Central limit theorems for local network statistics

Abstract

Subgraph counts-in particular the number of occurrences of small shapes such as triangles-characterize properties of random networks, and as a result have seen wide use as network summary statistics. However, subgraphs are typically counted globally, and existing approaches fail to describe vertex-specific characteristics. On the other hand, rooted subgraph counts-counts focusing on any given vertex's neighborhoodare fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model which generalizes many popular statistical network models. This result enables a shift in the statistical analysis of large graphs, from estimating network summaries, to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that local friendship patterns are significant predictors of gender and race.

Limiting eigenvalue distribution of random matrices of Ihara zeta function of long-range percolation graphs

“…The current interest to the number of triangles in random graphs is motivated, among others, by applications in social networks (see, for instance, [19] and references therein). From another hand, related to X n ) can be thought as a number of closed walks of three steps performed over the edges of corresponding graph.…”