Relation between combinatorial Ricci curvature and Lin-Lu-Yau's Ricci Curvature on cell complexes

TL;DR

This paper compares combinatorial Ricci curvature on cell complexes with Lin-Lu-Yau's Ricci curvature on graphs, using coupling and Kantorovich duality to analyze their relationship.

Contribution

It introduces a comparison framework between two types of Ricci curvature on cell complexes and graphs, highlighting their connections through coupling and duality methods.

Findings

Establishes a relationship between combinatorial Ricci and LLY-Ricci curvature.

Provides a method to compare curvature notions on cell complexes and graphs.

Enhances understanding of curvature transfer between cell complexes and their associated graphs.

Abstract

In this paper we compare the combinatorial Ricci curvature on cell complexes and the LLY-Ricci curvature defined on graphs. A cell complex is correspondence to a graph such that the vertexes are cells and the edges are vectors on the cell complex. We compare this two Ricci curvature by the cooupling and Kantrovich duality.