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

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 Hölder 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 Hölder 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.

💡 Research Summary

This paper tackles a long‑standing gap in the theoretical analysis of message‑passing graph neural networks (MPNNs): existing frameworks either handle graphs of unbounded size but only for dense regimes, or they accommodate sparse graphs at the cost of restricting the graph size. The authors propose a unified approach that works for both sparse and dense graphs of any size by leveraging and extending recent graph limit theory known as graphop analysis.

A graphop models a graph as a bounded linear operator acting on Lᵖ(Ω) functions, where Ω represents the vertex space. Unlike graphons, which are kernel operators on L∞