Oscillatory Neural Dynamics over Sheaves

Jan-Willem Van Looy; Alessandro Trenta; Alessio Gravina; Alessio Borgi; Ferdinando Zanchetta; Pietro Liò; Davide Bacciu; Rita Fioresi

arXiv · 2610.10018

Oscillatory Neural Dynamics over Sheaves

Jan-Willem Van Looy·Alessandro Trenta·Alessio Gravina·Alessio Borgi·Ferdinando Zanchetta·Pietro Liò·Davide Bacciu·Rita Fioresi

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Abstract

Effective long-range propagation remains a central challenge in graph neural networks, as increasing a model's propagation depth does not guarantee that distant nodes effectively influence each other. Sheaf neural networks enrich graph propagation through matrix-valued transport between stalks; still, this expressivity alone does not automatically imply effective long-range communication. We introduce ONDA, a long-range graph learning framework based on operator-valued information waves. Stalk-valued representations evolve through second-order dynamics governed by learned sheaf transport operators, combining wave-like propagation with expressive local geometry. We characterize long-range influence through a stalk-wise sensitivity analysis and show that the cross-influence never vanishes. Across long-range propagation, severe graph bottlenecks, graph transfer, and heterophilic benchmarks, ONDA consistently improves over scalar wave propagation, diffusive sheaf baselines, and state-of-the-art models, demonstrating the benefit of coupling wave dynamics with matrix-valued transport.

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Jan-Willem Van Looy, Alessandro Trenta, Alessio Gravina, Alessio Borgi, Ferdinando Zanchetta, Pietro Liò, Davide Bacciu, Rita Fioresi. 2026-10-07. Oscillatory Neural Dynamics over Sheaves. https://arxiv.org/abs/2610.10018

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