We present an approach for analyzing message passing graph neural networks (MPNNs) based on an extension of graphon analysis to a so called graphon-signal analysis. A MPNN is a function that takes a graph and a signal on the graph (a graph-signal) and returns some value. Since the input space of MPNNs is non-Euclidean, i.e., graphs can be of any size and topology, properties such as generalization are less well understood for MPNNs than for Euclidean neural networks. We claim that one important missing ingredient in past work is a meaningful notion of graph-signal similarity measure, that endows the space of inputs to MPNNs with a regular structure. We present such a similarity measure, called the graphon-signal cut distance, which makes the space of all graph-signals a dense subset of a compact metric space -- the graphon-signal space. Informally, two deterministic graph-signals are close in cut distance if they ``look like'' they were sampled from the same random graph-signal model. Hence, our cut distance is a natural notion of graph-signal similarity, which allows comparing any pair of graph-signals of any size and topology. We prove that MPNNs are Lipschitz continuous functions over the graphon-signal metric space. We then give two applications of this result: 1) a generalization bound for MPNNs, and, 2) the stability of MPNNs to subsampling of graph-signals. Our results apply to any regular enough MPNN on any distribution of graph-signals, making the analysis rather universal.

翻译：本文提出了一种基于图论信号分析扩展至所谓图信号分析的方法，用于分析消息传递图神经网络（MPNNs）。MPNN是一种接收图及其信号（图信号）并返回数值的函数。由于MPNN的输入空间为非欧几里得空间，即图可具有任意尺寸与拓扑结构，因此与欧几里得神经网络相比，MPNN的泛化性等性质尚不明确。我们认为，过往研究缺失的关键要素在于缺乏有意义的图信号相似性度量，该度量能为MPNN的输入空间赋予规则结构。我们提出了一种名为"图信号切割距离"的相似性度量，该度量使所有图信号构成的集合成为紧度量空间（即图信号空间）的稠密子集。非正式地，若两个确定性图信号在切割距离下"看起来"像从同一随机图信号模型中采样所得，则它们彼此接近。因此，我们的切割距离是一种自然的图信号相似性概念，可比较任意尺寸与拓扑结构的图信号对。我们证明MPNNs在图信号度量空间上是Lipschitz连续函数，并给出该结果的两项应用：1）MPNN的泛化界；2）MPNN对图信号子采样的稳定性。该结论适用于任意足够规则的MPNN及任意图信号分布，具有广泛适用性。