Dynamic mean-field and dynamic cavity for diluted Ising systems

We compare dynamic mean-field and dynamic cavity as methods to describe the stationary states of dilute kinetic Ising models. We compute dynamic mean-field theory by expanding in interaction strength to third order, and compare to the restricted dynamic mean-field theory of M\'ezard and Sakellariou (2011), which strictly only holds for fully asymmetric networks. We show that in diluted networks the dynamic cavity method generally predicts magnetizations of individual spins better than both first order ("naive") and second order ("TAP") dynamic mean field theory.