Covariate Regularized Community Detection in Sparse Graphs

In this paper, we investigate community detection in networks in the presence of node covariates. In many instances, covariates and networks individually only give a partial view of the cluster structure. One needs to jointly infer the full cluster structure by considering both. In statistics, an emerging body of work has been focused on combining information from both the edges in the network and the node covariates to infer community memberships. However, so far the theoretical guarantees have been established in the dense regime, where the network can lead to perfect clustering under a broad parameter regime, and hence the role of covariates is often not clear. In this paper, we examine sparse networks in conjunction with finite dimensional sub-gaussian mixtures as covariates under moderate separation conditions. In this setting each individual source can only cluster a non-vanishing fraction of nodes correctly. We propose a simple optimization framework which provably improves clustering accuracy when the two sources carry partial information about the cluster memberships, and hence perform poorly on their own. Our optimization problem can be solved using scalable convex optimization algorithms. Using a variety of simulated and real data examples, we show that the proposed method outperforms other existing methodology.