This work establishes rigorous, novel and widely applicable stability guarantees and transferability bounds for graph convolutional networks -- without reference to any underlying limit object or statistical distribution. Crucially, utilized graph-shift operators (GSOs) are not necessarily assumed to be normal, allowing for the treatment of networks on both directed- and for the first time also undirected graphs. Stability to node-level perturbations is related to an 'adequate (spectral) covering' property of the filters in each layer. Stability to edge-level perturbations is related to Lipschitz constants and newly introduced semi-norms of filters. Results on stability to topological perturbations are obtained through recently developed mathematical-physics based tools. As an important and novel example, it is showcased that graph convolutional networks are stable under graph-coarse-graining procedures (replacing strongly-connected sub-graphs by single nodes) precisely if the GSO is the graph Laplacian and filters are regular at infinity. These new theoretical results are supported by corresponding numerical investigations.

翻译：本工作为图卷积网络建立了严格、新颖且广泛适用的稳定性保证与可迁移性界——无需借助任何底层极限对象或统计分布。关键之处在于，无需假设所用图移位算子(GSO)是正规的，从而首次同时允许对有向图和无向图上的网络进行处理。节点层面扰动的稳定性与每层滤波器的"充分（谱）覆盖"性质相关；边层面扰动的稳定性则与滤波器的Lipschitz常数及新引入的半范数相关。通过基于近期发展的数学物理工具，获得了关于拓扑扰动稳定性的结果。作为一个重要且新颖的实例，本文展示了：当且仅当GSO为图拉普拉斯算子且滤波器在无穷远处正则时，图卷积网络在图粗粒化过程（将强连通子图替换为单个节点）下具有稳定性。这些新理论成果得到了相应数值实验的支持。