The Power of Two Matrices in Spectral Algorithms for Community Recovery

TL;DR

This paper demonstrates that using two matrices in spectral algorithms enhances community detection in graphs, achieving optimal recovery in the censored stochastic block model, surpassing single-matrix methods.

Contribution

It introduces a two-matrix spectral algorithm for community detection, proving its optimality and showing the limitations of single-matrix spectral algorithms in certain models.

Findings

Two-matrix spectral algorithms are optimal for community recovery.

Single-matrix spectral algorithms are generally suboptimal.

Conditions for the (sub)-optimality of spectral algorithms are provided.

Abstract

Spectral algorithms are some of the main tools in optimization and inference problems on graphs. Typically, the graph is encoded as a matrix and eigenvectors and eigenvalues of the matrix are then used to solve the given graph problem. Spectral algorithms have been successfully used for graph partitioning, hidden clique recovery and graph coloring. In this paper, we study the power of spectral algorithms using two matrices in a graph partitioning problem. We use two different matrices resulting from two different encodings of the same graph and then combine the spectral information coming from these two matrices. We analyze a two-matrix spectral algorithm for the problem of identifying latent community structure in large random graphs. In particular, we consider the problem of recovering community assignments exactly in the censored stochastic block model, where each edge status is revealed independently with some probability. We show that spectral algorithms based on two matrices are optimal and succeed in recovering communities up to the information theoretic threshold. Further, we show that for most choices of the parameters, any spectral algorithm based on one matrix is suboptimal. The latter observation is in contrast to our prior works (2022a, 2022b) which showed that for the symmetric Stochastic Block Model and the Planted Dense Subgraph problem, a spectral algorithm based on one matrix achieves the information theoretic threshold. We additionally provide more general geometric conditions for the (sub)-optimality of spectral algorithms.