On the Stability of Low Pass Graph Filter With a Large Number of Edge Rewires

TL;DR

This paper investigates the stability of low pass graph filters under large edge rewiring, revealing that stability depends on community structure perturbations and demonstrating convergence in stochastic block models.

Contribution

It provides a novel stability bound for low pass graph filters under extensive edge rewiring, extending previous results to large perturbations.

Findings

Stability depends on community structure perturbation.

Graph filter distance converges to zero in large stochastic block models.

Numerical simulations support theoretical results.

Abstract

Recently, the stability of graph filters has been studied as one of the key theoretical properties driving the highly successful graph convolutional neural networks (GCNs). The stability of a graph filter characterizes the effect of topology perturbation on the output of a graph filter, a fundamental building block for GCNs. Many existing results have focused on the regime of small perturbation with a small number of edge rewires. However, the number of edge rewires can be large in many applications. To study the latter case, this work departs from the previous analysis and proves a bound on the stability of graph filter relying on the filter's frequency response. Assuming the graph filter is low pass, we show that the stability of the filter depends on perturbation to the community structure. As an application, we show that for stochastic block model graphs, the graph filter distance converges to zero when the number of nodes approaches infinity. Numerical simulations validate our findings.