Graph and graphon neural network stability

TL;DR

This paper investigates the stability of graph neural networks (GNNs) using graphons, showing that GNNs are stable to graphon perturbations with bounds that improve as graph size increases, supported by theoretical analysis and experiments.

Contribution

It introduces a novel framework using graphons to analyze GNN stability and extends the theory to stochastic and deterministic graphs derived from graphons.

Findings

GNNs are stable to graphon perturbations with bounds decreasing asymptotically.

The stability bound improves as the size of the graph increases.

Experimental validation on movie recommendation demonstrates the theoretical results.

Abstract

Graph neural networks (GNNs) are learning architectures that rely on knowledge of the graph structure to generate meaningful representations of large-scale network data. GNN stability is thus important as in real-world scenarios there are typically uncertainties associated with the graph. We analyze GNN stability using kernel objects called graphons. Graphons are both limits of convergent graph sequences and generating models for deterministic and stochastic graphs. Building upon the theory of graphon signal processing, we define graphon neural networks and analyze their stability to graphon perturbations. We then extend this analysis by interpreting the graphon neural network as a generating model for GNNs on deterministic and stochastic graphs instantiated from the original and perturbed graphons. We observe that GNNs are stable to graphon perturbations with a stability bound that decreases asymptotically with the size of the graph. This asymptotic behavior is further demonstrated in an experiment of movie recommendation.