Unified Fourier bases for signals on random graphs with group symmetries

TL;DR

This paper develops a unified Fourier basis framework for signals on random graphs with group symmetries, leveraging graphons and representation theory, and analyzes the effects of non-uniform block sizes in stochastic block models.

Contribution

It extends the graphon-based GSP approach to stochastic block models derived from weighted Cayley graphs, and analyzes Fourier basis approximation under varying block size distributions.

Findings

Group Fourier basis closely approximates SBM Fourier basis when block sizes are nearly uniform.

Approximation error is quantified using matrix perturbation theory.

Partial Fourier basis information can be obtained even with highly non-uniform 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.