Stability of Graph Convolutional Neural Networks to Stochastic Perturbations
This work addresses the stability of GCNNs to random topological changes, which is crucial for applications like source localization and robot swarm control, but it is incremental as it extends existing deterministic analyses to stochastic perturbations.
The paper tackles the problem of analyzing the stability of graph convolutional neural networks (GCNNs) to stochastic graph perturbations, specifically link losses, and proves that the expected output difference is linearly bounded by the link loss probability, with numerical simulations validating the findings.
Graph convolutional neural networks (GCNNs) are nonlinear processing tools to learn representations from network data. A key property of GCNNs is their stability to graph perturbations. Current analysis considers deterministic perturbations but fails to provide relevant insights when topological changes are random. This paper investigates the stability of GCNNs to stochastic graph perturbations induced by link losses. In particular, it proves the expected output difference between the GCNN over random perturbed graphs and the GCNN over the nominal graph is upper bounded by a factor that is linear in the link loss probability. We perform the stability analysis in the graph spectral domain such that the result holds uniformly for any graph. This result also shows the role of the nonlinearity and the architecture width and depth, and allows identifying handle to improve the GCNN robustness. Numerical simulations on source localization and robot swarm control corroborate our theoretical findings.