更改

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

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

==How is it defined?==

==How is it defined?它是如何定义的？==

===Traditional definition传统定义===

===Traditional definition传统定义===

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>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：

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.

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.

(a { ji }-bar { a }){ sum { i neq }(a { ij }-bar { a })}(sum { i neq }(a { ij }-bar { a }) ^/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===

===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>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):

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

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

其中平均值 < math > bar { a } equiv frac { sum { i neq } 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>\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>\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.

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。

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。

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

The new definition of reciprocity gives an absolute quantity which directly allows one to distinguish between reciprocal (<math>\rho > 0</math>) and antireciprocal (<math>\rho < 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(< math > rho > 0 </math >)和反互惠网络(< math > rho < 0 </math >) ，相互联系比随机网络发生的频率更高、更少。

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。

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

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

如果所有的链接都是相互的，那么如果 r = 0，那么 rho = rho _ { min } </math > 。

如果所有的链接都是相互的，那么如果r=0, ρ=ρmin. 。

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

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

 

This is another advantage of using <math>\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.

This is another advantage of using <math>\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.

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

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

This is another advantage of using <math>\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.

This is another advantage of using <math>\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.

 

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

 

===Reciprocity in real social networks===

The reciprocity was analyzed in some real social networks by Gallos.<ref name="GallosRybski2012">{{cite journal|author=Gallos, Lazaros K.|author2=Rybski, Diego|author3=[[Fredrik Liljeros]]|author4=[[Shlomo Havlin]]|author5=Makse, Hernán A.|title=How People Interact in Evolving Online Affiliation Networks|journal=Physical Review X|volume=2|issue=3|year=2012|page=031014|issn=2160-3308|oclc=969762960|doi=10.1103/PhysRevX.2.031014|arxiv=1111.5534|s2cid=16905579}}</ref>

The reciprocity was analyzed in some real social networks by Gallos.<ref name="GallosRybski2012">{{cite journal|author=Gallos, Lazaros K.|author2=Rybski, Diego|author3=[[Fredrik Liljeros]]|author4=[[Shlomo Havlin]]|author5=Makse, Hernán A.|title=How People Interact in Evolving Online Affiliation Networks|journal=Physical Review X|volume=2|issue=3|year=2012|page=031014|issn=2160-3308|oclc=969762960|doi=10.1103/PhysRevX.2.031014|arxiv=1111.5534|s2cid=16905579}}</ref>