Graph theoretic derivation of mutual linearity for transient probabilities and hitting time distributions in Markov networks

TL;DR

This paper uses graph theory to extend the concept of mutual linearity in Markov networks to non-stationary regimes and hitting time densities, providing explicit combinatorial formulas.

Contribution

It introduces a graph theoretic derivation of mutual linearity for transient probabilities and hitting times, expanding previous stationary results.

Findings

Mutual linearity holds for non-stationary response ratios.

Explicit combinatorial expressions are derived for transient response ratios.

Mutual linearity extends to hitting time densities.

Abstract

For irreducible, time-homogeneous Markov networks, mutual linearity has recently been established for both occupation probabilities and network currents in the stationary regime as well as in the non-stationary regime in Laplace space. The derivation of this property for the stationary distribution utilized the Markov-chain tree theorem, which also allows for an explicit combinatorial expression of the response ratios under variation of a single transition rate. The extension of this result was proven at the trajectory level by employing the Doob-Meyer decomposition. By employing the all-minors matrix-tree theorem, we show that this property also follows from a graph theoretic formulation and derive explicit combinatorial expressions for the non-stationary response ratios. The stationary result follows as the long-time limit and we also show that the small-time asymptotics are entirely determined by minimal path distances in the underlying graph. Finally we use the graph theoretic approach to prove that mutual linearity also extends to hitting time densities.