (inevitability of missing data or links), fuzzy boundaries (subjectivity in deciding what to include) and dynamic changes (recognition that data is ever-changing) as the three primary problems with data analysis.&lt;br /&gt;
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数据本身存在固有的问题，包括完整性(或缺失性)和持续的改变。数据可能包含'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;“由于错误的收集或处理，以及当实体积极试图欺骗和 / 或隐瞒其行为，而造成的错误的遗漏和委托”&amp;lt;/font&amp;gt;''' “errors of omission and commission because of faulty collection or handling, and when entities are actively attempting to deceive and/or conceal their actions”。Sparro强调了数据分析中三个主要的问题，不完整性(数据或链接缺失的必然性)、模糊边界(边界确定的主观性)和动态变化(数据的持续变化性)。&lt;br /&gt;
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  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）存疑的地方也可以用绿色语法进行标注&lt;br /&gt;
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Once data is transformed into a usable format, open texture and cross referencing issues may arise. [[Open texture]] was defined by [[Friedrich Waismann|Waismann]] as the unavoidable uncertainty in meaning when empirical terms are used in different contexts.&amp;lt;ref&amp;gt;Friedrich Waismann, Verifiability (1945), p.2.&amp;lt;/ref&amp;gt; Uncertainty in meaning of terms presents problems when attempting to search and cross reference data from multiple sources.&amp;lt;ref&amp;gt;Lyons, D., [http://ssrn.com/abstract=212328 Open Texture and the Possibility of Legal Interpretation (2000)].&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Once data is transformed into a usable format, open texture and cross referencing issues may arise. Open texture was defined by Waismann as the unavoidable uncertainty in meaning when empirical terms are used in different contexts. Uncertainty in meaning of terms presents problems when attempting to search and cross reference data from multiple sources.&lt;br /&gt;
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一旦数据转换成可用的格式，开放纹理和交叉引用问题就会出现。Waismann将'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 开放纹理 Open Texture&amp;lt;/font&amp;gt;'''定义为在不同语境中使用经验词汇时不可避免的语义不确定性。当试图从多个数据源搜索和交叉引用数据时，术语含义的不确定性带来了问题。&lt;br /&gt;
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The primary method for resolving data analysis issues is reliance on [[domain knowledge]] from an expert. This is a very time-consuming and costly method of conducting link analysis and has inherent problems of its own. McGrath et al. conclude that the layout and presentation of a network diagram have a significant impact on the user’s “perceptions of the existence of groups in networks”.&amp;lt;ref&amp;gt;McGrath, C., Blythe, J., Krackhardt, D., [http://www.andrew.cmu.edu/user/cm3t/groups.html Seeing Groups in Graph Layouts].&amp;lt;/ref&amp;gt; Even using domain experts may result in differing conclusions as analysis may be subjective.&lt;br /&gt;
&lt;br /&gt;
The primary method for resolving data analysis issues is reliance on domain knowledge from an expert. This is a very time-consuming and costly method of conducting link analysis and has inherent problems of its own. McGrath et al. conclude that the layout and presentation of a network diagram have a significant impact on the user’s “perceptions of the existence of groups in networks”. Even using domain experts may result in differing conclusions as analysis may be subjective.&lt;br /&gt;
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目前，解决数据分析中这些问题的主要方法是依赖专家的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;领域知识 Domain Knowledge&amp;lt;/font&amp;gt;'''。如此进行链路分析是非常耗时和昂贵的，并且无法排除其自身固有的问题。麦格拉斯等人得出结论，网络图的分布和表示方式对用户的“对存在在网络中群体的感知”有重大影响。即使是领域内的专家也可能导致不同的结论，因为分析可能是很主观的。&lt;br /&gt;
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===Prosecution vs. crime prevention===&lt;br /&gt;
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Link analysis techniques have primarily been used for prosecution, as it is far easier to review historical data for patterns than it is to attempt to predict future actions.&lt;br /&gt;
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Link analysis techniques have primarily been used for prosecution, as it is far easier to review historical data for patterns than it is to attempt to predict future actions.&lt;br /&gt;
&lt;br /&gt;
目前，链接分析技术主要用于起诉，因为回顾历史数据以期从中获得模式，要比预测未来的行动容易得多。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Krebs demonstrated the use of an association matrix and link chart of the terrorist network associated with the 19 hijackers responsible for the [[September 11 attacks|September 11th attacks]] by mapping publicly available details made available following the attacks.&amp;lt;ref name=Krebs/&amp;gt; Even with the advantages of hindsight and publicly available information on people, places and transactions, it is clear that there is missing data.&lt;br /&gt;
&lt;br /&gt;
Krebs demonstrated the use of an association matrix and link chart of the terrorist network associated with the 19 hijackers responsible for the September 11th attacks by mapping publicly available details made available following the attacks. Even with the advantages of hindsight and publicly available information on people, places and transactions, it is clear that there is missing data.&lt;br /&gt;
&lt;br /&gt;
Krebs基于袭击后的详细公开资料进行绘图，演示了与9月11日袭击事件的19名劫机者有关的恐怖分子关系网的关联矩阵和链接图。即使有事后诸葛亮的优势--[[用户:Ryan|Ryan]]（[[用户讨论:Ryan|讨论]]），以及关于人员、地点和交易的公开可用信息，做出的结果图很明显仍然缺少数据。&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）即使有事后诸葛亮的优势  如果进行了意译应标注讨论出来&lt;br /&gt;
&lt;br /&gt;
  &lt;br /&gt;
Alternatively, Picarelli argued that use of link analysis techniques could have been used to identify and potentially prevent illicit activities within the [[Aum Shinrikyo]] network.&amp;lt;ref&amp;gt;Picarelli, J. T., [http://kdl.cs.umass.edu/events/aila1998/picarelli.ps Transnational Threat Indications and Warning: The Utility of Network Analysis, Military and Intelligence Analysis Group].&amp;lt;/ref&amp;gt; “We must be careful of ‘guilt by association’. Being linked to a terrorist does not prove guilt – but it does invite investigation.”&amp;lt;ref name=Krebs/&amp;gt; Balancing the legal concepts of [[probable cause]], [[right to privacy]] and [[freedom of association]] become challenging when reviewing potentially sensitive data with the objective to prevent crime or illegal activity that has not yet occurred.&lt;br /&gt;
&lt;br /&gt;
Alternatively, Picarelli argued that use of link analysis techniques could have been used to identify and potentially prevent illicit activities within the Aum Shinrikyo network. “We must be careful of ‘guilt by association’. Being linked to a terrorist does not prove guilt – but it does invite investigation.” Balancing the legal concepts of probable cause, right to privacy and freedom of association become challenging when reviewing potentially sensitive data with the objective to prevent crime or illegal activity that has not yet occurred.&lt;br /&gt;
&lt;br /&gt;
另外，Picarelli认为，使用链路分析技术可以用来查明并有可能防止奥姆真理教的非法活动。“我们必须小心‘牵连犯罪’。与恐怖分子有联系并不能证明有罪——但确实得进行调查。” 在审查较为敏感的数据以防止尚未发生的犯罪或非法活动时，如何同时不违背合理依据、隐私权和结社自由等法律概念将变得很困难。&lt;br /&gt;
&lt;br /&gt;
==Proposed solutions==&lt;br /&gt;
&lt;br /&gt;
There are four categories of proposed link analysis solutions:&amp;lt;ref&amp;gt;Schroeder et al., Automated Criminal Link Analysis Based on Domain Knowledge, Journal of the American Society for Information Science and Technology, 58:6 (842), 2007.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
There are four categories of proposed link analysis solutions:&lt;br /&gt;
&lt;br /&gt;
有四类拟议的链路分析解决方案:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
# [[Heuristic|Heuristic-based]]&lt;br /&gt;
&lt;br /&gt;
 Heuristic-based&lt;br /&gt;
&lt;br /&gt;
基于启发式的&lt;br /&gt;
&lt;br /&gt;
# Template-based&lt;br /&gt;
&lt;br /&gt;
 Template-based&lt;br /&gt;
&lt;br /&gt;
基于模板的&lt;br /&gt;
&lt;br /&gt;
# Similarity-based&lt;br /&gt;
&lt;br /&gt;
 Similarity-based&lt;br /&gt;
&lt;br /&gt;
基于相似性的&lt;br /&gt;
&lt;br /&gt;
# [[Statistical]]&lt;br /&gt;
&lt;br /&gt;
 Statistical&lt;br /&gt;
&lt;br /&gt;
统计方法&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Heuristic-based tools utilize decision rules that are distilled from expert knowledge using structured data. Template-based tools employ [[Natural language processing|Natural Language Processing (NLP)]] to extract details from [[unstructured data]] that are matched to pre-defined templates. Similarity-based approaches use weighted [[Score (statistics)|scoring]] to compare attributes and identify potential links. Statistical approaches identify potential links based on lexical statistics.&lt;br /&gt;
&lt;br /&gt;
Heuristic-based tools utilize decision rules that are distilled from expert knowledge using structured data. Template-based tools employ Natural Language Processing (NLP) to extract details from unstructured data that are matched to pre-defined templates. Similarity-based approaches use weighted scoring to compare attributes and identify potential links. Statistical approaches identify potential links based on lexical statistics.&lt;br /&gt;
&lt;br /&gt;
基于启发式的工具运用从专家知识中提取出来的决策规则对结构化数据进行操作。基于模板的工具使用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;自然语言处理 Natural Language Processing&amp;lt;/font&amp;gt;'''从非结构化数据中提取与预定义模板匹配的细节。基于相似度的方法使用加权评分来比较属性和识别潜在的链接。统计方法基于词汇统计识别潜在的链接。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===CrimeNet explorer===&lt;br /&gt;
&lt;br /&gt;
J.J. Xu and H. Chen propose a framework for automated network analysis and visualization called CrimeNet Explorer.&amp;lt;ref name=Xu&amp;gt;Xu, J.J. &amp;amp; Chen, H., CrimeNet Explorer: A Framework for Criminal Network Knowledge Discovery, ACM Transactions on Information Systems, 23(2), April 2005, pp. 201-226.&amp;lt;/ref&amp;gt; This framework includes the following elements:&lt;br /&gt;
&lt;br /&gt;
J.J. Xu and H. Chen propose a framework for automated network analysis and visualization called CrimeNet Explorer. This framework includes the following elements:&lt;br /&gt;
&lt;br /&gt;
J.J.Xu和H.Chen 提出了一个自动化网络分析和可视化的框架，叫做 CrimeNet Explorer。这一框架包括以下内容:&lt;br /&gt;
&lt;br /&gt;
* Network Creation through a concept space approach that uses “[[Co-occurrence networks|co-occurrence]] weight to measure the frequency with which two words or phrases appear in the same document. The more frequently two words or phrases appear together, the more likely it will be that they are related”.&amp;lt;ref name=Xu/&amp;gt;&lt;br /&gt;
通过概念空间方法创建网络，该方法使用“共现网络”来衡量两个单词或短语在同一文档中出现的频率。两个单词或短语在一起出现的频率越高，它们关联的可能性就越大。&lt;br /&gt;
&lt;br /&gt;
* Network Partition using “hierarchical clustering to partition a network into subgroups based on relational strength”.&amp;lt;ref name=Xu/&amp;gt;&lt;br /&gt;
网络分区通过“根据关系强度的分层聚类，将网络划分为子组”而实现。&lt;br /&gt;
&lt;br /&gt;
* Structural Analysis through “three centrality measures (degree, betweenness, and closeness) to identify central members in a given subgroup.&amp;lt;ref name=Xu/&amp;gt; CrimeNet Explorer employed [[Dijkstra's algorithm|Dijkstra’s shortest-path algorithm]] to calculate the betweenness and closeness from a single node to all other nodes in the subgroup.&lt;br /&gt;
通过“三种中心性度量（度，中间性和紧密性）来识别给定子集中的中心成员”进行结构分析。CrimeNet Explorer使用Dijkstra的最短路径算法来计算从单个节点到子组中所有其他节点的间隔度和紧密度。&lt;br /&gt;
&lt;br /&gt;
* Network Visualization using Torgerson’s metric [[Multidimensional scaling|multidimensional scaling (MDS)]] algorithm.&lt;br /&gt;
使用Torgerson的度量多维标度（MDS）算法进行网络可视化。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using  tags which will then appear here automatically --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using  tags which will then appear here automatically --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
!-参见 http: / / en.wikipedia. org / wiki / wikipedia: 如何使用标记创建引用的脚注，这些标记将自动出现在这里 -- &lt;br /&gt;
&lt;br /&gt;
{{reflist|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==External links==&lt;br /&gt;
&lt;br /&gt;
