Strongly Consistent Community Detection in Popularity Adjusted Block Models

TL;DR

This paper introduces novel spectral clustering algorithms for the Popularity Adjusted Block Model, achieving strong consistency and improved accuracy in community detection, validated through simulations and real data.

Contribution

It develops the Thresholded Cosine Spectral Clustering and refined algorithms that ensure strong consistency in community detection under the PABM, addressing previous challenges.

Findings

The one-step Refined TCSC achieves strong consistency in label recovery.

The two-step Refined TCSC accelerates error convergence with small samples.

The proposed community number selection method outperforms existing approaches.

Abstract

The Popularity Adjusted Block Model (PABM) provides a flexible framework for community detection in network data by allowing heterogeneous node popularity across communities. However, this flexibility increases model complexity and raises key unresolved challenges, particularly in effectively adapting spectral clustering techniques and efficiently achieving strong consistency in label recovery. To address these challenges, we first propose the Thresholded Cosine Spectral Clustering (TCSC) algorithm and establish its weak consistency under the PABM. We then introduce the one-step Refined TCSC algorithm and prove that it achieves strong consistency under the PABM, correctly recovering all community labels with high probability. We further show that the two-step Refined TCSC accelerates clustering error convergence, especially with small sample sizes. Additionally, we propose a data-driven approach for selecting the number of communities, which outperforms existing methods under the PABM. The effectiveness and robustness of our methods are validated through extensive simulations and real-world applications.