A Matrix Optimization Method for Blind Extraction of External Equitable Partitions from Low Pass Graph Signals

TL;DR

This paper introduces BE-EEPs, a novel matrix optimization approach for extracting external equitable partitions from low pass graph signals, with theoretical error bounds and comparative algorithm analysis.

Contribution

It presents the first method linking low pass graph signals to EEPs, using nonnegative orthogonality matrix decomposition and multiple algorithms.

Findings

Iterative Lagrangian and K-means outperform the exact penalty method under strong signals.

All three algorithms perform similarly with weak signals.

Numerical experiments confirm the effectiveness of the proposed method.

Abstract

Seeking the external equitable partitions (EEPs) of networks under unknown structures is an emerging problem in network analysis. The special structure of EEPs has found widespread applications in many fields such as cluster synchronization and consensus dynamics. While most literature focuses on utilizing the special structural properties of EEPs for network studies, there has been little work on the extraction of EEPs or their connection with graph signals. In this paper, we address the interesting connection between low pass graph signals and EEPs, which, as far as we know, is the first time. We provide a method BE-EEPs for extracting EEPs from low pass graph signals and propose an optimization model, which is essentially a problem involving nonnegative orthogonality matrix decomposition. We derive theoretical error bounds for the performance of our proposed method under certain assumptions and apply three algorithms to solve the resulting model, including the K-means algorithm, the practical exact penalty method and the iterative Lagrangian approach. Numerical experiments verify the effectiveness of the proposed method. Under strong low pass graph signals, the iterative Lagrangian and K-means perform equally well, outperforming the exact penalty method. However, under complex weak low pass signals, all three perform equally well.