Tractable description of hydrodynamic limits of a class of interacting jump processes on sparse graphs

We consider dynamics of the empirical measure of vertex neighborhood states of Markov interacting jump processes on sparse random graphs, in a suitable asymptotic limit as the graph size goes to infinity. Under the assumption of a certain acyclic structure on single-particle transitions, we provide a tractable autonomous description of the evolution of this hydrodynamic limit in terms of a finite coupled system of ordinary differential equations. Key ingredients of the proof include a characterization of the hydrodynamic limit of the neighborhood empirical measure in terms of a certain local-field equation, well-posedness of its Markovian projection, and a Markov random field property of the time-marginals, which may be of independent interest. We also show how our results lead to principled approximations for classes of interacting jump processes and illustrate its efficacy via simulations on several examples, including an idealized model of seizure spread in the brain.