Network modularity in the presence of covariates

TL;DR

This paper develops a statistical framework to assess the significance of community structures in networks with covariates, enabling objective evaluation of modularity in various complex network models.

Contribution

It introduces limit theorems for network modularity with covariates, allowing for p-value computation and validation across diverse network types and regimes.

Findings

Provides a method to quantify the significance of community structure.

Validates the approach with benchmark network examples.

Applies the methodology to corporate email interaction networks.

Abstract

We characterize the large-sample properties of network modularity in the presence of covariates, under a natural and flexible nonparametric null model. This provides for the first time an objective measure of whether or not a particular value of modularity is meaningful. In particular, our results quantify the strength of the relation between observed community structure and the interactions in a network. Our technical contribution is to provide limit theorems for modularity when a community assignment is given by nodal features or covariates. These theorems hold for a broad class of network models over a range of sparsity regimes, as well as weighted, multi-edge, and power-law networks. This allows us to assign $p$-values to observed community structure, which we validate using several benchmark examples in the literature. We conclude by applying this methodology to investigate a multi-edge network of corporate email interactions.