Reconstruction in the Labeled Stochastic Block Model

TL;DR

This paper proves the impossibility of reconstruction below a certain threshold in the labeled stochastic block model and introduces methods to achieve reconstruction above this threshold, confirming a phase transition phenomenon.

Contribution

It proves one direction of a conjecture on reconstruction feasibility and introduces weighted graph techniques and algorithms for successful community detection above the threshold.

Findings

Reconstruction is impossible below the threshold.

Reconstruction is achievable above the threshold using new methods.

Phase transition occurs at the conjectured threshold.

Abstract

The labeled 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 labeled at random with probability depending on whether their two endpoints belong to the same community or not. It has been conjectured in \cite{Heimlicher12} 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 the threshold. 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 labeled stochastic block model and the labeled Erd\H{o}s-R\'enyi random graph model exhibits a phase transition at the conjectured reconstruction threshold.