Generalizations of Edge Overlap to Weighted and Directed Networks

With the increasing availability of behavioral data from diverse digital sources, such as social media sites and cell phones, it is now possible to obtain detailed information on the strength and directionality of social interactions in various settings. While most metrics used to characterize network structure have traditionally been defined for unweighted and undirected networks only, the richness of current network data calls for extending these metrics to weighted and directed networks. One fundamental metric, especially in social networks, is edge overlap, the proportion of friends shared by two connected individuals. Here we extend definitions of edge overlap to weighted and directed networks, and we present closed-form expressions for the mean and variance of each version for the classic \er random graph and its weighted and directed counterparts. We apply these results to social network data collected in rural villages, and we use our analytical results to quantify the extent to which the average edge overlap in the empirical social networks deviates from that of corresponding random graphs. Finally, we carry out comparisons across attribute categories including sex, caste, and age, finding that women tend to form more tightly clustered friendship circles than men, where the extent of overlap depends on the nature of social interaction in question.