互惠性

此词条暂由Henry翻译

Network science
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Graph Complex network Contagion Small-world Scale-free Community structure Percolation Evolution Controllability Graph drawing Social capital Link analysis Optimization Reciprocity Closure Homophily Transitivity Preferential attachment Balance theory Network effect Social influence
Network types
Informational (computing) Telecommunication Transport Social Scientific collaboration Biological Artificial neural Interdependent Semantic Spatial Dependency Flow on-Chip
Graphs
Features Clique Component Cut Cycle Data structure Edge Loop Neighborhood Path Vertex Adjacency list模板:\matrix Incidence list模板:\matrix Types Bipartite Complete Directed Hyper Multi Random Weighted
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Centrality Degree Betweenness Closeness PageRank Motif Clustering Degree distribution Assortativity Distance Modularity Efficiency
Models
Topology Random graph Erdős–Rényi Barabási–Albert Fitness model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG) Hyperbolic (HGN) Hierarchical Stochastic block Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR
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In network science, reciprocity is a measure of the likelihood of vertices in a directed network to be mutually linked. Like the clustering coefficient, scale-free degree distribution, or community structure, reciprocity is a quantitative measure used to study complex networks.

In network science, reciprocity is a measure of the likelihood of vertices in a directed network to be mutually linked. Like the clustering coefficient, scale-free degree distribution, or community structure, reciprocity is a quantitative measure used to study complex networks. 在网络科学中， Reciprocity互惠性是一种度量有向网络中顶点相互连接的可能性的方法。就像集聚系数、无标度分布或者社区结构一样，互惠性是一种用于研究复杂网络的定量度量。

Motivation动机

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, 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。

[math]\displaystyle{ r = \frac {L^{\lt -\gt }}{L} }[/math]

[ math > r = frac { l ^ { <-> }{ l } </math >

How is it defined?它是如何定义的？

Traditional definition传统定义

With this definition, [math]\displaystyle{ 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]\displaystyle{ L^{\lt -\gt } }[/math] to the total number of links L

[math]\displaystyle{ r = 0 }[/math] for a purely unidirectional one. Real networks have an intermediate value between 0 and 1.

对于一个单向的人来说。实际网络的中间值介于0和1之间。

[math]\displaystyle{ r = \frac {L^{\lt -\gt }}{L} }[/math]

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]\displaystyle{ r = 1 }[/math] is for a purely bidirectional network while

[math]\displaystyle{ 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]\displaystyle{ a_{ij} = 1 }[/math] if a link from i to j is there, and [math]\displaystyle{ a_{ij} = 0 }[/math] if not):

为了克服上述定义的缺陷，加拉舍利和洛弗雷多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. 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

[math]\displaystyle{ \rho \equiv \frac {\sum_{i \neq j} (a_{ij} - \bar{a}) (a_{ji} - \bar{a})}{\sum_{i \neq j} (a_{ij} - \bar{a})^2} }[/math],

(a { ji }-bar { a }){ sum { i neq }(a { ij }-bar { a })}(sum { i neq }(a { ij }-bar { a }) ^/math > ,

Garlaschelli and Loffredo's definition 加拉舍利和洛弗雷多的定义

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]\displaystyle{ a_{ij} = 1 }[/math] if a link from i to j is there, and [math]\displaystyle{ a_{ij} = 0 }[/math] if not):

where the average value [math]\displaystyle{ \bar{a} \equiv \frac {\sum_{i \neq j} a_{ij}} {N(N-1)} = \frac {L} {N(N-1)} }[/math].

