{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13505",
  "author_total": 1,
  "authors": [
    {
      "name": "Julian Voits"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2nmeegft05393.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13505",
  "fulltext_url": null,
  "id": "2nmeegft05393",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13505",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7221520318",
  "pdf_url": "https://arxiv.org/pdf/2610.13505",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2nmeegft05393.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:36Z",
      "native_id": "2610.13505",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13505"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Graph-theoretic framework for semi-Markov networks: transient and stationary occupation probabilities, hitting times and weak ergodicity breaking",
  "title_text": "Graph-theoretic framework for semi-Markov networks: transient and stationary occupation probabilities, hitting times and weak ergodicity breaking",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:34Z",
  "url": "https://research.draconitic.com/paper/2nmeegft05393",
  "venue": "arXiv",
  "versions": []
}