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第17行：
第17行： − 在真实的网络问题中，人们感兴趣的是确定顶点对之间出现双链路(方向相反)的可能性。这个问题对一些人来说是根本性的+ 第23行：
第23行： − 原因。首先，在传输信息或材料的网络(如电子邮件网络、万维网、世界贸易网络或维基百科)中，相互链接促进了传输过程。其次，在分析有向网络时，为了简单起见，人们通常将其视为无向网络; 因此，从互易性研究中获得的信息有助于估计有向网络被视为无向网络时引入的误差(例如，在测量集聚系数时)。最后，检测非平凡的互易模式可以揭示可能的机制和组织原则，形成观察网络的拓扑结构。+ − 第33行：
第32行： − + − 根据这个定义，< math > r = 1 </math > 是一个纯粹的双向网络，而+ 第51行：
第50行： − 然而，这种互惠的定义有一些缺陷。与纯随机网络相比，在顶点和边数相同的情况下，不能区分互易性的相对差别。来自互惠的有用信息不是价值本身，而是相互联系是否比偶然发生的频率高或低。此外，在含有自链路的网络中(链路起始和终止于同一顶点) ，在计算 l 时应该排除自链路。+ − + − 为了克服上述定义的缺陷，Garlaschelli 和 Loffredo 将互易性定义为一个有向图的邻接矩阵条目之间的相关系数(如果存在从 i 到 j 的链接，如果不存在，则相关系数为 a { ij } = 0 </math >) :+ − +
无编辑摘要
In real network problems, people are interested in determining the likelihood of occurring double links (with opposite directions) between vertex pairs. This problem is fundamental for several
In real network problems, people are interested in determining the likelihood of occurring double links (with opposite directions) between vertex pairs. This problem is fundamental for several
在实际的网络问题中，人们感兴趣的是确定顶点对之间发生双链接（方向相反）的可能性。这个问题对许多人来说都是根本问题
reasons. First, in the networks that transport information or material (such as email networks,<ref name=reciporcity2>{{cite journal | last1=Newman | first1=M. E. J. | last2=Forrest | first2=Stephanie | last3=Balthrop | first3=Justin | title=Email networks and the spread of computer viruses | journal=Physical Review E | publisher=American Physical Society (APS) | volume=66 | issue=3 | date=2002-09-10 | issn=1063-651X | doi=10.1103/physreve.66.035101 | page=035101(R)| pmid=12366169 }}</ref> World Wide Web (WWW),<ref name=reciporcity3>{{cite journal | last1=Albert | first1=Réka | last2=Jeong | first2=Hawoong | last3=Barabási | first3=Albert-László | title=Diameter of the World-Wide Web | journal=Nature | volume=401 | issue=6749 | year=1999 | issn=0028-0836 | doi=10.1038/43601 | pages=130–131| arxiv=cond-mat/9907038 | s2cid=4419938 }}</ref> World Trade Web,<ref name=reciporcity4>{{cite journal | last1=Garlaschelli | first1=Diego | last2=Loffredo | first2=Maria I. | title=Fitness-Dependent Topological Properties of the World Trade Web | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=93 | issue=18 | date=2004-10-28 | issn=0031-9007 | doi=10.1103/physrevlett.93.188701 | page=188701| pmid=15525215 | arxiv=cond-mat/0403051 | s2cid=16367275 }}</ref> or Wikipedia<ref name=reciporcity6>{{cite journal | last1=Zlatić | first1=V. | last2=Božičević | first2=M. | last3=Štefančić | first3=H. | last4=Domazet | first4=M. | title=Wikipedias: Collaborative web-based encyclopedias as complex networks | journal=Physical Review E | volume=74 | issue=1 | date=2006-07-24 | issn=1539-3755 | doi=10.1103/physreve.74.016115 | page=016115| pmid=16907159 | arxiv=physics/0602149 | s2cid=3388193 }}</ref> ), mutual links facilitate the transportation process. Second, when analyzing directed networks, people often treat them as undirected ones for simplicity; therefore, the information obtained from reciprocity studies helps to estimate the error introduced when a directed network is treated as undirected (for example, when measuring the [[clustering coefficient]]). Finally, detecting nontrivial patterns of reciprocity can reveal possible mechanisms and organizing principles that shape the observed network's topology.<ref name="gl04"/>
