WEAK KANTOROVICH DIFFERENCE AND ASSOCIATED RICCI CURVATURE OF HYPERGRAPHS

Abstract

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 ($\text{wIKTU}$ 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 and the hypergraph Laplacian satisfies a certain property, then $\mathcal{C}(x,y)$ coincides with the $\text{wIKTU}$ curvature along $x,y$.