Stochastic pair approximation treatment of the noisy voter model

Abstract

We present a full stochastic description of the pair approximation scheme to study binary-state dynamics on heterogeneous networks. Within this general approach, we obtain a set of equations for the dynamical correlations, fluctuations and finite-size effects, as well as for the temporal evolution of all relevant variables. We test this scheme for a prototypical model of opinion dynamics known as the noisy voter model that has a finite-size critical point. Using a closure approach based on a system size expansion around a stochastic dynamical attractor we obtain very accurate results, as compared with numerical simulations, for stationary and time-dependent quantities whether below, within or above the critical region. We also show that finite-size effects in complex networks cannot be captured, as often suggested, by merely replacing the actual system size N by an effective network dependent size N eff . IntroductionFrom classical problems in statistical physics [1, 2] to questions in biology and ecology [3][4][5], and over to the spreading of opinions and diseases in social systems [6-10], stochastic binary-state models have been widely used to study the emergence of collective phenomena in systems of stochastically interacting components. In general, these components are modeled as binary-state variables -spin up or down-sitting at the nodes of a network whose links represent the possible interactions among them. While initial research focused on the limiting cases of a well-mixed population, where each of the components is allowed to interact with any other, and regular lattice structures, later works turned to more complex and heterogeneous topologies [11][12][13][14]. A most important insight derived from these more recent works is that the macroscopic dynamics of the system can be greatly affected by the particular topology of the underlying network. In the case of systems with critical behavior, different network characteristics have been shown to have a significant impact on the critical values of the model parameters [15][16][17], such as the critical temperature of the Ising model [18][19][20] and the epidemic threshold in models of infectious disease transmission [21][22][23][24][25]. Thus, the identification of the particular network characteristics that have an impact on the dynamics of these models, as well as the quantification of their effect, are of paramount importance.The first theoretical treatments introduced for the study of stochastic, binary-state dynamics on networks relied on a global-state approach [2, 26], i.e. they focused on a single variable-for example, the number of nodes in one of the two possible states-assumed to represent the whole state of the system. In order to write a master equation for this global-state variable, some approximation is required to move from the individual particle transition rates defining the model to some effective transition rates depending only on the chosen global variable. Within this global-state approach, the effective-field approximation assumes...

Multistate voter model with imperfect copying

“…As mentioned in section IV A, the 2-state case of our MSVM is equivalent to the noisy voter model studied in [14]. This suggests that it might be possible to map the MSVM with S > 2 states to the 2-state noisy voter model studied in [14,18] by finding appropriate copying and noise rates that depend on η, and then use known results on these studied models to derive the scaling relations obtained in this article. We have become aware of a recent unpublished article [31] that investigates multi-state noisy voter models where imitation and mutation (noise) events occur at respective rates r ji and ǫ ji that may depend on states j and i.…”

Consensus and diversity in multistate noisy voter models

“…It is an obvious challenge to extend the results we have obtained to multi-state noisy voter models on lattices and more complex graphs. For example, it would be interesting to see if and how our ideas can be combined with pair-approximation methods used recently for noisy voter models in [25]. A further limitation of our approach consists in its inability to capture correlations between species.…”

Statistical Physics Of Opinion Formation: Is it a SPOOF?

“…It has been argued that independence plays a role similar to temperature [38] or noise. Indeed, one kind of the voter model with this type of nonconformity is frequently called the noisy voter model [14,[80][81][82].…”

“…Another version of pair approximation that divides nodes into degree classes and keeps track of correlations between them has been proposed to study noisy voter model in Ref. [81].…”