  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) recently obtained
in other publications. Our derivation clarifies some of the hypotheses and
approximations that originally lead to the CME, considered now as the first
order of a more general technique. We tested the new 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 new equations perform similarly to the best-known approaches
in the literature. In the latter, they outperform the well-known Pair Quenched
Mean-Field Approximation.
