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Moved page from wikipedia:en:Degree distribution (history)
此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
{{Network Science}}
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.
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.
在图形和网络的研究中，网络中一个节点的度就是它与其他节点的连接数，度的分布就是整个网络中这些度的概率分布。
==Definition==
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.
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.
网络中一个节点的程度(有时错误地称为连通性)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，那么节点有两个不同的度，一个度是传入边的数量，另一个度是传出边的数量。
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''<sub>''k''</sub> of them have degree ''k'', we have ''P''(''k'') = ''n''<sub>''k''</sub>/''n''.
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<sub>k</sub> of them have degree k, we have P(k) = n<sub>k</sub>/n.
将网络的度分布 p (k)定义为 k 度网络中所有节点的分数，如果一个网络中有 n 个节点，且其中 n 个节点的 k 度为 n，则 p (k) = n < sub > k </sub >/n。
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''.
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.
同样的信息有时也以累积度分布、度小于 k 的节点比例、甚至是互补累积度分布的形式出现，如果把 c 看作累积度分布，则为度大于或等于 k (1-c)的节点比例;。C 的补语。
== Observed degree distributions ==
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'':
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:
学位分布在研究实际网络(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(erd s-Rényi 模型)随机图，其中每个 n 个节点都以概率 p (或1-p)独立连接(或不独立连接) ，其二项分布为 k:
:<math>
<math>
《数学》
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},
选择 k } p ^ k (1-p) ^ { n-1-k } ,
</math>
</math>
数学
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree <math>\langle k\rangle=p(n-1)</math> 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 "hubs", 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]]: <math>
(or Poisson in the limit of large n, if the average degree <math>\langle k\rangle=p(n-1)</math> 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 "hubs", 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: <math>
(如果平均度 < math > langle k rangle = p (n-1) </math > 保持不变)。然而，现实世界中的大多数网络的度分布与此大相径庭。大多数节点是高度右倾斜的，这意味着大多数节点的度数较低，但少数节点，即所谓的“枢纽” ，度数较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有度分布，近似遵循一个幂定律: < math >
P(k)\sim k^{-\gamma}
P(k)\sim k^{-\gamma}
P (k) sim k ^ {-gamma }
</math>, where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties<ref name="BA">{{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 }}</ref><ref name="AB">{{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 }}</ref><ref name="Doro">{{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 }}</ref><ref name="PSY">{{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}}</ref>. 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]].<ref>{{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}}</ref>
</math>, 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.
</math>, where γ is a constant.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，最近有一些基于现实数据集的研究声称，尽管大多数观测到的网络具有厚尾度分布，但它们偏离了无标度分布。
== Excess degree distribution ==
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.<ref name=":0">{{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}}</ref> In other words, it is the distribution of outgoing links from a node reached by following a link.
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.
过度度分布是一个节点通过跟随一条边到达的其他边数的概率分布。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。
Suppose a network has a degree distribution <math>
Suppose a network has a degree distribution <math>
假设一个网络具有度分布
P(k)
P(k)
P (k)
</math>, 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 <math>
</math>, 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 <math>
</math > ，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 < math > 的概率
k
k
K
</math> neighbors is not given by <math>
</math> neighbors is not given by <math>
邻居不是由数学给出的
P(k)
P(k)
P (k)
</math>. 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 <math>
</math>. 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 <math>
数学。原因在于，无论何时在异质网路节点中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达滚刀。这些节点具有度 < math > 的真实概率
k
k
K
</math> is <math>
</math> is <math>
是“ math”
q(k)
q(k)
Q (k)
</math> 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<ref name=":0" />:
</math> 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:
</math > ，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，超额度分布为:
<math>
<math>
《数学》
q(k) = \frac{k+1}{\langle k \rangle}P(k+1),
q(k) = \frac{k+1}{\langle k \rangle}P(k+1),
Q (k) = frac { k + 1}{ langle k rangle } p (k + 1) ,
</math>
</math>
数学
where <math>
where <math>
在哪里
{\langle k \rangle}
{\langle k \rangle}
我不知道该怎么办
</math> 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:
</math> 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:
</math > 是模型的平均度(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组件:
<math>
<math>
《数学》
\sum_k kq(k) > 1 \Rightarrow {\langle k^2 \rangle}-2{\langle k \rangle}>0
\sum_k kq(k) > 1 \Rightarrow {\langle k^2 \rangle}-2{\langle k \rangle}>0
1 right tarrow { langle k ^ 2 rangle }-2{ langle k rangle } > 0
</math>
</math>
数学
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.<ref name=":0" />
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.
