A generalized hypothesis test for community structure and homophily in networks

Networks continue to be of great interest to statisticians, with an emphasis on community detection. Less work, however, has addressed this question: given some network, does it exhibit meaningful community structure? We propose to answer this question in a principled manner by framing it as a statistical hypothesis in terms of a formal and general homophily metric. Homophily is a well-studied network property where intra-community edges are more likely than between-community edges. We use the homophily metric to identify and distinguish between three concepts: nominal, collateral, and intrinsic homophily. We propose a simple and interpretable test statistic leveraging this homophily parameter and formulate both asymptotic and bootstrap-based rejection thresholds. We prove its asymptotic properties and demonstrate it outperforms benchmark methods on both simulated and real world data. Furthermore, the proposed method yields rich, provocative insights on four classic data sets; namely, that meany well-studied networks do not actually have intrinsic homophily.