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.

翻译：本文研究了图卷积神经网络（GCNs）在底层图拓扑结构发生随机小扰动（即有限数量的边增删）条件下的稳定性问题。我们推导了未扰动与扰动GCNs输出期望差异的一个新颖上界。该上界显式依赖于拉普拉斯矩阵特征对扰动的幅度，而扰动幅度又显式取决于具体插入或删除的边。随后，我们定量刻画了特定边扰动对网络稳定性的影响。借助小扰动分析工具，我们将该上界表示为封闭的近似形式，以增强结果的可解释性，且无需计算任何扰动后的移位算子。最后，我们通过数值实验验证了所提出上界的有效性。