# 互惠性

In network science, reciprocity is a measure of the likelihood of vertices in a directed network to be mutually linked.[1] 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

reasons. First, in the networks that transport information or material (such as email networks,[2] World Wide Web (WWW),[3] World Trade Web,[4] or Wikipedia[5] ), 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.[1]

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.

$\displaystyle{ r = \frac {L^{\lt -\gt }}{L} }$

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

## How is it defined?它是如何定义的？

With this definition, $\displaystyle{ r = 1 }$ is for a purely bidirectional network while

A traditional way to define the reciprocity r is using the ratio of the number of links pointing in both directions $\displaystyle{ L^{\lt -\gt } }$ to the total number of links L [6]

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

$\displaystyle{ r = \frac {L^{\lt -\gt }}{L} }$

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.

With this definition, $\displaystyle{ r = 1 }$ is for a purely bidirectional network while

$\displaystyle{ r = 0 }$ 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 ($\displaystyle{ a_{ij} = 1 }$ if a link from i to j is there, and $\displaystyle{ a_{ij} = 0 }$ if not):

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

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

(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 ($\displaystyle{ a_{ij} = 1 }$ if a link from i to j is there, and $\displaystyle{ a_{ij} = 0 }$ if not):

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

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

$\displaystyle{ \bar{a} }$ 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 $\displaystyle{ \bar{a} \equiv \frac {\sum_{i \neq j} a_{ij}} {N(N-1)} = \frac {L} {N(N-1)} }$.

The definition can be written in the following simple form:

$\displaystyle{ \bar{a} }$ 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.

$\displaystyle{ \rho = \frac {r - \bar{a}} {1- \bar{a}} }$

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 ($\displaystyle{ \rho \gt 0 }$) and antireciprocal ($\displaystyle{ \rho \lt 0 }$) networks, with mutual links occurring more and less often than random respectively.

$\displaystyle{ \rho = \frac {r - \bar{a}} {1- \bar{a}} }$

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

$\displaystyle{ \rho_{min} \equiv \frac {- \bar{a}} {1- \bar{a}} }$

1-bar { a } </math >

The new definition of reciprocity gives an absolute quantity which directly allows one to distinguish between reciprocal ($\displaystyle{ \rho \gt 0 }$) and antireciprocal ($\displaystyle{ \rho \lt 0 }$) 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 $\displaystyle{ \rho }$, 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, $\displaystyle{ \rho = 1 }$; if r=0, $\displaystyle{ \rho = \rho_{min} }$.

$\displaystyle{ \rho_{min} \equiv \frac {- \bar{a}} {1- \bar{a}} }$

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

This is another advantage of using $\displaystyle{ \rho }$, 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.[7]

Category:Computer networking

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

1. Diego Garlaschelli; Loffredo, Maria I. (December 2004). "Patterns of Link Reciprocity in Directed Networks". Physical Review Letters. American Physical Society. 93 (26): 268701. arXiv:cond-mat/0404521. doi:10.1103/PhysRevLett.93.268701. PMID 15698035. Unknown parameter |s2cid= ignored (help)
2. Newman, M. E. J.; Forrest, Stephanie; Balthrop, Justin (2002-09-10). "Email networks and the spread of computer viruses". Physical Review E. American Physical Society (APS). 66 (3): 035101(R). doi:10.1103/physreve.66.035101. ISSN 1063-651X. PMID 12366169.
3. Albert, Réka; Jeong, Hawoong; Barabási, Albert-László (1999). "Diameter of the World-Wide Web". Nature. 401 (6749): 130–131. arXiv:cond-mat/9907038. doi:10.1038/43601. ISSN 0028-0836. Unknown parameter |s2cid= ignored (help)
4. Garlaschelli, Diego; Loffredo, Maria I. (2004-10-28). "Fitness-Dependent Topological Properties of the World Trade Web". Physical Review Letters. American Physical Society (APS). 93 (18): 188701. arXiv:cond-mat/0403051. doi:10.1103/physrevlett.93.188701. ISSN 0031-9007. PMID 15525215. Unknown parameter |s2cid= ignored (help)
5. Zlatić, V.; Božičević, M.; Štefančić, H.; Domazet, M. (2006-07-24). "Wikipedias: Collaborative web-based encyclopedias as complex networks". Physical Review E. 74 (1): 016115. arXiv:physics/0602149. doi:10.1103/physreve.74.016115. ISSN 1539-3755. PMID 16907159. Unknown parameter |s2cid= ignored (help)
6. Newman, M. E. J.; Forrest, Stephanie; Balthrop, Justin (2002-09-10). "Email networks and the spread of computer viruses". Physical Review E. American Physical Society (APS). 66 (3): 035101(R). doi:10.1103/physreve.66.035101. ISSN 1063-651X. PMID 12366169.
7. Gallos, Lazaros K.; Rybski, Diego; Fredrik Liljeros; Shlomo Havlin; Makse, Hernán A. (2012). "How People Interact in Evolving Online Affiliation Networks". Physical Review X. 2 (3): 031014. arXiv:1111.5534. doi:10.1103/PhysRevX.2.031014. ISSN 2160-3308. OCLC 969762960. Unknown parameter |s2cid= ignored (help)