Stable and Transferable Hyper-Graph Neural Networks

TL;DR

This paper introduces Hyper-graph Expansion Neural Networks (HENN) for processing signals on hypergraphs, providing new theoretical bounds on their stability and transferability, and demonstrating superior performance in transfer tasks.

Contribution

The paper presents the first stability and transferability bounds for hypergraph neural networks, linking spectral similarity of graphs to GNN performance and extending transferability results.

Findings

HENN outperforms existing models in transferability tasks.

Stability improves with larger graphs and spectral similarity.

Theoretical bounds relate GNN stability to spectral properties.

Abstract

We introduce an architecture for processing signals supported on hypergraphs via graph neural networks (GNNs), which we call a Hyper-graph Expansion Neural Network (HENN), and provide the first bounds on the stability and transferability error of a hypergraph signal processing model. To do so, we provide a framework for bounding the stability and transferability error of GNNs across arbitrary graphs via spectral similarity. By bounding the difference between two graph shift operators (GSOs) in the positive semi-definite sense via their eigenvalue spectrum, we show that this error depends only on the properties of the GNN and the magnitude of spectral similarity of the GSOs. Moreover, we show that existing transferability results that assume the graphs are small perturbations of one another, or that the graphs are random and drawn from the same distribution or sampled from the same graphon can be recovered using our approach. Thus, both GNNs and our HENNs (trained using normalized Laplacians as graph shift operators) will be increasingly stable and transferable as the graphs become larger. Experimental results illustrate the importance of considering multiple graph representations in HENN, and show its superior performance when transferability is desired.