* {{cite book | last1 = Bartolini | first1 = I | last2 = Ciaccia | first2 = P. | title = Imagination: Accurate Image Annotation Using Link-Analysis Techniques | citeseerx = 10.1.1.63.2453 }}&lt;br /&gt;
&lt;br /&gt;
* [https://fightfinancialcrimes.com/2015/12/17/link-analysis-and-fraud/ Link Analysis and Crime - An Examination]&lt;br /&gt;
&lt;br /&gt;
* [https://www.ncbi.nlm.nih.gov/pubmed/9039449 Elink Schuurman MW, Srisaenpang S, Pinitsoontorn S, Bijleveld I, Vaeteewoothacharn K, Methapat C., The rapid village survey in tuberculosis control, Tuber Lung Dis. 1996 Dec;77(6):549-54.]&lt;br /&gt;
&lt;br /&gt;
* [https://www.cs.cmu.edu/~gunhee/publish/cvpr08_gunhee.pdf Gunhee, K., Faloutsos, C, Hebert, M, Unsupervised Modeling of Object Categories Using Link Analysis Techniques.]&lt;br /&gt;
&lt;br /&gt;
* [https://web.archive.org/web/20110928070620/http://www.blythe.org/Intelligence/readme/72sum McGehee, R., Intelligence Report.]&lt;br /&gt;
&lt;br /&gt;
* [https://web.archive.org/web/20061107040453/http://www.hsaj.org/pages/volume2/issue2/pdfs/2.2.8.pdf Ressler, S., Social Network Analysis as an Approach to Combat Terrorism: Past, Present and Future Research.]&lt;br /&gt;
&lt;br /&gt;
* [http://www.rff.com/link_analysis.htm RFFlow, How to Draw a Link Analysis Diagram.]&lt;br /&gt;
&lt;br /&gt;
* [http://www-03.ibm.com/software/products/en/analysts-notebook-premium IBM i2 Analyst's Notebook Premium]&lt;br /&gt;
&lt;br /&gt;
* {{cite journal | last1 = Silberschatz | first1 = A. | title = What Makes Patterns Interesting in Knowledge Discovery Systems | journal = IEEE Transactions on Knowledge and Data Engineering | volume = 8 | issue = 6 | pages = 970–974 | citeseerx = 10.1.1.53.2780 | year = 1996 | doi = 10.1109/69.553165 }}&lt;br /&gt;
&lt;br /&gt;
* [http://www.intelcenter.com/trlaf.html Terrorist &amp;amp; Rebel Link Analysis Feed (TR-LAF)]&lt;br /&gt;
&lt;br /&gt;
* [http://kt.ijs.si/dunja/LinkKDD2006/ Workshop on Link Analysis: Dynamics and Static of Large Networks (LinkKDD2006) August 20, 2006]&lt;br /&gt;
&lt;br /&gt;
* [http://www.cluemaker.com ClueMaker]&lt;br /&gt;
&lt;br /&gt;
* [http://www.datawalk.com Data Walk]&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Link analysis]]. Its edit history can be viewed at [[链路分析/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E9%93%BE%E8%B7%AF%E5%88%86%E6%9E%90&amp;diff=12550</id>
		<title>链路分析</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E9%93%BE%E8%B7%AF%E5%88%86%E6%9E%90&amp;diff=12550"/>
		<updated>2020-08-15T02:02:11Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
In [[network theory]], '''link analysis''' is a [[data-analysis]] technique used to evaluate relationships (connections) between nodes. Relationships may be identified among various types of nodes (objects), including [[organization]]s, [[people]] and [[Financial transaction|transactions]]. Link analysis has been used for investigation of criminal activity ([[fraud detection]], [[counterterrorism]], and [[Intelligence (information gathering)|intelligence]]), [[Computer security|computer security analysis]], [[search engine optimization]], [[market research]], [[medical research]], and art.&lt;br /&gt;
&lt;br /&gt;
In network theory, link analysis is a data-analysis technique used to evaluate relationships (connections) between nodes. Relationships may be identified among various types of nodes (objects), including organizations, people and transactions. Link analysis has been used for investigation of criminal activity (fraud detection, counterterrorism, and intelligence), computer security analysis, search engine optimization, market research, medical research, and art.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 网络理论 Network Theory&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 链路分析 Link Analysis&amp;lt;/font&amp;gt;'''是一种用于评估节点之间关系(连接)的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 数据分析 Data Analysis&amp;lt;/font&amp;gt;'''技术。该技术可以鉴别各种类型节点(对象)之间的关系，包括组织、人群和市场交易双方。链路分析已被应用于诸多领域，如打击犯罪活动(如欺诈侦查、反恐和情报)、计算机安全分析、搜索引擎优化、市场调查、医学研究和艺术。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Knowledge discovery==&lt;br /&gt;
知识发现&lt;br /&gt;
&lt;br /&gt;
[[Knowledge discovery]] is an [[Iteration|iterative]] and [[interactive]] process used to [[Identification (information)|identify]], analyze and visualize patterns in data.&amp;lt;ref&amp;gt;{{cite web|url=https://www.torproject.org/about/overview.html.en|title=Tor Project: Overview|first=The Tor Project|last=Inc.|publisher=}}&amp;lt;/ref&amp;gt; Network analysis, link analysis and [[social network analysis]] are all methods of knowledge discovery, each a corresponding subset of the prior method. Most knowledge discovery methods follow these steps (at the highest level):&amp;lt;ref&amp;gt;Ahonen, H., [http://www.cs.helsinki.fi/u/hahonen/features.txt Features of Knowledge Discovery Systems].&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Knowledge discovery is an iterative and interactive process used to identify, analyze and visualize patterns in data. Network analysis, link analysis and social network analysis are all methods of knowledge discovery, each a corresponding subset of the prior method. Most knowledge discovery methods follow these steps (at the highest level):&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 知识的发现 Knowledge Discovery&amp;lt;/font&amp;gt;'''，是指不断地识别、分析和可视化数据中的内在模式，这是一个持续迭代和交互的过程。网络分析、链路分析和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会网络分析 Social Network Analysis&amp;lt;/font&amp;gt;'''都是知识发现的方法，它们都是属于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 先验方法 Prior Method&amp;lt;/font&amp;gt;'''。大多数知识发现方法遵循以下几个步骤('''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;在最高级别&amp;lt;/font&amp;gt;''' at the highest level) :&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）遇到拿不准的专业名词翻译 可从cnki翻译助手检索一下该名词，下为链接http://dict.cnki.net/dict_result.aspx  选取引用量较高的释义进行翻译 譬如Knowledge Discovery 可以暂时定为知识发现  后续专家审校进行反馈再核实其准不准确&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
# [[Data processing]]&lt;br /&gt;
&lt;br /&gt;
 Data processing&lt;br /&gt;
&lt;br /&gt;
数据处理&lt;br /&gt;
&lt;br /&gt;
# [[Data transformation|Transformation]]&lt;br /&gt;
&lt;br /&gt;
 Transformation&lt;br /&gt;
&lt;br /&gt;
数据转变&lt;br /&gt;
&lt;br /&gt;
# [[Data analysis|Analysis]]&lt;br /&gt;
&lt;br /&gt;
 Analysis&lt;br /&gt;
&lt;br /&gt;
数据分析&lt;br /&gt;
&lt;br /&gt;
# [[Data visualization|Visualization]]&lt;br /&gt;
&lt;br /&gt;
 Visualization&lt;br /&gt;
&lt;br /&gt;
数据可视化&lt;br /&gt;
&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）这里很棒 补充出“数据”这个限定词&lt;br /&gt;
&lt;br /&gt;
Data gathering and processing requires access to data and has several inherent issues, including [[information overload]] and data errors. Once data is collected, it will need to be transformed into a format that can be effectively used by both human and computer analyzers. Manual or computer-generated visualizations tools may be mapped from the data, including network charts. Several algorithms exist to help with analysis of data – [[Dijkstra's algorithm|Dijkstra’s algorithm]], [[breadth-first search]], and [[depth-first search]].&lt;br /&gt;
&lt;br /&gt;
Data gathering and processing requires access to data and has several inherent issues, including information overload and data errors. Once data is collected, it will need to be transformed into a format that can be effectively used by both human and computer analyzers. Manual or computer-generated visualizations tools may be mapped from the data, including network charts. Several algorithms exist to help with analysis of data – Dijkstra’s algorithm, breadth-first search, and depth-first search.&lt;br /&gt;
&lt;br /&gt;
数据的收集和处理是首先进行的过程，但此过程存在一些固有的问题，包括'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;信息超载 Information Overload&amp;lt;/font&amp;gt;'''和数据错误等。在数据被收集后，它将转换成一种人和计算机分析程序都能有效使用的格式。之后基于数据，可使用计算机生成的或人工操作的可视化工具进行作图（如网络图）。目前有几种算法可以帮助人类进行数据分析-'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Dijkstra算法&amp;lt;/font&amp;gt;'''，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;广度优先搜索 Breadth-First Search&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 深度优先搜索 Depth-First Search&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Link analysis focuses on analysis of relationships among nodes through [[Visualization (computer graphics)|visualization methods]] ([[network charts]], association matrix). Here is an example of the relationships that may be mapped for crime investigations:&amp;lt;ref name=Krebs&amp;gt;Krebs, V. E. 2001, [http://vlado.fmf.uni-lj.si/pub/networks/doc/Seminar/Krebs.pdf Mapping networks of terrorist cells] {{webarchive|url=https://web.archive.org/web/20110720000539/http://vlado.fmf.uni-lj.si/pub/networks/doc/Seminar/Krebs.pdf |date=2011-07-20 }}, Connections 24, 43–52.&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
Link analysis focuses on analysis of relationships among nodes through visualization methods (network charts, association matrix). Here is an example of the relationships that may be mapped for crime investigations: &lt;br /&gt;
&lt;br /&gt;
链路分析主要通过可视化方法(网络图、关联矩阵)分析节点之间的关系。这里有一个基于罪犯和社会各部分关系绘制网图的例子:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
&lt;br /&gt;
{ | class“ wikitable”&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Relationship/Network !! Data Sources&lt;br /&gt;
&lt;br /&gt;
! Relationship/Network !! Data Sources&lt;br /&gt;
&lt;br /&gt;
!关系 / 网络! ！数据来源&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
| 1. Trust || Prior contacts in family, neighborhood, school, military, club or organization. Public and court records. Data may only be available in suspect's native country.&lt;br /&gt;
&lt;br /&gt;
| 1. Trust || Prior contacts in family, neighborhood, school, military, club or organization. Public and court records. Data may only be available in suspect's native country.&lt;br /&gt;
&lt;br /&gt;
| 1.信任 | 嫌疑人在家庭、社区、学校、军队、俱乐部或组织中已有的联系。公开信息及法庭纪录。以及只能在嫌疑人本国获得的数据。&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
| 2. Task || Logs and records of phone calls, electronic mail, chat rooms, instant messages, Web site visits. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
&lt;br /&gt;
| 2. Task || Logs and records of phone calls, electronic mail, chat rooms, instant messages, Web site visits. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 2.任务 | | 电话、电子邮件、聊天室、即时消息、网站访问的日志和记录。出入境纪录。'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;人类智能: 会议评论和公共活动的出席。&amp;lt;/font&amp;gt;'''Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 3. Money &amp;amp; Resources || Bank account and money transfer records. Pattern and location of credit card use. Prior court records. Human intelligence: observation of visits to alternate banking resources such as [[Hawala]].&lt;br /&gt;
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| 3. Money &amp;amp; Resources || Bank account and money transfer records. Pattern and location of credit card use. Prior court records. Human intelligence: observation of visits to alternate banking resources such as Hawala.&lt;br /&gt;
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| 3.资金和资源 | 银行账户和汇款记录。信用卡使用地点及使用习惯。以前的法庭记录。'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;人类智能: 访问其他银行资源的观察，如 Hawala。&amp;lt;/font&amp;gt;'''Human intelligence: observation of visits to alternate banking resources such as Hawala.&lt;br /&gt;
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| 4. Strategy &amp;amp; Goals || Web sites. Videos and encrypted disks delivered by courier. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 4. Strategy &amp;amp; Goals || Web sites. Videos and encrypted disks delivered by courier. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 4.策略与目标 | 网站。由快递公司递送的视频和加密光盘。出入境纪录。'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;人类智能: 会议评论和公共活动的出席。&amp;lt;/font&amp;gt;'''Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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Link analysis is used for 3 primary purposes:&amp;lt;ref name=&amp;quot;Link Analysis Workbench&amp;quot;&amp;gt;[https://docs.google.com/viewer?a=v&amp;amp;q=cache:R4_7k-udkxMJ:www.fas.org/irp/eprint/law.pdf+%22link+analysis%22&amp;amp;hl=en&amp;amp;gl=uk&amp;amp;pid=bl&amp;amp;srcid=ADGEESjUVRJFP2O2eMGzSuW4WtGgujUHVeuUYDQ9wqOkJUeVMHOwpagPa65ypDM5Fgma2AlwDyFBblZwXNMoYDCHwCQuTQvK-HnmqbW-z5A-MOwMKLy8nk6_uHLa0CiUAql-kAilAYcd&amp;amp;sig=AHIEtbRgGVNE4PMZ0a2vipzlDTd6OLX4fA Link Analysis Workbench], Air Force Research Laboratory Information Directorate, Rome Research Site, Rome, New York, September 2004.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Link analysis is used for 3 primary purposes:&lt;br /&gt;
