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第54行： − 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]]. + − The size of a network can refer to the number of nodes <math>N</math> or, less commonly, the number of edges <math>E</math> which (for connected graphs with no multi-edges) can range from <math>N-1</math> (a tree) to <math>E_{\max}</math> (a complete graph). In the case of a simple graph (a network in which at most one (undirected) edge exists between each pair of vertices, and in which no vertices connect to themselves), we have <math>E_{\max}=\tbinom N2=N(N-1)/2</math>; for directed graphs (with no self-connected nodes), <math>E_{\max}=N(N-1)</math>; for directed graphs with self-connections allowed, <math>E_{\max}=N^2</math>. In the circumstance of a graph within which multiple edges may exist between a pair of vertices, <math>E_{\max}=\infty</math>. + − The density <math>D</math> of a network is defined as a ratio of the number of edges <math>E</math> to the number of possible edges in a network with <math>N</math> nodes, given (in the case of simple graphs) by the [[binomial coefficient]] <math>\tbinom N2</math>, giving <math>D =\frac{E-(N-1)}{Emax - (N-1)} = \frac{2(E-N+1)}{N(N-3)+2}</math>+ − Another possible equation is <math>D = \frac{T-2N+2}{N(N-3)+2},</math> whereas the ties <math>T</math> are unidirectional (Wasserman & Faust 1994).<ref>http://psycnet.apa.org/journals/prs/9/4/172/</ref> This gives a better overview over the network density, because unidirectional relationships can be measured. − 网络的密度 <math>D</math> 通常定义为在具有 <math>N</math> 个节点的网络中，连边数量 <math>E</math> 与可能的连边数的比率，例如在简单图中，由二项式系数 <math>\tbinom N2</math> 计算得出，则 <math>D =\frac{E-(N-1)}{Emax - (N-1)} = \frac{2(E-N+1)}{N(N-3)+2}</math>。另一个可能的等式为<math>D = \frac{T-2N+2}{N(N-3)+2}</math> ，而联系<math>T</math> 是单向的（Wasserman & Faust 1994）。这为网络密度提供了更好的概述，因为单向关系是可测量的。 − The density <math>D</math> of a network, where there is no intersection between edges, is defined as a ratio of the number of edges <math>E</math> to the number of possible edges in a network with <math>N</math> nodes, given by a graph with no intersecting edges <math>(E_{\max}=3N-6)</math>, giving <math>D = \frac{E-N+1}{2N-5}.</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>.+ − 一个节点所连接的边数被称为节点的度 <math>k</math> 。与网络密度密切相关的是平均度， <math>\langle k\rangle = \tfrac{2E}{N}</math> （或者在一些有向图中， <math>\langle k\rangle = \tfrac{E}{N}</math>，前一个式子中多了一个2倍关系，是因为无向图中的每一条边形成了两个不同顶点的度）。在[[ER随机图模型]](<math>G(N,p)</math>) 中，我们可以计算<math>\langle k \rangle </math> 的期望值（等于随机点的期望值 <math>k</math>）：一个随机点在网络中有 <math>N-1</math> 个其他可用顶点两两相连，相连的概率为<math>p</math>。因此，<math>\mathbb{E}[\langle k \rangle]=\mathbb{E}[k]= p(N-1)</math>。 − 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. + − As another means of measuring network graphs, we can define the diameter of a network as the longest of all the calculated shortest paths in a network. It is the shortest distance between the two most distant nodes in the network. In other words, once the shortest path length from every node to all other nodes is calculated, the diameter is the longest of all the calculated path lengths. The diameter is representative of the linear size of a network. If node A-B-C-D are connected, going from A->D this would be the diameter of 3 (3-hops, 3-links). − The clustering coefficient is a measure of an "all-my-friends-know-each-other" property. This is sometimes described as the friends of my friends are my friends. More