On the Stability of Graph Convolutional Neural Networks: A Probabilistic Perspective
Ning Zhang, Henry Kenlay, Li Zhang, Mihai Cucuringu, Xiaowen Dong
TL;DR
This work addresses the stability of graph convolutional neural networks under graph perturbations by proposing a probabilistic, distribution-aware framework that ties embedding perturbations to the second-moment statistics of graph signals. It derives an exact expression for the expected perturbation of graph filters and a tight upper bound for multilayer GCNNs, highlighting how stability depends on the second-moment matrix $\\mathbf{K}$ and the filter perturbation $\\mathbf{E}_g$. The authors provide a structural interpretation of perturbations, validate the theory with a distribution-aware adversarial attack, Prob-PGD, and demonstrate that distribution-aware perturbations can cause larger embedding changes and greater downstream performance degradation than worst-case approaches. This framework enables incorporating data distribution into stability analyses, informing more realistic robustness assessments and potential defenses in graph-based learning systems.
Abstract
Graph convolutional neural networks (GCNNs) have emerged as powerful tools for analyzing graph-structured data, achieving remarkable success across diverse applications. However, the theoretical understanding of the stability of these models, i.e., their sensitivity to small changes in the graph structure, remains in rather limited settings, hampering the development and deployment of robust and trustworthy models in practice. To fill this gap, we study how perturbations in the graph topology affect GCNN outputs and propose a novel formulation for analyzing model stability. Unlike prior studies that focus only on worst-case perturbations, our distribution-aware formulation characterizes output perturbations across a broad range of input data. This way, our framework enables, for the first time, a probabilistic perspective on the interplay between the statistical properties of the node data and perturbations in the graph topology. We conduct extensive experiments to validate our theoretical findings and demonstrate their benefits over existing baselines, in terms of both representation stability and adversarial attacks on downstream tasks. Our results demonstrate the practical significance of the proposed formulation and highlight the importance of incorporating data distribution into stability analysis.