Aggregation and Truncation of Reversible Markov Chains Modulo State Renaming

Abstract

The theory of time-reversibility has been widely used to derive the expressions of the invariant measures and, consequently, of the equilibrium distributions for a large class of Markov chains which found applications in optimisation problems, computer science, physics, and bioinformatics. One of the key-properties of reversible models is that the truncation of a reversible Markov chain is still reversible. In this work we consider a more general notion of reversibility, i.e., the reversibility modulo state renaming, called \(\rho \)-reversibility, and show that some of the properties of reversible chains cannot be straightforwardly extended to \(\rho \)-reversible ones. Among these properties, we show that in general the truncation of the state space of a \(\rho \)-reversible chain is not \(\rho \)-reversible. Hence, we derive further conditions that allow the formulation of the well-known properties of reversible chains for \(\rho \)-reversible Markov chains. Finally, we study the properties of the state aggregation in \(\rho \)-reversible chains and prove that there always exists a state aggregation that associates a \(\rho \)-reversible process with a reversible one.