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。
== The Generating Functions Method ==
[[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, <math>
Generating functions can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, <math>
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，< math >
P(k)
P(k)
P (k)
</math> and <math>
</math> and <math>
[ math ]和[ math ]
q(k)
q(k)
Q (k)
</math> respectively, it is possible to write two power series in the following forms:
</math> respectively, it is possible to write two power series in the following forms:
分别以下列形式编写两个幂级数是可能的:
<math>
<math>
《数学》
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k
0(x) = textstyle sum { k }显示样式 p (k) x ^ k
</math> and <math>
</math> and <math>
[ math ]和[ math ]
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}
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}
1(x) = textstyle sum { k } displaystyle q (k) x ^ k = textstyle sum { k } displastyle frac { k }{ langle k rangle } p (k) x ^ { k-1}
</math>
</math>
数学
<math>
<math>
《数学》
G_1(x)
G_1(x)
G _ 1(x)
</math> can also be obtained from derivatives of <math>
</math> can also be obtained from derivatives of <math>
</math > 也可以从 < math > 的导数得到
G_0(x)
G_0(x)
G _ 0(x)
</math>:
</math>:
(/math)
<math>
<math>
《数学》
G_1(x) = \frac{G'_0(x)}{G'_0(1)}
G_1(x) = \frac{G'_0(x)}{G'_0(1)}
1(x) = frac { g’ _ 0(x)}{ g’ _ 0(1)}
</math>
</math>
数学
If we know the generating function for a probability distribution <math>
If we know the generating function for a probability distribution <math>
如果我们知道美国母函数的一个概率分布
P(k)
P(k)
P (k)
</math> then we can recover the values of <math>
</math> then we can recover the values of <math>
然后我们就可以恢复
P(k)
P(k)
P (k)
</math> by differentiating:
</math> by differentiating:
通过区分:
<math>
<math>
《数学》
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}
= frac {1}{ k! }{ operatorname { d } ^ k! g over operatorname { d } ! x ^ k } bigvert { x = 0}
</math>
</math>
数学
Some properties, e.g. the moments, can be easily calculated from <math>
Some properties, e.g. the moments, can be easily calculated from <math>
有些物业，例如。时刻，可以很容易地计算从 < math >
G_0(x)
G_0(x)
G _ 0(x)
</math> and its derivatives:
</math> and its derivatives:
和它的衍生物:
*<math>
{\langle k \rangle} = G'_0(1)
{\langle k \rangle} = G'_0(1)
{ langle k rangle } = g’ _ 0(1)
</math>
</math>
数学
*<math>
{\langle k^2 \rangle} = G''_0(1) + G'_0(1)
{\langle k^2 \rangle} = G_0(1) + G'_0(1)
{ langle k ^ 2 rangle } = g _ 0(1) + g’ _ 0(1)
</math>
</math>
数学
And in general<ref name=":0" />:
And in general:
一般来说:
* <math>
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}
{ langle k ^ m rangle } = Biggl [{ bigg (operatorname { x }{ operatorname { d } ! over operatorname { dx } ！[} Biggl) ^ m } g _ 0(x) Biggl ] _ { x = 1}
</math>
</math>
数学
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], <math>
For Poisson-distributed random networks, such as the ER graph, <math>
对于泊松分布的随机网络，如 ER 图，< math >
G_1(x) = G_0(x)
G_1(x) = G_0(x)
G1(x) = g0(x)
</math>, 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 <math>
</math>, 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 <math>
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 < math > 生成的
G_0(x)
G_0(x)
G _ 0(x)
</math> and <math>
</math> and <math>
[ math ]和[ math ]
G_0(G_1(x))
G_0(G_1(x))
G _ 0(g _ 1(x))
</math>. By extension, the distribution of <math>
</math>. By extension, the distribution of <math>
数学。通过扩展，< math > 的分布
m
m
M
</math>-th neighbors is generated by:
</math>-th neighbors is generated by:
这个邻居是由以下几个因素产生的:
<math>
<math>
《数学》
G_0\bigl(G_1(...G_1(x)...)\bigr)
G_0\bigl(G_1(...G_1(x)...)\bigr)
G _ 0 bigl (g _ 1(... g _ 1(x) ...) bigr)
</math>, with <math>
</math>, with <math>
[ math ] ，with < math >
m-1
m-1
M-1
</math> iterations of the function <math>
</math> iterations of the function <math>
函数的迭代
G_1
G_1
1
</math> acting on itself.<ref name=":1">{{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}}</ref>
</math> acting on itself.
自我作用。
The average number of 1st neighbors, <math>
The average number of 1st neighbors, <math>
第一个邻居的平均数量
c_1
c_1
C _ 1
</math>, is <math>
</math>, is <math>
[ math ] ，is < math >
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}
{ langle k rangle } = { dG _ 0(x) over dx } | { x = 1}
</math> and the average number of 2nd neighbors is: <math>
</math> and the average number of 2nd neighbors is: <math>
第二个邻居的平均数量是:
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)
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)
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)
</math>
</math>
数学
== Degree distribution for directed networks ==
[[File:Enwiki-degree-distribution.png|thumb|right|320px|In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales)]]
In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales)
维基百科超链接图(对数尺度)的 In/out 度分布
In a directed network, each node has some in-degree <math>
In a directed network, each node has some in-degree <math>
在有向网络中，每个节点都有一定程度的“数学”
k_{
k_{
这是一个很好的例子
in}
in}
开始吧
</math> and some out-degree <math>
</math> and some out-degree <math>
还有一些超过学位的
k_{out}
k_{out}
我不知道你在说什么
</math> which are the number of links which have run into and out of that node respectfully. If <math>
</math> which are the number of links which have run into and out of that node respectfully. If <math>
</math > 是指以尊敬的方式进出该节点的链接数量。如果 < math >
P(k_{in}, k_{out})
P(k_{in}, k_{out})
P (k _ { in } ，k _ { out })
</math> is the probability that a randomly chosen node has in-degree <math>
</math> is the probability that a randomly chosen node has in-degree <math>
是一个随机选择的节点在程度上
k_{
k_{
这是一个很好的例子
in}
in}
开始吧
</math> and out-degree <math>
</math> and out-degree <math>
“数学”和“学位”
k_{out}
k_{out}
我不知道你在说什么
</math> then the generating function assigned to this [[joint probability distribution]] can be written with two valuables <math>
</math> then the generating function assigned to this joint probability distribution can be written with two valuables <math>
那么分配给这个母函数的联合分布可以用两个贵重物品来写
x
x
X
</math> and <math>
</math> and <math>
[ math ]和[ math ]
y
y
Y
</math> as:
</math> as:
作为:
<math>
<math>
《数学》
\mathcal{G}(x,y) = \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .
