Community Detection on Inhomogeneous Multilayer Networks with Extreme Sparsity

TL;DR

This paper introduces MARS-CD, a novel spectral method for layer-specific community detection in extremely sparse, inhomogeneous multilayer networks, with theoretical guarantees and practical applications.

Contribution

The paper develops the first theoretical guarantees for multilayer community detection in sparse networks and proposes a regularized spectral method that incorporates layer-specific information.

Findings

MARS-CD outperforms existing methods in simulations.

Theoretical guarantees for community recovery are established.

Application to food trading networks reveals interpretable community structures.

Abstract

We study layer-specific community detection in an $L$-layer network $\{A^{(l)}\}_{l\in[L]}$ on a common set of $n$ nodes. Because modern networks are constructed from multi-modal data or with different contexts, the community labels $\pi^{(l)}\in[K]^n$ are layer-dependent and the degree heterogeneity parameters $\theta_i^{(l)}$ vary widely across nodes and layers. The inhomogeneity and extreme sparsity raise a challenge for classical community detection methods. We propose a multilayer-assisted regularized spectral method (MARS-CD) to address this challenge. For layer $l$, MARS-CD first constructs $X^{(l)}$ from the remaining layers, so that the problem is transformed into a network-with-covariates clustering problem on $(A^{(l)}, X^{(l)})$. Then we recover $\pi^{(l)}$ by NAC in Hu and Wang (2024) that allows misalignment. The key component is to construct $X^{(l)}$, where we stack regularized embeddings. Building upon this, we establish the first theoretical guarantees for the quality of $X^{(l)}$ under multilayer networks with extreme sparsity. These further lead to weak and strong consistency for recovering $\pi^{(l)}$. We further develop an optional label alignment step to interpret the shared community structure across layers. Simulations demonstrate the superior performance of our MARS-CD method. Applying MARS-CD to international food trading networks provides an interpretable product-specific community structure.