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

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