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There are efficient schemes for directly computing the relevant probability distributions in the context of linear multi-stage clonal expansion (MSCE) models. However, these schemes have not been generalised to models on arbitrary graphs, forcing the use of either full stochastic simulations or mean-field approximations, which can become inaccurate at late times or old ages. Here, we present a numerical integration scheme for directly computing survival probabilities of a first-order birth-death process on an arbitrary directed graph, without the use of stochastic simulations. As a concrete application, we show that this new numerical method can be used to infer the parameters of an example graphical model from simulated data.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1012784","type":"journal-article","created":{"date-parts":[[2026,8,4]],"date-time":"2026-08-04T17:29:22Z","timestamp":1785864562000},"page":"e1012784","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["A fast numerical integration scheme for clonal expansion processes on 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