Oscillatory Neural Dynamics over Sheaves
Abstract Overview
This paper introduces ONDA, a sheaf neural network framework for long-range graph learning that combines second-order oscillatory dynamics with learned matrix-valued sheaf transport operators. Node representations evolve as stalk-valued states under a sheaf Laplacian, so propagation is shaped both by wave-like dynamics and by learned local geometric transformations on edges. The paper also provides a theoretical sensitivity analysis showing how source-to-target influence decomposes over paths and spectral components, with the key claim that cross-node influence in connected graphs does not vanish under the conservative oscillatory dynamics. Empirically, the method is evaluated on long-range propagation, graph bottleneck transfer, graph transfer, and heterophilic benchmarks against oscillatory, sheaf-based, and standard graph baselines.
Novelty
The distinctive contribution is the integration of oscillatory second-order wave dynamics into sheaf neural networks, yielding operator-valued information waves rather than scalar graph propagation. The work also contributes a stalk-wise sensitivity analysis that characterizes long-range influence through pathwise and spectral forms and argues that connected nodes retain non-vanishing cross-influence.
Results
ONDA achieves the best reported MAE on the ECHO-Synth SSSP and eccentricity tasks, while remaining competitive on diameter prediction. On the Barbell benchmark, it nearly solves the task for N=10 and N=20 and is the only reported method meeting the stated solving criterion at N=50. In the graph transfer experiments, it attains the lowest error at almost all tested distances across line, ring, and crossed-ring topologies.
Key Points
- ONDA models graph propagation through a sheaf Laplacian-driven second-order dynamical system, combining wave persistence with matrix-valued transport between stalks.
- The theoretical analysis gives pathwise and spectral expressions for source-to-target sensitivity and shows that, for connected nodes, oscillatory cross-influence does not converge to zero.
- Experiments on ECHO-Synth, Barbell, and graph transfer benchmarks indicate stronger long-range and bottleneck information transfer than scalar wave, diffusive sheaf, and several other graph learning baselines.