Noisy Voter Model Research Articles

Delayed interactions in the noisy voter model through the periodic polling mechanism

We investigate the effects of delayed interactions on the stationary distribution of the noisy voter model. We assume that the delayed interactions occur through the periodic polling mechanism and replace the original instantaneous two-agent interactions. In our analysis, we require that the polling period aligns with the delay in announcing poll outcomes. As expected, when the polling period is relatively short, the model with delayed interactions is almost equivalent to the original model. As the polling period increases, oscillatory behavior emerges, but the model with delayed interactions still converges to stationary distribution. The stationary distribution resembles a Beta-binomial distribution, with its shape parameters scaling with the polling period. The observed scaling behavior is non-monotonic. Namely, the shape parameters peak at some intermediate polling period.

Polarization-induced stress in the noisy voter model

A new model for the dynamics of opinion formation is proposed and analyzed at the mean-field level. It can be regarded as a generalization of the noisy voter model in which agents update their binary states by copying others and by an intrinsic mechanism affected by the degree of polarization in the system. It also takes into account whether the agents enhance or reduce their intrinsic mechanism upon increasing polarization. Four phases or shapes of the steady-state probability of a fraction of agents in a given state are found (unimodal, bimodal, W and M). In the unimodal (resp. bimodal) phase, the copying (resp. intrinsic) mechanism is globally dominant, while in the W (resp. M) phase the copying (resp. intrinsic) mechanism is the relevant one close to the consensus states while it reduces its influence as approaching coexistence. In the thermodynamic limit, the bimodal and W phases disappear, while the unimodal and M phases prevail. The theoretical results, obtained analytically from the master equation, and the numerical simulations are in good agreement.

First-passage distributions of an asymmetric noisy voter model.

This paper explores the first-passage times in an asymmetric noisy voter model through analytical methods. The noise in the model leads to bistable behavior, and the asymmetry arises from heterogeneous rates for spontaneous switching. We obtain exact analytical expressions for the probability distribution for two different initial conditions, first-passage times for switching transitions and first return times to a stable state for all system sizes, offering a deeper understanding of the model's dynamics. Additionally, we derive exact expressions for the mean switching time, mean return time, and their mean square variants. The findings are verified through numerical simulations. To enhance clarity regarding the model's behavior, we also provide approximate solutions, emphasizing the parameter dependence of first-passage times in the small switching parameter regime. An interesting result in this regime is that while the mean switching time in the leading order is independent of system size, the mean return time depends inversely on system size. This study not only advances our analytical understanding of the asymmetric noisy voter model but also establishes a framework for exploring similar phenomena in social and biological systems where the noisy voter model is applicable.

Model of opinion dynamics caused by information pressure in multi-agent system with stochastic activation of links

The dynamic properties of a modified nonlinear noisy voter model on a square lattice are studied in the paper. The introduced modifications are as follows. First, the changes of the binary variable (opinion) at the voters (agents) are caused by avalanche-like perturbations of the system. Secondly, the structure of inter-agent links is not static. Its temporal evolution is due to a new characteristic of the agent called ‘activity’. It determines the probability for the agent to be linked with its nearest neighbors at a given time moment. In addition we introduce a binary variable that changes randomly in time (an ‘external opinion’). According to the proposed rules for opinion changes, an agent that unlinked to neighbors changes its opinion to a current value of the external opinion, regardless of the opinions of other agents. A linked agent can copy the opinion of its neighbor during an avalanche process. Analytically and numerically, we show that the agents’ ‘activity’ distribution and the time-averaged value of the external opinion completely determines the mode of opinion dynamics. The phase diagram for the model under consideration is obtained. In the case of large values of averaged agents’ ‘activity’ the system switches between two consensus states spending the most of time in the consensus with shared opinion corresponding to the prevailing value of the external opinion. For small value of averaged ‘activity’ the system tends to the state where the opposite opinions coexist, but agents predominate with opinion corresponding to sign of the time-averaged value of the external opinion. We demonstrate that the resulting model reflects the main features of the behavior of real multi-agent systems where the external information inflows into the system and, spreading among agents, drives opinion dynamics.

Partisan voter model: Stochastic description and noise-induced transitions.

