Stability of Graph Convolutional Neural Networks through the lens of small perturbation analysis
This work addresses stability issues for GCNs in applications like social networks or recommendation systems, but it is incremental as it builds on existing perturbation analysis.
The authors tackled the problem of stability in Graph Convolutional Neural Networks (GCNs) under random small perturbations in graph topology, deriving a novel bound on the expected output difference that depends on eigenpair perturbations and edge changes, and numerically evaluated its effectiveness.
In this work, we study the problem of stability of Graph Convolutional Neural Networks (GCNs) under random small perturbations in the underlying graph topology, i.e. under a limited number of insertions or deletions of edges. We derive a novel bound on the expected difference between the outputs of unperturbed and perturbed GCNs. The proposed bound explicitly depends on the magnitude of the perturbation of the eigenpairs of the Laplacian matrix, and the perturbation explicitly depends on which edges are inserted or deleted. Then, we provide a quantitative characterization of the effect of perturbing specific edges on the stability of the network. We leverage tools from small perturbation analysis to express the bounds in closed, albeit approximate, form, in order to enhance interpretability of the results, without the need to compute any perturbed shift operator. Finally, we numerically evaluate the effectiveness of the proposed bound.