\mathcal{G}(x,y) = \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .
数学{ g }(x，y) = sum _ { k _ { in } ，k _ { out }显示样式 p ({ k _ { in } ，k _ { out }) x ^ { k _ { in } y ^ { k _ { out }。
</math>
</math>
数学
Since every link in a directed network must leave some node and enter another, the net average number of links entering
Since every link in a directed network must leave some node and enter another, the net average number of links entering
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，因此进入的链路的净平均数
a node is zero. Therefore,
a node is zero. Therefore,
一个节点是零。所以,
<math>
<math>
《数学》
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0
Langle { k _ { in }-k _ { out } rangle = sum _ { k _ { in } ，k _ { out } displays (k _ { in }-k _ { out }) p ({ k _ { in } ，k _ { out }}) = 0
</math>,
</math>,
数学,
which implies that, the generation function must satisfy:
which implies that, the generation function must satisfy:
这意味着，生成函数必须满足:
<math>
<math>
《数学》
{\partial \mathcal{G}\over\partial x}\vert _{x,y=1} = {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,
{\partial \mathcal{G}\over\partial x}\vert _{x,y=1} = {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,
{ partial mathcal { g } over partial x } vert _ { x，y = 1} = { partial mathcal { g } over partial y } vert _ { x，y = 1} = c,
</math>
</math>
数学
where <math>
where <math>
在哪里
c
c
C
</math> is the mean degree (both in and out) of the nodes in the network; <math>
</math> is the mean degree (both in and out) of the nodes in the network; <math>
是网络中节点的平均度(内部和外部)
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c.
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c.
长角 = 长角 = 长角。
</math>
</math>
数学
Using the function <math>
Using the function <math>
使用函数 < math >
\mathcal{G}(x,y)
\mathcal{G}(x,y)
数学{ g }(x，y)
</math>, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. <math>
</math>, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. <math>
</math > ，我们可以再次找到进/出度分布和进/出超量度分布的生成函数，如前所述。《数学》
G^{in}_0(x)
G^{in}_0(x)
0(x)
</math> can be defined as generating functions for the number of arriving links at a randomly chosen node, and <math>
</math> can be defined as generating functions for the number of arriving links at a randomly chosen node, and <math>
可以将 </math > 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 < math >
G^{in}_1(x)
G^{in}_1(x)
1(x)
</math>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 <math>
</math>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 <math>
</math > 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 < math >
G^{out}_0(y)
G^{out}_0(y)
0(y)
</math> and <math>
</math> and <math>
[ math ]和[ math ]
G^{out}_1(y)
G^{out}_1(y)
1(y)
</math> for the number leaving such a node:<ref name=":1" />
</math> for the number leaving such a node:
离开这样一个节点的数字:
* <math>
G^{in}_0(x) = \mathcal{G}(x,1)
G^{in}_0(x) = \mathcal{G}(x,1)
0(x) = mathcal { g }(x，1)
</math>
</math>
数学
* <math>
G^{in}_1(x) = \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1}
G^{in}_1(x) = \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1}
1(x) = frac {1}{ c }{ partial mathcal { g } over partial x } vert _ { y = 1}
</math>
</math>
数学
* <math>
G^{out}_0(y) = \mathcal{G}(1,y)
G^{out}_0(y) = \mathcal{G}(1,y)
0(y) = mathcal { g }(1，y)
</math>
</math>
数学
* <math>
G^{out}_1(y) = \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1}
G^{out}_1(y) = \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1}
1(y) = frac {1}{ c }{ partial mathcal { g } over partial y } vert _ { x = 1}
</math>
</math>
数学
Here, the average number of 1st neighbors, <math>
Here, the average number of 1st neighbors, <math>
这里，第一个邻居的平均数量，< math >
c
c
C
</math>, or as previously introduced as <math>
</math>, or as previously introduced as <math>
或者像之前介绍的那样
c_1
c_1
C _ 1
</math>, is <math>
</math>, is <math>
[ math ] ，is < math >
{\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} = {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}
{\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} = {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}
{ partial mathcal { g } over partial x } biggl vert _ { x，y = 1} = { partial mathcal { g } over partial y } biggl vert _ { x，y = 1}
</math> and the average number of 2nd neighbors reachable from a randomly chosen node is given by: <math>
</math> and the average number of 2nd neighbors reachable from a randomly chosen node is given by: <math>
从一个随机选择的节点上可达到的第二邻居的平均数是:
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}
C _ 2 = g _ 1’(1) g’ _ 0(1) = {{部分 ^ 2数学{ g }/部分 x 部分 y } biggl vert _ { x，y = 1}
</math>. 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 <math>
</math>. 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 <math>
数学。这些也是第一和第二邻居的数量，从这些随机节点可以达到，因为这些方程显然是对称的 < math >
x
x
X
</math> and <math>
</math> and <math>
[ math ]和[ math ]
y
y
Y
</math>.<ref name=":1" />
</math>.
