Mitigating Over-squashing without Rewiring: A Sheaf Effective Resistance Perspective

2026-10-03

AI Summary

This study addresses the oversquashing bottleneck in graph neural networks, wherein excessive compression impedes the capture of long-range dependencies. Drawing upon cellular sheaf theory, it introduces the concept of layer effective resistance to mitigate this issue by adjusting layer structures rather than modifying graph topology. Theoretically, the authors demonstrate that Jacobian sensitivity is bounded above by layer effective resistance. Building on this insight, they propose a FlatNSD message-passing variant and neural sheaf diffusion techniques to suppress oversquashing without graph rewiring. The resulting models achieve superior performance on stress-test benchmarks and can implicitly learn to regulate total layer effective resistance. This work establishes a novel paradigm for long-range modeling in graph networks by decoupling information flow optimization from topological alterations.

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.

Problem

• Graph Neural Networks
• Over-squashing
• Long-range dependencies
• Effective resistance
• Cellular sheaves

Innovation

• Over-squashing
• Sheaf Effective Resistance
• Graph Neural Networks
• Cellular Sheaves
• Neural Sheaf Diffusion