We give a comprehensive mean-field analysis of the partisan voter model (PVM) and report analytical results for exit probabilities, fixation times, and the quasistationary distribution. In addition, and similarly to the noisy voter model, we introduce a noisy version of the PVM, named the noisy partisan voter model (NPVM), which accounts for the preferences of each agent for the two possible states, as well as for idiosyncratic spontaneous changes of state. We find that the finite-size noise-induced transition of the noisy voter model is modified in the NPVM leading to the emergence of intermediate phases that were absent in the standard version of the noisy voter model, as well as to both continuous and discontinuous transitions.

Global information from local observations of the noisy voter model on a graph

Global information from local observations of the noisy voter model on a graph

Anomalous diffusion and long-range memory in the scaled voter model.

We analyze the scaled voter model, which is a generalization of the noisy voter model with time-dependent herding behavior. We consider the case when the intensity of herding behavior grows as a power-law function of time. In this case, the scaled voter model reduces to the usual noisy voter model, but it is driven by the scaled Brownian motion. We derive analytical expressions for the time evolution of the first and second moments of the scaled voter model. In addition, we have derived an analytical approximation of the first passage time distribution. By numerical simulation, we confirm our analytical results as well as showing that the model exhibits long-range memory indicators despite being a Markov model. The proposed model has steady-state distribution consistent with the bounded fractional Brownian motion, thus we expect it to be a good substitute model for the bounded fractional Brownian motion.

Noisy voter model: Explicit expressions for finite system size.

Urn models are classic stochastic models that have been used to describe a diverse kind of complex systems. Voter and Ehrenfest's models are very well-known urn models. An opinion model that combines these two models is presented in this work and it is used to study a noisy voter model. In particular, at each temporal step, an Ehrenfest's model step is done with probability α or a voter step is done with probability 1-α. The parameter α plays the role of noise. By performing a spectral analysis, it is possible to obtain explicit expressions for the order parameter, susceptibility, and Binder's fourth-order cumulant. Recursive expressions in terms of the dual Hahn polynomials are given for first passage and return distributions to consensus and the equal coexistence of opinions. In the cases where they follow power-law distributions, their exponents are computed. This model has a pseudocritical noise value that depends on the system size; a discussion about thermodynamic limits is given.

Analytical and Numerical Treatment of Continuous Ageing in the Voter Model

The conventional voter model is modified so that an agent’s switching rate depends on the ‘age’ of the agent—that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. The thinning algorithm of Lewis and Shedler can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age-dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of a spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.

Noisy multistate voter model for flocking in finite dimensions.

We study a model for the collective behavior of self-propelled particles subject to pairwise copying interactions and noise. Particles move at a constant speed v on a two-dimensional space and, in a single step of the dynamics, each particle adopts the direction of motion of a randomly chosen neighboring particle within a distance R=1, with the addition of a perturbation of amplitude η (noise). We investigate how the global level of particles' alignment (order) is affected by their motion and the noise amplitude η. In the static case scenario v=0 where particles are fixed at the sites of a square lattice and interact with their first neighbors, we find that for any noise η>0 the system reaches a steady state of complete disorder in the thermodynamic limit, while for η=0 full order is eventually achieved for a system with any number of particles N. Therefore, the model displays a transition at zero noise when particles are static, and thus there are no ordered steady states for a finite noise (η>0). We show that the finite-size transition noise vanishes with N as η_{c}^{1D}∼N^{-1} and η_{c}^{2D}∼(NlnN)^{-1/2} in one- and two-dimensional lattices, respectively, which is linked to known results on the behavior of a type of noisy voter model for catalytic reactions. When particles are allowed to move in the space at a finite speed v>0, an ordered phase emerges, characterized by a fraction of particles moving in a similar direction. The system exhibits an order-disorder phase transition at a noise amplitude η_{c}>0 that is proportional to v, and that scales approximately as η_{c}∼v(-lnv)^{-1/2} for v≪1. These results show that the motion of particles is able to sustain a state of global order in a system with voter-like interactions.

Species exclusion and coexistence in a noisy voter model with a competition-colonization tradeoff.

