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 is the ratio of observed directed edges to possible directed connections (i.e., edge density).

Since and , the above equation is rewritten as

(2.41)

This equation indicates that the novel definition of link reciprocity is an extended version of the conventional definition of link reciprocity r_d.

For ρ > 0, the network represents link reciprocity. Mutual edges are significantly observed, indicating symmetric relationship. This property is prominently found in the world trade web (i.e., international import–export networks). This result is in agreement with the empirical knowledge that import–export relationships tend to be symmetric because international problems occur due to asymmetric relationships.

On the other hand, for ρ < 0, the unidirectional edges are dominant, suggesting an asymmetric relationship. Food webs, in particular, exhibit antireciprocity. This result is also consistent with our assumption. There are several examples of asymmetric relationships in food webs (e.g., wolves prey on rabbits, but rabbits never eat wolves).