An approach to non-equilibrium Markov chains through cycle matrices

TL;DR

This paper introduces a graph-theoretic framework using cycle matrices to analyze non-equilibrium properties in Markov chains, drawing parallels with quantum systems and providing a basis for describing non-equilibrium states.

Contribution

It proposes cycle matrices as a novel basis for understanding non-equilibrium in Markov chains, extending graph-theoretic methods.

Findings

Kernel of the incidence matrix is isomorphic to anti-symmetric matrices with zero row sum.

Cycle matrices form a basis for matrices describing non-equilibrium.

Provides a new mathematical tool for analyzing non-equilibrium Markov processes.

Abstract

Analogously to the quantum case considered in Cruz-de-la-Rosa and Guerrero-Poblete (Open Syst. Inf. Dyn. 32, 2550005, 2025), this work proposes a graph-theoretic approach to studying non-equilibrium properties in Markov chains. We prove that the kernel of the incidence matrix associated with the interaction graph of the chain, which consists of cycles, is isomorphic to the space of anti-symmetric matrices with rows sum to zero. The main contribution of this work is the introduction of the called cycle matrices, which constitute a basis for the space of matrices that describe the non-equilibrium.