Consistency of spectral clustering for directed network community detection

Directed networks appear in various areas, such as biology, sociology, physiology and computer science. In this paper, we construct a spectral clustering method based on the singular decomposition of the adjacency matrix to detect community in directed stochastic block model (DiSBM). By considering a sparsity parameter, under mild conditions, we show the proposed approach can consistently recover hidden row and column communities for different scaling of degrees. By considering the degree heterogeneity of both row and column nodes, we further modify the proposed method and establish a theoretical framework for directed degree corrected stochastic block model (DiDCSBM), and also show the consistency of the modified method for this case. Our theoretical results under DiSBM and DiDCSBM provide some innovations on some special directed networks, such as directed network with balanced clusters, directed network with nodes enjoying similar degrees, and the directed Erd\"os-R\'enyi graph. Furthermore, the theoretical results under DiDCSBM are consistent with those under DiSBM.