A Note on Graphon-Signal Analysis of Graph Neural Networks

TL;DR

This paper refines and extends the graphon-signal analysis framework for graph neural networks, addressing practical limitations and broadening applicability to multidimensional signals, non-symmetric graphons, and more robust generalization bounds.

Contribution

It introduces several key extensions to existing graphon-signal analysis, including multidimensional signals, readout Lipschitz continuity, improved bounds, and non-symmetric graphon analysis.

Findings

Extended analysis to multidimensional graphon-signals.

Improved generalization bounds using robustness techniques.

Analyzed non-symmetric graphons and kernels.

Abstract

A recent paper, ``A Graphon-Signal Analysis of Graph Neural Networks'', by Levie, analyzed message passing graph neural networks (MPNNs) by embedding the input space of MPNNs, i.e., attributed graphs (graph-signals), to a space of attributed graphons (graphon-signals). Based on extensions of standard results in graphon analysis to graphon-signals, the paper proved a generalization bound and a sampling lemma for MPNNs. However, there are some missing ingredients in that paper, limiting its applicability in practical settings of graph machine learning. In the current paper, we introduce several refinements and extensions to existing results that address these shortcomings. In detail, 1) we extend the main results in the paper to graphon-signals with multidimensional signals (rather than 1D signals), 2) we extend the Lipschitz continuity to MPNNs with readout with respect to cut distance (rather than MPNNs without readout with respect to cut metric), 3) we improve the generalization bound by utilizing robustness-type generalization bounds, and 4) we extend the analysis to non-symmetric graphons and kernels.