Reciprocity (network science)

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

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.

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.

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):

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

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

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

The definition can be written in the following simple form:

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

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

If all the links occur in reciprocal pairs, ${\displaystyle \rho =1}$; if r=0, ${\displaystyle \rho =\rho _{min}}$. ${\displaystyle \rho _{min}\equiv {\frac {-{\bar {a}}}{1-{\bar {a}}}}}$

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.