Charting the Replica Symmetric Phase

Random graph models and associated inference problems such as the stochastic block model play an eminent role in computer science, discrete mathematics and statistics. Based on nonrigorous arguments physicists predicted the existence of a generic phase transition that separates a “replica symmetric phase” where statistical inference is impossible from a phase where the detection of the “ground truth” is information-theoretically possible. In this paper we prove a contiguity result that shows that detectability is indeed impossible within the replica-symmetric phase for a broad class of models. In particular, this implies the detectability conjecture for the disassortative stochastic block model from [Decelle et al.: Phys. Rev. E 2011]. Additionally, we investigate key features of the replica symmetric phase such as the nature of point-to-set correlations (‘reconstruction’). 1998 ACM Subject Classification G.2 Discrete Mathematics, G.3 Probability and Statistics