Weak Kantorovich difference and associated Ricci curvature of hypergraphs (Q6621597)

A hypergraph is a natural generalization of a graph. A graph describes binary relations by connecting two vertices with an edge, while in a hypergraph one can connect three or more vertices with a hyperedge.\N\NRicci curvature is one of the most important quantities in Riemannian geometry, and its generalization to discrete spaces has been studied widely in recent years. Several types of discrete Ricci curvature have been studied, including Forman-type and Bakry-Émery-type Ricci curvatures.\N\N\textit{Y. Ollivier} [J. Funct. Anal. 256, No. 3, 810--864 (2009; Zbl 1181.53015)] and \textit{Y. Lin} et al. [Tôhoku Math. J. (2) 63, No. 4, 605--627 (2011; Zbl 1237.05204)] established the theory of graph Ricci curvature (LLY curvature) via optimal transport on graphs. \textit{M. Ikeda} et al. [``Coarse Ricci curvature of hypergraphs and its generalization'', Preprint, \url{arXiv:2102.00698}] introduced a new distance called the Kantorovich difference on hypergraphs and generalized the LLY curvature to hypergraphs (IKTU curvature).\N\NAs the LLY curvature can be represented by the graph Laplacian by \textit{F. Münch} and \textit{R. K. Wojciechowski} [Adv. Math. 356, Article ID 106759, 45 p. (2019; Zbl 1426.8205)], {M. Ikeda} et al. [loc. cit.] conjectured that the IKTU curvature has a similar expression in terms of the hypergraph Laplacian.\N\NHere, the author introduces a variant of the Kantorovich difference inspired by the above conjecture and study the Ricci curvature associated with this distance (wIKTU curvature). Moreover, for hypergraphs with a specific structure, the authors analyze a quantity \(C(x; y)\) at two distinct vertices \(x, y\) defined by using the hypergraph Laplacian. If the resolvent operator converges uniformly to the identity and the hypergraph Laplacian satisfies a certain property, then \(C(x; y)\) coincides with the wIKTU curvature along \(x, y\).\N\NThe paper gives a beautiful survey and contains interesting results. Young researchers will get motivation from this work.