Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications
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arXiv
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| Format: | Preprint |
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2025
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| author | de Souza, Danillo Barros Teodomiro, Jonatas Santos, Fernando A. N. Ding, Mengjun Sun, Weiqiang Desroches, Mathieu Jost, Jürgen Rodrigues, Serafim |
| author_facet | de Souza, Danillo Barros Teodomiro, Jonatas Santos, Fernando A. N. Ding, Mengjun Sun, Weiqiang Desroches, Mathieu Jost, Jürgen Rodrigues, Serafim |
| contents | Discrete Forman-Ricci curvature (FRC) is an efficient tool that characterizes essential geometrical features and associated transitions of real-world networks, extending seamlessly to higher-dimensional computations in simplicial complexes. In this article, we provide two major advancements: First, we give a decomposition for FRC, enabling local computations of FRC. Second, we construct a set-theoretical proof enabling an efficient algorithm for the local computation of FRC in Vietoris-Rips (VR) complexes.Strikingly, this approach reveals critical information and geometric insights often overlooked by conventional classification techniques. Our findings open new avenues for geometric computations in VR complexes and highlight an essential yet under-explored aspect of data classification: the geometry underpinning statistical patterns. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications de Souza, Danillo Barros Teodomiro, Jonatas Santos, Fernando A. N. Ding, Mengjun Sun, Weiqiang Desroches, Mathieu Jost, Jürgen Rodrigues, Serafim Geometric Topology Computational Geometry Discrete Mathematics Data Structures and Algorithms Combinatorics 05C85, 52C99, 90C35, 62R40, 68W99, 68T09 Discrete Forman-Ricci curvature (FRC) is an efficient tool that characterizes essential geometrical features and associated transitions of real-world networks, extending seamlessly to higher-dimensional computations in simplicial complexes. In this article, we provide two major advancements: First, we give a decomposition for FRC, enabling local computations of FRC. Second, we construct a set-theoretical proof enabling an efficient algorithm for the local computation of FRC in Vietoris-Rips (VR) complexes.Strikingly, this approach reveals critical information and geometric insights often overlooked by conventional classification techniques. Our findings open new avenues for geometric computations in VR complexes and highlight an essential yet under-explored aspect of data classification: the geometry underpinning statistical patterns. |
| title | Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications |
| topic | Geometric Topology Computational Geometry Discrete Mathematics Data Structures and Algorithms Combinatorics 05C85, 52C99, 90C35, 62R40, 68W99, 68T09 |
| url | https://arxiv.org/abs/2504.21601 |