Phase Transition in the Generalized Stochastic Block Model

TL;DR

This paper analyzes the spectral properties of the generalized stochastic block model, revealing a phase transition in community detection capabilities through spectral analysis and eigenvalue behavior.

Contribution

It introduces a BBP-type transition analysis for the largest eigenvalue in GSBM, providing precise formulas for specific models and advancing understanding of spectral phase transitions.

Findings

Identifies a BBP-type phase transition in GSBM eigenvalues

Provides explicit formulas for eigenvalues in certain models

Establishes spectral gap corresponding to community detectability

Abstract

We study the problem of detecting the community structure from the generalized stochastic block model (GSBM). Based on the analysis of the Stieljtes transform of the empirical spectral distribution, we prove a BBP-type transition for the largest eigenvalue of the GSBM. For specific models such as a hidden community model and an unbalanced stochastic model, we provide precise formulas for the two largest eigenvalues, establishing the gap in the BBP-type transition.