A Graphop Analysis of Graph Neural Networks on Sparse Graphs: Generalization and Universal Approximation

TL;DR

This paper introduces a unified graph metric framework that encompasses both sparse and dense graphs, leading to improved universal approximation and generalization results for message passing neural networks.

Contribution

It extends graph limit theory using graphop analysis to unify the study of MPNNs on all graph sizes, enhancing theoretical guarantees.

Findings

Unified metric for all graph sizes improves analysis.

Enhanced universal approximation theorems.

Stronger generalization bounds for MPNNs.

Abstract

Generalization and approximation capabilities of message passing graph neural networks (MPNNs) are often studied by defining a compact metric on a space of input graphs under which MPNNs are H\"older continuous. Such analyses are of two varieties: 1) when the metric space includes graphs of unbounded sizes, the theory is only appropriate for dense graphs, and, 2) when studying sparse graphs, the metric space only includes graphs of uniformly bounded size. In this work, we present a unified approach, defining a compact metric on the space of graphs of all sizes, both sparse and dense, under which MPNNs are H\"older continuous. This leads to more powerful universal approximation theorems and generalization bounds than previous works. The theory is based on, and extends, a recent approach to graph limit theory called graphop analysis.