Bias-adjusted spectral clustering in multi-layer stochastic block models

TL;DR

This paper introduces a bias-adjusted spectral clustering method for multi-layer stochastic block models that effectively aggregates community signals across sparse layers, supported by novel probabilistic bounds and demonstrated on synthetic and gene co-expression data.

Contribution

It proposes a new bias-removal spectral clustering technique for multi-layer SBMs that handles sparsity and introduces novel tail bounds for matrix quadratic forms.

Findings

Method outperforms existing clustering approaches in sparse regimes.

Bias removal is essential for accurate community detection in very sparse layers.

Effective in synthetic data and gene co-expression network analysis.

Abstract

We consider the problem of estimating common community structures in multi-layer stochastic block models, where each single layer may not have sufficient signal strength to recover the full community structure. In order to efficiently aggregate signal across different layers, we argue that the sum-of-squared adjacency matrices contain sufficient signal even when individual layers are very sparse. Our method uses a bias-removal step that is necessary when the squared noise matrices may overwhelm the signal in the very sparse regime. The analysis of our method relies on several novel tail probability bounds for matrix linear combinations with matrix-valued coefficients and matrix-valued quadratic forms, which may be of independent interest. The performance of our method and the necessity of bias removal is demonstrated in synthetic data and in microarray analysis about gene co-expression networks.