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 non-linear deterministic equations is assumed to characterize its dynamics and stationary properties. We develop in this work the stochastic formalism of the different compartmental approaches, these are: approximate master equation (AME), pair approximation (PA) and heterogeneous mean field (HMF), in descending order of accuracy. Using different system-size expansions of a general master equation, we are able to obtain approximate solutions of the fluctuations and finite-size corrections of the global state. On the one hand, far from criticality, the deviations from the deterministic solution are well captured by a Gaussian distribution whose properties we derive, including its correlation matrix and corrections to the average values. On the other hand, close to a critical point there are non-Gaussian statistical features that can be described by the finite-size scaling functions of the models. We show how to obtain the scaling functions departing only from the theory of the different approximations. We apply the techniques for a wide variety of binary-state models in different contexts, such as epidemic, opinion and ferromagnetic models.

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• Pair approximation for the q-voter models with quenched disorder on networks
• Opinion formation on social networks with algorithmic bias: Dynamics and bias imbalance
• Deterministic and stochastic cooperation transitions in evolutionary games on networks
• The $q$-neighbor Ising model on multiplex networks with partial overlap of nodes
• Interplay between exogenous triggers and endogenous behavioral changes in contagion processes on social networks