  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.
