Liouville property and non-negative Ollivier curvature on graphs

TL;DR

This paper proves that graphs with non-negative Ollivier curvature have the Liouville property, meaning all bounded harmonic functions are constant, and also enhances understanding of measure concentration under positive curvature.

Contribution

It establishes the Liouville property for graphs with non-negative Ollivier curvature and improves previous results on measure concentration for positively curved graphs.

Findings

Graphs with non-negative Ollivier curvature have only constant bounded harmonic functions.

Enhanced bounds on measure concentration under positive Ollivier curvature.

Extended previous results on Ollivier curvature effects on graph properties.

Abstract

For graphs with non-negative Ollivier curvature, we prove the Liouville property, i.e., every bounded harmonic function is constant. Moreover, we improve Ollivier's results on concentration of the measure under positive Ollivier curvature.