为了克服上述定义的缺陷，加拉舍利和洛弗雷多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}。如果存在从i到j的链接，a{ij}=1，如果不存在，a{ij}=0：其中平均值a¯≡∑i≠jaijN(N−1)=LN(N−1)。

[math]\displaystyle{ \rho \equiv \frac {\sum_{i \neq j} (a_{ij} - \bar{a}) (a_{ji} - \bar{a})}{\sum_{i \neq j} (a_{ij} - \bar{a})^2} }[/math],

[math]\displaystyle{ \bar{a} }[/math] measures the ratio of observed to possible directed links (link density), and self-linking loops are now excluded from L because of i not equal to j.

bar{a}测量观察到的与可能的有向链路的比率(链路密度) ，自链路现在被排除在l之外，因为 i 不等于 j。

where the average value [math]\displaystyle{ \bar{a} \equiv \frac {\sum_{i \neq j} a_{ij}} {N(N-1)} = \frac {L} {N(N-1)} }[/math].

The definition can be written in the following simple form:

定义可以用以下简单的形式写出:

[math]\displaystyle{ \bar{a} }[/math] measures the ratio of observed to possible directed links (link density), and self-linking loops are now excluded from L because of i not equal to j.

[math]\displaystyle{ \rho = \frac {r - \bar{a}} {1- \bar{a}} }[/math]

1-bar { a }} </math >

The definition can be written in the following simple form:

The new definition of reciprocity gives an absolute quantity which directly allows one to distinguish between reciprocal ([math]\displaystyle{ \rho \gt 0 }[/math]) and antireciprocal ([math]\displaystyle{ \rho \lt 0 }[/math]) networks, with mutual links occurring more and less often than random respectively.

互惠的新定义给出了一个绝对量，这个绝对量直接允许人们区分互惠网络ρ>0和反互惠网络ρ<0,相互联系比随机网络发生的频率更高、更少。

[math]\displaystyle{ \rho = \frac {r - \bar{a}} {1- \bar{a}} }[/math]

If all the links occur in reciprocal pairs, [math]\displaystyle{ \rho = 1 }[/math]; if r=0, [math]\displaystyle{ \rho = \rho_{min} }[/math].

如果所有的链接都是相互的，那么如果r=0, ρ=ρmin. 。

[math]\displaystyle{ \rho_{min} \equiv \frac {- \bar{a}} {1- \bar{a}} }[/math]

1-bar { a } </math >

The new definition of reciprocity gives an absolute quantity which directly allows one to distinguish between reciprocal ([math]\displaystyle{ \rho \gt 0 }[/math]) and antireciprocal ([math]\displaystyle{ \rho \lt 0 }[/math]) networks, with mutual links occurring more and less often than random respectively. 互惠的新定义给出了一个绝对量，这个绝对量直接允许人们区分互惠网络ρ>0和反互惠网络ρ<0,相互联系比随机网络发生的频率更高、更少。

This is another advantage of using [math]\displaystyle{ \rho }[/math], because it incorporates the idea that complete antireciprocal is more statistical significant in the networks with larger density, while it has to be regarded as a less pronounced effect in sparser networks. 这是使用ρ的另一个优点，因为它包含了这样一个思想，即在密度较大的网络中，完全反精确更具统计意义，而在稀疏网络中，它则被视为不太明显的效果。

If all the links occur in reciprocal pairs, [math]\displaystyle{ \rho = 1 }[/math]; if r=0, [math]\displaystyle{ \rho = \rho_{min} }[/math].

[math]\displaystyle{ \rho_{min} \equiv \frac {- \bar{a}} {1- \bar{a}} }[/math]

The reciprocity was analyzed in some real social networks by Gallos.

在一些真实的社会网络中，Gallos 对这种互惠关系进行了分析。

This is another advantage of using [math]\displaystyle{ \rho }[/math], because it incorporates the idea that complete antireciprocal is more statistical significant in the networks with larger density, while it has to be regarded as a less pronounced effect in sparser networks. 这是使用ρ的另一个优点，因为它包含了这样一个思想，即在密度较大的网络中，完全反精确更具统计意义，而在稀疏网络中，它则被视为不太明显的效果。

Reciprocity in real social networks真实社会网络中的互惠性

The reciprocity was analyzed in some real social networks by Gallos.

加洛斯分析了现实社会网络中的互惠性。

Category:Computer networking

类别: 计算机网络

This page was moved from wikipedia:en:Reciprocity (network science). Its edit history can be viewed at 互惠性/edithistory