precisely, the clustering coefficient of a node is the ratio of existing links connecting a node's neighbors to each other to the maximum possible number of such links. The clustering coefficient for the entire network is the average of the clustering coefficients of all the nodes. A high clustering coefficient for a network is another indication of a [[Small-world experiment|small world]]. − − − The clustering coefficient of the <math>i</math>'th node is − :<math>C_i = {2e_i\over k_i{(k_i - 1)}}\,,</math> − where <math>k_i</math> is the number of neighbours of the <math>i</math>'th node, and <math>e_i</math> is the number of connections between these neighbours. The maximum possible number of connections between neighbors is, then, − :<math>{\binom {k}{2}} = {{k(k-1)}\over 2}\,.</math> 第107行：
第96行： − − From a probabalistic standpoint, the expected local clustering coefficient is the likelihood of a link existing between two arbitrary neighbors of the same node. − The way in which a network is connected plays a large part into how networks are analyzed and interpreted. Networks are classified in four different categories: + − + − − * ''全连接图'': 完全连接的网络，其中所有的节点之间都相互连接。这样的网络是对称的，每个节点都有其他所有节点的入边和出边。 − − − * ''Giant Component'': A single connected component which contains most of the nodes in the network. − + + − * ''Weakly Connected Component'': A collection of nodes in which there exists a path from any node to any other, ignoring directionality of the edges. − − * ''弱连通子图'': 不考虑边的方向的情况下，任意两个节点之间都存在路径的子图。 − − − * ''Strongly Connected Component'': A collection of nodes in which there exists a ''directed'' path from any node to any other. − − * ''强连通子图'': 任意两个节点间都存在有向路径的子图。 − Centrality indices produce rankings which seek to identify the most important nodes in a network model. Different centrality indices encode different contexts for the word "importance." The [[betweenness centrality]], for example, considers a node highly important if it form bridges between many other nodes. The [[Centrality#Eigenvector centrality|eigenvalue centrality]], in contrast, considers a node highly important if many other highly important nodes link to it. Hundreds of such measures have been proposed in the literature. − − Centrality indices are only accurate for identifying the most central nodes. The measures are seldom, if ever, meaningful for the remainder of network nodes. − <ref name="Lawyer2015"> − {{cite journal |last1= Lawyer |first1= Glenn |title= Understanding the spreading power of all nodes in a network| journal=Scientific Reports − |volume= 5 |pages= 8665 |date=March 2015 − |number=O8665 − | doi=10.1038/srep08665 − |pmid= 25727453 |pmc= 4345333 |bibcode= 2015NatSR...5E8665L |arxiv=1405.6707}} − </ref> − <ref name="Sikic2013"> − {{cite journal − |last1 = Sikic − |first1=Mile − |last2=Lancic − |first2=Alen − |last3= Antulov-Fantulin − |first3=Nino − |last4=Stefancic − |first4=Hrvoje − | title = Epidemic centrality -- is there an underestimated epidemic impact of network peripheral nodes? − | journal = European Physical Journal B − |date=October 2013 − | volume = 86 − | pages = 440 − | number = 10 − | doi=10.1140/epjb/e2013-31025-5 − |arxiv=1110.2558 − |bibcode=2013EPJB...86..440S − }} − </ref> − Also, their indications are only accurate within their assumed context for importance, and tend to "get it wrong" for other contexts.<ref name="Borgatti2005"> − {{cite journal |last1= Borgatti |first1= Stephen P.