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Unified Fourier bases for signals on random graphs with group symmetries

Mahya Ghandehari, Jeannette Janssen, Silo Murphy

TL;DR

This work addresses instance-independent graph signal processing on graphs drawn from stochastic block models by leveraging graphon theory. It demonstrates how to compute a Fourier basis for SBM samples from the SBM’s model and weighted-probability matrices, and shows that, for nearly uniform block sizes, the group-Fourier basis from Cayley structure provides a close approximation, with rigorous bounds via Davis–Kahan perturbation theory. The paper then extends to nonuniform block sizes, presenting a reduced-dimensional framework that partially recovers the SBM Fourier basis through a smaller matrix, revealing when symmetry-based bases remain informative. An explicit Z_5 example and transfer experiments illustrate the practical effectiveness and robustness of the approach, highlighting its potential for scalable, structure-aware GSP across families of graphs. Overall, the SBM-driven Fourier framework promises stable, interpretable signal processing on large, structured networks with group-like symmetries, even under perturbations of block sizes.

Abstract

We consider a recently proposed approach to graph signal processing (GSP) based on graphons. We show how the graphon-based approach to GSP applies to graphs sampled from a stochastic block model derived from a weighted Cayley graph. When SBM block sizes are equal, a nice Fourier basis can be derived from the representation theory of the underlying group. We explore how the SBM Fourier basis is affected when block sizes are not uniform. When block sizes are nearly uniform, we demonstrate that the group Fourier basis closely approximates the SBM Fourier basis. More specifically, we quantify the approximation error using matrix perturbation theory. When block sizes are highly non-uniform, the group-based Fourier basis can no longer be used. However, we show that partial information regarding the SBM Fourier basis can still be obtained from the underlying group.

Unified Fourier bases for signals on random graphs with group symmetries

TL;DR

This work addresses instance-independent graph signal processing on graphs drawn from stochastic block models by leveraging graphon theory. It demonstrates how to compute a Fourier basis for SBM samples from the SBM’s model and weighted-probability matrices, and shows that, for nearly uniform block sizes, the group-Fourier basis from Cayley structure provides a close approximation, with rigorous bounds via Davis–Kahan perturbation theory. The paper then extends to nonuniform block sizes, presenting a reduced-dimensional framework that partially recovers the SBM Fourier basis through a smaller matrix, revealing when symmetry-based bases remain informative. An explicit Z_5 example and transfer experiments illustrate the practical effectiveness and robustness of the approach, highlighting its potential for scalable, structure-aware GSP across families of graphs. Overall, the SBM-driven Fourier framework promises stable, interpretable signal processing on large, structured networks with group-like symmetries, even under perturbations of block sizes.

Abstract

We consider a recently proposed approach to graph signal processing (GSP) based on graphons. We show how the graphon-based approach to GSP applies to graphs sampled from a stochastic block model derived from a weighted Cayley graph. When SBM block sizes are equal, a nice Fourier basis can be derived from the representation theory of the underlying group. We explore how the SBM Fourier basis is affected when block sizes are not uniform. When block sizes are nearly uniform, we demonstrate that the group Fourier basis closely approximates the SBM Fourier basis. More specifically, we quantify the approximation error using matrix perturbation theory. When block sizes are highly non-uniform, the group-based Fourier basis can no longer be used. However, we show that partial information regarding the SBM Fourier basis can still be obtained from the underlying group.
Paper Structure (19 sections, 12 theorems, 56 equations, 4 figures, 1 table)

This paper contains 19 sections, 12 theorems, 56 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main contribution
  3. Organization of the paper
  4. Notations and background
  5. Graphons and the stochastic block model
  6. The graph/graphon Fourier transform
  7. The graphon Fourier transform for the SBM
  8. SBM and model matrix
  9. Computing the SBM Fourier basis
  10. Fourier basis for SBM with group symmetries
  11. ${\mathbb{G}}$-SBM with uniform measure
  12. General robustness results under block size perturbation
  13. ${\mathbb{G}}$-SBM with general measure
  14. Example: ${\mathbb{Z}}_5$-SBMs with different block sizes
  15. Comparing the graph Fourier basis with the SBM Fourier basis
  16. ...and 4 more sections

Key Result

Theorem 2.9

(Ghanehari-Janssen-Kalyaniwalla) \newlabelthm:convergence0 Let $\{G_n\}$ be a graph sequence converging to a graphon $w$ in cut norm, and $\{Y_n\}$ be a sequence of graph signals on each of the $G_n$, such that the corresponding sequence of step-functions $\{f_{Y_n}\}_n$ converges to a graphon sign where each $\phi^n_i$ is an eigenvector of $G_n$ corresponding to $\lambda^n_i$, represented as a s

Figures (4)

  • Figure 1: Graphon $w$ representing SBM, shown in the middle , taking value 0.8 on gray and 0.2 on white cells. Sampled graph on 60 vertices (left) and one on 600 vertices (right) are shown. For graphs, the vertices have been placed on $[0,1]$, and edges are represented by black pixels.
  • Figure 1: Comparison of transferred character basis with SBM Fourier basis for non-uniform measures
  • Figure 1: Algorithm to compute the SBM-driven Fourier transform of a signal $X$ on a graph sampled from ${\rm{SBM}}(A,\mu,N)$.
  • Figure 2: Comparison of graphon with 1 perturbed block versus 2 perturbed blocks

Theorems & Definitions (31)

  • Remark 2.1
  • Definition 2.2
  • Example 2.3: Graphs and matrices as graphons
  • Example 2.4: SBMs as graphons
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9
  • Definition 3.1
  • ...and 21 more