Multi-Frequency Joint Community Detection and Phase Synchronization

TL;DR

This paper introduces new algorithms for joint community detection and phase synchronization that leverage a multi-frequency structure, significantly improving accuracy and recovery compared to existing methods.

Contribution

It proposes two novel algorithms based on the MLE formulation that utilize multi-frequency information for improved community detection and phase estimation.

Findings

Algorithms outperform state-of-the-art methods in accuracy.

Significant improvement in exact cluster recovery.

Enhanced phase angle estimation accuracy.

Abstract

This paper studies the joint community detection and phase synchronization problem on the \textit{stochastic block model with relative phase}, where each node is associated with an unknown phase angle. This problem, with a variety of real-world applications, aims to recover the cluster structure and associated phase angles simultaneously. We show this problem exhibits a \textit{``multi-frequency''} structure by closely examining its maximum likelihood estimation (MLE) formulation, whereas existing methods are not originated from this perspective. To this end, two simple yet efficient algorithms that leverage the MLE formulation and benefit from the information across multiple frequencies are proposed. The former is a spectral method based on the novel multi-frequency column-pivoted QR factorization. The factorization applied to the top eigenvectors of the observation matrix provides key information about the cluster structure and associated phase angles. The second approach is an iterative multi-frequency generalized power method, where each iteration updates the estimation in a matrix-multiplication-then-projection manner. Numerical experiments show that our proposed algorithms significantly improve the ability of exactly recovering the cluster structure and the accuracy of the estimated phase angles, compared to state-of-the-art algorithms.