Weak Kantorovich difference and associated Ricci curvature of hypergraphs

TL;DR

This paper introduces a new variant of the Kantorovich difference to define Ricci curvature on hypergraphs, connecting it with the hypergraph Laplacian and analyzing its properties under certain conditions.

Contribution

The paper proposes a novel variant of the Kantorovich difference for hypergraphs and studies the associated Ricci curvature, extending the theory from graphs to hypergraphs.

Findings

Defined a new $ ext{ extsf{wIKTU}}$ curvature for hypergraphs.

Showed the curvature relates to the hypergraph Laplacian under certain conditions.

Analyzed the convergence of a quantity $ ext{ extsf{C}}(x,y)$ to the curvature.

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 M\"unch--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 $x,y$.