  We study the dynamic cavity method for dilute kinetic Ising models with
synchronous update rules. For the parallel update rule we find for fully
asymmetric models that the dynamic cavity equations reduce to a Markovian
dynamics of the (time-dependent) marginal probabilities. For the random
sequential update rule, also an instantiation of a synchronous update rule, we
find on the other hand that the dynamic cavity equations do not reduce to a
Markovian dynamics, unless an additional assumption of time factorization is
introduced. For symmetric models we show that a fixed point of ordinary Belief
propagation is also a fixed point of the dynamic cavity equations in the time
factorized approximation. For clarity, the conclusions of the paper are
formulated as three lemmas.