数学。
== See also ==
* [[Graph theory]]
* [[Complex network]]
* [[Scale-free network]]
* [[Random graph]]
* [[Structural cut-off]]
== References ==
{{Reflist}}
* {{cite journal
| last = Albert
| last = Albert
| last = Albert
| first = R.
| first = R.
| 第一 = r。
|author2=Barabasi, A.-L.
|author2=Barabasi, A.-L.
2 = Barabasi，A.-L.
| title = Statistical mechanics of complex networks
| title = Statistical mechanics of complex networks
| title = 复杂网络的统计力学
| journal = Reviews of Modern Physics
| journal = Reviews of Modern Physics
| 杂志 = 现代物理学评论
| volume = 74
| volume = 74
74
| pages = 47–97
| pages = 47–97
| 页数 = 47-97
| year = 2002
| year = 2002
2002年
| issue = 1
| issue = 1
1
| arxiv = cond-mat/0106096
| arxiv = cond-mat/0106096
| arxiv = cond-mat/0106096
| doi = 10.1103/RevModPhys.74.47
| doi = 10.1103/RevModPhys.74.47
10.1103/RevModPhys. 74.47
| bibcode=2002RvMP...74...47A
| bibcode=2002RvMP...74...47A
2002RvMP... 74... 47A
}}
}}
}}
* {{cite journal
| last = Dorogovtsev
| last = Dorogovtsev
| last = Dorogovtsev
| first = S.
| first = S.
| 第一 = s。
|author2=Mendes, J. F. F.
|author2=Mendes, J. F. F.
2 = Mendes，J.f.
| title = Evolution of networks
| title = Evolution of networks
| title = 网络的进化
| journal = Advances in Physics
| journal = Advances in Physics
| 杂志 = 物理进展
| volume = 51
| volume = 51
51
| pages = 1079–1187
| pages = 1079–1187
| pages = 1079-1187
| year = 2002
| year = 2002
2002年
| arxiv = cond-mat/0106144
| arxiv = cond-mat/0106144
| arxiv = cond-mat/0106144
| doi = 10.1080/00018730110112519
| doi = 10.1080/00018730110112519
10.1080/00018730110112519
| issue = 4
| issue = 4
第四期
|bibcode = 2002AdPhy..51.1079D }}
|bibcode = 2002AdPhy..51.1079D }}
2002/adphy. . 51.1079 d }
* {{cite journal
|last=Newman
|last=Newman
纽曼
|first=M. E. J.
|first=M. E. J.
| first = m.E.j.
|title=The structure and function of complex networks
|title=The structure and function of complex networks
| title = 复杂网络的结构和功能
|journal=SIAM Review
|journal=SIAM Review
| journal = SIAM Review
|volume=45
|volume=45
45
|pages=167–256
|pages=167–256
| 页 = 167-256
|year=2003
|year=2003
2003年
|doi=10.1137/S003614450342480
|doi=10.1137/S003614450342480
| doi = 10.1137/S003614450342480
|issue=2
|issue=2
2
|arxiv=cond-mat/0303516
|arxiv=cond-mat/0303516
|arxiv=cond-mat/0303516
|bibcode=2003SIAMR..45..167N
|bibcode=2003SIAMR..45..167N
| bibcode = 2003siaml. . 45. . 167 n
}}
}}
}}
* {{cite book
| title= Complex Networks: Structure, Robustness and Function
| title= Complex Networks: Structure, Robustness and Function
复杂网络: 结构、健壮性和功能
| author= Shlomo Havlin
| author= Shlomo Havlin
作者: Shlomo Havlin
| author-link= Shlomo Havlin
| author-link= Shlomo Havlin
| 作者链接 = Shlomo Havlin
| author2= Reuven Cohen| last-author-amp= yes
| author2= Reuven Cohen| last-author-amp= yes
2 = Reuven Cohen | last-author-amp = yes
| year= 2010
| year= 2010
2010年
| url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php
| url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php
| publisher= Cambridge University Press
| publisher= Cambridge University Press
剑桥大学出版社
}}
}}
}}
[[Category:Graph theory]]
Category:Graph theory
范畴: 图论
[[Category:Graph invariants]]
Category:Graph invariants
类别: 图形不变量
[[Category:Network theory]]
Category:Network theory
范畴: 网络理论
<noinclude>
<small>This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]</small></noinclude>
[[Category:待整理页面]]
{{Network Science}}
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.
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.
在图形和网络的研究中，网络中一个节点的度就是它与其他节点的连接数，度的分布就是整个网络中这些度的概率分布。
==Definition==
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.
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.
网络中一个节点的程度(有时错误地称为连通性)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，那么节点有两个不同的度，一个度是传入边的数量，另一个度是传出边的数量。
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''<sub>''k''</sub> of them have degree ''k'', we have ''P''(''k'') = ''n''<sub>''k''</sub>/''n''.
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<sub>k</sub> of them have degree k, we have P(k) = n<sub>k</sub>/n.
将网络的度分布 p (k)定义为 k 度网络中所有节点的分数，如果一个网络中有 n 个节点，且其中 n 个节点的 k 度为 n，则 p (k) = n < sub > k </sub >/n。
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''.
