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

Stability of Graph Neural Networks to Relative Perturbations