Network modularity in the presence of covariates

We characterize the large-sample properties of network modularity in the presence of covariates, under a natural and flexible 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 for weighted, multiedge, and power-law networks. This allows us to assign $p$-values to observed community structure, which we validate using several benchmark examples from the literature. We conclude by applying this methodology to investigate a multiedge network of corporate email interactions.