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A Dynamic Mode Decomposition Approach for Decentralized Spectral Clustering of Graphs

Hongyu Zhu, Stefan Klus, Tuhin Sahai

TL;DR

The paper addresses scalable decentralized spectral clustering for large graphs by eliminating reliance on FFT-based local frequency estimation in the wave equation clustering. It introduces a local Dynamic Mode Decomposition (DMD) step at each node, leveraging the Koopman operator to recover Laplacian eigenvalues $\lambda_j$ and eigenvectors from time-series data, with the mapping $\lambda_j = (2 - e^{i\omega_j} - e^{-i\omega_j})/c^2$. The approach is shown to be provably equivalent to spectral clustering, robust to irrational frequency relations, and to require significantly fewer wave equation updates than FFT-based methods, achieving comparable clustering across line, karate club, synthetic, and Facebook graphs. This yields accurate, scalable, and fully decentralized clustering, with potential applicability to broader graph spectral analysis and distributed computation tasks.

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.

A Dynamic Mode Decomposition Approach for Decentralized Spectral Clustering of Graphs

TL;DR

The paper addresses scalable decentralized spectral clustering for large graphs by eliminating reliance on FFT-based local frequency estimation in the wave equation clustering. It introduces a local Dynamic Mode Decomposition (DMD) step at each node, leveraging the Koopman operator to recover Laplacian eigenvalues and eigenvectors from time-series data, with the mapping . The approach is shown to be provably equivalent to spectral clustering, robust to irrational frequency relations, and to require significantly fewer wave equation updates than FFT-based methods, achieving comparable clustering across line, karate club, synthetic, and Facebook graphs. This yields accurate, scalable, and fully decentralized clustering, with potential applicability to broader graph spectral analysis and distributed computation tasks.

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.
Paper Structure (12 sections, 3 theorems, 23 equations, 12 figures, 1 table, 2 algorithms)

This paper contains 12 sections, 3 theorems, 23 equations, 12 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Decentralized approach
  3. Spectral clustering
  4. Wave equation based clustering method
  5. Dynamic Mode Decomposition for one-dimensional data
  6. Numerical results
  7. A line graph
  8. Zachary Karate club graph
  9. A synthetic network
  10. A Facebook network
  11. Comparison with FFT based wave equation clustering
  12. Conclusions

Key Result

Lemma 1

Given the initial condition $\mathbf{u}(-1)=\mathbf{u}(0)$ and $0<c<\sqrt{2}$, the solution of the wave equation eq:discrete_wave can be written as where $p_{j}=(1+i\tan(\omega_j/2))/2$, $q_{j}=(1-i\tan(\omega_j/2))/2$.

Figures (12)

  • Figure 1: FFT of $[\mathbf{u}_i(1),...,\mathbf{u}_i(T_{max})]$ for Karate graph.
  • Figure 2: Spectral clustering of a line graph with a weak connection.
  • Figure 3: Clustering of the line graph shown in Figure \ref{['fig:line_graph']}. The signs indicate clustering.
  • Figure 4: DMD eigenvalues and modes $\hat{\phi}_j$ at Node 1 for wave propagation on the line graph, j=0 (blue), 1 (orange), 2 (green), 3 (red) and 4 (purple). The exact eigenvalues are the eigenvalues of the matrix $\mathbf{M}$ defined in \ref{['eq:matrix_wave']}.
  • Figure 5: The second eigenvector of the Karate graph shown in Figure \ref{['fig:karate_graph']}. The signs of the eigenvector entries indicate assignment of the corresponding node into one of two clusters.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Proposition 1
  • proof