On the Stability of Low Pass Graph Filter With a Large Number of Edge Rewires
This work addresses a theoretical gap in graph neural networks by analyzing filter stability for large perturbations, which is incremental but relevant for applications with significant graph changes.
The paper tackles the stability of low-pass graph filters under large numbers of edge rewires, proving a bound based on the filter's frequency response and showing that stability depends on perturbation to community structure, with numerical simulations validating the findings.
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.