Degree Bounds for a Minimal Markov Basis for the Three-State Toric Homogeneous Markov Chain Model

TL;DR

This paper investigates the structure of the three-state toric homogeneous Markov chain model, providing bounds on the associated polytope's vertices, analyzing semigroup normality, and conjecturing stabilization and degree bounds for Markov bases.

Contribution

It introduces degree bounds for a minimal Markov basis in the three-state model and explores properties like polytope vertex bounds, semigroup normality, and stabilization conjectures.

Findings

Bound on the number of vertices of the associated polytope independent of time

Conjecture on the stabilization of the polytope's f-vector

Analysis of the normality of the semigroup and conjectural degree bounds

Abstract

We study the three state toric homogeneous Markov chain model and three special cases of it, namely: (i) when the initial state parameters are constant, (ii) without self-loops, and (iii) when both cases are satisfied at the same time. Using as a key tool a directed multigraph associated to the model, the state-graph, we give a bound on the number of vertices of the polytope associated to the model which does not depend on the time. Based on our computations, we also conjecture the stabilization of the f-vector of the polytope, analyze the normality of the semigroup, give conjectural bounds on the degree of the Markov bases.