|year= 2005 |title= Centrality and Network Flow |journal=Social Networks |volume= 27|issue= |pages= 55–71|doi=10.1016/j.socnet.2004.11.008 |url= |citeseerx= 10.1.1.387.419}} − </ref> For example, imagine two separate communities whose only link is an edge between the most junior member of each community. Since any transfer from one community to the other must go over this link, the two junior members will have high betweenness centrality. But, since they are junior, (presumably) they have few connections to the "important" nodes in their community, meaning their eigenvalue centrality would be quite low. 第210行：
第151行： − The concept of centrality in the context of static networks was extended, based on empirical and theoretical research, to dynamic centrality<ref name="dynamic1">{{cite journal | last1 = Braha | first1 = D. | last2 = Bar-Yam | first2 = Y. | year = 2006 | title = From Centrality to Temporary Fame: Dynamic Centrality in Complex Networks | url = | journal = Complexity | volume = 12 | issue = 2| pages = 59–63 | doi=10.1002/cplx.20156| arxiv = physics/0611295 | bibcode = 2006Cmplx..12b..59B }}</ref> in the context of time-dependent and temporal networks.<ref name="dynamic2">{{cite journal | last1 = Hill | first1 = S.A. | last2 = Braha | first2 = D. | year = 2010 | title = Dynamic Model of Time-Dependent Complex Networks | url = | journal = Physical Review E | volume = 82 | issue = 4| page = 046105 | doi=10.1103/physreve.82.046105| pmid = 21230343 | arxiv = 0901.4407 | bibcode = 2010PhRvE..82d6105H }}</ref><ref name="dynamic3">Gross, T. and Sayama, H. (Eds.). 2009. ''Adaptive Networks: Theory, Models and Applications.'' Springer.</ref><ref name="dynamic4">Holme, P. and Saramäki, J. 2013. ''Temporal Networks.'' Springer.</ref>+ − 基于经验和理论的研究，静态网络上定义的中心性的概念可以拓展到时序网络中的动态中心性。<ref name="dynamic1">{{cite journal | last1 = Braha | first1 = D. | last2 = Bar-Yam | first2 = Y. | year = 2006 | title = From Centrality to Temporary Fame: Dynamic Centrality in Complex Networks | url = | journal = Complexity | volume = 12 | issue = 2| pages = 59–63 | doi=10.1002/cplx.20156| arxiv = physics/0611295 | bibcode = 2006Cmplx..12b..59B }}</ref><ref name="dynamic2">{{cite journal | last1 = Hill | first1 = S.A. | last2 = Braha | first2 = D. | year = 2010 | title = Dynamic Model of Time-Dependent Complex Networks | url = | journal = Physical Review E | volume = 82 | issue = 4| page = 046105 | doi=10.1103/physreve.82.046105| pmid = 21230343 | arxiv = 0901.4407 | bibcode = 2010PhRvE..82d6105H }}</ref><ref name="dynamic3">Gross, T. and Sayama, H. (Eds.). 2009. ''Adaptive Networks: Theory, Models and Applications.'' Springer.</ref><ref name="dynamic4">Holme, P. and Saramäki, J. 2013. ''Temporal Networks.'' Springer.</ref> − Limitations to centrality measures have led to the development of more general measures. − Two examples are − the '''accessibility''', which uses the diversity of random walks to measure how accessible the rest of the network is from a given start node,<ref name="Travencolo2008">{{ cite journal| last1=Travençolo |first1=B. A. N. |last2=da F. Costa| first2 =L.| title=Accessibility in complex networks| journal=Physics Letters A| year=2008|volume=373|issue=1|pages=89–95 |doi=10.1016/j.physleta.2008.10.069 |bibcode=2008PhLA..373...89T }}</ref> − and the '''expected force''', derived from the expected value of the [[force of infection]] generated by a node.<ref name="Lawyer2015"/> − Both of these measures can be meaningfully computed from the structure of the network alone.
