Obtaining the long-term behavior of Master equations with finite state space from the structure of the associated state transition network

TL;DR

This paper establishes a graph-theoretic criterion for the uniqueness of the stationary state in finite-state Master equations, linking minimal absorbing sets to linearly independent stationary solutions.

Contribution

It provides a direct, complete derivation of a necessary and sufficient condition for stationary state uniqueness using the structure of the state transition network.

Findings

Minimal absorbing sets correspond to linearly independent stationary states.

The criterion for uniqueness is both necessary and sufficient.

Graph theory offers a clear framework for analyzing long-term behavior.

Abstract

The Master equation describes the time evolution of the probabilities of a system with a discrete state space. This time evolution approaches for long times a stationary state that will in general depend on the initial probability distribution. Conditions under which the stationary state is unique are usually given as remarks appended to more comprehensive theories in the mathematical literature. We provide a direct and complete derivation of a necessary and sufficient criterion for when this steady state is unique. We translate this problem into the language of graph theory and show that there is a one-to-one correspondence between minimal absorbing sets within the state-transition network and linearly independent stationary states of the Master equation.