  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 new model, the multi-layer mixed
membership stochastic co-block model (multi-layer MM-ScBM), to model
multi-layer directed networks in which nodes can belong to multiple
communities. 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 are conducted to validate these theoretical findings. We also apply
our method to one real-world multi-layer directed network, gaining insightful
results.
