Reconstruction in the Labelled Stochastic Block Model

Abstract

The labelled stochastic block model is a random graph model representing networks with community structure and interactions of multiple types. In its simplest form, it consists of two communities of approximately equal size, and the edges are drawn and labelledat random with probability depending on whether their two endpoints belong to the same community or not. It has been conjectured in [1] that correlated reconstruction (i.e., identification of a partition correlated with the true partition into the underlying communities) would be feasible if and only if a model parameter exceeds a threshold. We prove one half of this conjecture, i.e., reconstruction is impossible when below thethreshold. In the positive direction, we introduce a weighted graph to exploit the label information. With a suitable choice of weight function, we show that when above the threshold by a specific constant, reconstruction is achieved by (1) minimum bisection, (2) a semidefinite relaxation of minimum bisection, and (3) a spectral method combined with removal of edges incident to vertices of high degree. Furthermore, we show that hypothesis testing between the labelled stochastic block model and the labelled Erdős-Renyi random graph model exhibits a phase transition at the conjectured reconstruction threshold.