# Ollivier-Ricci idleness functions of graphs

## TL;DR

This paper investigates how the Ollivier-Ricci curvature of graphs varies with idleness, revealing its concave, piecewise linear nature with limited segments, and explores its behavior under Cartesian products of regular graphs.

## Contribution

It characterizes the idleness function of Ollivier-Ricci curvature as concave and piecewise linear with few segments, and determines its behavior for Cartesian products of regular graphs.

## Key findings

- Idleness function is concave and piecewise linear with at most 3 segments.
- In regular graphs, the idleness function has at most 2 segments.
- Idleness function of Cartesian product of regular graphs is determined by the factors.

## Abstract

We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most $3$ linear parts, with at most $2$ linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartesian product of two regular graphs is completely determined by the idleness functions of the factors.