→网络性质
== 网络性质 ==
== 网络性质 ==
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。这些网络性质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可以在[[图论术语表]]''Glossary of graph theory''中找到相关定义。
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。这些网络性质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可以在[[图论术语表]]''Glossary of graph theory''中找到相关定义。
=== 规模 ===
=== 规模 ===
网络的规模可以指节点的个数<math>N</math>，或者，少数情况下，连边的数量<math>E</math>（对于没有重边的连通图），连边的数量<math>E</math>一般从<math>N-1</math> （看做是一个树）到<math>E_{\max}</math> （看做是一个完全图）不等。在简单图（网络中在每对节点之间至多存在一条(无向)边，并且没有节点连向自己）的例子中，可以计算<math>E_{\max}=\tbinom N2=N(N-1)/2</math>；对于有向图（没有自环节点）而言，<math>E_{\max}=N(N-1)</math>；对于有向图且允许存在自环节点的，<math>E_{\max}=N^2</math>。而对于一对节点之间存在重边的情况，<math>E_{\max}=\infty</math>。
网络的规模可以指节点的个数<math>N</math>，或者，少数情况下，连边的数量<math>E</math>（对于没有重边的连通图），连边的数量<math>E</math>一般从<math>N-1</math> （看做是一个树）到<math>E_{\max}</math> （看做是一个完全图）不等。在简单图（网络中在每对节点之间至多存在一条(无向)边，并且没有节点连向自己）的例子中，可以计算<math>E_{\max}=\tbinom N2=N(N-1)/2</math>；对于有向图（没有自环节点）而言，<math>E_{\max}=N(N-1)</math>；对于有向图且允许存在自环节点的，<math>E_{\max}=N^2</math>。而对于一对节点之间存在重边的情况，<math>E_{\max}=\infty</math>。
=== 密度 ===
=== 密度 ===
网络的密度 <math>D</math> 通常定义为在具有 <math>N</math> 个节点的网络中，连边数量 <math>E</math> 与可能的连边数的比率，例如在简单图中，由二项式系数 <math>\tbinom N2</math> 计算得出，则 <math>D =\frac{E-(N-1)}{Emax - (N-1)} = \frac{2(E-N+1)}{N(N-3)+2}</math>。另一个可能的等式为<math>D = \frac{T-2N+2}{N(N-3)+2}</math> ，而联系<math>T</math> 是单向的（Wasserman & Faust 1994）。这为网络密度提供了更好的概述，因为单向关系是可测量的。
=== 平面网络密度 ===
=== 平面网络密度 ===
在连边之间没有交集的情况下，网络的密度 <math>D</math> 被定义为在具有 <math>N</math> 个节点的网络中，连边数量 <math>E</math> 与可能的连边数的比率，由没有交集连边的图给出 <math>(E_{\max}=3N-6)</math>，则 <math>D = \frac{E-N+1}{2N-5}.</math>
在连边之间没有交集的情况下，网络的密度 <math>D</math> 被定义为在具有 <math>N</math> 个节点的网络中，连边数量 <math>E</math> 与可能的连边数的比率，由没有交集连边的图给出 <math>(E_{\max}=3N-6)</math>，则 <math>D = \frac{E-N+1}{2N-5}.</math>
=== 平均度 ===
=== 平均度 ===
一个节点所连接的边数被称为节点的度 <math>k</math> 。与网络密度密切相关的是平均度， <math>\langle k\rangle = \tfrac{2E}{N}</math> （或者在一些有向图中， <math>\langle k\rangle = \tfrac{E}{N}</math>，前一个式子中多了一个2倍关系，是因为无向图中的每一条边形成了两个不同顶点的度）。在[[ER随机图模型]](<math>G(N,p)</math>) 中，我们可以计算<math>\langle k \rangle </math> 的期望值（等于随机点的期望值 <math>k</math>）：一个随机点在网络中有 <math>N-1</math> 个其他可用顶点两两相连，相连的概率为<math>p</math>。因此，<math>\mathbb{E}[\langle k \rangle]=\mathbb{E}[k]= p(N-1)</math>。