&lt;br /&gt;
链接分析主要有3个作用:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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# Find matches in data for known patterns of interest;&lt;br /&gt;
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 Find matches in data for known patterns of interest;&lt;br /&gt;
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在数据中寻找有意义的已知模式;&lt;br /&gt;
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# Find anomalies where known patterns are violated;&lt;br /&gt;
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 Find anomalies where known patterns are violated;&lt;br /&gt;
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发现与已知模式不相符的异常数据;&lt;br /&gt;
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# Discover new patterns of interest (social network analysis, [[data mining]]).&lt;br /&gt;
&lt;br /&gt;
 Discover new patterns of interest (social network analysis, data mining).&lt;br /&gt;
&lt;br /&gt;
发现有意义的新模式(社会网络分析、数据挖掘)。&lt;br /&gt;
&lt;br /&gt;
==History==&lt;br /&gt;
&lt;br /&gt;
Klerks categorized link analysis tools into 3 generations.&amp;lt;ref&amp;gt;{{cite journal | last = Klerks | first = P. | year = 2001 | title = The network paradigm applied to criminal organizations: Theoretical nitpicking or a relevant doctrine for investigators? Recent developments in the Netherlands | citeseerx = 10.1.1.129.4720 | journal = Connections | volume = 24 | pages = 53–65 }}&amp;lt;/ref&amp;gt; The first generation was introduced in 1975 as the Anacpapa Chart of Harper and Harris.&amp;lt;ref&amp;gt;Harper and Harris, The Analysis of Criminal Intelligence, Human Factors and Ergonomics Society Annual Meeting Proceedings, 19(2), 1975, pp. 232-238.&amp;lt;/ref&amp;gt; This method requires that a domain expert review data files, identify associations by constructing an association matrix, create a link chart for visualization and finally analyze the network chart to identify patterns of interest. This method requires extensive domain knowledge and is extremely time-consuming when reviewing vast amounts of data.&lt;br /&gt;
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[[File:图1：Association Matrix 关联矩阵.png|thumb|Association Matrix 关联矩阵]]&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）点击编辑后会发现[[File: 开头关于图片的语法  该图注需要进行翻译且遵循格式【图1：英文原文＋翻译内容】&lt;br /&gt;
&lt;br /&gt;
Klerks categorized link analysis tools into 3 generations. The first generation was introduced in 1975 as the Anacpapa Chart of Harper and Harris. This method requires that a domain expert review data files, identify associations by constructing an association matrix, create a link chart for visualization and finally analyze the network chart to identify patterns of interest. This method requires extensive domain knowledge and is extremely time-consuming when reviewing vast amounts of data.&lt;br /&gt;
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Klerks把链接分析工具分为三代。第一代是由哈珀和哈里斯在1975年引入的，阿纳卡帕图。这种方法需要一个领域内的专家来审查数据文件，通过构造一个关联矩阵来识别关联，然后创建一个可视化的链路图，最后通过分析网络图来识别有意义的模式。这种方法需要广泛的领域知识，且因要人工审查大量数据，所以非常耗时。&lt;br /&gt;
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In addition to the association matrix, the activities matrix can be used to produce actionable information, which has practical value and use to law-enforcement. The activities matrix, as the term might imply, centers on the actions and activities of people with respect to locations. Whereas the association matrix focuses on the relationships between people, organizations, and/or properties. The distinction between these two types of matrices, while minor, is nonetheless significant in terms of the output of the analysis completed or rendered.&amp;lt;ref&amp;gt;{{cite web|url=http://www.globalsecurity.org/military/library/policy/army/fm/3-07-22/app-f.htm|title=FMI 3-07.22 Appendix F Intelligence Analysis Tools and Indicators|first=John|last=Pike|publisher=}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[https://rdl.train.army.mil/catalog/view/100.ATSC/41449AB4-E8E0-46C4-8443-E4276B6F0481-1274576841878/3-24/appb.htm Social Network Analysis and Other Analytical Tools] {{webarchive|url=https://web.archive.org/web/20140308233614/https://rdl.train.army.mil/catalog/view/100.ATSC/41449AB4-E8E0-46C4-8443-E4276B6F0481-1274576841878/3-24/appb.htm |date=2014-03-08 }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite web|url=http://www.nasa.gov/audience/foreducators/topnav/materials/listbytype/Aeronautics_Activity_Matrices.html|title=Aeronautics Educator Guide - Activity Matrices|first=Rebecca Whitaker|last=MSFC|date=10 July 2009|publisher=}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[https://rdl.train.army.mil/catalog/view/100.ATSC/0EF89CA1-2680-4782-B103-D2F5DC941188-1274309335668/7-98-1/chap2l6.htm Personality/Activity Matrix] {{webarchive|url=https://web.archive.org/web/20140308234135/https://rdl.train.army.mil/catalog/view/100.ATSC/0EF89CA1-2680-4782-B103-D2F5DC941188-1274309335668/7-98-1/chap2l6.htm |date=2014-03-08 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In addition to the association matrix, the activities matrix can be used to produce actionable information, which has practical value and use to law-enforcement. The activities matrix, as the term might imply, centers on the actions and activities of people with respect to locations. Whereas the association matrix focuses on the relationships between people, organizations, and/or properties. The distinction between these two types of matrices, while minor, is nonetheless significant in terms of the output of the analysis completed or rendered.&lt;br /&gt;
&lt;br /&gt;
除了关联矩阵外，活动矩阵也可用于生成对执法活动具有实用和使用价值的可操作的信息。正如这个术语可能暗示的那样，活动矩阵关注的是人们基于地点的行动和活动。而关联矩阵关注的是人，组织和/或属性之间的关系。这两类矩阵之间的区别虽然很小，但就已完成或经过分析的数据来看，区别还是很重要的。&lt;br /&gt;
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Second generation tools consist of automatic graphics-based analysis tools such as IBM i2 Analyst’s Notebook, Netmap, [[ClueMaker]] and Watson. These tools offer the ability to automate the construction and updates of the link chart once an association matrix is manually created, however, analysis of the resulting charts and graphs still requires an expert with extensive domain knowledge.&lt;br /&gt;
&lt;br /&gt;
Second generation tools consist of automatic graphics-based analysis tools such as IBM i2 Analyst’s Notebook, Netmap, ClueMaker and Watson. These tools offer the ability to automate the construction and updates of the link chart once an association matrix is manually created, however, analysis of the resulting charts and graphs still requires an expert with extensive domain knowledge.&lt;br /&gt;
&lt;br /&gt;
第二代工具包括基于图形的自动分析工具，如 IBM i 2 Analyst’ s Notebook、 Netmap、 ClueMaker 和 Watson。在手动创建关联矩阵的情况下，这些工具提供了自动构建和更新链接图表的能力。然而，对结果图表的分析仍然需要在该领域具有广泛知识的专家。&lt;br /&gt;
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The third generation of link-analysis tools like DataWalk allow the automatic visualization of linkages between elements in a data set, that can then serve as the canvas for further exploration or manual updates.&lt;br /&gt;
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The third generation of link-analysis tools like DataWalk allow the automatic visualization of linkages between elements in a data set, that can then serve as the canvas for further exploration or manual updates.&lt;br /&gt;
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像 DataWalk 这样的第三代链接分析工具能自动可视化数据集中元素之间的链接，其结果可允许进一步的人工探索与改进。&lt;br /&gt;
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==Applications==&lt;br /&gt;
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* [[ViCAP|FBI Violent Criminal Apprehension Program (ViCAP)]]&lt;br /&gt;
联邦调查局暴力刑事逮捕程序&lt;br /&gt;
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* Iowa State Sex Crimes Analysis System&lt;br /&gt;
爱荷华州性犯罪分析系统&lt;br /&gt;
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* Minnesota State Sex Crimes Analysis System (MIN/SCAP)&lt;br /&gt;
明尼苏达州性犯罪分析系统&lt;br /&gt;
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* Washington State Homicide Investigation Tracking System (HITS)&amp;lt;ref&amp;gt;{{cite web|url=http://www.atg.wa.gov/HITS.aspx |title=Archived copy |accessdate=2010-10-31 |url-status=dead |archiveurl=https://web.archive.org/web/20101021005202/http://atg.wa.gov/HITS.aspx |archivedate=2010-10-21 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
华盛顿州凶杀案调查追踪系统&lt;br /&gt;
&lt;br /&gt;
* New York State Homicide Investigation &amp;amp; Lead Tracking (HALT)&lt;br /&gt;
纽约州凶杀案调查和线索追踪系统&lt;br /&gt;
&lt;br /&gt;
* New Jersey Homicide Evaluation &amp;amp; Assessment Tracking (HEAT)&amp;lt;ref&amp;gt;{{cite web|url=http://www.state.nj.us/njsp/divorg/invest/invest.html |title=Archived copy |accessdate=2010-10-31 |url-status=dead |archiveurl=https://web.archive.org/web/20090325004722/http://www.state.nj.us/njsp/divorg/invest/invest.html |archivedate=2009-03-25 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
新泽西州凶杀案评估与测评跟踪系统&lt;br /&gt;
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* Pennsylvania State ATAC Program.&lt;br /&gt;
宾夕法尼亚州ATAC程序&lt;br /&gt;
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* Violent Crime Linkage Analysis System (ViCLAS)&amp;lt;ref&amp;gt;{{cite web|url=http://www.rcmp-grc.gc.ca/tops-opst/bs-sc/viclas-salvac-eng.htm |title=Archived copy |accessdate=2010-10-31 |url-status=dead |archiveurl=https://web.archive.org/web/20101202144141/http://www.rcmp-grc.gc.ca/tops-opst/bs-sc/viclas-salvac-eng.htm |archivedate=2010-12-02 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
暴力犯罪联系分析系统&lt;br /&gt;
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==Issues with link analysis==&lt;br /&gt;
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===Information overload===&lt;br /&gt;
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With the vast amounts of data and information that are stored electronically, users are confronted with multiple unrelated sources of information available for analysis. Data analysis techniques are required to make effective and efficient use of the data. Palshikar classifies data analysis techniques into two categories – ([[Statistics|statistical]] [[Models of computation|models]], [[Time series analysis|time-series analysis]], [[Clustering coefficient|clustering]] and [[Statistical classification|classification]], matching algorithms to detect anomalies) and [[Artificial intelligence|artificial intelligence (AI)]] techniques (data mining, [[expert systems]], [[pattern recognition]], [[Machine learning|machine learning techniques]], [[neural network]]s).&amp;lt;ref&amp;gt;Palshikar, G. K., [http://www.intelligententerprise.com//020528/509feat3_1.jhtml The Hidden Truth], Intelligent Enterprise, May 2002.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
With the vast amounts of data and information that are stored electronically, users are confronted with multiple unrelated sources of information available for analysis. Data analysis techniques are required to make effective and efficient use of the data. Palshikar classifies data analysis techniques into two categories – (statistical models, time-series analysis, clustering and classification, matching algorithms to detect anomalies) and artificial intelligence (AI) techniques (data mining, expert systems, pattern recognition, machine learning techniques, neural networks).&lt;br /&gt;
