Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918155119493120 |
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| author | Hickok, Abigail Blumberg, Andrew J. |
| author_facet | Hickok, Abigail Blumberg, Andrew J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04936 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures Hickok, Abigail Blumberg, Andrew J. Discrete Mathematics Computational Geometry Social and Information Networks Machine Learning 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. |
| title | Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures |
| topic | Discrete Mathematics Computational Geometry Social and Information Networks Machine Learning |
| url | https://arxiv.org/abs/2510.04936 |