Spectral co-clustering in multi-layer directed networks

Modern network analysis often involves multi-layer network data in which the nodes are aligned, and the edges on each layer represent one of the multiple relations among the nodes. Current literature on multi-layer network data is mostly limited to undirected relations. However, direct relations are more common and may introduce extra information. This study focuses on community detection (or clustering) in multi-layer directed networks. To take into account the asymmetry, a novel spectral-co-clustering-based algorithm is developed to detect co-clusters, which capture the sending patterns and receiving patterns of nodes, respectively. Specifically, the eigendecomposition of the debiased sum of Gram matrices over the layer-wise adjacency matrices is computed, followed by the k-means, where the sum of Gram matrices is used to avoid possible cancellation of clusters caused by direct summation. Theoretical analysis of the algorithm under the multi-layer stochastic co-block model is provided, where the common assumption that the cluster number is coupled with the rank of the model is relaxed. After a systematic analysis of the eigenvectors of the population version algorithm, the misclassification rates are derived, which show that multi-layers would bring benefits to the clustering performance. The experimental results of simulated data corroborate the theoretical predictions, and the analysis of a real-world trade network dataset provides interpretable results.