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由于大量数据和信息以电子形式存储，用户可能会面临拥有多种不相关的信息来源却不知如何分析的难题。数据分析技术的使用可以帮助有效和高效地利用数据。Palshikar 将数据分析技术分为两大类(统计模型、时间序列分析、聚类分类、异常检测匹配算法)和人工智能(AI)技术(数据挖掘、专家系统、模式识别、机器学习技术、神经网络)。&lt;br /&gt;
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Bolton &amp;amp; Hand define statistical data analysis as either supervised or unsupervised methods.&amp;lt;ref&amp;gt;Bolton, R. J. &amp;amp; Hand, D. J., Statistical Fraud Detection: A Review, Statistical Science, 2002, 17(3), pp. 235-255.&amp;lt;/ref&amp;gt; [[Supervised learning|Supervised learning methods]] require that rules are defined within the system to establish what is expected or unexpected behavior. [[Unsupervised learning|Unsupervised learning methods]] review data in comparison to the norm and detect statistical outliers. Supervised learning methods are limited in the scenarios that can be handled as this method requires that training rules are established based on previous patterns. Unsupervised learning methods can provide detection of broader issues, however, may result in a higher false-positive ratio if the behavioral norm is not well established or understood.&lt;br /&gt;
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Bolton &amp;amp; Hand define statistical data analysis as either supervised or unsupervised methods. Supervised learning methods require that rules are defined within the system to establish what is expected or unexpected behavior. Unsupervised learning methods review data in comparison to the norm and detect statistical outliers. Supervised learning methods are limited in the scenarios that can be handled as this method requires that training rules are established based on previous patterns. Unsupervised learning methods can provide detection of broader issues, however, may result in a higher false-positive ratio if the behavioral norm is not well established or understood.&lt;br /&gt;
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Bolton &amp;amp; Hand 将统计数据分析定义为有监督或无监督的方法。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;监督式学习方法 Supervised Learning Methods&amp;lt;/font&amp;gt;'''要求在系统中有明确的规则来指出什么是预期行为，什么是意外行为。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;非监督式学习方法 Unsupervised Learning Methods&amp;lt;/font&amp;gt;'''在审视数据时，通过将数据与正常值的比较，来发现统计异常值。监督式学习方法能处理的场景是有限的，因为这种方法需要基于以前的模式建立训练规则。非监督式学习方法可以对更广泛的问题采取进攻。但是，如果数据的行为规范没有很好的建立或被机器理解 --[[用户:Ryan|Ryan]]（[[用户讨论:Ryan|讨论]]）该句存疑，其结果可能会导致较高的假阳性率（本身不是正常值，但识别为正常值，说明算法预测了“正确”或“有”的判断，但却判断错误了）。&lt;br /&gt;
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Data itself has inherent issues including integrity (or lack of) and continuous changes. Data may contain “errors of omission and commission because of faulty collection or handling, and when entities are actively attempting to deceive and/or conceal their actions”.&amp;lt;ref name=&amp;quot;Link Analysis Workbench&amp;quot;/&amp;gt; Sparrow&amp;lt;ref&amp;gt;Sparrow M.K. 1991. Network Vulnerabilities and Strategic Intelligence in Law Enforcement’, [[International Journal of Intelligence and Counterintelligence]] Vol. 5 #3.&amp;lt;/ref&amp;gt; highlights incompleteness (inevitability of missing data or links), fuzzy boundaries (subjectivity in deciding what to include) and dynamic changes (recognition that data is ever-changing) as the three primary problems with data analysis.&amp;lt;ref name=Krebs/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Data itself has inherent issues including integrity (or lack of) and continuous changes. Data may contain “errors of omission and commission because of faulty collection or handling, and when entities are actively attempting to deceive and/or conceal their actions”. Sparrow highlights incompleteness (inevitability of missing data or links), fuzzy boundaries (subjectivity in deciding what to include) and dynamic changes (recognition that data is ever-changing) as the three primary problems with data analysis.&lt;br /&gt;
&lt;br /&gt;
数据本身存在固有的问题，包括完整性(或缺失性)和持续的改变。数据可能包含'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;“由于错误的收集或处理，以及当实体积极试图欺骗和 / 或隐瞒其行为，而造成的错误的遗漏和委托”&amp;lt;/font&amp;gt;''' “errors of omission and commission because of faulty collection or handling, and when entities are actively attempting to deceive and/or conceal their actions”。Sparro强调了数据分析中三个主要的问题，不完整性(数据或链接缺失的必然性)、模糊边界(边界确定的主观性)和动态变化(数据的持续变化性)。&lt;br /&gt;
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  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）存疑的地方也可以用绿色语法进行标注&lt;br /&gt;
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Once data is transformed into a usable format, open texture and cross referencing issues may arise. [[Open texture]] was defined by [[Friedrich Waismann|Waismann]] as the unavoidable uncertainty in meaning when empirical terms are used in different contexts.&amp;lt;ref&amp;gt;Friedrich Waismann, Verifiability (1945), p.2.&amp;lt;/ref&amp;gt; Uncertainty in meaning of terms presents problems when attempting to search and cross reference data from multiple sources.&amp;lt;ref&amp;gt;Lyons, D., [http://ssrn.com/abstract=212328 Open Texture and the Possibility of Legal Interpretation (2000)].&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Once data is transformed into a usable format, open texture and cross referencing issues may arise. Open texture was defined by Waismann as the unavoidable uncertainty in meaning when empirical terms are used in different contexts. Uncertainty in meaning of terms presents problems when attempting to search and cross reference data from multiple sources.&lt;br /&gt;
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一旦数据转换成可用的格式，开放纹理和交叉引用问题就会出现。Waismann将'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 开放纹理 Open Texture&amp;lt;/font&amp;gt;'''定义为在不同语境中使用经验词汇时不可避免的语义不确定性。当试图从多个数据源搜索和交叉引用数据时，术语含义的不确定性带来了问题。&lt;br /&gt;
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The primary method for resolving data analysis issues is reliance on [[domain knowledge]] from an expert. This is a very time-consuming and costly method of conducting link analysis and has inherent problems of its own. McGrath et al. conclude that the layout and presentation of a network diagram have a significant impact on the user’s “perceptions of the existence of groups in networks”.&amp;lt;ref&amp;gt;McGrath, C., Blythe, J., Krackhardt, D., [http://www.andrew.cmu.edu/user/cm3t/groups.html Seeing Groups in Graph Layouts].&amp;lt;/ref&amp;gt; Even using domain experts may result in differing conclusions as analysis may be subjective.&lt;br /&gt;
&lt;br /&gt;
The primary method for resolving data analysis issues is reliance on domain knowledge from an expert. This is a very time-consuming and costly method of conducting link analysis and has inherent problems of its own. McGrath et al. conclude that the layout and presentation of a network diagram have a significant impact on the user’s “perceptions of the existence of groups in networks”. Even using domain experts may result in differing conclusions as analysis may be subjective.&lt;br /&gt;
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目前，解决数据分析中这些问题的主要方法是依赖专家的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;领域知识 Domain Knowledge&amp;lt;/font&amp;gt;'''。如此进行链路分析是非常耗时和昂贵的，并且无法排除其自身固有的问题。麦格拉斯等人得出结论，网络图的分布和表示方式对用户的“对存在在网络中群体的感知”有重大影响。即使是领域内的专家也可能导致不同的结论，因为分析可能是很主观的。&lt;br /&gt;
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===Prosecution vs. crime prevention===&lt;br /&gt;
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Link analysis techniques have primarily been used for prosecution, as it is far easier to review historical data for patterns than it is to attempt to predict future actions.&lt;br /&gt;
&lt;br /&gt;
Link analysis techniques have primarily been used for prosecution, as it is far easier to review historical data for patterns than it is to attempt to predict future actions.&lt;br /&gt;
&lt;br /&gt;
目前，链接分析技术主要用于起诉，因为回顾历史数据以期从中获得模式，要比预测未来的行动容易得多。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Krebs demonstrated the use of an association matrix and link chart of the terrorist network associated with the 19 hijackers responsible for the [[September 11 attacks|September 11th attacks]] by mapping publicly available details made available following the attacks.&amp;lt;ref name=Krebs/&amp;gt; Even with the advantages of hindsight and publicly available information on people, places and transactions, it is clear that there is missing data.&lt;br /&gt;
&lt;br /&gt;
Krebs demonstrated the use of an association matrix and link chart of the terrorist network associated with the 19 hijackers responsible for the September 11th attacks by mapping publicly available details made available following the attacks. Even with the advantages of hindsight and publicly available information on people, places and transactions, it is clear that there is missing data.&lt;br /&gt;
&lt;br /&gt;
Krebs基于袭击后的详细公开资料进行绘图，演示了与9月11日袭击事件的19名劫机者有关的恐怖分子关系网的关联矩阵和链接图。即使有事后诸葛亮的优势--[[用户:Ryan|Ryan]]（[[用户讨论:Ryan|讨论]]），以及关于人员、地点和交易的公开可用信息，做出的结果图很明显仍然缺少数据。&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）即使有事后诸葛亮的优势  如果进行了意译应标注讨论出来&lt;br /&gt;
&lt;br /&gt;
  &lt;br /&gt;
Alternatively, Picarelli argued that use of link analysis techniques could have been used to identify and potentially prevent illicit activities within the [[Aum Shinrikyo]] network.&amp;lt;ref&amp;gt;Picarelli, J. T., [http://kdl.cs.umass.edu/events/aila1998/picarelli.ps Transnational Threat Indications and Warning: The Utility of Network Analysis, Military and Intelligence Analysis Group].&amp;lt;/ref&amp;gt; “We must be careful of ‘guilt by association’. Being linked to a terrorist does not prove guilt – but it does invite investigation.”&amp;lt;ref name=Krebs/&amp;gt; Balancing the legal concepts of [[probable cause]], [[right to privacy]] and [[freedom of association]] become challenging when reviewing potentially sensitive data with the objective to prevent crime or illegal activity that has not yet occurred.&lt;br /&gt;
&lt;br /&gt;
Alternatively, Picarelli argued that use of link analysis techniques could have been used to identify and potentially prevent illicit activities within the Aum Shinrikyo network. “We must be careful of ‘guilt by association’. Being linked to a terrorist does not prove guilt – but it does invite investigation.” Balancing the legal concepts of probable cause, right to privacy and freedom of association become challenging when reviewing potentially sensitive data with the objective to prevent crime or illegal activity that has not yet occurred.&lt;br /&gt;
&lt;br /&gt;
另外，Picarelli认为，使用链路分析技术可以用来查明并有可能防止奥姆真理教的非法活动。“我们必须小心‘牵连犯罪’。与恐怖分子有联系并不能证明有罪——但确实得进行调查。” 在审查较为敏感的数据以防止尚未发生的犯罪或非法活动时，如何同时不违背合理依据、隐私权和结社自由等法律概念将变得很困难。&lt;br /&gt;
&lt;br /&gt;
==Proposed solutions==&lt;br /&gt;
&lt;br /&gt;
There are four categories of proposed link analysis solutions:&amp;lt;ref&amp;gt;Schroeder et al., Automated Criminal Link Analysis Based on Domain Knowledge, Journal of the American Society for Information Science and Technology, 58:6 (842), 2007.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
There are four categories of proposed link analysis solutions:&lt;br /&gt;
&lt;br /&gt;
有四类拟议的链路分析解决方案:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
# [[Heuristic|Heuristic-based]]&lt;br /&gt;
&lt;br /&gt;
 Heuristic-based&lt;br /&gt;
&lt;br /&gt;
基于启发式的&lt;br /&gt;
&lt;br /&gt;
# Template-based&lt;br /&gt;
&lt;br /&gt;
 Template-based&lt;br /&gt;
&lt;br /&gt;
基于模板的&lt;br /&gt;
&lt;br /&gt;
# Similarity-based&lt;br /&gt;
&lt;br /&gt;
 Similarity-based&lt;br /&gt;
&lt;br /&gt;
基于相似性的&lt;br /&gt;
&lt;br /&gt;
# [[Statistical]]&lt;br /&gt;
&lt;br /&gt;
 Statistical&lt;br /&gt;
&lt;br /&gt;
统计方法&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Heuristic-based tools utilize decision rules that are distilled from expert knowledge using structured data. Template-based tools employ [[Natural language processing|Natural Language Processing (NLP)]] to extract details from [[unstructured data]] that are matched to pre-defined templates. Similarity-based approaches use weighted [[Score (statistics)|scoring]] to compare attributes and identify potential links. Statistical approaches identify potential links based on lexical statistics.&lt;br /&gt;
