Li-Yau inequality on finite graphs via non-linear curvature dimension conditions

TL;DR

This paper introduces a new non-linear curvature-dimension inequality for finite graphs, leading to Li-Yau and Harnack inequalities, and generalizes curvature concepts from manifolds to graphs.

Contribution

It develops a novel non-linear calculus and curvature-dimension inequality that unify curvature notions on graphs and manifolds, enabling new gradient and Harnack inequalities.

Findings

Proves a logarithmic Li-Yau inequality on finite graphs.

Shows Ricci-flat graphs have non-negative curvature in this framework.

Derives various Li-Yau type gradient estimates and Harnack inequalities.

Abstract

We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li-Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and \'Emery. In the case of manifolds, the new calculus and the new curvature-dimension inequality coincide with the common ones. In the case of graphs, they coincide in a limit. In this sense, the new curvature-dimension inequality gives a more general concept of curvature on graphs and on manifolds. We show that Ricci-flat graphs have a non-negative curvature in this sense. Moreover, a variety of non-logarithmic Li-Yau type gradient estimates can be obtained by using the new Bakry-\'Emery type calculus. Furthermore, we use these Li-Yau inequalities to derive Harnack inequalities on graphs.