Stability of Graph Neural Networks to Relative Perturbations

TL;DR

This paper analyzes how graph neural networks (GNNs) respond to changes in the underlying graph structure, proving stability bounds and highlighting their ability to discriminate features on high eigenvalues, with practical experiments on recommendation systems.

Contribution

It provides theoretical bounds on GNN stability to graph perturbations and explains their discriminative power, supported by experiments.

Findings

GNN outputs are bounded by the relative change in graph topology.

GNNs can discriminate features on high eigenvalues, unlike linear filters.

Experiments demonstrate stability benefits in recommendation systems.

Abstract

Graph neural networks (GNNs), consisting of a cascade of layers applying a graph convolution followed by a pointwise nonlinearity, have become a powerful architecture to process signals supported on graphs. Graph convolutions (and thus, GNNs), rely heavily on knowledge of the graph for operation. However, in many practical cases the GSO is not known and needs to be estimated, or might change from training time to testing time. In this paper, we are set to study the effect that a change in the underlying graph topology that supports the signal has on the output of a GNN. We prove that graph convolutions with integral Lipschitz filters lead to GNNs whose output change is bounded by the size of the relative change in the topology. Furthermore, we leverage this result to show that the main reason for the success of GNNs is that they are stable architectures capable of discriminating features on high eigenvalues, which is a feat that cannot be achieved by linear graph filters (which are either stable or discriminative, but cannot be both). Finally, we comment on the use of this result to train GNNs with increased stability and run experiments on movie recommendation systems.