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.

中文翻译：

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腔主方程：随机图中铁磁模型的平均值和不动点

空腔主方程 (CME) 是通常的主方程的闭合方案，表示连续时间中离散变量的动态。在这项工作中，我们探索了随机图中铁磁模型的 CME。我们首先推导出描述系统平均磁化动力学的 CME 的平均方程。我们表明，数值结果与蒙特卡洛模拟非常好。然后，我们证明了 CME 的平稳状态可以用类似 BP 的方程很好地描述（独立于让系统趋向平稳状态的动态规则）。如果还假设稳态由玻尔兹曼分布很好地描述，则这些方程可以完全重写为空腔方程的不动点解。