From random point processes to hierarchical Cavity Master Equations for the stochastic dynamics of disordered systems in Random Graphs: Ising models and epidemics

TL;DR

This paper derives a hierarchical set of master equations from random point process theory to better model the stochastic dynamics of discrete systems on random graphs, improving upon existing methods.

Contribution

It introduces a hierarchical approximation framework that generalizes the Cavity Master Equation, providing more accurate modeling of dynamics in disordered systems on random graphs.

Findings

The new equations perform comparably to leading approaches for Ising and spin-glass models.

They outperform the Pair Quenched Mean-Field Approximation in epidemic modeling.

The derivation clarifies the assumptions behind the original CME.

Abstract

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.