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Compound birth-death processes are widely used to model the age-incidence curves of many cancers. 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.