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{{Network Science}}

{{Network Science}}

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.

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.

==Definition==

==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 (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.

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。

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。

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_k of them have degree k, we have P(k) = n_k/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_k of them have degree k, we have P(k) = n_k/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。

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_k''个节点的度值为''k''，则 ''P''(''k'') = ''n''_''k''/''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。

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。

== Observed degree distributions ==

== 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'':

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:

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:

</math>

</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>

(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 >  

(如果平均度 < math > langle k rangle = p (n-1) </math > 保持不变，也会出现在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: < math >  

P(k)\sim k^{-\gamma}

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<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. 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。

</math>,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。

== Excess degree distribution ==

== 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.<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.

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.

Suppose a network has a degree distribution <math>

Suppose a network has a degree distribution <math>

Suppose a network has a degree distribution <math>

Suppose a network has a degree distribution  

 

假设一个网络具有度分布<math>P(k)</math >，

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

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 > k</math>个邻居的概率

k

k

</math> neighbors is not given by <math>

</math> neighbors is not given by <math>

P(k)

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>

</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

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< math >k</math>的真实概率

k

k

</math> is <math>

</math> is <math>

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:

</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:

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:

</math>

</math>

 

where <math>

where <math>

 

{\langle k \rangle}

{\langle k \rangle}

{\langle k \rangle}

{\langle k \rangle}

 

</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> 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:

这里<math>{\langle k \rangle}</math > 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:

<math>

<math>

\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  

</math>

</math>

 

== The Generating Functions Method ==

== 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>

[[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>

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 >  

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)</math>和<math>q(k)</math>

 

P(k)

P(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:

可以以下列形式写出两个幂级数:

<math>

<math>

 

G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k  

G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k  

G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k  

G_0(x) = \textstyle \sum_{k} \displaystyle P(k)x^k  

 

</math> and <math>

</math> and <math>

</math>

</math>

 

</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)</math > 也可以从 < math > 的导数得到

 

G_0(x)

G_0(x)

If we know the generating function for a probability distribution <math>

If we know the generating function for a probability distribution <math>

P(k)

P(k)

</math> then we can recover the values of <math>

</math> then we can recover the values of <math>

 

P(k)

P(k)

</math> by differentiating:

</math> by differentiating:

 

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 >  

一些性质，例如，时刻，可以很容易地计算依据 < math > G0(x)</math>和它的导数:

G_0(x)  

G_0(x)  

</math> and its derivatives:

</math> and its derivatives:

 

For Poisson-distributed random networks, such as the 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 >  

对于泊松分布的随机网络，如 ER 图，< math > G1(x) = G0(x)</math>

G_1(x) = G_0(x)  

G_1(x) = G_0(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

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 > G0(x)</math>和<math>G0(G1(x))</math>生成的

G_0(x)  

G_0(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

继续扩展，< math > m-th</math>的邻居的分布

m

m

The average number of 1st neighbors, <math>

The average number of 1st neighbors, <math>

c_1

c_1

== Degree distribution for directed networks ==

== Degree distribution for directed networks ==

In a directed network, each node has some in-degree <math>

In a directed network, each node has some in-degree <math>

In a directed network, each node has some in-degree <math>

In a directed network, each node has some in-degree <math>

k_{

k_{

in}

in}

 

</math> and some out-degree <math>

</math> and some out-degree <math>

</math> and some out-degree <math>

</math> and some out-degree <math>

k_{out}

k_{out}

k_{out}

k_{out}

 

</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> 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>

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>

k_{

k_{

</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>

x

x

</math> as:

</math> as:

 

<math>

<math>

 

\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}} .

\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}} .

</math>  

</math>  

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

a node is zero. Therefore,  

a node is zero. Therefore,  

一个节点是零。所以,

所以,

<math>

<math>

 

\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}} \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  

 

</math>,

</math>,

</math>,

</math>,

 

where <math>

where <math>

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)

C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.  

\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c.  

\langle{k_{in}}\rangle = \langle{k_{out}}\rangle = c.  

 

</math>

</math>

Using the function <math>

Using the function <math>

使用函数 < math >  

使用函数 < math > \mathcal{G}(x,y)</math>

\mathcal{G}(x,y)

\mathcal{G}(x,y)

\mathcal{G}(x,y)

\mathcal{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>, 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》

</math > ，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。

G^{in}_0(x)  

G^{in}_0(x)  

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 >  

这里，第一个邻居的平均数量C，< math >  

c

c

</math>, or as previously introduced as <math>

</math>, or as previously introduced as <math>

c_1

c_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) ={\partial^2 \mathcal{G}\over\partial x\partial 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>

</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 >  

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 < math >  

x

x

* [[Graph theory]]

* [[Graph theory]]

 

* [[Complex network]]

* [[Complex network]]

 

* [[Scale-free network]]

* [[Scale-free network]]

 

* [[Random graph]]

* [[Random graph]]

 

* [[Structural cut-off]]

* [[Structural cut-off]]