&lt;br /&gt;
Heuristic-based tools utilize decision rules that are distilled from expert knowledge using structured data. Template-based tools employ Natural Language Processing (NLP) to extract details from unstructured data that are matched to pre-defined templates. Similarity-based approaches use weighted scoring to compare attributes and identify potential links. Statistical approaches identify potential links based on lexical statistics.&lt;br /&gt;
&lt;br /&gt;
基于启发式的工具运用从专家知识中提取出来的决策规则对结构化数据进行操作。基于模板的工具使用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;自然语言处理 Natural Language Processing&amp;lt;/font&amp;gt;'''从非结构化数据中提取与预定义模板匹配的细节。基于相似度的方法使用加权评分来比较属性和识别潜在的链接。统计方法基于词汇统计识别潜在的链接。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===CrimeNet explorer===&lt;br /&gt;
&lt;br /&gt;
J.J. Xu and H. Chen propose a framework for automated network analysis and visualization called CrimeNet Explorer.&amp;lt;ref name=Xu&amp;gt;Xu, J.J. &amp;amp; Chen, H., CrimeNet Explorer: A Framework for Criminal Network Knowledge Discovery, ACM Transactions on Information Systems, 23(2), April 2005, pp. 201-226.&amp;lt;/ref&amp;gt; This framework includes the following elements:&lt;br /&gt;
&lt;br /&gt;
J.J. Xu and H. Chen propose a framework for automated network analysis and visualization called CrimeNet Explorer. This framework includes the following elements:&lt;br /&gt;
&lt;br /&gt;
J.J.Xu和H.Chen 提出了一个自动化网络分析和可视化的框架，叫做 CrimeNet Explorer。这一框架包括以下内容:&lt;br /&gt;
&lt;br /&gt;
* Network Creation through a concept space approach that uses “[[Co-occurrence networks|co-occurrence]] weight to measure the frequency with which two words or phrases appear in the same document. The more frequently two words or phrases appear together, the more likely it will be that they are related”.&amp;lt;ref name=Xu/&amp;gt;&lt;br /&gt;
通过概念空间方法创建网络，该方法使用“共现网络”来衡量两个单词或短语在同一文档中出现的频率。两个单词或短语在一起出现的频率越高，它们关联的可能性就越大。&lt;br /&gt;
&lt;br /&gt;
* Network Partition using “hierarchical clustering to partition a network into subgroups based on relational strength”.&amp;lt;ref name=Xu/&amp;gt;&lt;br /&gt;
网络分区通过“根据关系强度的分层聚类，将网络划分为子组”而实现。&lt;br /&gt;
&lt;br /&gt;
* Structural Analysis through “three centrality measures (degree, betweenness, and closeness) to identify central members in a given subgroup.&amp;lt;ref name=Xu/&amp;gt; CrimeNet Explorer employed [[Dijkstra's algorithm|Dijkstra’s shortest-path algorithm]] to calculate the betweenness and closeness from a single node to all other nodes in the subgroup.&lt;br /&gt;
通过“三种中心性度量（度，中间性和紧密性）来识别给定子集中的中心成员”进行结构分析。CrimeNet Explorer使用Dijkstra的最短路径算法来计算从单个节点到子组中所有其他节点的间隔度和紧密度。&lt;br /&gt;
&lt;br /&gt;
* Network Visualization using Torgerson’s metric [[Multidimensional scaling|multidimensional scaling (MDS)]] algorithm.&lt;br /&gt;
使用Torgerson的度量多维标度（MDS）算法进行网络可视化。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using  tags which will then appear here automatically --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using  tags which will then appear here automatically --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
!-参见 http: / / en.wikipedia. org / wiki / wikipedia: 如何使用标记创建引用的脚注，这些标记将自动出现在这里 -- &lt;br /&gt;
&lt;br /&gt;
{{reflist|colwidth=30em}}&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==External links==&lt;br /&gt;
&lt;br /&gt;
* {{cite book | last1 = Bartolini | first1 = I | last2 = Ciaccia | first2 = P. | title = Imagination: Accurate Image Annotation Using Link-Analysis Techniques | citeseerx = 10.1.1.63.2453 }}&lt;br /&gt;
&lt;br /&gt;
* [https://fightfinancialcrimes.com/2015/12/17/link-analysis-and-fraud/ Link Analysis and Crime - An Examination]&lt;br /&gt;
&lt;br /&gt;
* [https://www.ncbi.nlm.nih.gov/pubmed/9039449 Elink Schuurman MW, Srisaenpang S, Pinitsoontorn S, Bijleveld I, Vaeteewoothacharn K, Methapat C., The rapid village survey in tuberculosis control, Tuber Lung Dis. 1996 Dec;77(6):549-54.]&lt;br /&gt;
&lt;br /&gt;
* [https://www.cs.cmu.edu/~gunhee/publish/cvpr08_gunhee.pdf Gunhee, K., Faloutsos, C, Hebert, M, Unsupervised Modeling of Object Categories Using Link Analysis Techniques.]&lt;br /&gt;
&lt;br /&gt;
* [https://web.archive.org/web/20110928070620/http://www.blythe.org/Intelligence/readme/72sum McGehee, R., Intelligence Report.]&lt;br /&gt;
&lt;br /&gt;
* [https://web.archive.org/web/20061107040453/http://www.hsaj.org/pages/volume2/issue2/pdfs/2.2.8.pdf Ressler, S., Social Network Analysis as an Approach to Combat Terrorism: Past, Present and Future Research.]&lt;br /&gt;
&lt;br /&gt;
* [http://www.rff.com/link_analysis.htm RFFlow, How to Draw a Link Analysis Diagram.]&lt;br /&gt;
&lt;br /&gt;
* [http://www-03.ibm.com/software/products/en/analysts-notebook-premium IBM i2 Analyst's Notebook Premium]&lt;br /&gt;
&lt;br /&gt;
* {{cite journal | last1 = Silberschatz | first1 = A. | title = What Makes Patterns Interesting in Knowledge Discovery Systems | journal = IEEE Transactions on Knowledge and Data Engineering | volume = 8 | issue = 6 | pages = 970–974 | citeseerx = 10.1.1.53.2780 | year = 1996 | doi = 10.1109/69.553165 }}&lt;br /&gt;
&lt;br /&gt;
* [http://www.intelcenter.com/trlaf.html Terrorist &amp;amp; Rebel Link Analysis Feed (TR-LAF)]&lt;br /&gt;
&lt;br /&gt;
* [http://kt.ijs.si/dunja/LinkKDD2006/ Workshop on Link Analysis: Dynamics and Static of Large Networks (LinkKDD2006) August 20, 2006]&lt;br /&gt;
&lt;br /&gt;
* [http://www.cluemaker.com ClueMaker]&lt;br /&gt;
&lt;br /&gt;
* [http://www.datawalk.com Data Walk]&lt;br /&gt;
&lt;br /&gt;
[[Category:Network theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Network theory&lt;br /&gt;
&lt;br /&gt;
范畴: 网络理论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Link analysis]]. Its edit history can be viewed at [[链路分析/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E9%93%BE%E8%B7%AF%E5%88%86%E6%9E%90&amp;diff=12549</id>
		<title>链路分析</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E9%93%BE%E8%B7%AF%E5%88%86%E6%9E%90&amp;diff=12549"/>
		<updated>2020-08-15T02:00:07Z</updated>

		<summary type="html">&lt;p&gt;Ryan：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;本词条由Ryan初步翻译&lt;br /&gt;
&lt;br /&gt;
In [[network theory]], '''link analysis''' is a [[data-analysis]] technique used to evaluate relationships (connections) between nodes. Relationships may be identified among various types of nodes (objects), including [[organization]]s, [[people]] and [[Financial transaction|transactions]]. Link analysis has been used for investigation of criminal activity ([[fraud detection]], [[counterterrorism]], and [[Intelligence (information gathering)|intelligence]]), [[Computer security|computer security analysis]], [[search engine optimization]], [[market research]], [[medical research]], and art.&lt;br /&gt;
&lt;br /&gt;
In network theory, link analysis is a data-analysis technique used to evaluate relationships (connections) between nodes. Relationships may be identified among various types of nodes (objects), including organizations, people and transactions. Link analysis has been used for investigation of criminal activity (fraud detection, counterterrorism, and intelligence), computer security analysis, search engine optimization, market research, medical research, and art.&lt;br /&gt;
&lt;br /&gt;
在'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 网络理论 Network Theory&amp;lt;/font&amp;gt;'''中，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 链路分析 Link Analysis&amp;lt;/font&amp;gt;'''是一种用于评估节点之间关系(连接)的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 数据分析 Data Analysis&amp;lt;/font&amp;gt;'''技术。该技术可以鉴别各种类型节点(对象)之间的关系，包括组织、人群和市场交易双方。链路分析已被应用于诸多领域，如打击犯罪活动(如欺诈侦查、反恐和情报)、计算机安全分析、搜索引擎优化、市场调查、医学研究和艺术。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
==Knowledge discovery==&lt;br /&gt;
知识发现&lt;br /&gt;
&lt;br /&gt;
[[Knowledge discovery]] is an [[Iteration|iterative]] and [[interactive]] process used to [[Identification (information)|identify]], analyze and visualize patterns in data.&amp;lt;ref&amp;gt;{{cite web|url=https://www.torproject.org/about/overview.html.en|title=Tor Project: Overview|first=The Tor Project|last=Inc.|publisher=}}&amp;lt;/ref&amp;gt; Network analysis, link analysis and [[social network analysis]] are all methods of knowledge discovery, each a corresponding subset of the prior method. Most knowledge discovery methods follow these steps (at the highest level):&amp;lt;ref&amp;gt;Ahonen, H., [http://www.cs.helsinki.fi/u/hahonen/features.txt Features of Knowledge Discovery Systems].&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Knowledge discovery is an iterative and interactive process used to identify, analyze and visualize patterns in data. Network analysis, link analysis and social network analysis are all methods of knowledge discovery, each a corresponding subset of the prior method. Most knowledge discovery methods follow these steps (at the highest level):&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 知识的发现 Knowledge Discovery&amp;lt;/font&amp;gt;'''，是指不断地识别、分析和可视化数据中的内在模式，这是一个持续迭代和交互的过程。网络分析、链路分析和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;社会网络分析 Social Network Analysis&amp;lt;/font&amp;gt;'''都是知识发现的方法，它们都是属于'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 先验方法 Prior Method&amp;lt;/font&amp;gt;'''。大多数知识发现方法遵循以下几个步骤('''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;在最高级别&amp;lt;/font&amp;gt;''' at the highest level) :&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）遇到拿不准的专业名词翻译 可从cnki翻译助手检索一下该名词，下为链接http://dict.cnki.net/dict_result.aspx  选取引用量较高的释义进行翻译 譬如Knowledge Discovery 可以暂时定为知识发现  后续专家审校进行反馈再核实其准不准确&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
# [[Data processing]]&lt;br /&gt;
&lt;br /&gt;
 Data processing&lt;br /&gt;
&lt;br /&gt;
数据处理&lt;br /&gt;
&lt;br /&gt;
# [[Data transformation|Transformation]]&lt;br /&gt;
&lt;br /&gt;
 Transformation&lt;br /&gt;
&lt;br /&gt;
数据转变&lt;br /&gt;
&lt;br /&gt;
# [[Data analysis|Analysis]]&lt;br /&gt;
&lt;br /&gt;
 Analysis&lt;br /&gt;
&lt;br /&gt;
数据分析&lt;br /&gt;
&lt;br /&gt;
# [[Data visualization|Visualization]]&lt;br /&gt;
&lt;br /&gt;
 Visualization&lt;br /&gt;
&lt;br /&gt;
数据可视化&lt;br /&gt;
&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）这里很棒 补充出“数据”这个限定词&lt;br /&gt;
&lt;br /&gt;
Data gathering and processing requires access to data and has several inherent issues, including [[information overload]] and data errors. Once data is collected, it will need to be transformed into a format that can be effectively used by both human and computer analyzers. Manual or computer-generated visualizations tools may be mapped from the data, including network charts. Several algorithms exist to help with analysis of data – [[Dijkstra's algorithm|Dijkstra’s algorithm]], [[breadth-first search]], and [[depth-first search]].&lt;br /&gt;
&lt;br /&gt;
Data gathering and processing requires access to data and has several inherent issues, including information overload and data errors. Once data is collected, it will need to be transformed into a format that can be effectively used by both human and computer analyzers. Manual or computer-generated visualizations tools may be mapped from the data, including network charts. Several algorithms exist to help with analysis of data – Dijkstra’s algorithm, breadth-first search, and depth-first search.&lt;br /&gt;
&lt;br /&gt;