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.
同样的信息有时也以累积度分布、度小于 k 的节点比例、甚至是互补累积度分布的形式出现，如果把 c 看作累积度分布，则为度大于或等于 k (1-c)的节点比例;。C 的补语。
== Observed degree distributions ==
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'':
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:
学位分布在研究实际网络(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(erd s-Rényi 模型)随机图，其中每个 n 个节点都以概率 p (或1-p)独立连接(或不独立连接) ，其二项分布为 k:
:<math>
<math>
《数学》
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},
P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},
选择 k } p ^ k (1-p) ^ { n-1-k } ,
</math>
</math>
数学
(or [[Poisson distribution|Poisson]] in the limit of large ''n'', if the average degree <math>\langle k\rangle=p(n-1)</math> 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 "hubs", 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]]: <math>
(or Poisson in the limit of large n, if the average degree <math>\langle k\rangle=p(n-1)</math> 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 "hubs", 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: <math>
(如果平均度 < math > langle k rangle = p (n-1) </math > 保持不变)。然而，现实世界中的大多数网络的度分布与此大相径庭。大多数节点是高度右倾斜的，这意味着大多数节点的度数较低，但少数节点，即所谓的“枢纽” ，度数较高。一些网络，特别是互联网、万维网和一些社交网络，被认为具有度分布，近似遵循一个幂定律: < math >
P(k)\sim k^{-\gamma}
P(k)\sim k^{-\gamma}
P (k) sim k ^ {-gamma }
</math>, where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties<ref name="BA">{{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 }}</ref><ref name="AB">{{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 }}</ref><ref name="Doro">{{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 }}</ref><ref name="PSY">{{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}}</ref>. 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]].<ref>{{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}}</ref>
</math>, 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.
</math>, where γ is a constant.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，最近有一些基于现实数据集的研究声称，尽管大多数观测到的网络具有厚尾度分布，但它们偏离了无标度分布。
== Excess degree distribution ==
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.<ref name=":0">{{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}}</ref> In other words, it is the distribution of outgoing links from a node reached by following a link.
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.
过度度分布是一个节点通过跟随一条边到达的其他边数的概率分布。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。
Suppose a network has a degree distribution <math>
Suppose a network has a degree distribution <math>
假设一个网络具有度分布
P(k)
P(k)
P (k)
</math>, 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 <math>
</math>, 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 <math>
</math > ，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 < math > 的概率
k
k
K
</math> neighbors is not given by <math>
</math> neighbors is not given by <math>
邻居不是由数学给出的
P(k)
P(k)
P (k)
</math>. 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 <math>
</math>. 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 <math>
数学。原因在于，无论何时在异质网路节点中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达滚刀。这些节点具有度 < math > 的真实概率
k
k
K
</math> is <math>
</math> is <math>
是“ math”
q(k)
q(k)
Q (k)
</math> 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<ref name=":0" />:
</math> 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:
</math > ，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，超额度分布为:
<math>
<math>
《数学》
q(k) = \frac{k+1}{\langle k \rangle}P(k+1),
q(k) = \frac{k+1}{\langle k \rangle}P(k+1),
Q (k) = frac { k + 1}{ langle k rangle } p (k + 1) ,
</math>
</math>
数学
where <math>
where <math>
在哪里
{\langle k \rangle}
{\langle k \rangle}
我不知道该怎么办
</math> 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:
</math> 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:
</math > 是模型的平均度(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组件:
<math>
<math>
《数学》
\sum_k kq(k) > 1 \Rightarrow {\langle k^2 \rangle}-2{\langle k \rangle}>0
\sum_k kq(k) > 1 \Rightarrow {\langle k^2 \rangle}-2{\langle k \rangle}>0
1 right tarrow { langle k ^ 2 rangle }-2{ langle k rangle } > 0
</math>
</math>
数学
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.<ref name=":0" />
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.
