A new transport distance and its associated Ricci curvature of hypergraphs

TL;DR

This paper introduces a novel transport distance tailored for hypergraphs, generalizes Ricci curvature concepts for hypergraphs, and derives geometric bounds, offering new tools for hypergraph analysis.

Contribution

It proposes a new transport distance for hypergraphs and extends Ricci curvature notions, providing a foundation for further geometric and combinatorial studies.

Findings

Derived a Bonnet-Myers type estimate for hypergraphs.

Introduced a new transport distance applicable to graphs and hypergraphs.

Generalized Lin-Lu-Yau curvature to hypergraphs.

Abstract

The coarse Ricci curvature of graphs introduced by Ollivier as well as its modification by Lin-Lu-Yau have been studied from various aspects. In this paper, we propose a new transport distance appropriate for hypergraphs and study a generalization of Lin-Lu-Yau type curvature of hypergraphs. As an application, we derive a Bonnet-Myers type estimate for hypergraphs under a lower Ricci curvature bound associated with our transport distance. We remark that our transport distance is new even for graphs and worthy of further study.