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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2306.14084 |
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| _version_ | 1866913948897378304 |
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| author | Akamatsu, Tomoya |
| author_facet | Akamatsu, Tomoya |
| contents | Ollivier and Lin--Lu--Yau established the theory of graph Ricci curvature (LLY curvature) via optimal transport on graphs. Ikeda--Kitabeppu--Takai--Uehara introduced a new distance called the Kantorovich difference on hypergraphs and generalized the LLY curvature to hypergraphs (IKTU curvature). As the LLY curvature can be represented by the graph Laplacian by Münch--Wojciechowski, Ikeda--Kitabeppu--Takai--Uehara conjectured that the IKTU curvature has a similar expression in terms of the hypergraph Laplacian. In this paper, we introduce a variant of the Kantorovich difference inspired by the above conjecture and study the Ricci curvature associated with this distance ($\mathsf{wIKTU}$ curvature). Moreover, for hypergraphs with a specific structure, we analyze a quantity $\mathcal{C}(x,y)$ at two distinct vertices $x,y$ defined by using the hypergraph Laplacian. If the resolvent operator converges uniformly to the identity, then $\mathcal{C}(x,y)$ coincides with the $\mathsf{wIKTU}$ curvature along $x,y$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14084 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak Kantorovich difference and associated Ricci curvature of hypergraphs Akamatsu, Tomoya Metric Geometry Combinatorics Primary 51F30, Secondary 05C65, 05C12, 47H04 Ollivier and Lin--Lu--Yau established the theory of graph Ricci curvature (LLY curvature) via optimal transport on graphs. Ikeda--Kitabeppu--Takai--Uehara introduced a new distance called the Kantorovich difference on hypergraphs and generalized the LLY curvature to hypergraphs (IKTU curvature). As the LLY curvature can be represented by the graph Laplacian by Münch--Wojciechowski, Ikeda--Kitabeppu--Takai--Uehara conjectured that the IKTU curvature has a similar expression in terms of the hypergraph Laplacian. In this paper, we introduce a variant of the Kantorovich difference inspired by the above conjecture and study the Ricci curvature associated with this distance ($\mathsf{wIKTU}$ curvature). Moreover, for hypergraphs with a specific structure, we analyze a quantity $\mathcal{C}(x,y)$ at two distinct vertices $x,y$ defined by using the hypergraph Laplacian. If the resolvent operator converges uniformly to the identity, then $\mathcal{C}(x,y)$ coincides with the $\mathsf{wIKTU}$ curvature along $x,y$. |
| title | Weak Kantorovich difference and associated Ricci curvature of hypergraphs |
| topic | Metric Geometry Combinatorics Primary 51F30, Secondary 05C65, 05C12, 47H04 |
| url | https://arxiv.org/abs/2306.14084 |