Unified Fourier bases for signals on random graphs with group symmetries

2024-06-10

AI Summary

This work addresses Fourier analysis on stochastic block model (SBM) graphs leveraging group symmetry, specifically investigating the relationship between the SBM’s graph Fourier basis and the underlying group’s Fourier basis under non-uniform block sizes. Method: We develop the first theoretical framework—grounded in group representation theory—that characterizes the spectral structure of SBMs with heterogeneous block sizes. Our approach quantitatively bounds the approximation error incurred when using the group Fourier basis to approximate the SBM graph Fourier basis. Contribution/Results: We prove that the two bases align with controllable error when block sizes are approximately uniform; remarkably, even under severe block-size imbalance, key spectral structural properties of the SBM remain recoverable. This work establishes the first systematic bridge among graph signal processing, graphon theory, and group harmonic analysis—achieving theoretical unification without sacrificing computational tractability. It introduces a novel paradigm for spectral modeling of symmetric random graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Explores Fourier bases for signals on random graphs with group symmetries.
• Analyzes impact of non-uniform block sizes on SBM Fourier basis.
• Quantifies approximation error using matrix perturbation theory.

Innovation

• Graphon-based approach for graph signal processing
• Fourier basis derived from group representation theory
• Matrix perturbation theory quantifies approximation error