  Stability of graph neural networks (GNNs) characterizes how GNNs react to
graph perturbations and provides guarantees for architecture performance in
noisy scenarios. This paper develops a self-regularized graph neural network
(SR-GNN) solution that improves the architecture stability by regularizing the
filter frequency responses in the graph spectral domain. The SR-GNN considers
not only the graph signal as input but also the eigenvectors of the underlying
graph, where the signal is processed to generate task-relevant features and the
eigenvectors to characterize the frequency responses at each layer. We train
the SR-GNN by minimizing the cost function and regularizing the maximal
frequency response close to one. The former improves the architecture
performance, while the latter tightens the perturbation stability and
alleviates the information loss through multi-layer propagation. We further
show the SR-GNN preserves the permutation equivariance, which allows to explore
the internal symmetries of graph signals and to exhibit transference on similar
graph structures. Numerical results with source localization and movie
recommendation corroborate our findings and show the SR-GNN yields a comparable
performance with the vanilla GNN on the unperturbed graph but improves
substantially the stability.
