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 about the structure, strength, and directionality of social interactions in varied settings. While most metrics of 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 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 present closed-form expressions for the mean and variance of each version for the Erdos-Renyi random graph and its weighted and directed counterparts. We apply these results to social network data collected in rural villages in southern Karnataka, India. We use our analytical results to quantify the extent to which the average overlap of the empirical social network deviates from that of corresponding random graphs and compare the values of overlap across networks. Our novel definitions allow the calculation of edge overlap for more complex networks and our derivations provide a statistically rigorous way for comparing edge overlap across networks.