Bundle Neural Networks for message diffusion on graphs

TL;DR

This paper introduces Bundle Neural Networks (BuNN), a novel GNN architecture that uses message diffusion over vector bundles to overcome over-smoothing and over-squashing, achieving state-of-the-art results on graph tasks.

Contribution

The paper proposes BuNN, a new GNN model based on message diffusion over vector bundles, providing universal expressivity and improved scalability over traditional message passing neural networks.

Findings

BuNN mitigates over-smoothing and over-squashing issues.

BuNN achieves state-of-the-art performance on multiple benchmarks.

Theoretical proof of universal node-level expressivity.

Abstract

The dominant paradigm for learning on graph-structured data is message passing. Despite being a strong inductive bias, the local message passing mechanism suffers from pathological issues such as over-smoothing, over-squashing, and limited node-level expressivity. To address these limitations we propose Bundle Neural Networks (BuNN), a new type of GNN that operates via message diffusion over flat vector bundles - structures analogous to connections on Riemannian manifolds that augment the graph by assigning to each node a vector space and an orthogonal map. A BuNN layer evolves the features according to a diffusion-type partial differential equation. When discretized, BuNNs are a special case of Sheaf Neural Networks (SNNs), a recently proposed MPNN capable of mitigating over-smoothing. The continuous nature of message diffusion enables BuNNs to operate on larger scales of the graph and, therefore, to mitigate over-squashing. Finally, we prove that BuNN can approximate any feature transformation over nodes on any (potentially infinite) family of graphs given injective positional encodings, resulting in universal node-level expressivity. We support our theory via synthetic experiments and showcase the strong empirical performance of BuNNs over a range of real-world tasks, achieving state-of-the-art results on several standard benchmarks in transductive and inductive settings.