=== 平均最短路径长度（特征路径长度） ===
=== 平均最短路径长度（特征路径长度） ===
平均最短路径长度的计算方法是找到所有节点对之间的最短路径，并取其长度所有路径的平均值（其长度为路径中包含的中间边的数目，即图中两个顶点 <math>u,v</math> 之间的距离<math>d_{u,v}</math>）。这向我们展示了从网络中的一个成员到另一个成员所需的平均步数。期望平均最短路径长度（即平均最短路径长度的总体均值）作为随机网络模型的顶点数 <math>N</math> 的函数的行为定义了该模型是否表现出小世界效应；如果它变为 <math>O(\ln N)</math> ，则该模型生成小世界网络。对于比对数更快的增长，该模型不会产生小世界。<math>O(\ln\ln N)</math>的特例是超小世界效应。
平均最短路径长度的计算方法是找到所有节点对之间的最短路径，并取其长度所有路径的平均值（其长度为路径中包含的中间边的数目，即图中两个顶点 <math>u,v</math> 之间的距离<math>d_{u,v}</math>）。这向我们展示了从网络中的一个成员到另一个成员所需的平均步数。期望平均最短路径长度（即平均最短路径长度的总体均值）作为随机网络模型的顶点数 <math>N</math> 的函数的行为定义了该模型是否表现出小世界效应；如果它变为 <math>O(\ln N)</math> ，则该模型生成小世界网络。对于比对数更快的增长，该模型不会产生小世界。<math>O(\ln\ln N)</math>的特例是超小世界效应。
=== 网络直径 ===
=== 网络直径 ===
网络直径指的是网络中所有计算出来的最短路径中的最大值，它是另一种测量网络图的方法。它是网络中两个最远节点之间的最短距离。换言之，只要计算出来网络中每个节点到其他所有节点的最短路径长度，直径就是所有计算出的路径长度中最长的。网络直径代表网络的线性规模。假如网络的节点以A-B-C-D的方式连接，那么从A->D的路径长度3（3个跳跃，3个连接）就是这个网络的直径。
网络直径指的是网络中所有计算出来的最短路径中的最大值，它是另一种测量网络图的方法。它是网络中两个最远节点之间的最短距离。换言之，只要计算出来网络中每个节点到其他所有节点的最短路径长度，直径就是所有计算出的路径长度中最长的。网络直径代表网络的线性规模。假如网络的节点以A-B-C-D的方式连接，那么从A->D的路径长度3（3个跳跃，3个连接）就是这个网络的直径。
===聚集系数===
===聚集系数===
聚集系数是表示“我所有的朋友都互相认识”这样一种性质的指标。有时候也被表述为“我朋友的朋友也是我的朋友”。更准确地说，节点的聚类系数是该节点的相邻节点之间的现有连接数与其最大可能连接数之比。整个网络的聚集系数是所有节点的聚集系数的平均值。网络的高聚集系数是[[小世界实验|小世界]]的另一个标志。
聚集系数是表示“我所有的朋友都互相认识”这样一种性质的指标。有时候也被表述为“我朋友的朋友也是我的朋友”。更准确地说，节点的聚类系数是该节点的相邻节点之间的现有连接数与其最大可能连接数之比。整个网络的聚集系数是所有节点的聚集系数的平均值。网络的高聚集系数是[[小世界实验|小世界]]的另一个标志。
:<math>{\binom {k}{2}} = {{k(k-1)}\over 2}\,.</math>
:<math>{\binom {k}{2}} = {{k(k-1)}\over 2}\,.</math>
从概率的角度来看，期望的局部聚集系数是同一节点的任意两个相邻节点之间存在连接的可能性。
从概率的角度来看，期望的局部聚集系数是同一节点的任意两个相邻节点之间存在连接的可能性。
=== 连通性 ===
=== 连通性 ===
在分析和理解网络的过程中，网络的连接方式有着很重要的作用。网络可以分为以下四类：
在分析和理解网络的过程中，网络的连接方式有着很重要的作用。网络可以分为以下四类：
* 全连接图: 完全连接的网络，其中所有的节点之间都相互连接。这样的网络是对称的，每个节点都有其他所有节点的入边和出边。
* ''Clique''/''Complete Graph'': a completely connected network, where all nodes are connected to every other node. These networks are symmetric in that all nodes have in-links and out-links from all others.
