A Markov chain is reversible if there exist positive weights on its state-transition graph, symmetric between any two states, such that the chain moves from a state to a neighbor with probability proportional to the connecting weight; every reversible chain can be represented this way, and its stationary distribution is proportional to each state's total outgoing weight, derivable without matrix inversion via the detailed-balance (reversibility) equation. Non-reversible chains lack this shortcut and generally require solving the full stationary-distribution equation; Google PageRank is a canonical non-reversible chain in which states are web pages, transition probabilities are derived from hyperlink structure (diluted by out-degree) mixed with a small uniform "teleportation" probability to guarantee irreducibility and aperiodicity, and page importance is the resulting unique stationary distribution. This belongs to the theory of Markov chains within probability, specifically stationary distributions, reversibility/detailed balance, and irreducibility conditions guaranteeing existence and uniqueness of the limiting distribution.
A Markov chain is reversible if there exist positive weights on its state-transition graph, symmetric between any two states, such that the chain moves from a state to a neighbor with probability pro…