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。
== The Generating Functions Method ==
[[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, <math>
Generating functions can be used to calculate different properties of random networks. Given the degree distribution and the excess degree distribution of some network, <math>
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，< math >
P(k)
P(k)
P (k)
</math> and <math>
</math> and <math>
[ math ]和[ math ]
q(k)
q(k)
Q (k)
</math> respectively, it is possible to write two power series in the following forms:
</math> respectively, it is possible to write two power series in the following forms:
分别以下列形式编写两个幂级数是可能的:
<math>
<math>
《数学》
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k
G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k
0(x) = textstyle sum { k }显示样式 p (k) x ^ k
</math> and <math>
</math> and <math>
[ math ]和[ math ]
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}
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}
1(x) = textstyle sum { k } displaystyle q (k) x ^ k = textstyle sum { k } displastyle frac { k }{ langle k rangle } p (k) x ^ { k-1}
</math>
</math>
数学
<math>
<math>
《数学》
G_1(x)
G_1(x)
G _ 1(x)
</math> can also be obtained from derivatives of <math>
</math> can also be obtained from derivatives of <math>
</math > 也可以从 < math > 的导数得到
G_0(x)
G_0(x)
G _ 0(x)
</math>:
</math>:
(/math)
<math>
<math>
《数学》
G_1(x) = \frac{G'_0(x)}{G'_0(1)}
G_1(x) = \frac{G'_0(x)}{G'_0(1)}
1(x) = frac { g’ _ 0(x)}{ g’ _ 0(1)}
</math>
</math>
数学
If we know the generating function for a probability distribution <math>
If we know the generating function for a probability distribution <math>
如果我们知道美国母函数的一个概率分布
P(k)
P(k)
P (k)
</math> then we can recover the values of <math>
</math> then we can recover the values of <math>
然后我们就可以恢复
P(k)
P(k)
P (k)
</math> by differentiating:
</math> by differentiating:
通过区分:
<math>
<math>
《数学》
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}
P(k) = \frac{1}{k!} {\operatorname{d}^k\!G\over\operatorname{d}\!x^k}\biggl \vert _{x=0}
= frac {1}{ k! }{ operatorname { d } ^ k! g over operatorname { d } ! x ^ k } bigvert { x = 0}
</math>
</math>
数学
Some properties, e.g. the moments, can be easily calculated from <math>
Some properties, e.g. the moments, can be easily calculated from <math>
有些物业，例如。时刻，可以很容易地计算从 < math >
G_0(x)
G_0(x)
G _ 0(x)
</math> and its derivatives:
</math> and its derivatives:
和它的衍生物:
*<math>
{\langle k \rangle} = G'_0(1)
{\langle k \rangle} = G'_0(1)
{ langle k rangle } = g’ _ 0(1)
</math>
</math>
数学
*<math>
{\langle k^2 \rangle} = G''_0(1) + G'_0(1)
{\langle k^2 \rangle} = G_0(1) + G'_0(1)
{ langle k ^ 2 rangle } = g _ 0(1) + g’ _ 0(1)
</math>
</math>
数学
And in general<ref name=":0" />:
And in general:
一般来说:
* <math>
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}
{\langle k^m \rangle} = \Biggl[{\bigg(\operatorname{x}{\operatorname{d}\!\over\operatorname{dx}\!}\biggl)^m}G_0(x)\Biggl]_{x=1}
{ langle k ^ m rangle } = Biggl [{ bigg (operatorname { x }{ operatorname { d } ! over operatorname { dx } ！[} Biggl) ^ m } g _ 0(x) Biggl ] _ { x = 1}
</math>
</math>
数学
For [[Poisson distribution|Poisson]]-distributed random networks, such as the [[Erdős–Rényi model|ER graph]], <math>
For Poisson-distributed random networks, such as the ER graph, <math>
对于泊松分布的随机网络，如 ER 图，< math >
G_1(x) = G_0(x)
G_1(x) = G_0(x)
G1(x) = g0(x)
</math>, 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 <math>
</math>, 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 <math>
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 < math > 生成的
G_0(x)
G_0(x)
G _ 0(x)
</math> and <math>
</math> and <math>
[ math ]和[ math ]
G_0(G_1(x))
G_0(G_1(x))
G _ 0(g _ 1(x))
</math>. By extension, the distribution of <math>
</math>. By extension, the distribution of <math>
数学。通过扩展，< math > 的分布
m
m
M
</math>-th neighbors is generated by:
</math>-th neighbors is generated by:
这个邻居是由以下几个因素产生的:
<math>
<math>
《数学》
G_0\bigl(G_1(...G_1(x)...)\bigr)
G_0\bigl(G_1(...G_1(x)...)\bigr)
G _ 0 bigl (g _ 1(... g _ 1(x) ...) bigr)
</math>, with <math>
</math>, with <math>
[ math ] ，with < math >
m-1
m-1
M-1
</math> iterations of the function <math>
</math> iterations of the function <math>
函数的迭代
G_1
G_1
1
</math> acting on itself.<ref name=":1">{{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}}</ref>
</math> acting on itself.
自我作用。
The average number of 1st neighbors, <math>
The average number of 1st neighbors, <math>
第一个邻居的平均数量
c_1
c_1
C _ 1
</math>, is <math>
</math>, is <math>
[ math ] ，is < math >
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}
{\langle k \rangle} = {dG_0(x)\over dx}|_{x=1}
{ langle k rangle } = { dG _ 0(x) over dx } | { x = 1}
</math> and the average number of 2nd neighbors is: <math>
</math> and the average number of 2nd neighbors is: <math>
第二个邻居的平均数量是:
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)
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)
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)
</math>
</math>
数学
== Degree distribution for directed networks ==
[[File:Enwiki-degree-distribution.png|thumb|right|320px|In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales)]]
In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales)
维基百科超链接图(对数尺度)的 In/out 度分布
In a directed network, each node has some in-degree <math>
In a directed network, each node has some in-degree <math>
在有向网络中，每个节点都有一定程度的“数学”
k_{
k_{
这是一个很好的例子
in}
in}
开始吧
</math> and some out-degree <math>
</math> and some out-degree <math>
还有一些超过学位的
k_{out}
k_{out}
我不知道你在说什么
</math> which are the number of links which have run into and out of that node respectfully. If <math>
</math> which are the number of links which have run into and out of that node respectfully. If <math>
</math > 是指以尊敬的方式进出该节点的链接数量。如果 < math >
P(k_{in}, k_{out})
P(k_{in}, k_{out})
P (k _ { in } ，k _ { out })
</math> is the probability that a randomly chosen node has in-degree <math>
</math> is the probability that a randomly chosen node has in-degree <math>
是一个随机选择的节点在程度上
k_{
k_{
这是一个很好的例子
in}
in}
开始吧
</math> and out-degree <math>
</math> and out-degree <math>
“数学”和“学位”
k_{out}
k_{out}
我不知道你在说什么
</math> then the generating function assigned to this [[joint probability distribution]] can be written with two valuables <math>
</math> then the generating function assigned to this joint probability distribution can be written with two valuables <math>
那么分配给这个母函数的联合分布可以用两个贵重物品来写
x
x
X
</math> and <math>
</math> and <math>
[ math ]和[ math ]
y
y
Y
</math> as:
</math> as:
作为:
<math>
<math>
《数学》
\mathcal{G}(x,y) = \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .
