Maximum Likelihood Estimation on Stochastic Blockmodels for Directed Graph Clustering

This paper studies the directed graph clustering problem through the lens of statistics, where we formulate clustering as estimating underlying communities in the directed stochastic block model (DSBM). We conduct the maximum likelihood estimation (MLE) on the DSBM and thereby ascertain the most probable community assignment given the observed graph structure. In addition to the statistical point of view, we further establish the equivalence between this MLE formulation and a novel flow optimization heuristic, which jointly considers two important directed graph statistics: edge density and edge orientation. Building on this new formulation of directed clustering, we introduce two efficient and interpretable directed clustering algorithms, a spectral clustering algorithm and a semidefinite programming based clustering algorithm. We provide a theoretical upper bound on the number of misclustered vertices of the spectral clustering algorithm using tools from matrix perturbation theory. We compare, both quantitatively and qualitatively, our proposed algorithms with existing directed clustering methods on both synthetic and real-world data, thus providing further ground to our theoretical contributions.