数据的收集和处理是首先进行的过程，但此过程存在一些固有的问题，包括'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;信息超载 Information Overload&amp;lt;/font&amp;gt;'''和数据错误等。在数据被收集后，它将转换成一种人和计算机分析程序都能有效使用的格式。之后基于数据，可使用计算机生成的或人工操作的可视化工具进行作图（如网络图）。目前有几种算法可以帮助人类进行数据分析-'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;Dijkstra算法&amp;lt;/font&amp;gt;'''，'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;广度优先搜索 Breadth-First Search&amp;lt;/font&amp;gt;'''和'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 深度优先搜索 Depth-First Search&amp;lt;/font&amp;gt;'''。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Link analysis focuses on analysis of relationships among nodes through [[Visualization (computer graphics)|visualization methods]] ([[network charts]], association matrix). Here is an example of the relationships that may be mapped for crime investigations:&amp;lt;ref name=Krebs&amp;gt;Krebs, V. E. 2001, [http://vlado.fmf.uni-lj.si/pub/networks/doc/Seminar/Krebs.pdf Mapping networks of terrorist cells] {{webarchive|url=https://web.archive.org/web/20110720000539/http://vlado.fmf.uni-lj.si/pub/networks/doc/Seminar/Krebs.pdf |date=2011-07-20 }}, Connections 24, 43–52.&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
Link analysis focuses on analysis of relationships among nodes through visualization methods (network charts, association matrix). Here is an example of the relationships that may be mapped for crime investigations: &lt;br /&gt;
&lt;br /&gt;
链路分析主要通过可视化方法(网络图、关联矩阵)分析节点之间的关系。这里有一个基于罪犯和社会各部分关系绘制网图的例子:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
&lt;br /&gt;
{ | class“ wikitable”&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
! Relationship/Network !! Data Sources&lt;br /&gt;
&lt;br /&gt;
! Relationship/Network !! Data Sources&lt;br /&gt;
&lt;br /&gt;
!关系 / 网络! ！数据来源&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
| 1. Trust || Prior contacts in family, neighborhood, school, military, club or organization. Public and court records. Data may only be available in suspect's native country.&lt;br /&gt;
&lt;br /&gt;
| 1. Trust || Prior contacts in family, neighborhood, school, military, club or organization. Public and court records. Data may only be available in suspect's native country.&lt;br /&gt;
&lt;br /&gt;
| 1.信任 | 嫌疑人在家庭、社区、学校、军队、俱乐部或组织中已有的联系。公开信息及法庭纪录。以及只能在嫌疑人本国获得的数据。&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
|-&lt;br /&gt;
&lt;br /&gt;
| 2. Task || Logs and records of phone calls, electronic mail, chat rooms, instant messages, Web site visits. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
&lt;br /&gt;
| 2. Task || Logs and records of phone calls, electronic mail, chat rooms, instant messages, Web site visits. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 2.任务 | | 电话、电子邮件、聊天室、即时消息、网站访问的日志和记录。出入境纪录。'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;人类智能: 会议评论和公共活动的出席。&amp;lt;/font&amp;gt;'''Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 3. Money &amp;amp; Resources || Bank account and money transfer records. Pattern and location of credit card use. Prior court records. Human intelligence: observation of visits to alternate banking resources such as [[Hawala]].&lt;br /&gt;
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| 3. Money &amp;amp; Resources || Bank account and money transfer records. Pattern and location of credit card use. Prior court records. Human intelligence: observation of visits to alternate banking resources such as Hawala.&lt;br /&gt;
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| 3.资金和资源 | 银行账户和汇款记录。信用卡使用地点及使用习惯。以前的法庭记录。'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;人类智能: 访问其他银行资源的观察，如 Hawala。&amp;lt;/font&amp;gt;'''Human intelligence: observation of visits to alternate banking resources such as Hawala.&lt;br /&gt;
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| 4. Strategy &amp;amp; Goals || Web sites. Videos and encrypted disks delivered by courier. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 4. Strategy &amp;amp; Goals || Web sites. Videos and encrypted disks delivered by courier. Travel records. Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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| 4.策略与目标 | 网站。由快递公司递送的视频和加密光盘。出入境纪录。'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;人类智能: 会议评论和公共活动的出席。&amp;lt;/font&amp;gt;'''Human intelligence: observation of meetings and attendance at common events.&lt;br /&gt;
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Link analysis is used for 3 primary purposes:&amp;lt;ref name=&amp;quot;Link Analysis Workbench&amp;quot;&amp;gt;[https://docs.google.com/viewer?a=v&amp;amp;q=cache:R4_7k-udkxMJ:www.fas.org/irp/eprint/law.pdf+%22link+analysis%22&amp;amp;hl=en&amp;amp;gl=uk&amp;amp;pid=bl&amp;amp;srcid=ADGEESjUVRJFP2O2eMGzSuW4WtGgujUHVeuUYDQ9wqOkJUeVMHOwpagPa65ypDM5Fgma2AlwDyFBblZwXNMoYDCHwCQuTQvK-HnmqbW-z5A-MOwMKLy8nk6_uHLa0CiUAql-kAilAYcd&amp;amp;sig=AHIEtbRgGVNE4PMZ0a2vipzlDTd6OLX4fA Link Analysis Workbench], Air Force Research Laboratory Information Directorate, Rome Research Site, Rome, New York, September 2004.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Link analysis is used for 3 primary purposes:&lt;br /&gt;
&lt;br /&gt;
链接分析主要有3个作用:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
# Find matches in data for known patterns of interest;&lt;br /&gt;
&lt;br /&gt;
 Find matches in data for known patterns of interest;&lt;br /&gt;
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在数据中寻找有意义的已知模式;&lt;br /&gt;
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# Find anomalies where known patterns are violated;&lt;br /&gt;
&lt;br /&gt;
 Find anomalies where known patterns are violated;&lt;br /&gt;
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发现与已知模式不相符的异常数据;&lt;br /&gt;
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# Discover new patterns of interest (social network analysis, [[data mining]]).&lt;br /&gt;
&lt;br /&gt;
 Discover new patterns of interest (social network analysis, data mining).&lt;br /&gt;
&lt;br /&gt;
发现有意义的新模式(社会网络分析、数据挖掘)。&lt;br /&gt;
&lt;br /&gt;
==History==&lt;br /&gt;
&lt;br /&gt;
Klerks categorized link analysis tools into 3 generations.&amp;lt;ref&amp;gt;{{cite journal | last = Klerks | first = P. | year = 2001 | title = The network paradigm applied to criminal organizations: Theoretical nitpicking or a relevant doctrine for investigators? Recent developments in the Netherlands | citeseerx = 10.1.1.129.4720 | journal = Connections | volume = 24 | pages = 53–65 }}&amp;lt;/ref&amp;gt; The first generation was introduced in 1975 as the Anacpapa Chart of Harper and Harris.&amp;lt;ref&amp;gt;Harper and Harris, The Analysis of Criminal Intelligence, Human Factors and Ergonomics Society Annual Meeting Proceedings, 19(2), 1975, pp. 232-238.&amp;lt;/ref&amp;gt; This method requires that a domain expert review data files, identify associations by constructing an association matrix, create a link chart for visualization and finally analyze the network chart to identify patterns of interest. This method requires extensive domain knowledge and is extremely time-consuming when reviewing vast amounts of data.&lt;br /&gt;
&lt;br /&gt;
[[File:图1：Association Matrix 关联矩阵.png|thumb|Association Matrix 关联矩阵]]&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）点击编辑后会发现[[File: 开头关于图片的语法  该图注需要进行翻译且遵循格式【图1：英文原文＋翻译内容】&lt;br /&gt;
&lt;br /&gt;
Klerks categorized link analysis tools into 3 generations. The first generation was introduced in 1975 as the Anacpapa Chart of Harper and Harris. This method requires that a domain expert review data files, identify associations by constructing an association matrix, create a link chart for visualization and finally analyze the network chart to identify patterns of interest. This method requires extensive domain knowledge and is extremely time-consuming when reviewing vast amounts of data.&lt;br /&gt;
&lt;br /&gt;
Klerks把链接分析工具分为三代。第一代是由哈珀和哈里斯在1975年引入的，阿纳卡帕图。这种方法需要一个领域内的专家来审查数据文件，通过构造一个关联矩阵来识别关联，然后创建一个可视化的链路图，最后通过分析网络图来识别有意义的模式。这种方法需要广泛的领域知识，且因要人工审查大量数据，所以非常耗时。&lt;br /&gt;
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In addition to the association matrix, the activities matrix can be used to produce actionable information, which has practical value and use to law-enforcement. The activities matrix, as the term might imply, centers on the actions and activities of people with respect to locations. Whereas the association matrix focuses on the relationships between people, organizations, and/or properties. The distinction between these two types of matrices, while minor, is nonetheless significant in terms of the output of the analysis completed or rendered.&amp;lt;ref&amp;gt;{{cite web|url=http://www.globalsecurity.org/military/library/policy/army/fm/3-07-22/app-f.htm|title=FMI 3-07.22 Appendix F Intelligence Analysis Tools and Indicators|first=John|last=Pike|publisher=}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[https://rdl.train.army.mil/catalog/view/100.ATSC/41449AB4-E8E0-46C4-8443-E4276B6F0481-1274576841878/3-24/appb.htm Social Network Analysis and Other Analytical Tools] {{webarchive|url=https://web.archive.org/web/20140308233614/https://rdl.train.army.mil/catalog/view/100.ATSC/41449AB4-E8E0-46C4-8443-E4276B6F0481-1274576841878/3-24/appb.htm |date=2014-03-08 }}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite web|url=http://www.nasa.gov/audience/foreducators/topnav/materials/listbytype/Aeronautics_Activity_Matrices.html|title=Aeronautics Educator Guide - Activity Matrices|first=Rebecca Whitaker|last=MSFC|date=10 July 2009|publisher=}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[https://rdl.train.army.mil/catalog/view/100.ATSC/0EF89CA1-2680-4782-B103-D2F5DC941188-1274309335668/7-98-1/chap2l6.htm Personality/Activity Matrix] {{webarchive|url=https://web.archive.org/web/20140308234135/https://rdl.train.army.mil/catalog/view/100.ATSC/0EF89CA1-2680-4782-B103-D2F5DC941188-1274309335668/7-98-1/chap2l6.htm |date=2014-03-08 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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In addition to the association matrix, the activities matrix can be used to produce actionable information, which has practical value and use to law-enforcement. The activities matrix, as the term might imply, centers on the actions and activities of people with respect to locations. Whereas the association matrix focuses on the relationships between people, organizations, and/or properties. The distinction between these two types of matrices, while minor, is nonetheless significant in terms of the output of the analysis completed or rendered.&lt;br /&gt;
&lt;br /&gt;
除了关联矩阵外，活动矩阵也可用于生成对执法活动具有实用和使用价值的可操作的信息。正如这个术语可能暗示的那样，活动矩阵关注的是人们基于地点的行动和活动。而关联矩阵关注的是人，组织和/或属性之间的关系。这两类矩阵之间的区别虽然很小，但就已完成或经过分析的数据来看，区别还是很重要的。&lt;br /&gt;
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Second generation tools consist of automatic graphics-based analysis tools such as IBM i2 Analyst’s Notebook, Netmap, [[ClueMaker]] and Watson. These tools offer the ability to automate the construction and updates of the link chart once an association matrix is manually created, however, analysis of the resulting charts and graphs still requires an expert with extensive domain knowledge.&lt;br /&gt;
&lt;br /&gt;
Second generation tools consist of automatic graphics-based analysis tools such as IBM i2 Analyst’s Notebook, Netmap, ClueMaker and Watson. These tools offer the ability to automate the construction and updates of the link chart once an association matrix is manually created, however, analysis of the resulting charts and graphs still requires an expert with extensive domain knowledge.&lt;br /&gt;
&lt;br /&gt;
第二代工具包括基于图形的自动分析工具，如 IBM i 2 Analyst’ s Notebook、 Netmap、 ClueMaker 和 Watson。在手动创建关联矩阵的情况下，这些工具提供了自动构建和更新链接图表的能力。然而，对结果图表的分析仍然需要在该领域具有广泛知识的专家。&lt;br /&gt;
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The third generation of link-analysis tools like DataWalk allow the automatic visualization of linkages between elements in a data set, that can then serve as the canvas for further exploration or manual updates.&lt;br /&gt;
&lt;br /&gt;
The third generation of link-analysis tools like DataWalk allow the automatic visualization of linkages between elements in a data set, that can then serve as the canvas for further exploration or manual updates.&lt;br /&gt;
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像 DataWalk 这样的第三代链接分析工具能自动可视化数据集中元素之间的链接，其结果可允许进一步的人工探索与改进。&lt;br /&gt;
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==Applications==&lt;br /&gt;
&lt;br /&gt;
