Recovering Unbalanced Communities in the Stochastic Block Model With Application to Clustering with a Faulty Oracle

TL;DR

This paper introduces a simple SVD-based algorithm for community detection in unbalanced stochastic block models, improving theoretical guarantees and demonstrating near-optimal recovery of large clusters even with many small ones, with applications to faulty oracle clustering.

Contribution

It provides a novel SVD-based method for unbalanced SBM community recovery, removing previous size assumptions and improving theoretical bounds.

Findings

Algorithm nearly optimal under planted clique conjecture

Capable of detecting all large clusters in faulty oracle model

Outperforms previous methods in handling many small clusters

Abstract

The stochastic block model (SBM) is a fundamental model for studying graph clustering or community detection in networks. It has received great attention in the last decade and the balanced case, i.e., assuming all clusters have large size, has been well studied. However, our understanding of SBM with unbalanced communities (arguably, more relevant in practice) is still limited. In this paper, we provide a simple SVD-based algorithm for recovering the communities in the SBM with communities of varying sizes. We improve upon a result of Ailon, Chen and Xu [ICML 2013; JMLR 2015] by removing the assumption that there is a large interval such that the sizes of clusters do not fall in, and also remove the dependency of the size of the recoverable clusters on the number of underlying clusters. We further complement our theoretical improvements with experimental comparisons. Under the planted clique conjecture, the size of the clusters that can be recovered by our algorithm is nearly optimal (up to poly-logarithmic factors) when the probability parameters are constant. As a byproduct, we obtain an efficient clustering algorithm with sublinear query complexity in a faulty oracle model, which is capable of detecting all clusters larger than $\tilde{\Omega}({\sqrt{n}})$, even in the presence of $\Omega(n)$ small clusters in the graph. In contrast, previous efficient algorithms that use a sublinear number of queries are incapable of recovering any large clusters if there are more than $\tilde{\Omega}(n^{2/5})$ small clusters.