Lin-Lu-Yau Ricci curvature on hypergraphs

TL;DR

This paper extends the Lin-Lu-Yau Ricci curvature framework to hypergraphs, establishing theoretical properties and a Bonnet-Myers-type theorem, thereby enhancing tools for analyzing hypergraph geometry and structure.

Contribution

It introduces a unified LLY Ricci curvature definition for hypergraphs, proving key properties and a Bonnet-Myers-type theorem, expanding geometric analysis tools to hypergraph structures.

Findings

Established upper bounds and monotonicity of curvature parameter $_$

Proved a Bonnet-Myers-type theorem for hypergraphs

Extended graph Ricci curvature concepts to hypergraphs

Abstract

In this paper, we introduce a unified framework for defining Lin-Lu-Yau (LLY) Ricci curvature on both undirected and directed hypergraphs. By establishing upper bounds and monotonicity properties for the parameterized curvature $\kappa_\alpha$, we justify the well-posedness and compatibility of our definitions. Furthermore, we prove a Bonnet-Myers-type theorem for hypergraphs, which highlights the potential of LLY Ricci curvature in hypergraph analysis, particularly in studying geometric and structural properties. Our results extends the foundational definitions of graph Ricci curvature by Ollivier and Lin-Lu-Yau to the hypergraph setting.