Limitless stability for Graph Convolutional Networks

TL;DR

This paper provides new, rigorous stability guarantees for graph convolutional networks that apply to both directed and undirected graphs, including stability under various perturbations and graph-coarse-graining.

Contribution

It introduces the first stability bounds for GCNs on directed graphs and under graph-coarse-graining, without relying on statistical distribution assumptions.

Findings

GCNs are stable under node and edge perturbations with spectral and Lipschitz conditions.

Stability under graph-coarse-graining is achieved when using the graph Laplacian and regular filters.

Numerical experiments support the theoretical stability guarantees.

Abstract

This work establishes rigorous, novel and widely applicable stability guarantees and transferability bounds for graph convolutional networks -- without reference to any underlying limit object or statistical distribution. Crucially, utilized graph-shift operators (GSOs) are not necessarily assumed to be normal, allowing for the treatment of networks on both undirected- and for the first time also directed graphs. Stability to node-level perturbations is related to an 'adequate (spectral) covering' property of the filters in each layer. Stability to edge-level perturbations is related to Lipschitz constants and newly introduced semi-norms of filters. Results on stability to topological perturbations are obtained through recently developed mathematical-physics based tools. As an important and novel example, it is showcased that graph convolutional networks are stable under graph-coarse-graining procedures (replacing strongly-connected sub-graphs by single nodes) precisely if the GSO is the graph Laplacian and filters are regular at infinity. These new theoretical results are supported by corresponding numerical investigations.