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.
Highlights
Binary-state models on complex network are a very general theoretical framework to study the effect of interactions in the dynamics of a population of individuals
We can distinguish between two types of approaches depending on the variables that one chooses to describe the system: (i) individual basedapproaches [22,23,24,25,26], where the “spin” or state of each node of the network is considered as an independent variable, (ii) compartmental approaches [11,27,28,29,30,31,32], where nodes sharing the same topological property such as, for example, the number of neighbors in the network, are aggregated in a single variable, being this an integer number
In Appendix A, we show the expressions of the matrices involved in the van Kampen and Kramers-Moyal expansions, while Appendix B contains the details of the expansion around the critical point
Summary
Binary-state models on complex network are a very general theoretical framework to study the effect of interactions in the dynamics of a population of individuals. Even a very weak fluctuation may eventually drive the system to a completely different final state after some time steps For this reason, the model shows heavy finite size effects even for large system sizes and a description of the stochastic dynamics is of crucial important to understand it. The method that we will use to obtain the theoretical scaling functions is a similar system-size expansion of the master equation, but with an anomalous scaling with system size [75,76] These techniques will be applied to several models of interest on different network types, comparing altogether the different levels of approximation and accuracy of compartmental approaches. One of the interesting properties of ρ for binary-state models is that it can be used as an alternative to m or mS to measure the level of order or agreement on one of the options, a situation in which ρ approaches zero, independently of the option
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