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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==

定义

定义

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 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''。

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。

同样的信息有时也以'''<font color="#ff8000">累积度分布函数 Cumulative Degree Distribution</font>'''（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'''<font color="#ff8000">互补累积度分布函数 Complementary Cumulative Degree Distribution</font>'''（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 ==

观察度分布

观察度分布

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:

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:

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)'''<font color="#ff8000">随机图 Random Graph</font>'''，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'''<font color="#ff8000">二项分布 Binomial Distribution</font>'''的度值为''k'':

 

 

 

:<math>

 

<math>

 

 

P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},

 

</math>

<math>\P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},</math>