Algorithmic detectability threshold of the stochastic block model

TL;DR

This paper analyzes the detectability threshold of the stochastic block model when using EM with belief propagation, accounting for practical scenarios where model parameters are unknown or inaccurately learned.

Contribution

It derives the algorithmic detectability threshold for EM with BP without assuming known model parameters, extending analysis beyond the Nishimori condition.

Findings

Threshold differs from the Nishimori-based one when parameters are learned inaccurately.

Analysis applies to general modular structures, not just community detection.

Provides insights into the limits of parameter learning in stochastic block models.

Abstract

The assumption that the values of model parameters are known or correctly learned, i.e., the Nishimori condition, is one of the requirements for the detectability analysis of the stochastic block model in statistical inference. In practice, however, there is no example demonstrating that we can know the model parameters beforehand, and there is no guarantee that the model parameters can be learned accurately. In this study, we consider the expectation--maximization (EM) algorithm with belief propagation (BP) and derive its algorithmic detectability threshold. Our analysis is not restricted to the community structure, but includes general modular structures. Because the algorithm cannot always learn the planted model parameters correctly, the algorithmic detectability threshold is qualitatively different from the one with the Nishimori condition.