reasons. First, in the networks that transport information or material (such as email networks,<ref name=reciporcity2>{{cite journal | last1=Newman | first1=M. E. J. | last2=Forrest | first2=Stephanie | last3=Balthrop | first3=Justin | title=Email networks and the spread of computer viruses | journal=Physical Review E | publisher=American Physical Society (APS) | volume=66 | issue=3 | date=2002-09-10 | issn=1063-651X | doi=10.1103/physreve.66.035101 | page=035101(R)| pmid=12366169 }}</ref> World Wide Web (WWW),<ref name=reciporcity3>{{cite journal | last1=Albert | first1=Réka | last2=Jeong | first2=Hawoong | last3=Barabási | first3=Albert-László | title=Diameter of the World-Wide Web | journal=Nature | volume=401 | issue=6749 | year=1999 | issn=0028-0836 | doi=10.1038/43601 | pages=130–131| arxiv=cond-mat/9907038 | s2cid=4419938 }}</ref> World Trade Web,<ref name=reciporcity4>{{cite journal | last1=Garlaschelli | first1=Diego | last2=Loffredo | first2=Maria I. | title=Fitness-Dependent Topological Properties of the World Trade Web | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=93 | issue=18 | date=2004-10-28 | issn=0031-9007 | doi=10.1103/physrevlett.93.188701 | page=188701| pmid=15525215 | arxiv=cond-mat/0403051 | s2cid=16367275 }}</ref> or Wikipedia<ref name=reciporcity6>{{cite journal | last1=Zlatić | first1=V. | last2=Božičević | first2=M. | last3=Štefančić | first3=H. | last4=Domazet | first4=M. | title=Wikipedias: Collaborative web-based encyclopedias as complex networks | journal=Physical Review E | volume=74 | issue=1 | date=2006-07-24 | issn=1539-3755 | doi=10.1103/physreve.74.016115 | page=016115| pmid=16907159 | arxiv=physics/0602149 | s2cid=3388193 }}</ref> ), mutual links facilitate the transportation process. Second, when analyzing directed networks, people often treat them as undirected ones for simplicity; therefore, the information obtained from reciprocity studies helps to estimate the error introduced when a directed network is treated as undirected (for example, when measuring the [[clustering coefficient]]). Finally, detecting nontrivial patterns of reciprocity can reveal possible mechanisms and organizing principles that shape the observed network's topology.<ref name="gl04"/>
reasons. First, in the networks that transport information or material (such as email networks, World Wide Web (WWW), World Trade Web, or Wikipedia ), mutual links facilitate the transportation process. Second, when analyzing directed networks, people often treat them as undirected ones for simplicity; therefore, the information obtained from reciprocity studies helps to estimate the error introduced when a directed network is treated as undirected (for example, when measuring the clustering coefficient). Finally, detecting nontrivial patterns of reciprocity can reveal possible mechanisms and organizing principles that shape the observed network's topology.
reasons. First, in the networks that transport information or material (such as email networks, World Wide Web (WWW), World Trade Web, or Wikipedia ), mutual links facilitate the transportation process. Second, when analyzing directed networks, people often treat them as undirected ones for simplicity; therefore, the information obtained from reciprocity studies helps to estimate the error introduced when a directed network is treated as undirected (for example, when measuring the clustering coefficient). Finally, detecting nontrivial patterns of reciprocity can reveal possible mechanisms and organizing principles that shape the observed network's topology.
