Graph-theoretic framework for semi-Markov networks: transient and stationary occupation probabilities, hitting times and weak ergodicity breaking

Source abstract

Semi-Markov networks provide a natural framework for describing stochastic transitions between coarse-grained states with non-exponential dwell times, with applications ranging from biophysics and neuroscience to epidemiology and finance. Here, it is shown that finite semi-Markov networks admit explicit combinatorial representations of their Laplace-transformed occupation probabilities and hitting-time distributions as sums over weighted spanning trees and forests of an augmented graph. These representations yield asymptotic results, including a generalization of the Markov Chain Tree Theorem to semi-Markov networks, in which the rate constants replaced by pseudo-rates given by the inverse mean sojourn time weighted by the exit probability through this channel, as well as the long-time scaling of hitting-time survival probabilities in the presence of infinite-mean sojourn times. They also extend recently derived combinatorial formulas for hitting-time moments to the semi-Markov setting. In particular, the mean hitting-time formula coincides with that of a Markov network with the corresponding pseudo-rates, which also yields a short proof of the constancy of Kemeny's constant for finite-mean semi-Markov networks. Finally, we use the same framework to show that weak ergodicity breaking occurs generically for time-additive state observables that are nonconstant on the support of the limiting distribution in heavy-tailed semi-Markov networks.