Analyzing the stability of graph neural networks (GNNs) under topological perturbations is key to understanding their transferability and the role of each architecture component. However, stability has been investigated only for particular architectures, questioning whether it holds for a broader spectrum of GNNs or only for a few instances. To answer this question, we study the stability of EdgeNet: a general GNN framework that unifies more than twenty solutions including the convolutional and attention-based classes, as well as graph isomorphism networks and hybrid architectures. We prove that all GNNs within the EdgeNet framework are stable to topological perturbations. By studying the effect of different EdgeNet categories on the stability, we show that GNNs with fewer degrees of freedom in their parameter space, linked to a lower representational capacity, are more stable. The key factor yielding this trade-off is the eigenvector misalignment between the EdgeNet parameter matrices and the graph shift operator. For example, graph convolutional neural networks that assign a single scalar per signal shift (hence, with a perfect alignment) are more stable than the more involved node or edge-varying counterparts. Extensive numerical results corroborate our theoretical findings and highlight the role of different architecture components in the trade-off.

验证了在图神经网络中引入拓扑扰动的稳定性，针对不同结构的EdgeNet框架证明其稳定性，以及参数空间的自由度与代表能力之间的关系和边缘网络参数矩阵与图移动算子的特征向量错配程度对稳定性的影响。

图神经网络中稳定性与表示能力之间的权衡