原因。首先，在传输信息或材料的网络（如电子邮件网络、万维网（WWW）、世界贸易网或维基百科）中，相互联系促进了传输过程。其次，在分析有向网络时，为了简单起见，人们通常将其视为无向网络；因此，从互惠性研究中获得的信息有助于估计将有向网络视为无向网络时引入的误差（例如，在测量聚类系数时）。最后，检测非凡的互惠模式可以揭示形成观察到的网络拓扑的可能机制和组织原则。
==How is it defined?==
==How is it defined?==
===Traditional definition===
===Traditional definition传统定义===
With this definition, <math>r = 1</math> is for a purely bidirectional network while
With this definition, <math>r = 1</math> is for a purely bidirectional network while
根据这个定义，r = 1是一个纯粹的双向网络，而
A traditional way to define the reciprocity r is using the ratio of the number of links pointing in both directions <math>L^{<->}</math> to the total number of links L <ref name=reciporcity5>{{cite journal | last1=Newman | first1=M. E. J. | last2=Forrest | first2=Stephanie | last3=Balthrop | first3=Justin | title=Email networks and the spread of computer viruses | journal=Physical Review E | publisher=American Physical Society (APS) | volume=66 | issue=3 | date=2002-09-10 | issn=1063-651X | doi=10.1103/physreve.66.035101 | page=035101(R)| pmid=12366169 }}</ref>
A traditional way to define the reciprocity r is using the ratio of the number of links pointing in both directions <math>L^{<->}</math> to the total number of links L <ref name=reciporcity5>{{cite journal | last1=Newman | first1=M. E. J. | last2=Forrest | first2=Stephanie | last3=Balthrop | first3=Justin | title=Email networks and the spread of computer viruses | journal=Physical Review E | publisher=American Physical Society (APS) | volume=66 | issue=3 | date=2002-09-10 | issn=1063-651X | doi=10.1103/physreve.66.035101 | page=035101(R)| pmid=12366169 }}</ref>
However, this definition of reciprocity has some defects. It cannot tell the relative difference of reciprocity compared with purely random network with the same number of vertices and edges. The useful information from reciprocity is not the value itself, but whether mutual links occur more or less often than expected by chance. Besides, in those networks containing self-linking loops (links starting and ending at the same vertex), the self-linking loops should be excluded when calculating L.
However, this definition of reciprocity has some defects. It cannot tell the relative difference of reciprocity compared with purely random network with the same number of vertices and edges. The useful information from reciprocity is not the value itself, but whether mutual links occur more or less often than expected by chance. Besides, in those networks containing self-linking loops (links starting and ending at the same vertex), the self-linking loops should be excluded when calculating L.
然而，这种互惠的定义也有一些缺陷。与具有相同顶点和边数的纯随机网络相比，它无法分辨互惠性的相对差异。从互惠中得到的有用信息不是价值本身，而是相互联系发生的频率是否比偶然预期的要高。此外，在含有自联环的网络中（在同一顶点开始和结束的链接），计算L时应排除自联环
With this definition, <math>r = 1</math> is for a purely bidirectional network while
With this definition, <math>r = 1</math> is for a purely bidirectional network while
<math>r = 0 </math> for a purely unidirectional one. Real networks have an intermediate value between 0 and 1.
<math>r = 0 </math> for a purely unidirectional one. Real networks have an intermediate value between 0 and 1.
根据这个定义，r=1表示纯双向网络，r=0表示纯单向的。实际网络的中间值介于0和1之间。
In order to overcome the defects of the above definition, Garlaschelli and Loffredo defined reciprocity as the correlation coefficient between the entries of the adjacency matrix of a directed graph (<math>a_{ij} = 1</math> if a link from i to j is there, and <math>a_{ij} = 0</math> if not):
In order to overcome the defects of the above definition, Garlaschelli and Loffredo defined reciprocity as the correlation coefficient between the entries of the adjacency matrix of a directed graph (<math>a_{ij} = 1</math> if a link from i to j is there, and <math>a_{ij} = 0</math> if not):
为了克服上述定义的缺陷，Garlaschelli和Loffredo将互惠性定义为有向图的邻接矩阵项之间的相关系数a{ij}=1如果存在从i到j的链接，如果不存在，a{ij}=0：
However, this definition of reciprocity has some defects. It cannot tell the relative difference of reciprocity compared with purely random network with the same number of vertices and edges. The useful information from reciprocity is not the value itself, but whether mutual links occur more or less often than expected by chance. Besides, in those networks containing self-linking loops (links starting and ending at the same vertex), the self-linking loops should be excluded when calculating L.
However, this definition of reciprocity has some defects. It cannot tell the relative difference of reciprocity compared with purely random network with the same number of vertices and edges. The useful information from reciprocity is not the value itself, but whether mutual links occur more or less often than expected by chance. Besides, in those networks containing self-linking loops (links starting and ending at the same vertex), the self-linking loops should be excluded when calculating L.
然而，这种互惠的定义也有一些缺陷。与具有相同顶点和边数的纯随机网络相比，它无法分辨互惠性的相对差异。从互惠中得到的有用信息不是价值本身，而是相互联系发生的频率是否比偶然预期的要高。此外，在含有自联环的网络中（在同一顶点开始和结束的链接），计算L时应排除自联环