Bundle Neural Network for message diffusion on graphs
Jacob Bamberger Federico Barbero Xiaowen Dong Michael M. Bronstein
The dominant paradigm for learning on graphs is message passing. Despite being a strong inductive bias, the local message passing mechanism faces challenges such as over-smoothing, over-squashing, and limited expressivity. To address these issues, we introduce Bundle Neural Networks (BuNNs), a novel graph neural network architecture that operates via *message diffusion* on *flat vector bundles* — geometrically inspired structures that assign to each node a vector space and an orthogonal map. A BuNN layer evolves node features through a diffusion-type partial differential equation, where its discrete form acts as a special case of the recently introduced Sheaf Neural Network (SNN), effectively alleviating over-smoothing. The continuous nature of message diffusion enables BuNNs to operate at larger scales, reducing over-squashing. We establish the universality of BuNNs in approximating feature transformations on infinite families of graphs with injective positional encodings, marking the first positive expressivity result of its kind. We support our claims with formal analysis and synthetic experiments. Empirically, BuNNs perform strongly on heterophilic and long-range tasks, which demonstrates their robustness on a diverse range of challenging real-world tasks.
graph neural network, sheaf neural network, geometric deep learning, algebraic topology, vector bundles, expressivity
We propose Bundle Neural Networks, a new type of Graph Neural Network that operates via message diffusion, a continuous version of message-passing that allows to mitigate over-smoothing, over-squashing, and is provably expressive.