An approach to non-equilibrium Markov chains through cycle matrices

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.