On extreme points of the diffusion polytope

TL;DR

This paper investigates the structure of the diffusion polytope in graph-based diffusion problems, characterizing its extreme points for various graph types and initial conditions, with implications for physical and optimization problems.

Contribution

It provides a detailed analysis of the extreme points of the diffusion polytope for path and cyclic graphs, extending known results from complete graphs, and describes its topology in specific cases.

Findings

Diffusion polytope for $P_n$ is an $n$-dimensional hypercube with increasing initial populations.

Characterization of extreme points varies with graph topology and initial conditions.

The study enhances understanding of feasible regions in physical and optimization problems.

Abstract

We consider a class of diffusion problems defined on simple graphs in which the populations at any two vertices may be averaged if they are connected by an edge. The diffusion polytope is the convex hull of the set of population vectors attainable using finite sequences of these operations. A number of physical problems have linear programming solutions taking the diffusion polytope as the feasible region, e.g. the free energy that can be removed from plasma using waves, so there is a need to describe and enumerate its extreme points. We review known results for the case of the complete graph $K_n$, and study a variety of problems for the path graph $P_n$ and the cyclic graph $C_n$. We describe the different kinds of extreme points that arise, and identify the diffusion polytope in a number of simple cases. In the case of increasing initial populations on $P_n$ the diffusion polytope is topologically an $n$-dimensional hypercube.