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. AMPNN 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.

In this section, we provide background information in probability theory, and focus on random processes and concentration of measure inequalities.

Wecanseethis as the slope of the update function changes (middle row of Figure 1), these green lines correspond tothelocations givenbythearrowsinthetoprow.

Learning object-centric representations of complex scenes is a promising step towards enabling efficient abstract reasoning from low-levelperceptual features. Yet, most deep learning approaches learn distributed representations that do not capture the compositional properties of natural scenes.