* [[ViCAP|FBI Violent Criminal Apprehension Program (ViCAP)]]&lt;br /&gt;
联邦调查局暴力刑事逮捕程序&lt;br /&gt;
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* Iowa State Sex Crimes Analysis System&lt;br /&gt;
爱荷华州性犯罪分析系统&lt;br /&gt;
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* Minnesota State Sex Crimes Analysis System (MIN/SCAP)&lt;br /&gt;
明尼苏达州性犯罪分析系统&lt;br /&gt;
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* Washington State Homicide Investigation Tracking System (HITS)&amp;lt;ref&amp;gt;{{cite web|url=http://www.atg.wa.gov/HITS.aspx |title=Archived copy |accessdate=2010-10-31 |url-status=dead |archiveurl=https://web.archive.org/web/20101021005202/http://atg.wa.gov/HITS.aspx |archivedate=2010-10-21 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
华盛顿州凶杀案调查追踪系统&lt;br /&gt;
&lt;br /&gt;
* New York State Homicide Investigation &amp;amp; Lead Tracking (HALT)&lt;br /&gt;
纽约州凶杀案调查和线索追踪系统&lt;br /&gt;
&lt;br /&gt;
* New Jersey Homicide Evaluation &amp;amp; Assessment Tracking (HEAT)&amp;lt;ref&amp;gt;{{cite web|url=http://www.state.nj.us/njsp/divorg/invest/invest.html |title=Archived copy |accessdate=2010-10-31 |url-status=dead |archiveurl=https://web.archive.org/web/20090325004722/http://www.state.nj.us/njsp/divorg/invest/invest.html |archivedate=2009-03-25 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
新泽西州凶杀案评估与测评跟踪系统&lt;br /&gt;
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* Pennsylvania State ATAC Program.&lt;br /&gt;
宾夕法尼亚州ATAC程序&lt;br /&gt;
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* Violent Crime Linkage Analysis System (ViCLAS)&amp;lt;ref&amp;gt;{{cite web|url=http://www.rcmp-grc.gc.ca/tops-opst/bs-sc/viclas-salvac-eng.htm |title=Archived copy |accessdate=2010-10-31 |url-status=dead |archiveurl=https://web.archive.org/web/20101202144141/http://www.rcmp-grc.gc.ca/tops-opst/bs-sc/viclas-salvac-eng.htm |archivedate=2010-12-02 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
暴力犯罪联系分析系统&lt;br /&gt;
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==Issues with link analysis==&lt;br /&gt;
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===Information overload===&lt;br /&gt;
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With the vast amounts of data and information that are stored electronically, users are confronted with multiple unrelated sources of information available for analysis. Data analysis techniques are required to make effective and efficient use of the data. Palshikar classifies data analysis techniques into two categories – ([[Statistics|statistical]] [[Models of computation|models]], [[Time series analysis|time-series analysis]], [[Clustering coefficient|clustering]] and [[Statistical classification|classification]], matching algorithms to detect anomalies) and [[Artificial intelligence|artificial intelligence (AI)]] techniques (data mining, [[expert systems]], [[pattern recognition]], [[Machine learning|machine learning techniques]], [[neural network]]s).&amp;lt;ref&amp;gt;Palshikar, G. K., [http://www.intelligententerprise.com//020528/509feat3_1.jhtml The Hidden Truth], Intelligent Enterprise, May 2002.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
With the vast amounts of data and information that are stored electronically, users are confronted with multiple unrelated sources of information available for analysis. Data analysis techniques are required to make effective and efficient use of the data. Palshikar classifies data analysis techniques into two categories – (statistical models, time-series analysis, clustering and classification, matching algorithms to detect anomalies) and artificial intelligence (AI) techniques (data mining, expert systems, pattern recognition, machine learning techniques, neural networks).&lt;br /&gt;
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由于大量数据和信息以电子形式存储，用户可能会面临拥有多种不相关的信息来源却不知如何分析的难题。数据分析技术的使用可以帮助有效和高效地利用数据。Palshikar 将数据分析技术分为两大类(统计模型、时间序列分析、聚类分类、异常检测匹配算法)和人工智能(AI)技术(数据挖掘、专家系统、模式识别、机器学习技术、神经网络)。&lt;br /&gt;
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Bolton &amp;amp; Hand define statistical data analysis as either supervised or unsupervised methods.&amp;lt;ref&amp;gt;Bolton, R. J. &amp;amp; Hand, D. J., Statistical Fraud Detection: A Review, Statistical Science, 2002, 17(3), pp. 235-255.&amp;lt;/ref&amp;gt; [[Supervised learning|Supervised learning methods]] require that rules are defined within the system to establish what is expected or unexpected behavior. [[Unsupervised learning|Unsupervised learning methods]] review data in comparison to the norm and detect statistical outliers. Supervised learning methods are limited in the scenarios that can be handled as this method requires that training rules are established based on previous patterns. Unsupervised learning methods can provide detection of broader issues, however, may result in a higher false-positive ratio if the behavioral norm is not well established or understood.&lt;br /&gt;
&lt;br /&gt;
Bolton &amp;amp; Hand define statistical data analysis as either supervised or unsupervised methods. Supervised learning methods require that rules are defined within the system to establish what is expected or unexpected behavior. Unsupervised learning methods review data in comparison to the norm and detect statistical outliers. Supervised learning methods are limited in the scenarios that can be handled as this method requires that training rules are established based on previous patterns. Unsupervised learning methods can provide detection of broader issues, however, may result in a higher false-positive ratio if the behavioral norm is not well established or understood.&lt;br /&gt;
&lt;br /&gt;
Bolton &amp;amp; Hand 将统计数据分析定义为有监督或无监督的方法。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;监督式学习方法 Supervised Learning Methods&amp;lt;/font&amp;gt;'''要求在系统中有明确的规则来指出什么是预期行为，什么是意外行为。'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;非监督式学习方法 Unsupervised Learning Methods&amp;lt;/font&amp;gt;'''在审视数据时，通过将数据与正常值的比较，来发现统计异常值。监督式学习方法能处理的场景是有限的，因为这种方法需要基于以前的模式建立训练规则。非监督式学习方法可以对更广泛的问题采取进攻。但是，如果数据的行为规范没有很好的建立或被机器理解 --[[用户:Ryan|Ryan]]（[[用户讨论:Ryan|讨论]]）该句存疑，其结果可能会导致较高的假阳性率（本身不是正常值，但识别为正常值，说明算法预测了“正确”或“有”的判断，但却判断错误了）。&lt;br /&gt;
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Data itself has inherent issues including integrity (or lack of) and continuous changes. Data may contain “errors of omission and commission because of faulty collection or handling, and when entities are actively attempting to deceive and/or conceal their actions”.&amp;lt;ref name=&amp;quot;Link Analysis Workbench&amp;quot;/&amp;gt; Sparrow&amp;lt;ref&amp;gt;Sparrow M.K. 1991. Network Vulnerabilities and Strategic Intelligence in Law Enforcement’, [[International Journal of Intelligence and Counterintelligence]] Vol. 5 #3.&amp;lt;/ref&amp;gt; highlights incompleteness (inevitability of missing data or links), fuzzy boundaries (subjectivity in deciding what to include) and dynamic changes (recognition that data is ever-changing) as the three primary problems with data analysis.&amp;lt;ref name=Krebs/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Data itself has inherent issues including integrity (or lack of) and continuous changes. Data may contain “errors of omission and commission because of faulty collection or handling, and when entities are actively attempting to deceive and/or conceal their actions”. Sparrow highlights incompleteness (inevitability of missing data or links), fuzzy boundaries (subjectivity in deciding what to include) and dynamic changes (recognition that data is ever-changing) as the three primary problems with data analysis.&lt;br /&gt;
&lt;br /&gt;
数据本身存在固有的问题，包括完整性(或缺失性)和持续的改变。数据可能包含'''&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;“由于错误的收集或处理，以及当实体积极试图欺骗和 / 或隐瞒其行为，而造成的错误的遗漏和委托”&amp;lt;/font&amp;gt;''' “errors of omission and commission because of faulty collection or handling, and when entities are actively attempting to deceive and/or conceal their actions”。Sparro强调了数据分析中三个主要的问题，不完整性(数据或链接缺失的必然性)、模糊边界(边界确定的主观性)和动态变化(数据的持续变化性)。&lt;br /&gt;
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  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）存疑的地方也可以用绿色语法进行标注&lt;br /&gt;
&lt;br /&gt;
Once data is transformed into a usable format, open texture and cross referencing issues may arise. [[Open texture]] was defined by [[Friedrich Waismann|Waismann]] as the unavoidable uncertainty in meaning when empirical terms are used in different contexts.&amp;lt;ref&amp;gt;Friedrich Waismann, Verifiability (1945), p.2.&amp;lt;/ref&amp;gt; Uncertainty in meaning of terms presents problems when attempting to search and cross reference data from multiple sources.&amp;lt;ref&amp;gt;Lyons, D., [http://ssrn.com/abstract=212328 Open Texture and the Possibility of Legal Interpretation (2000)].&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Once data is transformed into a usable format, open texture and cross referencing issues may arise. Open texture was defined by Waismann as the unavoidable uncertainty in meaning when empirical terms are used in different contexts. Uncertainty in meaning of terms presents problems when attempting to search and cross reference data from multiple sources.&lt;br /&gt;
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一旦数据转换成可用的格式，开放纹理和交叉引用问题就会出现。Waismann将'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt; 开放纹理 Open Texture&amp;lt;/font&amp;gt;'''定义为在不同语境中使用经验词汇时不可避免的语义不确定性。当试图从多个数据源搜索和交叉引用数据时，术语含义的不确定性带来了问题。&lt;br /&gt;
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The primary method for resolving data analysis issues is reliance on [[domain knowledge]] from an expert. This is a very time-consuming and costly method of conducting link analysis and has inherent problems of its own. McGrath et al. conclude that the layout and presentation of a network diagram have a significant impact on the user’s “perceptions of the existence of groups in networks”.&amp;lt;ref&amp;gt;McGrath, C., Blythe, J., Krackhardt, D., [http://www.andrew.cmu.edu/user/cm3t/groups.html Seeing Groups in Graph Layouts].&amp;lt;/ref&amp;gt; Even using domain experts may result in differing conclusions as analysis may be subjective.&lt;br /&gt;
&lt;br /&gt;
The primary method for resolving data analysis issues is reliance on domain knowledge from an expert. This is a very time-consuming and costly method of conducting link analysis and has inherent problems of its own. McGrath et al. conclude that the layout and presentation of a network diagram have a significant impact on the user’s “perceptions of the existence of groups in networks”. Even using domain experts may result in differing conclusions as analysis may be subjective.&lt;br /&gt;
&lt;br /&gt;
目前，解决数据分析中这些问题的主要方法是依赖专家的'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;领域知识 Domain Knowledge&amp;lt;/font&amp;gt;'''。如此进行链路分析是非常耗时和昂贵的，并且无法排除其自身固有的问题。麦格拉斯等人得出结论，网络图的分布和表示方式对用户的“对存在在网络中群体的感知”有重大影响。即使是领域内的专家也可能导致不同的结论，因为分析可能是很主观的。&lt;br /&gt;
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===Prosecution vs. crime prevention===&lt;br /&gt;
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Link analysis techniques have primarily been used for prosecution, as it is far easier to review historical data for patterns than it is to attempt to predict future actions.&lt;br /&gt;
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Link analysis techniques have primarily been used for prosecution, as it is far easier to review historical data for patterns than it is to attempt to predict future actions.&lt;br /&gt;
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目前，链接分析技术主要用于起诉，因为回顾历史数据以期从中获得模式，要比预测未来的行动容易得多。&lt;br /&gt;
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Krebs demonstrated the use of an association matrix and link chart of the terrorist network associated with the 19 hijackers responsible for the [[September 11 attacks|September 11th attacks]] by mapping publicly available details made available following the attacks.&amp;lt;ref name=Krebs/&amp;gt; Even with the advantages of hindsight and publicly available information on people, places and transactions, it is clear that there is missing data.&lt;br /&gt;
