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The Cavity Master Equation: average and fixed point of the ferromagnetic model in random graphs

The Cavity Master Equation (CME) is a closure scheme to the usual Master Equation representing the dynamics of discrete variables in continuous time. In this work we explore the CME for a ferromagnetic model in a random graph. We first derive and average equation of the CME that describes the dynamics of mean magnetization of the system. We show that the numerical results compare remarkably well with the Monte Carlo simulations. Then, we show that the stationary state of the CME is well described by BP-like equations (independently of the dynamic rules that let the system towards the stationary state). These equations may be rewritten exactly as the fixed point solutions of the Cavity Equation if one also assumes that the stationary state is well described by a Boltzmann distribution.