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== Network properties ==

== Network properties ==

Often, networks have certain attributes that can be calculated to analyze the properties & characteristics of the network. The behavior of these network properties often define [[network model]]s and can be used to analyze how certain models contrast to each other. Many of the definitions for other terms used in network science can be found in [[Glossary of graph theory]].

Often, networks have certain attributes that can be calculated to analyze the properties & characteristics of the network. The behavior of these network properties often define [[network model]]s and can be used to analyze how certain models contrast to each other. Many of the definitions for other terms used in network science can be found in [[Glossary of graph theory]].

通常，网络具有一些可计算的属性来分析网络的性质和特征。 这些网络属性的特征通常被定义为[[网络模型]] ，可用于对比分析不同模型之间的差异。 网络科学中使用的许多其他术语的定义可以在[[图论术语表]]''Glossary of graph theory''中找到。

通常，网络具有一些可计算的属性来分析网络的性质和特征。 这些网络属性的特征通常被定义为[[网络模型]] ，可用于对比分析不同模型之间的差异。 网络科学中使用的许多其他术语的定义可以在[[图论术语表]]''Glossary of graph theory''中找到。

=== 密度 Density ===

=== 密度 Density ===

网络的密度 <math>D</math> 通常定义为连边数量<math>E</math>和节点数量<math>N</math> 的比值，给定二分系数binomial coefficient（如简单图中） <math>\tbinom N2</math>，假定<math>D =\frac{E-(N-1)}{Emax - (N-1)} = \frac{2(E-N+1)}{N(N-3)+2}</math>，另外一个可能的等式就是<math>D = \frac{T-2N+2}{N(N-3)+2}</math> ，而<math>T</math> 节是没有方向的(Wasserman & Faust 1994)。这为网络密度提供了更好的概述，因为可以度量单向关系。

网络的密度 <math>D</math> 通常定义为网络中已有的连边数量<math>E</math>与<math>N</math> 都连接的所有可能边数的的比值，例如在简单图中，所有可能的连边总数可由二项式系数binomial coefficient计算得到 <math>\tbinom N2</math>，所以网络的密度<math>D =\frac{E-(N-1)}{Emax - (N-1)} = \frac{2(E-N+1)}{N(N-3)+2}</math>，另外一个可能的等式就是<math>D = \frac{T-2N+2}{N(N-3)+2}</math> ，而<math>T</math> 节是没有方向的(Wasserman & Faust 1994)。这为网络密度提供了更好的概述，因为可以度量单向关系。

=== Planar Network Density ===

=== Planar Network Density ===

=== Average degree ===

=== Average degree ===

The degree <math>k</math> of a node is the number of edges connected to it. Closely related to the density of a network is the average degree, <math>\langle k\rangle = \tfrac{2E}{N}</math> (or, in the case of directed graphs, <math>\langle k\rangle = \tfrac{E}{N}</math>, the former factor of 2 arising from each edge in an undirected graph contributing to the degree of two distinct vertices). In the [[Erdős–Rényi model | ER random graph model]] (<math>G(N,p)</math>) we can compute the expected value of  <math>\langle k \rangle </math> (equal to the expected value of <math>k</math> of an arbitrary vertex): a random vertex has <math>N-1</math> other vertices in the network available, and with probability <math>p</math>, connects to each. Thus, <math>\mathbb{E}[\langle k \rangle]=\mathbb{E}[k]= p(N-1)</math>.

The degree <math>k</math> of a node is the number of edges connected to it. Closely related to the density of a network is the average degree, <math>\langle k\rangle = \tfrac{2E}{N}</math> (or, in the case of directed graphs, <math>\langle k\rangle = \tfrac{E}{N}</math>, the former factor of 2 arising from each edge in an undirected graph contributing to the degree of two distinct vertices). In the [[Erdős–Rényi model | ER random graph model]] (<math>G(N,p)</math>) we can compute the expected value of  <math>\langle k \rangle </math> (equal to the expected value of <math>k</math> of an arbitrary vertex): a random vertex has <math>N-1</math> other vertices in the network available, and with probability <math>p</math>, connects to each. Thus, <math>\mathbb{E}[\langle k \rangle]=\mathbb{E}[k]= p(N-1)</math>.

=== Average shortest path length (or characteristic path length) ===

=== Average shortest path length (or characteristic path length) ===

The average shortest path length is calculated by finding the shortest path between all pairs of nodes, and taking the average over all paths of the length thereof (the length being the number of intermediate edges contained in the path, i.e., the distance <math>d_{u,v}</math> between the two vertices <math>u,v</math> within the graph). This shows us, on average, the number of steps it takes to get from one member of the network to another. The behavior of the expected average shortest path length (that is, the ensemble average of the average shortest path length) as a function of the number of vertices <math>N</math> of a random network model defines whether that model exhibits the small-world effect; if it scales as <math>O(\ln N)</math>, the model generates small-world nets. For faster-than-logarithmic growth, the model does not produce small worlds. The special case of <math>O(\ln\ln N)</math> is known as ultra-small world effect.

The average shortest path length is calculated by finding the shortest path between all pairs of nodes, and taking the average over all paths of the length thereof (the length being the number of intermediate edges contained in the path, i.e., the distance <math>d_{u,v}</math> between the two vertices <math>u,v</math> within the graph). This shows us, on average, the number of steps it takes to get from one member of the network to another. The behavior of the expected average shortest path length (that is, the ensemble average of the average shortest path length) as a function of the number of vertices <math>N</math> of a random network model defines whether that model exhibits the small-world effect; if it scales as <math>O(\ln N)</math>, the model generates small-world nets. For faster-than-logarithmic growth, the model does not produce small worlds. The special case of <math>O(\ln\ln N)</math> is known as ultra-small world effect.

=== Diameter of a network ===

=== Diameter of a network ===