Central limit theorems for local network statistics

TL;DR

This paper develops central limit theorems for rooted subgraph counts in inhomogeneous random graphs, enabling vertex-specific network analysis and linking local structures to vertex attributes.

Contribution

It derives the asymptotic distribution of rooted subgraph counts, facilitating analysis of local network properties and their relation to vertex covariates.

Findings

Rooted subgraph counts follow a joint normal distribution asymptotically.

Local friendship patterns significantly predict gender and race.

Method applies to various inhomogeneous random graph models.

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 neighborhood - are 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.