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Krebs demonstrated the use of an association matrix and link chart of the terrorist network associated with the 19 hijackers responsible for the September 11th attacks by mapping publicly available details made available following the attacks. Even with the advantages of hindsight and publicly available information on people, places and transactions, it is clear that there is missing data.&lt;br /&gt;
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Krebs基于袭击后的详细公开资料进行绘图，演示了与9月11日袭击事件的19名劫机者有关的恐怖分子关系网的关联矩阵和链接图。即使有事后诸葛亮的优势--[[用户:Ryan|Ryan]]（[[用户讨论:Ryan|讨论]]）该句有待讨论，以及关于人员、地点和交易的公开可用信息，做出的结果图很明显仍然缺少数据。&lt;br /&gt;
  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）即使有事后诸葛亮的优势  如果进行了意译应标注讨论出来&lt;br /&gt;
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Alternatively, Picarelli argued that use of link analysis techniques could have been used to identify and potentially prevent illicit activities within the [[Aum Shinrikyo]] network.&amp;lt;ref&amp;gt;Picarelli, J. T., [http://kdl.cs.umass.edu/events/aila1998/picarelli.ps Transnational Threat Indications and Warning: The Utility of Network Analysis, Military and Intelligence Analysis Group].&amp;lt;/ref&amp;gt; “We must be careful of ‘guilt by association’. Being linked to a terrorist does not prove guilt – but it does invite investigation.”&amp;lt;ref name=Krebs/&amp;gt; Balancing the legal concepts of [[probable cause]], [[right to privacy]] and [[freedom of association]] become challenging when reviewing potentially sensitive data with the objective to prevent crime or illegal activity that has not yet occurred.&lt;br /&gt;
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Alternatively, Picarelli argued that use of link analysis techniques could have been used to identify and potentially prevent illicit activities within the Aum Shinrikyo network. “We must be careful of ‘guilt by association’. Being linked to a terrorist does not prove guilt – but it does invite investigation.” Balancing the legal concepts of probable cause, right to privacy and freedom of association become challenging when reviewing potentially sensitive data with the objective to prevent crime or illegal activity that has not yet occurred.&lt;br /&gt;
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另外，Picarelli认为，使用链路分析技术可以用来查明并有可能防止奥姆真理教的非法活动。“我们必须小心‘牵连犯罪’。与恐怖分子有联系并不能证明有罪——但确实得进行调查。” 在审查较为敏感的数据以防止尚未发生的犯罪或非法活动时，如何同时不违背合理依据、隐私权和结社自由等法律概念将变得很困难。&lt;br /&gt;
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==Proposed solutions==&lt;br /&gt;
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There are four categories of proposed link analysis solutions:&amp;lt;ref&amp;gt;Schroeder et al., Automated Criminal Link Analysis Based on Domain Knowledge, Journal of the American Society for Information Science and Technology, 58:6 (842), 2007.&amp;lt;/ref&amp;gt;&lt;br /&gt;
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There are four categories of proposed link analysis solutions:&lt;br /&gt;
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有四类拟议的链路分析解决方案:&lt;br /&gt;
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# [[Heuristic|Heuristic-based]]&lt;br /&gt;
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 Heuristic-based&lt;br /&gt;
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基于启发式的&lt;br /&gt;
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# Template-based&lt;br /&gt;
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 Template-based&lt;br /&gt;
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基于模板的&lt;br /&gt;
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# Similarity-based&lt;br /&gt;
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 Similarity-based&lt;br /&gt;
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基于相似性的&lt;br /&gt;
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# [[Statistical]]&lt;br /&gt;
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 Statistical&lt;br /&gt;
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统计方法&lt;br /&gt;
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Heuristic-based tools utilize decision rules that are distilled from expert knowledge using structured data. Template-based tools employ [[Natural language processing|Natural Language Processing (NLP)]] to extract details from [[unstructured data]] that are matched to pre-defined templates. Similarity-based approaches use weighted [[Score (statistics)|scoring]] to compare attributes and identify potential links. Statistical approaches identify potential links based on lexical statistics.&lt;br /&gt;
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Heuristic-based tools utilize decision rules that are distilled from expert knowledge using structured data. Template-based tools employ Natural Language Processing (NLP) to extract details from unstructured data that are matched to pre-defined templates. Similarity-based approaches use weighted scoring to compare attributes and identify potential links. Statistical approaches identify potential links based on lexical statistics.&lt;br /&gt;
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基于启发式的工具运用从专家知识中提取出来的决策规则对结构化数据进行操作。基于模板的工具使用'''&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;自然语言处理 Natural Language Processing&amp;lt;/font&amp;gt;'''从非结构化数据中提取与预定义模板匹配的细节。基于相似度的方法使用加权评分来比较属性和识别潜在的链接。统计方法基于词汇统计识别潜在的链接。&lt;br /&gt;
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===CrimeNet explorer===&lt;br /&gt;
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J.J. Xu and H. Chen propose a framework for automated network analysis and visualization called CrimeNet Explorer.&amp;lt;ref name=Xu&amp;gt;Xu, J.J. &amp;amp; Chen, H., CrimeNet Explorer: A Framework for Criminal Network Knowledge Discovery, ACM Transactions on Information Systems, 23(2), April 2005, pp. 201-226.&amp;lt;/ref&amp;gt; This framework includes the following elements:&lt;br /&gt;
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J.J. Xu and H. Chen propose a framework for automated network analysis and visualization called CrimeNet Explorer. This framework includes the following elements:&lt;br /&gt;
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J.J.Xu和H.Chen 提出了一个自动化网络分析和可视化的框架，叫做 CrimeNet Explorer。这一框架包括以下内容:&lt;br /&gt;
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* Network Creation through a concept space approach that uses “[[Co-occurrence networks|co-occurrence]] weight to measure the frequency with which two words or phrases appear in the same document. The more frequently two words or phrases appear together, the more likely it will be that they are related”.&amp;lt;ref name=Xu/&amp;gt;&lt;br /&gt;
通过概念空间方法创建网络，该方法使用“共现网络”来衡量两个单词或短语在同一文档中出现的频率。两个单词或短语在一起出现的频率越高，它们关联的可能性就越大。&lt;br /&gt;
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* Network Partition using “hierarchical clustering to partition a network into subgroups based on relational strength”.&amp;lt;ref name=Xu/&amp;gt;&lt;br /&gt;
网络分区通过“根据关系强度的分层聚类，将网络划分为子组”而实现。&lt;br /&gt;
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* Structural Analysis through “three centrality measures (degree, betweenness, and closeness) to identify central members in a given subgroup.&amp;lt;ref name=Xu/&amp;gt; CrimeNet Explorer employed [[Dijkstra's algorithm|Dijkstra’s shortest-path algorithm]] to calculate the betweenness and closeness from a single node to all other nodes in the subgroup.&lt;br /&gt;
通过“三种中心性度量（度，中间性和紧密性）来识别给定子集中的中心成员”进行结构分析。CrimeNet Explorer使用Dijkstra的最短路径算法来计算从单个节点到子组中所有其他节点的间隔度和紧密度。&lt;br /&gt;
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* Network Visualization using Torgerson’s metric [[Multidimensional scaling|multidimensional scaling (MDS)]] algorithm.&lt;br /&gt;
使用Torgerson的度量多维标度（MDS）算法进行网络可视化。&lt;br /&gt;
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==References==&lt;br /&gt;
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&amp;lt;!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using  tags which will then appear here automatically --&amp;gt;&lt;br /&gt;
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&amp;lt;!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using  tags which will then appear here automatically --&amp;gt;&lt;br /&gt;
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!-参见 http: / / en.wikipedia. org / wiki / wikipedia: 如何使用标记创建引用的脚注，这些标记将自动出现在这里 -- &lt;br /&gt;
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{{reflist|colwidth=30em}}&lt;br /&gt;
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==External links==&lt;br /&gt;
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* {{cite book | last1 = Bartolini | first1 = I | last2 = Ciaccia | first2 = P. | title = Imagination: Accurate Image Annotation Using Link-Analysis Techniques | citeseerx = 10.1.1.63.2453 }}&lt;br /&gt;
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* [https://fightfinancialcrimes.com/2015/12/17/link-analysis-and-fraud/ Link Analysis and Crime - An Examination]&lt;br /&gt;
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* [https://www.ncbi.nlm.nih.gov/pubmed/9039449 Elink Schuurman MW, Srisaenpang S, Pinitsoontorn S, Bijleveld I, Vaeteewoothacharn K, Methapat C., The rapid village survey in tuberculosis control, Tuber Lung Dis. 1996 Dec;77(6):549-54.]&lt;br /&gt;
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* [https://www.cs.cmu.edu/~gunhee/publish/cvpr08_gunhee.pdf Gunhee, K., Faloutsos, C, Hebert, M, Unsupervised Modeling of Object Categories Using Link Analysis Techniques.]&lt;br /&gt;
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* [https://web.archive.org/web/20110928070620/http://www.blythe.org/Intelligence/readme/72sum McGehee, R., Intelligence Report.]&lt;br /&gt;
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* [https://web.archive.org/web/20061107040453/http://www.hsaj.org/pages/volume2/issue2/pdfs/2.2.8.pdf Ressler, S., Social Network Analysis as an Approach to Combat Terrorism: Past, Present and Future Research.]&lt;br /&gt;
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* [http://www.rff.com/link_analysis.htm RFFlow, How to Draw a Link Analysis Diagram.]&lt;br /&gt;
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* [http://www-03.ibm.com/software/products/en/analysts-notebook-premium IBM i2 Analyst's Notebook Premium]&lt;br /&gt;
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* {{cite journal | last1 = Silberschatz | first1 = A. | title = What Makes Patterns Interesting in Knowledge Discovery Systems | journal = IEEE Transactions on Knowledge and Data Engineering | volume = 8 | issue = 6 | pages = 970–974 | citeseerx = 10.1.1.53.2780 | year = 1996 | doi = 10.1109/69.553165 }}&lt;br /&gt;
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* [http://www.intelcenter.com/trlaf.html Terrorist &amp;amp; Rebel Link Analysis Feed (TR-LAF)]&lt;br /&gt;
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* [http://kt.ijs.si/dunja/LinkKDD2006/ Workshop on Link Analysis: Dynamics and Static of Large Networks (LinkKDD2006) August 20, 2006]&lt;br /&gt;
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* [http://www.cluemaker.com ClueMaker]&lt;br /&gt;
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* [http://www.datawalk.com Data Walk]&lt;br /&gt;
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[[Category:Network theory]]&lt;br /&gt;
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Category:Network theory&lt;br /&gt;
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范畴: 网络理论&lt;br /&gt;
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&amp;lt;noinclude&amp;gt;&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Link analysis]]. Its edit history can be viewed at [[链路分析/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Ryan</name></author>
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