{"ID":2859948,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.04936","arxiv_id":"2510.04936","title":"Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures","abstract":"We study the relationship between discrete analogues of Ricci and scalar curvature that are defined for point clouds and graphs. In the discrete setting, Ricci curvature is replaced by Ollivier-Ricci curvature. Scalar curvature can be computed as the trace of Ricci curvature for a Riemannian manifold; this motivates a new definition of a scalar version of Ollivier-Ricci curvature. We show that our definition converges to scalar curvature for nearest neighbor graphs obtained by sampling from a manifold. We also prove some new results about the convergence of Ollivier-Ricci curvature to Ricci curvature.","short_abstract":"We study the relationship between discrete analogues of Ricci and scalar curvature that are defined for point clouds and graphs. In the discrete setting, Ricci curvature is replaced by Ollivier-Ricci curvature. Scalar curvature can be computed as the trace of Ricci curvature for a Riemannian manifold; this motivates a...","url_abs":"https://arxiv.org/abs/2510.04936","url_pdf":"https://arxiv.org/pdf/2510.04936v1","authors":"[\"Abigail Hickok\",\"Andrew J. Blumberg\"]","published":"2025-10-06T15:43:07Z","proceeding":"cs.DM","tasks":"[\"cs.DM\",\"cs.CG\",\"cs.SI\",\"stat.ML\"]","methods":"[]","has_code":false}