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

Wenshun Teng, Qingna Li

TL;DR

This work addresses blind extraction of external equitable partitions (EEPs) from graphs with unknown topology by exploiting low-pass graph signals. It introduces BE-EEPs, a matrix-optimization framework that uses the top $r$ eigenvectors of the sample covariance $\widehat{\Sigma}$ and enforces nonnegative orthogonality via a normalized indicator matrix $\widehat{H}$ to recover the partition. An error-bound analysis links the covariance estimation error $\|\widehat{\Sigma}-\Sigma\|_2$ to the clustering-cost gap, and three solvers—K-means, EP4Orth+, and Projective Semi-NMF—are developed to solve the resulting problem. Numerical experiments on a synthetic hierarchical graph show that PSNMF and K-means perform best under strong low-pass conditions while all three methods perform similarly under weak low-pass signals, confirming the practical viability of BE-EEPs for blind EEP extraction.

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.

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

TL;DR

This work addresses blind extraction of external equitable partitions (EEPs) from graphs with unknown topology by exploiting low-pass graph signals. It introduces BE-EEPs, a matrix-optimization framework that uses the top eigenvectors of the sample covariance and enforces nonnegative orthogonality via a normalized indicator matrix to recover the partition. An error-bound analysis links the covariance estimation error to the clustering-cost gap, and three solvers—K-means, EP4Orth+, and Projective Semi-NMF—are developed to solve the resulting problem. Numerical experiments on a synthetic hierarchical graph show that PSNMF and K-means perform best under strong low-pass conditions while all three methods perform similarly under weak low-pass signals, confirming the practical viability of BE-EEPs for blind EEP extraction.

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.

Paper Structure

This paper contains 16 sections, 5 theorems, 36 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. External Equitable Partitions of a Graph
  4. Graph Signal Model
  5. Blind Extraction with Low Pass Graph Filters
  6. Problem Statement
  7. Connections between EEP and low pass graph filters
  8. A Function Related to Structural Eigenvectors and an Indicator Matrix
  9. The Optimization Model
  10. Error Bound Analysis
  11. Three Methods for Solving \ref{['F2']}
  12. Method One
  13. Method Two
  14. Method Three
  15. Numerical Results
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $G$ be a graph, and $\pi$ be a partition of $V(G)$ encoded by the indicator matrix $H$. The following results hold.

Figures (6)

  • Figure 2.1: The original graph $G$ with $n=11$ vertices with an EEP $\pi$ into three cells (indicated with colors) and its associated quotient graph $G/\pi$.
  • Figure 2.2: A graph signal on the vertices of graph $G$ in \ref{['fig-quotient']}. The height of each blue bar represents the magnitude of the signal values.
  • Figure 6.1: The cost function $F_{c}$ and the group accuracy rate $\gamma$ of K-means, PSNMF and EP4Orth+ for strong low pass graph signals.
  • Figure 6.2: The cost function $F_{c}$ and the group accuracy rate $\gamma$ of K-means, PSNMF and EP4Orth+ for weak low pass graph signals.
  • Figure 6.3: The facet stacked bar chart of EEP extraction from strong low pass graph signals using three methods.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Proposition 2.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Example 2.2
  • Proposition 3.1
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.2
  • Theorem 4.1
  • ...and 1 more