Discovering overlapping communities in multi-layer directed networks

Community detection in multi-layer undirected networks has attracted considerable attention in recent years. However, multi-layer directed networks are common in the real world, and existing community detection methods often either ignore the asymmetric structure in multi-layer directed networks or assume that every node solely belongs to a single community, significantly limiting their applicability to overlapping multi-layer directed networks, where nodes can belong to multiple communities simultaneously. To fill this gap, this article explores the challenging problem of detecting overlapping communities in multi-layer directed networks. Our goal is to understand the underlying asymmetric overlapping community structure by analyzing the mixed memberships of nodes. We introduce a novel multi-layer mixed membership stochastic co-block model (multi-layer MM-ScBM) to model overlapping multi-layer directed networks. We develop a spectral procedure to estimate nodes' memberships in both sending and receiving patterns. Our method uses a successive projection algorithm on a few leading eigenvectors of two debiased aggregation matrices. To our knowledge, this is the first work to detect asymmetric overlapping communities in multi-layer directed networks. We demonstrate the consistent estimation properties of our method by providing per-node error rates under the multi-layer MM-ScBM framework. Our theoretical analysis reveals that increasing the overall sparsity, the number of nodes, or the number of layers can improve the accuracy of overlapping community detection. Extensive numerical experiments validate these theoretical findings. We also apply our method to one real-world multi-layer directed network, gaining insightful results.