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{{Redirect|Yule process|the type of birth process|Simple birth process}}

{{Redirect|Yule process|the type of birth process|Simple birth process}}

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.  "Preferential attachment" is only the most recent of many names that have been given to such processes.  They are also referred to under the names "Yule process", "cumulative advantage", "the rich get richer", and, less correctly, the "Matthew effect".  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.

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.  "Preferential attachment" is only the most recent of many names that have been given to such processes.  They are also referred to under the names "Yule process", "cumulative advantage", "the rich get richer", and, less correctly, the "Matthew effect".  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.

==Definition==

==Definition 定义==

A preferential attachment process is a [[stochastic process|stochastic]] [[urn problem|urn process]], meaning a process in which discrete units of wealth, usually called "balls", are added in a random or partly random fashion to a set of objects or containers, usually called "urns".  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.

A preferential attachment process is a [[stochastic process|stochastic]] [[urn problem|urn process]], meaning a process in which discrete units of wealth, usually called "balls", are added in a random or partly random fashion to a set of objects or containers, usually called "urns".  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.

A preferential attachment process is a stochastic urn process, meaning a process in which discrete units of wealth, usually called "balls", are added in a random or partly random fashion to a set of objects or containers, usually called "urns".  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.

A preferential attachment process is a stochastic urn process, meaning a process in which discrete units of wealth, usually called "balls", are added in a random or partly random fashion to a set of objects or containers, usually called "urns".  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.

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 ("urns") 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 ("balls") 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.

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 ("urns") 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 ("balls") 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.

优先依附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，首先由 Yule 研究，是一个线性优先附着过程，因为属增加新物种的速度是线性的数量，他们已经有。

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。

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 k₀ balls and further balls are added to urns at a rate proportional to the number k that they already have plus a constant a > −k₀.  With these definitions, the fraction P(k) of urns having k balls in the limit of long time is given by

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 k₀ balls and further balls are added to urns at a rate proportional to the number k that they already have plus a constant a > −k₀.  With these definitions, the fraction P(k) of urns having k balls in the limit of long time is given by

线性优先连接过程中，已知的骨灰瓮数量增加，产生一个分布的球在骨灰瓮下面的所谓圣诞节分布。在最一般的形式的过程中，球添加到系统中的整体速度为米每个新的瓮新球。每个新创建的骨灰瓮都以 k < sub > 0 </sub > 球开始，进一步的骨灰瓮球被添加到骨灰瓮中，其速度与骨灰瓮的数量 k 成正比，再加上一个常数 a >-k < sub > 0 </sub > 。利用这些定义，给出了长时间极限下具有 k 个球的骨灰瓮的分数 p (k)

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 k₀ 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''&nbsp;>&nbsp;−''k''₀，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)

P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_0+a,\gamma-1)},

<math> P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_0+a,\gamma-1)},</math>

P (k) = { mathrm { b }(k + a，gamma)在 mathrm { b }(k _ 0 + a，gamma-1)}上,

<math> P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_0+a,\gamma-1)},</math>

 

 

</math>

 

for k&nbsp;≥&nbsp;k₀ (and zero otherwise), where B(x,&nbsp;y) is the Euler beta function:

for k&nbsp;≥&nbsp;k₀ (and zero otherwise), where B(x,&nbsp;y) is the Euler beta function:

对于 k ≥ k < sub > 0 </sub > (否则为0) ，其中 b (x，y)是 Euler beta 函数:

对于k&nbsp;≥&nbsp;k₀(否则为0) ，其中 b (x，y)是 Euler beta 函数:

 

 

{ b }(x，y) = { Gamma (x) Gamma (y) over Gamma (x + y)} ,

<math> \mathrm{B}(x,y)={\Gamma(x)\Gamma(y)\over\Gamma(x+y)},</math>

</math>

<math> \mathrm{B}(x,y)={\Gamma(x)\Gamma(y)\over\Gamma(x+y)},</math>

Gamma = 2 + { k _ 0 + a/m }.

<math>\gamma=2 + {k_0 + a\over m}.</math>

</math>

<math>\gamma=2 + {k_0 + a\over m}.</math>

The beta function behaves asymptotically as B(x,&nbsp;y)&nbsp;~&nbsp;x^−y for large x and fixed y, which implies that for large values of k we have

The beta function behaves asymptotically as B(x,&nbsp;y)&nbsp;~&nbsp;x^−y for large x and fixed y, which implies that for large values of k we have

对于大 x 和固定 y，β 函数表现为 b (x，y) ~ x < sup >-y </sup > ，这意味着对于 k 的大值，我们有一个渐近的 β 函数

对于较大的x和固定的y，β函数表现为B(''x'',&nbsp;''y'')&nbsp;~&nbsp;''x''^−''y''，这意味着对于k的大值，我们有一个渐近的β函数

P(k) \propto k^{-\gamma}.

<math> P(k) \propto k^{-\gamma}.</math>

In other words, the preferential attachment process generates a "long-tailed" 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.

In other words, the preferential attachment process generates a "long-tailed" 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.

