arXiv:2608.06405 2026 Geometry 1 ideas extracted · analyzed Aug 31, 2026
The paper gives a discrete Gauss–Bonnet construction for a graph viewed through its simplicial structure: a vertex curvature is computed entirely from simplex counts in its unit sphere, and the curvatures sum exactly to the Euler characteristic. This creates a topology-aware, locally computable signal with a global conservation law, rather than an arbitrary graph statistic. A practical neural-network transfer is to use curvature as an attention or message-passing bias and to regularize graph representations with topology-derived features. The strongest initial test is on graph classification or node classification datasets where clique topology, not only degree, carries label information.
Compute each graph node's discrete curvature from the numbers of simplices in its neighbor-induced unit sphere, then inject this scalar into message-passing or attention logits. Add an optional topology-aware feature channel so that nodes with identical degree but different local clique structure receive different representations.