Exact Community Recovery under Side Information: Optimality of Spectral Algorithms

TL;DR

This paper demonstrates that a simple spectral algorithm can achieve the information-theoretic limits for exact community recovery in models with side information, unifying and extending previous results.

Contribution

It introduces an optimal spectral algorithm that incorporates side information for exact community recovery, matching the theoretical limits across various models.

Findings

Spectral algorithm achieves optimal recovery in models with side information.

The approach unifies understanding of spectral methods' optimality.

Algorithm mimics genie-aided estimators for label recovery.

Abstract

We study the problem of exact community recovery in general, two-community block models, in the presence of node-attributed $side$ $information$. We allow for a very general side information channel for node attributes, and for pairwise (edge) observations, consider both Bernoulli and Gaussian matrix models, capturing the Stochastic Block Model, Submatrix Localization, and $\mathbb{Z}_2$-Synchronization as special cases. A recent work of Dreveton et al. 2024 characterized the information-theoretic limit of a very general exact recovery problem with side information. In this paper, we show algorithmic achievability in the above important cases by designing a simple but optimal spectral algorithm that incorporates side information (when present) along with the eigenvectors of the pairwise observation matrix. Using the powerful tool of entrywise eigenvector analysis of Abbe et al. 2020, we show that our spectral algorithm can mimic the so called $genie$-$aided$ $estimators$, where the $i^{\mathrm{th}}$ genie-aided estimator optimally computes the estimate of the $i^{\mathrm{th}}$ label, when all remaining labels are revealed by a genie. This perspective provides a unified understanding of the optimality of spectral algorithms for various exact recovery problems in a recent line of work.