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

A method to approximately close the dynamic cavity equations for synchronous reversible dynamics on a locally tree-like topology is presented. The method builds on $(a)$ a graph expansion to eliminate loops from the normalizations of each step in the dynamics, and $(b)$ an assumption that a set of auxilary probability distributions on histories of pairs of spins mainly have dependencies that are local in time. The closure is then effectuated by projecting these probability distributions on $n$-step Markov processes. The method is shown in detail on the level of ordinary Markov processes ($n=1$), and outlined for higher-order approximations ($n>1$). Numerical validations of the technique are provided for the reconstruction of the transient and equilibrium dynamics of the kinetic Ising model on a random graph with arbitrary connectivity symmetry.