换句话说，优先连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依附行为感兴趣的主要原因: 物种分布和许多其他现象被经验地观察，遵循幂律，优先依附过程是解释这种行为的主要候选机制。优先连接被认为是一个可能的候选人，其中包括城市规模的分布，极其富有的个人的财富，以及万维网网页的链接数量。

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。

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 (k₀&nbsp;=&nbsp;1) and gains new species in direct proportion to the number it already has (a&nbsp;=&nbsp;0), and hence P(k)&nbsp;=&nbsp;B(k,&nbsp;γ)/B(k₀,&nbsp;γ&nbsp;−&nbsp;1) with γ=2&nbsp;+&nbsp;1/m.  Similarly the Price model for scientific citations corresponds to the case k₀&nbsp;=&nbsp;0, a&nbsp; =&nbsp;1 and the widely studied Barabási-Albert model corresponds to k₀&nbsp;=&nbsp;m, a&nbsp;=&nbsp;0.

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 (k₀&nbsp;=&nbsp;1) and gains new species in direct proportion to the number it already has (a&nbsp;=&nbsp;0), and hence P(k)&nbsp;=&nbsp;B(k,&nbsp;γ)/B(k₀,&nbsp;γ&nbsp;−&nbsp;1) with γ=2&nbsp;+&nbsp;1/m.  Similarly the Price model for scientific citations corresponds to the case k₀&nbsp;=&nbsp;0, a&nbsp; =&nbsp;1 and the widely studied Barabási-Albert model corresponds to k₀&nbsp;=&nbsp;m, a&nbsp;=&nbsp;0.

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 < sub > 0 </sub > = 1)开始，并且获得新的种(a = 0) ，因此 p (k) = b (k,)/b (k < sub > 0 </sub > ,-1) = 2 + 1/m。类似地，科学引用的价格模型对应于 k < sub > 0 </sub > = 0，a = 1，广泛研究的 Barabási-Albert 模型对应于 k < sub > 0 </sub > = m，a = 0。

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(k₀&nbsp;=&nbsp;1)开始，并且获得新的种(a&nbsp;=&nbsp;0) ，因此P(k)&nbsp;=&nbsp;B(k,&nbsp;γ)/B(k₀,&nbsp;γ&nbsp;−&nbsp;1)，γ=2&nbsp;+&nbsp;1/m. 。类似地，科学引文的价格模型对应于k₀&nbsp;=&nbsp;0，a&nbsp; =&nbsp;1 ，广泛研究的 Barabási-Albert 模型对应于 k₀&nbsp;=&nbsp;m, a&nbsp;=&nbsp;0.。

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: "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." (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.

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: "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." (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.

优先依恋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)优先连接过程不包括带走部分。然而，这一点可能没有实际意义，因为马太效应背后的科学洞察力在任何情况下都是完全不同的。定性地说，它的目的不是描述一种机械的乘法效应，如优先依恋，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的科学发现，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依恋(虽然肯定与优先依恋有关)是截然不同的。

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。

==History==

==History 历史==

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.<ref name=YulePhilTrans />  The process is sometimes called a "Yule process" 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.

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.<ref name=YulePhilTrans />  The process is sometimes called a "Yule process" 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.

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 "Yule process" 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.

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 "Yule process" 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.

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.

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.

大多数现代优先依恋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。

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。

The first application of preferential attachment to learned citations was given by Price in 1976.  (He referred to the process as a "cumulative advantage" 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.

The first application of preferential attachment to learned citations was given by Price in 1976.  (He referred to the process as a "cumulative advantage" 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.

普赖斯于1976年首次将优先附件应用于学术引文。(他把这个过程称为“累积优势”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尺度网络网络。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。普莱斯还将优先依附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。

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。

The application of preferential attachment to the growth of the World Wide Web was proposed by Barabási and Albert in 1999.

The application of preferential attachment to the growth of the World Wide Web was proposed by Barabási and Albert in 1999.

1999年，Barabási 和 Albert 提出了优先附件在万维网发展中的应用。

1999年，Barabási 和 Albert 提出了优先链接在万维网发展中的应用。

* [[Assortative mixing]]

* [[Assortative mixing]]

* [[BA model]]

* [[BA model]]

* [[Bose–Einstein condensation: a network theory approach]]

* [[Bose–Einstein condensation: a network theory approach]]

* [[Capital accumulation]]

* [[Capital accumulation]]

* [[Chinese restaurant process]]

* [[Chinese restaurant process]]

* [[Complex network]]

* [[Complex network]]

* [[Double jeopardy (marketing)]]

* [[Double jeopardy (marketing)]]

* [[Link-centric preferential attachment]]

* [[Link-centric preferential attachment]]

* [[Matthew effect (sociology)]]

* [[Matthew effect (sociology)]]

* [[Pitman–Yor process]]

* [[Pitman–Yor process]]

* [[Power law]]

* [[Power law]]

* [[Price's model]]

* [[Price's model]]

* [[Proof of stake]]

* [[Proof of stake]]

* [[Simon model]]

* [[Simon model]]

* [[Stochastic processes]]

* [[Stochastic processes]]

* [[Wealth condensation]]

* [[Wealth condensation]]

* [[Yule–Simon distribution]]

* [[Yule–Simon distribution]]

* [[Bibliogram]]

* [[Bibliogram]]

{{colend}}

{{colend}}