Finite size analysis of the detectability limit of the stochastic block model

TL;DR

This paper analyzes the finite-size detectability limits of the stochastic block model (SBM) using information theory, providing explicit formulas and revealing large parameter classes with equivalent detectability, including various important SBM variants.

Contribution

It introduces necessary conditions for finite-size detectability in SBM, distinguishes between average and instance-by-instance detectability, and applies these concepts to a broad class of models.

Findings

Existence of large parameter classes with equal detectability

Explicit formulas for finite-size detectability thresholds

Application to various SBM variants including symmetric and planted coloring models

Abstract

It has been shown in recent years that the stochastic block model (SBM) is sometimes undetectable in the sparse limit, i.e., that no algorithm can identify a partition correlated with the partition used to generate an instance, if the instance is sparse enough and infinitely large. In this contribution, we treat the finite case explicitly, using arguments drawn from information theory and statistics. We give a necessary condition for finite-size detectability in the general SBM. We then distinguish the concept of average detectability from the concept of instance-by-instance detectability and give explicit formulas for both definitions. Using these formulas, we prove that there exist large equivalence classes of parameters, where widely different network ensembles are equally detectable with respect to our definitions of detectability. In an extensive case study, we investigate the finite-size detectability of a simplified variant of the SBM, which encompasses a number of important models as special cases. These models include the symmetric SBM, the planted coloring model, and more exotic SBMs not previously studied. We conclude with three appendices, where we study the interplay of noise and detectability, establish a connection between our information-theoretic approach and random matrix theory, and provide proofs of some of the more technical results.