Approximate Master Equations for Dynamics on Complex Networks

Approximate Master Equations (AMEs) provide a powerful tool for studying dynamics on complex networksby accounting for not only the state of a focal node but also the states of its neighbors. This paper presents a detailedderivation of AMEs using the Susceptible-Infected-Susceptible (SIS) model, a prototypical epidemic model,following an introduction to both the standard and heterogeneous mean-field approximations. To demonstrate theeffectiveness of AMEs, we further apply the method to two three-state models: the Susceptible-Infected-Recovered(SIR) model and the Susceptible-Adopted-Susceptible (SAS) model. In both cases, AMEs yield results in excellentagreement with Monte Carlo simulations, accurately capturing not only the steady-state distributions of nodestates but also the transient dynamics. Notably, this paper provides a novel application of AMEs to the SAS model,which describes threshold-like adoption behavior in information diffusion. The analysis reveals a discontinuous phasetransition that critically depends on the initial fraction of adopters—a phenomenon not observed in the SIS model.Although AMEs offer high precision, they entail increased computational costs, especially on networks with broaddegree distributions. Developing efficient numerical methods and extending the framework to incorporate networkcycles remain important directions for future research.