We introduce an asymmetric noisy voter model to study the joint effect of immigration and a competition-dispersal tradeoff in the dynamics of two species competing for space in regular lattices. Individuals of one species can invade a nearest-neighbor site in the lattice, while individuals of the other species are able to invade sites at any distance but are less competitive locally, i.e., they establish with a probability g≤1. The model also accounts for immigration, modeled as an external noise that may spontaneously replace an individual at a lattice site by another individual of the other species. This combination of mechanisms gives rise to a rich variety of outcomes for species competition, including exclusion of either species, monostable coexistence of both species at different population proportions, and bistable coexistence with proportions of populations that depend on the initial condition. Remarkably, in the bistable phase, the system undergoes a discontinuous transition as the intensity of immigration overcomes a threshold, leading to a half loop dynamics associated to a cusp catastrophe, which causes the irreversible loss of the species with the shortest dispersal range.

Zealots in multistate noisy voter models.

The noisy voter model is a stylized representation of opinion dynamics. Individuals copy opinions from other individuals, and are subject to spontaneous state changes. In the case of two opinion states this model is known to have a noise-driven transition between a unimodal phase, in which both opinions are present, and a bimodal phase, in which one of the opinions dominates. The presence of zealots can remove the unimodal and bimodal phases in the model with two opinion states. Here we study the effects of zealots in noisy voter models with M>2 opinion states on complete interaction graphs. We find that the phase behavior diversifies, with up to six possible qualitatively different types of stationary states. The presence of zealots removes some of these phases, but not all. We analyze situations in which zealots affect the entire population, or only a fraction of agents, and show that this situation corresponds to a single-community model with a fractional number of zealots, further enriching the phase diagram. Our study is conducted analytically based on effective birth-death dynamics for the number of individuals holding a given opinion. Results are confirmed in numerical simulations.

Supportive interactions in the noisy voter model

Latane social impact theory predicts recruitment and supportive interactions being responsible for opinion formation. So far only recruitment interactions were considered in the voter models. Here we consider a noisy voter model with supportive interactions, which make voters less likely to change their opinions. This is similar to the voter models with freezing, but instead of interacting with their past selves voters get the support from their peers. We examine two different ways in which the support could be implemented: support deterring imitation as well as independence, support deterring imitation only. Both assumptions introduce strong drift into the model, which almost always overcomes the diffusion caused by the imitative behavior. The first assumption introduces strong attraction to a full consensus state, unless the support becomes too strong. The latter assumption promotes partial consensus, with surviving minority group.

Noisy voter model for the anomalous diffusion of parliamentary presence

We examine the parliamentary presence data of the 2008–2012 and 2012–2016 legislatures of the Lithuanian parliament. We consider the cumulative presence series of each individual representative in the data set. These series exhibit superdiffusive behavior. We propose a modified noisy voter model as a model for the parliamentary presence. We provide detailed analysis of the anomalous diffusion of the individual agent trajectories and show that the modified model is able to reproduce the empirical statistical properties.

Multistate voter model with imperfect copying.

The voter model with multiple states has found applications in areas as diverse as population genetics, opinion formation, species competition, and language dynamics, among others. In a single step of the dynamics, an individual chosen at random copies the state of a random neighbor in the population. In this basic formulation, it is assumed that the copying is perfect, and thus an exact copy of an individual is generated at each time step. Here, we introduce and study a variant of the multistate voter model in mean field that incorporates a degree of imperfection or error in the copying process, which leaves the states of the two interacting individuals similar but not exactly equal. This dynamics can also be interpreted as a perfect copying with the addition of noise: a minimalistic model for flocking. We found that the ordering properties of this multistate noisy voter model, measured by a parameter ψ in [0,1], depend on the amplitude η of the copying error or noise and the population size N. In the case of perfect copying η=0, the system reaches an absorbing configuration with complete order (ψ=1) for all values of N. However, for any degree of imperfection η>0, we show that the average value of ψ at the stationary state decreases with N as 〈ψ〉≃6/(π^{2}η^{2}N) for η≪1 and η^{2}N≳1, and thus the system becomes totally disordered in the thermodynamic limit N→∞. We also show that 〈ψ〉≃1-π^{2}/6η^{2}N in the vanishing small error limit η→0, which implies that complete order is never achieved for η>0. These results are supported by Monte Carlo simulations of the model, which allow to study other scenarios as well.

