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This research addresses the "over-squashing" bottleneck in Graph Neural Networks by introducing sheaf effective resistance, a metric that allows modulating information flow without altering graph topology. The proposed FlatNSD method demonstrates that adjusting the underlying sheaf can effectively mitigate representation collapse on long-range dependency benchmarks. This theoretical advance offers a novel architectural approach to enhancing GNN expressivity through algebraic adjustments rather than structural rewiring.
Mitigating Over-squashing without Rewiring: A Sheaf Effective Resistance Perspective
arXiv:2610.04157v1 Announce Type: cross Abstract: Graph Neural Networks (GNNs) often struggle to capture long-range dependencies due to over-squashing -- a phenomenon in which the repeated compression of node embeddings into finite-size messages causes representations to collapse. Over-squashing is most often diagnosed as a property of the graph topology, with effective resistance serving as a principled measure of the bottleneck. We provide a complementary view on the matter: building on cellular sheaves, we introduce sheaf effective resistance, a generalization of effective resistance that depends on the sheaf attached to the graph, and we prove that for flat vector bundles, the over-squashing sensitivity in the Jacobian sense is upper bounded by a quantity related to the sheaf effective resistance between the nodes. The bottleneck thus need not lie in the graph itself: it can be relocated, and reduced, by adjusting the sheaf. We instantiate this idea in FlatNSD, a simple message-passing variant of Neural Sheaf Diffusion, and show that it implicitly learns to modulate total sheaf effective resistance, performing well on benchmarks designed to stress over-squashing without altering the original graph topology.
Abstract:Graph Neural Networks (GNNs) often struggle to capture long-range dependencies due to over-squashing -- a phenomenon in which the repeated compression of node embeddings into finite-size messages causes representations to collapse. Over-squashing is most often diagnosed as a property of the graph topology, with effective resistance serving as a principled measure of the bottleneck. We provide a complementary view on the matter: building on cellular sheaves, we introduce sheaf effective resistance, a generalization of effective resistance that depends on the sheaf attached to the graph, and we prove that for flat vector bundles, the over-squashing sensitivity in the Jacobian sense is upper bounded by a quantity related to the sheaf effective resistance between the nodes. The bottleneck thus need not lie in the graph itself: it can be relocated, and reduced, by adjusting the sheaf. We instantiate this idea in FlatNSD, a simple message-passing variant of Neural Sheaf Diffusion, and show that it implicitly learns to modulate total sheaf effective resistance, performing well on benchmarks designed to stress over-squashing without altering the original graph topology.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2610.04157 [stat.ML]
(or arXiv:2610.04157v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.04157 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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