Bundle Neural Networks for message diffusion on graphs
<|reference_start|>Bundle Neural Networks for message diffusion on graphs: 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.<|reference_end|>
@article{bamberger2024bundle,
    title={Bundle Neural Networks for message diffusion on graphs},
    author={Jacob Bamberger, Federico Barbero, Xiaowen Dong, Michael Bronstein},
    journal={arXiv preprint arXiv:2405.15540},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.15540},
    primaryClass={cs.LG}
}