Community detection for directed weighted networks

\cite{rohe2016co} proposed Stochastic co-Blockmodel (ScBM) as a tool for detecting community structure of binary directed graph data in network studies. However, ScBM completely ignores node weight, and is unable to explain the block structure of directed weighted network which appears in various areas, such as biology, sociology, physiology and computer science. Here, to model directed weighted network, we introduce a Directed Distribution-Free model by releasing ScBM's distribution restriction. We also build an extension of the proposed model by considering variation of node degree. Our models do not require a specific distribution on generating elements of adjacency matrix but only a block structure on the expected adjacency matrix. Spectral algorithms with theoretical guarantee on consistent estimation of node label are presented to identify communities. Our proposed methods are illustrated by simulated and empirical examples.