Displacement convexity of entropy and related inequalities on graphs

TL;DR

This paper introduces a new concept of interpolating paths on finite graphs to establish convexity properties of entropy, leading to various inequalities that connect discrete graph settings with classical continuous results.

Contribution

It develops a novel framework for displacement convexity of entropy on graphs, deriving multiple inequalities and connecting discrete and continuous cases.

Findings

Proves displacement convexity of entropy on finite graphs.

Derives Prekopa-Leindler, Talagrand, HWI, and log-Sobolev inequalities in discrete settings.

Recovers classical Gaussian log-Sobolev inequality in the limit.

Abstract

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.