  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.