\mathcal{G}(x,y) = \sum_{k_{in},k_{out}} \displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}} .
数学{ g }(x，y) = sum _ { k _ { in } ，k _ { out }显示样式 p ({ k _ { in } ，k _ { out }) x ^ { k _ { in } y ^ { k _ { out }。
</math>
</math>
数学
Since every link in a directed network must leave some node and enter another, the net average number of links entering
Since every link in a directed network must leave some node and enter another, the net average number of links entering
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，因此进入的链路的净平均数
a node is zero. Therefore,
a node is zero. Therefore,
一个节点是零。所以,
<math>
<math>
《数学》
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0
\langle{k_{in}-k_{out}}\rangle =\sum_{k_{in},k_{out}} \displaystyle (k_{in}-k_{out})P({k_{in},k_{out}}) = 0
Langle { k _ { in }-k _ { out } rangle = sum _ { k _ { in } ，k _ { out } displays (k _ { in }-k _ { out }) p ({ k _ { in } ，k _ { out }}) = 0
</math>,
</math>,
数学,
which implies that, the generation function must satisfy:
which implies that, the generation function must satisfy:
这意味着，生成函数必须满足:
<math>
<math>
《数学》
{\partial \mathcal{G}\over\partial x}\vert _{x,y=1} = {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,
{\partial \mathcal{G}\over\partial x}\vert _{x,y=1} = {\partial \mathcal{G}\over\partial y}\vert _{x,y=1} = c,
{ partial mathcal { g } over partial x } vert _ { x，y = 1} = { partial mathcal { g } over partial y } vert _ { x，y = 1} = c,
</math>
</math>
数学
where <math>
where <math>
在哪里
c
c
C
</math> is the mean degree (both in and out) of the nodes in the network; <math>
</math> is the mean degree (both in and out) of the nodes in the network; <math>
是网络中节点的平均度(内部和外部)
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c.
\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c.
长角 = 长角 = 长角。
</math>
</math>
数学
Using the function <math>
Using the function <math>
使用函数 < math >
\mathcal{G}(x,y)
\mathcal{G}(x,y)
数学{ g }(x，y)
</math>, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. <math>
</math>, we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before. <math>
</math > ，我们可以再次找到进/出度分布和进/出超量度分布的生成函数，如前所述。《数学》
G^{in}_0(x)
G^{in}_0(x)
0(x)
</math> can be defined as generating functions for the number of arriving links at a randomly chosen node, and <math>
</math> can be defined as generating functions for the number of arriving links at a randomly chosen node, and <math>
可以将 </math > 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 < math >
G^{in}_1(x)
G^{in}_1(x)
1(x)
</math>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 <math>
</math>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 <math>
</math > 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 < math >
G^{out}_0(y)
G^{out}_0(y)
0(y)
</math> and <math>
</math> and <math>
[ math ]和[ math ]
G^{out}_1(y)
G^{out}_1(y)
1(y)
</math> for the number leaving such a node:<ref name=":1" />
</math> for the number leaving such a node:
离开这样一个节点的数字:
* <math>
G^{in}_0(x) = \mathcal{G}(x,1)
G^{in}_0(x) = \mathcal{G}(x,1)
0(x) = mathcal { g }(x，1)
</math>
</math>
数学
* <math>
G^{in}_1(x) = \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1}
G^{in}_1(x) = \frac{1}{c} {\partial \mathcal{G}\over\partial x}\vert _{y=1}
1(x) = frac {1}{ c }{ partial mathcal { g } over partial x } vert _ { y = 1}
</math>
</math>
数学
* <math>
G^{out}_0(y) = \mathcal{G}(1,y)
G^{out}_0(y) = \mathcal{G}(1,y)
0(y) = mathcal { g }(1，y)
</math>
</math>
数学
* <math>
G^{out}_1(y) = \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1}
G^{out}_1(y) = \frac{1}{c} {\partial \mathcal{G}\over\partial y}\vert _{x=1}
1(y) = frac {1}{ c }{ partial mathcal { g } over partial y } vert _ { x = 1}
</math>
</math>
数学
Here, the average number of 1st neighbors, <math>
Here, the average number of 1st neighbors, <math>
这里，第一个邻居的平均数量，< math >
c
c
C
</math>, or as previously introduced as <math>
</math>, or as previously introduced as <math>
或者像之前介绍的那样
c_1
c_1
C _ 1
</math>, is <math>
</math>, is <math>
[ math ] ，is < math >
{\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} = {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}
{\partial \mathcal{G}\over\partial x}\biggl \vert _{x,y=1} = {\partial \mathcal{G}\over\partial y}\biggl \vert _{x,y=1}
{ partial mathcal { g } over partial x } biggl vert _ { x，y = 1} = { partial mathcal { g } over partial y } biggl vert _ { x，y = 1}
</math> and the average number of 2nd neighbors reachable from a randomly chosen node is given by: <math>
</math> and the average number of 2nd neighbors reachable from a randomly chosen node is given by: <math>
从一个随机选择的节点上可达到的第二邻居的平均数是:
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}
c_2 = G_1'(1)G'_0(1) ={\partial^2 \mathcal{G}\over\partial x\partial y}\biggl \vert _{x,y=1}
C _ 2 = g _ 1’(1) g’ _ 0(1) = {{部分 ^ 2数学{ g }/部分 x 部分 y } biggl vert _ { x，y = 1}
</math>. 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 <math>
</math>. 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 <math>
数学。这些也是第一和第二邻居的数量，从这些随机节点可以达到，因为这些方程显然是对称的 < math >
x
x
X
</math> and <math>
</math> and <math>
[ math ]和[ math ]
y
y
Y
</math>.<ref name=":1" />
</math>.
