Strong Consistency of Spectral Clustering for Stochastic Block Models

TL;DR

This paper proves that spectral clustering methods can reliably identify communities in stochastic block models under certain conditions, ensuring almost sure correct classification.

Contribution

It establishes the strong consistency of spectral clustering for SBMs, including extensions to regularized and degree-corrected models, under weak assumptions.

Findings

Spectral clustering achieves almost sure correct community detection.

Extensions to regularized and degree-corrected SBMs are validated.

Performance demonstrated on simulated networks.

Abstract

In this paper we prove the strong consistency of several methods based on the spectral clustering techniques that are widely used to study the community detection problem in stochastic block models (SBMs). We show that under some weak conditions on the minimal degree, the number of communities, and the eigenvalues of the probability block matrix, the K-means algorithm applied to the eigenvectors of the graph Laplacian associated with its first few largest eigenvalues can classify all individuals into the true community uniformly correctly almost surely. Extensions to both regularized spectral clustering and degree-corrected SBMs are also considered. We illustrate the performance of different methods on simulated networks.