Stochastic pair approximation treatment of the noisy voter model

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 Neff.

The noisy voter model under the influence of contrarians

The influence of contrarians on the noisy voter model is studied at the mean-field level. The noisy voter model is a variant of the voter model where agents can adopt two opinions, optimistic or pessimistic, and can change them by means of an imitation (herding) and an intrinsic (noise) mechanisms. An ensemble of noisy voters undergoes a finite-size phase transition, upon increasing the relative importance of the noise to the herding, form a bimodal phase where most of the agents share the same opinion to a unimodal phase where almost the same fraction of agent are in opposite states. By the inclusion of contrarians we allow for some voters to adopt the opposite opinion of other agents (anti-herding). We first consider the case of only contrarians and show that the only possible steady state is the unimodal one. More generally, when voters and contrarians are present, we show that the bimodal-unimodal transition of the noisy voter model prevails only if the number of contrarians in the system is smaller than four, and their characteristic rates are small enough. For the number of contrarians bigger or equal to four, the voters and the contrarians can be seen only in the unimodal phase. Moreover, if the number of voters and contrarians, as well as the noise and herding rates, are of the same order, then the probability functions of the steady state are very well approximated by the Gaussian distribution.

Aging-induced continuous phase transition

Aging is considered as the property of the elements of a system to be less prone to change states as they get older. We incorporate aging into the noisy voter model, a stochastic model in which the agents modify their binary state by means of noise and pair-wise interactions. Interestingly, due to aging the system passes from a finite-size discontinuous transition between ordered (ferromagnetic) and disordered (paramagnetic) phases to a second order phase transition, well defined in the thermodynamic limit, belonging to the Ising universality class. We characterize it analytically by finding the stationary solution of an infinite set of mean field equations. The theoretical predictions are tested with extensive numerical simulations in low dimensional lattices and complex networks. We finally employ the aging properties to understand the symmetries broken in the phase transition.

Analytical and numerical study of the non-linear noisy voter model on complex networks.

We study the noisy voter model using a specific non-linear dependence of the rates that takes into account collective interaction between individuals. The resulting model is solved exactly under the all-to-all coupling configuration and approximately in some random network environments. In the all-to-all setup, we find that the non-linear interactions induce bona fide phase transitions that, contrary to the linear version of the model, survive in the thermodynamic limit. The main effect of the complex network is to shift the transition lines and modify the finite-size dependence, a modification that can be captured with the introduction of an effective system size that decreases with the degree heterogeneity of the network. While a non-trivial finite-size dependence of the moments of the probability distribution is derived from our treatment, mean-field exponents are nevertheless obtained in the thermodynamic limit. These theoretical predictions are well confirmed by numerical simulations of the stochastic process.

Zealots in the mean-field noisy voter model.

The influence of zealots on the noisy voter model is studied theoretically and numerically at the mean-field level. The noisy voter model is a modification of the voter model that includes a second mechanism for transitions between states: Apart from the original herding processes, voters may change their states because of an intrinsic noisy-in-origin source. By increasing the importance of the noise with respect to the herding, the system exhibits a finite-size phase transition from a quasiconsensus state, where most of the voters share the same opinion, to one with coexistence. Upon introducing some zealots, or voters with fixed opinion, the latter scenario may change significantly. We unveil new situations by carrying out a systematic numerical and analytical study of a fully connected network for voters, but allowing different voters to be directly influenced by different zealots. We show that this general system is equivalent to a system of voters without zealots, but with heterogeneous values of their parameters characterizing herding and noisy dynamics. We find excellent agreement between our analytical and numerical results. Noise and herding or zealotry acting together in the voter model yields a nontrivial mixture of the scenarios with the two mechanisms acting alone: It represents a situation where the global-local (noise-herding) competition is coupled to a symmetry breaking (zealots). In general, the zealotry enhances the effective noise of the system, which may destroy the original quasiconsensus state, and can introduce a bias towards the opinion of the majority of zealots, hence breaking the symmetry of the system and giving rise to new phases. In the most general case we find two different transitions: a discontinuous transition from an asymmetric bimodal phase to an extreme asymmetric phase and a second continuous transition from the extreme asymmetric phase to an asymmetric unimodal phase.