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A method to approximately close the dynamic cavity equations for synchronousreversible dynamics on a locally tree-like topology is presented. The methodbuilds on $(a)$ a graph expansion to eliminate loops from the normalizations ofeach step in the dynamics, and $(b)$ an assumption that a set of auxilaryprobability distributions on histories of pairs of spins mainly havedependencies that are local in time. The closure is then effectuated byprojecting these probability distributions on $n$-step Markov processes. Themethod is shown in detail on the level of ordinary Markov processes ($n=1$),and outlined for higher-order approximations ($n>1$). Numerical validations ofthe technique are provided for the reconstruction of the transient andequilibrium dynamics of the kinetic Ising model on a random graph witharbitrary connectivity symmetry.

Dynamic message-passing approach for kinetic spin models with reversible dynamics