Fundamental Limits of Spectral Clustering in Stochastic Block Models

TL;DR

This paper provides a rigorous theoretical analysis of spectral clustering's performance in sparse stochastic block models, showing it achieves exponentially small error rates with sharp bounds.

Contribution

It introduces a novel truncated  perturbation analysis and eigenvector truncation technique to precisely characterize spectral clustering in sparse networks.

Findings

Spectral clustering achieves exponentially small error rates in sparse SBMs.

The paper provides matching upper and lower bounds with identical exponents.

A new eigenvector truncation analysis is developed for sharper performance characterization.

Abstract

Spectral clustering has been widely used for community detection in network sciences. While its empirical successes are well-documented, a clear theoretical understanding, particularly for sparse networks where degrees are much smaller than $\log n$, remains unclear. In this paper, we address this significant gap by demonstrating that spectral clustering offers exponentially small error rates when applied to sparse networks under Stochastic Block Models. Our analysis provides sharp characterizations of its performance, backed by matching upper and lower bounds possessing an identical exponent with the same leading constant. The key to our results is a novel truncated $\ell_2$ perturbation analysis for eigenvectors, coupled with a new analysis idea of eigenvectors truncation.