Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures
- URL: http://arxiv.org/abs/2510.04936v1
- Date: Mon, 06 Oct 2025 15:43:07 GMT
- Title: Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures
- Abstract summary: In the discrete setting, Ricci curvature is replaced by Ollivier-Ricci curvature.<n>We show that our definition converges to scalar curvature for nearest neighbor graphs obtained by sampling from a manifold.
- Score: 4.891498111882615
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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