数学。
== See also ==
* [[Graph theory]]
* [[Complex network]]
* [[Scale-free network]]
* [[Random graph]]
* [[Structural cut-off]]
== References ==
{{Reflist}}
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| last = Albert
| last = Albert
| first = R.
| first = R.
| 第一 = r。
|author2=Barabasi, A.-L.
|author2=Barabasi, A.-L.
2 = Barabasi，A.-L.
| title = Statistical mechanics of complex networks
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| journal = Reviews of Modern Physics
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| 杂志 = 现代物理学评论
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| volume = 74
74
| pages = 47–97
| pages = 47–97
| 页数 = 47-97
| year = 2002
| year = 2002
2002年
| issue = 1
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1
| arxiv = cond-mat/0106096
| arxiv = cond-mat/0106096
| arxiv = cond-mat/0106096
| doi = 10.1103/RevModPhys.74.47
| doi = 10.1103/RevModPhys.74.47
10.1103/RevModPhys. 74.47
| bibcode=2002RvMP...74...47A
| bibcode=2002RvMP...74...47A
2002RvMP... 74... 47A
}}
}}
}}
* {{cite journal
| last = Dorogovtsev
| last = Dorogovtsev
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| first = S.
| first = S.
| 第一 = s。
|author2=Mendes, J. F. F.
|author2=Mendes, J. F. F.
2 = Mendes，J.f.
| title = Evolution of networks
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| title = 网络的进化
| journal = Advances in Physics
| journal = Advances in Physics
| 杂志 = 物理进展
| volume = 51
| volume = 51
51
| pages = 1079–1187
| pages = 1079–1187
| pages = 1079-1187
| year = 2002
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2002年
| arxiv = cond-mat/0106144
| arxiv = cond-mat/0106144
| arxiv = cond-mat/0106144
| doi = 10.1080/00018730110112519
| doi = 10.1080/00018730110112519
10.1080/00018730110112519
| issue = 4
| issue = 4
第四期
|bibcode = 2002AdPhy..51.1079D }}
|bibcode = 2002AdPhy..51.1079D }}
2002/adphy. . 51.1079 d }
* {{cite journal
|last=Newman
|last=Newman
纽曼
|first=M. E. J.
|first=M. E. J.
| first = m.E.j.
|title=The structure and function of complex networks
|title=The structure and function of complex networks
| title = 复杂网络的结构和功能
|journal=SIAM Review
|journal=SIAM Review
| journal = SIAM Review
|volume=45
|volume=45
45
|pages=167–256
|pages=167–256
| 页 = 167-256
|year=2003
|year=2003
2003年
|doi=10.1137/S003614450342480
|doi=10.1137/S003614450342480
| doi = 10.1137/S003614450342480
|issue=2
|issue=2
2
|arxiv=cond-mat/0303516
|arxiv=cond-mat/0303516
|arxiv=cond-mat/0303516
|bibcode=2003SIAMR..45..167N
|bibcode=2003SIAMR..45..167N
| bibcode = 2003siaml. . 45. . 167 n
}}
}}
}}
* {{cite book
| title= Complex Networks: Structure, Robustness and Function
| title= Complex Networks: Structure, Robustness and Function
复杂网络: 结构、健壮性和功能
| author= Shlomo Havlin
| author= Shlomo Havlin
作者: Shlomo Havlin
| author-link= Shlomo Havlin
| author-link= Shlomo Havlin
| 作者链接 = Shlomo Havlin
| author2= Reuven Cohen| last-author-amp= yes
| author2= Reuven Cohen| last-author-amp= yes
2 = Reuven Cohen | last-author-amp = yes
| year= 2010
| year= 2010
2010年
| url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php
| url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php
Http://havlin.biu.ac.il/shlomo%20havlin%20books_com_net.php
| publisher= Cambridge University Press
| publisher= Cambridge University Press
剑桥大学出版社
}}
}}
}}
[[Category:Graph theory]]
Category:Graph theory
范畴: 图论
[[Category:Graph invariants]]
Category:Graph invariants
类别: 图形不变量
[[Category:Network theory]]
Category:Network theory
范畴: 网络理论
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<small>This page was moved from [[wikipedia:en:Degree distribution]]. Its edit history can be viewed at [[度分布/edithistory]]</small></noinclude>
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