Long scale Ollivier-Ricci curvature of graphs

TL;DR

This paper investigates the properties of long scale Ollivier-Ricci curvature in graphs, revealing its concave, piecewise linear nature and analyzing curvature in Cartesian products of regular graphs.

Contribution

It extends previous work on short scale curvature to long scale, providing bounds and analyzing curvature behavior in graph products.

Findings

Curvature function is concave and piecewise linear with up to 3 segments.

Bounds are established for the lengths of the first and last linear parts.

Curvature behavior in Cartesian products of regular graphs is characterized.

Abstract

We study the long scale Ollivier-Ricci curvature of graphs as a function of the chosen idleness. As in the previous work on the short scale, we show that this idleness function is concave and piecewise linear with at most $3$ linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long scale curvature inside the Cartesian product of two regular graphs.