Graph-theoretic framework for semi-Markov networks: transient and stationary occupation probabilities, hitting times and weak ergodicity breaking
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.
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BibTeX
@misc{voits2026graphtheoretic,
title = {{Graph-theoretic framework for semi-Markov networks: transient and stationary occupation probabilities, hitting times and weak ergodicity breaking}},
author = {Julian Voits},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13505},
eprint = {2610.13505},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13505},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Graph-theoretic framework for semi-Markov networks: transient and stationary occupation probabilities, hitting times and weak ergodicity breaking
AU - Julian Voits
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13505
UR - https://arxiv.org/abs/2610.13505
AB - 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.
ER -