Mean field information Hessian matrices on graphs

TL;DR

This paper develops mean-field information Hessian matrices on finite graphs, defining a new metric space based on entropy functions and nonlinear weights, with applications including entropy inequalities.

Contribution

It introduces a novel framework for Hessian matrices on graphs using mean-field entropy concepts, extending optimal transport and curvature bounds to discrete structures.

Findings

Derived Hessian matrices for various energies on graphs

Defined mean-field Ricci curvature bounds via eigenvalues

Proved discrete Costa's entropy power inequalities

Abstract

We derive mean-field information Hessian matrices on finite graphs. The "information" refers to entropy functions on the probability simplex. And the "mean-field" means nonlinear weight functions of probabilities supported on graphs. These two concepts define a mean-field optimal transport type metric. In this metric space, we first derive Hessian matrices of energies on graphs, including linear, interaction energies, entropies. We name their smallest eigenvalues as mean-field Ricci curvature bounds on graphs. We next provide examples on two-point spaces and graph products. We last present several applications of the proposed matrices. E.g., we prove discrete Costa's entropy power inequalities on a two-point space.