Spectral Recovery in the Labeled SBM

TL;DR

This paper introduces a spectral algorithm for exact community detection in the Labeled Stochastic Block Model, achieving the theoretical limit in the sparse regime by leveraging eigenvectors from multiple matrices.

Contribution

It generalizes previous spectral methods to the Labeled SBM with multiple labels, using eigenvectors from L matrices for community recovery.

Findings

Achieves the information-theoretic threshold in the logarithmic-degree regime.

Extends spectral recovery results from the Censored SBM to the more general Labeled SBM.

Uses eigenvectors from multiple matrices, not just two, for improved community detection.

Abstract

We consider the problem of exact community recovery in the Labeled Stochastic Block Model (LSBM) with $k$ communities, where each pair of vertices is associated with a label from the set $\{0,1, \dots, L\}$. A pair of vertices from communities $i,j$ is given label $\ell$ with probability $p_{ij}^{(\ell)}$, and the goal is to recover the community partition. We propose a simple spectral algorithm for exact community recovery, and show that it achieves the information-theoretic threshold in the logarithmic-degree regime, under the assumption that the eigenvalues of certain parameter matrices are distinct and nonzero. Our results generalize recent work of Dhara, Gaudio, Mossel, and Sandon (2023), who showed that a spectral algorithm achieves the information-theoretic threshold in the Censored SBM, which is equivalent to the LSBM with $L = 2$. Interestingly, their algorithm uses eigenvectors from two matrix representations of the graph, while our algorithm uses eigenvectors from $L$ matrices.