A Spectral Framework for Graph Neural Operators: Convergence Guarantees and Tradeoffs

TL;DR

This paper introduces a spectral framework for analyzing graph neural operators using graphons, providing convergence guarantees and tradeoffs, and demonstrating empirical tightness on various graphs.

Contribution

It unifies convergence results for GNNs under different graphon assumptions within a common operator framework, enabling comparison and transferability analysis.

Findings

Convergence rates depend on graphon regularity assumptions.

The framework applies to synthetic and real-world graphs.

Empirical results validate theoretical convergence rates.

Abstract

Graphons, as limits of graph sequences, provide an operator-theoretic framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons induces convergence of the corresponding neural operators, enabling transferability analyses of graph neural networks (GNNs). This paper develops a unified spectral framework that brings together convergence results under different assumptions on the underlying graphon, including no regularity, global Lipschitz continuity, and piecewise-Lipschitz continuity. The framework places these results in a common operator setting, enabling direct comparison of their assumptions, convergence rates, and tradeoffs. We further illustrate the empirical tightness of these rates on synthetic and real-world graphs.