Statistical and Machine Learning Approaches for Network Analysis

2.10 Reciprocity

The direction of edges is a very interesting parameter because it critically influences system dynamics, although we did not consider it in the previous sections for the sake of simplicity. In particular, the link reciprocity is an important measure for characterizing the significance of symmetric relationships (i.e., mutual edges) in networks.

Conventionally, link reciprocity is expressed as the ratio (e.g., [41])

(2.39)

where and E_d correspond to the number of mutual edges and the total number of directed edges, respectively. It is clear that perfectly bidirectional and unidirectional networks show r_d = 1 and r_d = 0, respectively.

However, in order to evaluate the significance of mutual edges, the reciprocity r_d should be compared to the expected reciprocity estimated from random networks with the same number of nodes and edges. For instance, the frequent emergence of mutual edges is common in networks with the large number of edges.

To avoid this problem, Garlaschelli and Loffredo [42] proposed a novel definition of link reciprocity as the correlation coefficient between the entries of the adjacency matrix of a directed network:

(2.40)

where the average value ...