* 超大连通子图: 包含网络中大多数节点的单个连通子图。
* ''超大连通子图'': 包含网络中大多数节点的单个连通子图。
* 弱连通子图: 不考虑边的方向的情况下，任意两个节点之间都存在路径的子图。
* 强连通子图: 任意两个节点间都存在有向路径的子图。
===节点中心性===
===节点中心性===
中心性指标可以给出所有节点的一个排序，用来寻找网络模型中最重要的节点。在不同的“重要性”的含义下，中心性指标也可以是不同的。例如[[介数中心性]]，在它的定义下，如果一个节点和其他很多节点之间都有连接，那么这个节点就是很重要的。而[[中心性#本征向量中心性|本征值中心性]]则相反，如果某个节点有很多很重要的节点与之相连，那么这个节点就是很重要的。文献中给出了数百种这样的中心性的定义。
中心性指标可以给出所有节点的一个排序，用来寻找网络模型中最重要的节点。在不同的“重要性”的含义下，中心性指标也可以是不同的。例如[[介数中心性]]，在它的定义下，如果一个节点和其他很多节点之间都有连接，那么这个节点就是很重要的。而[[中心性#本征向量中心性|本征值中心性]]则相反，如果某个节点有很多很重要的节点与之相连，那么这个节点就是很重要的。文献中给出了数百种这样的中心性的定义。
中心性指数只能准确地识别最中心的节点，这些测量对其余的网络节点几乎没有意义。
中心性指数只能准确地识别最中心的节点，这些测量对其余的网络节点几乎没有意义。
基于经验和理论的研究，静态网络上定义的中心性的概念可以拓展到时序网络中的动态中心性。<ref name="dynamic1">{{cite journal | last1 = Braha | first1 = D. | last2 = Bar-Yam | first2 = Y. | year = 2006 | title = From Centrality to Temporary Fame: Dynamic Centrality in Complex Networks | url = | journal = Complexity | volume = 12 | issue = 2| pages = 59–63 | doi=10.1002/cplx.20156| arxiv = physics/0611295 | bibcode = 2006Cmplx..12b..59B }}</ref><ref name="dynamic2">{{cite journal | last1 = Hill | first1 = S.A. | last2 = Braha | first2 = D. | year = 2010 | title = Dynamic Model of Time-Dependent Complex Networks | url = | journal = Physical Review E | volume = 82 | issue = 4| page = 046105 | doi=10.1103/physreve.82.046105| pmid = 21230343 | arxiv = 0901.4407 | bibcode = 2010PhRvE..82d6105H }}</ref><ref name="dynamic3">Gross, T. and Sayama, H. (Eds.). 2009. ''Adaptive Networks: Theory, Models and Applications.'' Springer.</ref><ref name="dynamic4">Holme, P. and Saramäki, J. 2013. ''Temporal Networks.'' Springer.</ref>
===节点影响力===
===节点影响力===
中心性测量的局限性促使了更加常规的测量的发展。其中两个例子分别是'''可达性'''，它利用随机行走的多样性去度量从一个节点出发到网络其余部分的可到达性；<ref name="Travencolo2008">{{ cite journal| last1=Travençolo |first1=B. A. N. |last2=da F. Costa| first2 =L.| title=Accessibility in complex networks| journal=Physics Letters A| year=2008|volume=373|issue=1|pages=89–95 |doi=10.1016/j.physleta.2008.10.069 |bibcode=2008PhLA..373...89T }}</ref>
中心性测量的局限性促使了更加常规的测量的发展。其中两个例子分别是'''可达性'''，它利用随机行走的多样性去度量从一个节点出发到网络其余部分的可到达性；<ref name="Travencolo2008">{{ cite journal| last1=Travençolo |first1=B. A. N. |last2=da F. Costa| first2 =L.| title=Accessibility in complex networks| journal=Physics Letters A| year=2008|volume=373|issue=1|pages=89–95 |doi=10.1016/j.physleta.2008.10.069 |bibcode=2008PhLA..373...89T }}</ref>