A Dynamic Mode Decomposition Approach for Decentralized Spectral Clustering of Graphs

TL;DR

This paper introduces a decentralized spectral clustering method for graphs that leverages dynamic mode decomposition and the Koopman operator, offering increased robustness and efficiency over traditional FFT-based techniques.

Contribution

It presents a novel decentralized clustering algorithm that replaces FFT with DMD, improving robustness and reducing computational steps in spectral graph clustering.

Findings

DMD-based approach is more robust than FFT-based methods.

Requires 20 times fewer wave equation steps for accurate clustering.

Reduces relative error by orders of magnitude.

Abstract

We propose a novel robust decentralized graph clustering algorithm that is provably equivalent to the popular spectral clustering approach. Our proposed method uses the existing wave equation clustering algorithm that is based on propagating waves through the graph. However, instead of using a fast Fourier transform (FFT) computation at every node, our proposed approach exploits the Koopman operator framework. Specifically, we show that propagating waves in the graph followed by a local dynamic mode decomposition (DMD) computation at every node is capable of retrieving the eigenvalues and the local eigenvector components of the graph Laplacian, thereby providing local cluster assignments for all nodes. We demonstrate that the DMD computation is more robust than the existing FFT based approach and requires 20 times fewer steps of the wave equation to accurately recover the clustering information and reduces the relative error by orders of magnitude. We demonstrate the decentralized approach on a range of graph clustering problems.