From random point processes to hierarchical cavity master equations for stochastic dynamics of disordered systems in random graphs: Ising models and epidemics

We start from the theory of random point processes to derive 𝑛-point coupled master equations describing the continuous dynamics of discrete variables in random graphs. These equations constitute a hierarchical set of approximations that generalize and improve the cavity master equation (CME), a recently obtained closure for the usual master equation representing the dynamics. Our derivation clarifies some of the hypotheses and approximations that originally led to the CME, considered now as the first order of a more general technique. We tested the scheme in the dynamics of three models defined over diluted graphs: the Ising ferromagnet, the Viana-Bray spin-glass, and the susceptible-infectious-susceptible model for epidemics. In the first two, the equations perform similarly to the best-known approaches in literature. In the latter, they outperform the well-known pair quenched mean-field approximation.