  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.
