Binary-state dynamics on complex networks: Stochastic pair approximation and beyond

Abstract

Theoretical approaches to binary-state models on complex networks are generally restricted to infinite size systems, where a set of nonlinear deterministic equations is assumed to characterize its dynamical and stationary properties. We develop in this work the stochastic formalism of the different compartmental approaches, these are the approximate master equation (AME), pair approximation (PA), and heterogeneous mean-field (HMF), in descending order of accuracy. The stochastic formalism allows us to enlarge the range of validity and applicability of compartmental approaches. This includes (i) the possibility of studying the role of the size of the system in the different phenomena reproduced by the models together with a network structure, (ii) obtaining the finite-size scaling functions and critical exponents of the macroscopic quantities, and (iii) the extension of the rate description to a more general class of models.