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We introduce and apply a novel efficient method for the precise simulation ofstochastic dynamical processes on locally tree-like graphs. Networks withcycles are treated in the framework of the cavity method. Such modelscorrespond, for example, to spin-glass systems, Boolean networks, neuralnetworks, or other technological, biological, and social networks. Buildingupon ideas from quantum many-body theory, the new approach is based on a matrixproduct approximation of the so-called edge messages -- conditionalprobabilities of vertex variable trajectories. Computation costs and accuracycan be tuned by controlling the matrix dimensions of the matrix product edgemessages (MPEM) in truncations. In contrast to Monte Carlo simulations, thealgorithm has a better error scaling and works for both, single instances aswell as the thermodynamic limit. We employ it to examine prototypicalnon-equilibrium Glauber dynamics in the kinetic Ising model. Because of theabsence of cancellation effects, observables with small expectation values canbe evaluated accurately, allowing for the study of decay processes and temporalcorrelations.

A matrix product algorithm for stochastic dynamics on networks, applied to non-equilibrium Glauber dynamics