The noisy voter model is a stylised 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 behaviour 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 analyse 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.
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 quasi-consensus state, where most of the voters share the same opinion, to a one with coexistence. Upon intr…
The voter model has been extensive studied as an opinion dynamic model, and the role of the zealots has only been discussed recently. We introduce the adaptive voter model with zealots and show that the final distribution of the magnetism can be separated into two regions depending on the number of zealots as well as the probability of forming link. When the fraction of zealots is dominated in the population, the probability distribution of magnetism follows a Gaussian-like distribution and the relaxation time is population-size independent. When the population is dominated by the susceptible …
Stochastic Dynamics of the Multi-State Voter Model over a Network based on Interacting Cliques and Zealot Candidates
在一类由非重叠集团组成的复杂网络上研究了多状态选民模型的随机动力学，每个集团都有一个政治候选人，并通过Erdens-Rényi链路与其他成员相互作用.该模型的数值模拟解释的一个特设的平均场论，专门调整，以解决间/内集团的相互作用。在热力学极限的适当定义下（在增加网络大小的同时，代理的平均度保持固定），该模型被发现显示了由Alberto和Castellano发现的经验标度（Phys Rev Lett 99（13）：138701，2007），而选票分布大致类似于在巴西选举中观察到的。
The adaptive voter model has been widely studied as a conceptual model for opinion formation processes on time-evolving social networks. Past studies on the effect of zealots, i.e. nodes aiming to spread their fixed opinion throughout the system, only considered the voter model on a static network. Here, we extend the study of zealotry to the case of an adaptive network topology co-evolving with the state of the nodes and investigate opinion spreading induced by zealots depending on their initial density and connectedness. Numerical simulations reveal that below the fragmentation threshold a l…
We study the dynamics of the out-of-equilibrium nonlinear q-voter model with two types of susceptible voters and zealots, introduced in [EPL 113, 48001 (2016)]. In this model, each individual supports one of two parties and is either a susceptible voter of type $q_1$ or $q_2$, or is an inflexible zealot. At each time step, a $q_i$-susceptible voter ($i = 1,2$) consults a group of $q_i$ neighbors and adopts their opinion if all group members agree, while zealots are inflexible and never change their opinion. This model violates detailed balance whenever $q_1 \neq q_2$ and is characterized by tw…
In this paper, we study influence maximization in the voter model in the presence of biased voters (or zealots) on complex networks. Under what conditions should an external controller with finite budget who aims at maximizing its influence over the system target zealots? Our analysis, based on both analytical and numerical results, shows a rich diagram of preferences and degree-dependencies of allocations to zealots and normal agents varying with the budget. We find that when we have a large budget or for low levels of zealotry, optimal strategies should give larger allocations to zealots and…
We introduce an heterogeneous nonlinear $q$-voter model with zealots and two types of susceptible voters, and study its non-equilibrium properties when the population is finite and well mixed. In this two-opinion model, each individual supports one of two parties and is either a zealot or a susceptible voter of type $q_1$ or $q_2$. While here zealots never change their opinion, a $q_i$-susceptible voter ($i=1,2$) consults a group of $q_i$ neighbors at each time step, and adopts their opinion if all group members agree. We show that this model violates the detailed balance whenever $q_1 \neq q_…
We examine some agreement-dynamics models that are placed on directed random graphs. In such systems a fraction of sites $\exp(-z)$, where $z$ is the average degree, becomes permanently fixed or flickering. In the Voter model, which has no surface tension, such zealots or flickers freely spread their opinions and that makes the system disordered. For models with a surface tension, like the Ising model or the Naming Game model, their role is limited and such systems are ordered at large~$z$. However, when $z$ decreases, the density of zealots or flickers increases, and below a certain threshold…
We investigate the time evolution of the density of active links and of the entropy of the distribution of agents among opinions in multi-state voter models with all-to-all interaction and on uncorrelated networks. Individual realisations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the population remains in a meta-stable state. The density of active links and the entropy in these states varies from realisation to realisation. Making some simple assumptions we are able to analytically calculate the average density of active links and the ave…
We study memory dependent binary-state dynamics, focusing on the noisy-voter model. This is a non-Markovian process if we consider the set of binary states of the population as the description variables, or Markovian if we incorporate "age", related to the time one has spent holding the same state, as a part of the description. We show that, in some cases, the model can be reduced to an effective Markovian process, where the age distribution of the population rapidly equilibrates to a quasi-steady state, while the global state of the system is out of equilibrium. This effective Markovian proce…
Voter models are well known in the interdisciplinary community, yet they haven't been studied from the perspective of anomalous diffusion. In this paper we show that the original voter model exhibits ballistic regime. Non-linear transformations of the observation variable and time scale allows us to observe other regimes of anomalous diffusion as well as normal diffusion. We show that numerical simulation results coincide with derived analytical approximations describing the temporal evolution of the raw moments.
We introduce a generalized version of the noisy $q$-voter model, one of the most popular opinion dynamics models, in which voters can be in one of $s \ge 2$ states. As in the original binary $q$-voter model, which corresponds to $s=2$, at each update randomly selected voter can conform to its $q$ randomly chosen neighbors (copy their state) only if all $q$ neighbors are in the same state. Additionally, a voter can act independently, taking a randomly chosen state, which introduces disorder to the system. We consider two types of disorder: (1) annealed, which means that each voter can act indep…
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 networks 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 modificat…
In the standard $q$-voter model, a given agent can change its opinion only if there is a full consensus of the opposite opinion within a group of influence of size $q$. A more realistic extension is the threshold $q$-voter, where a minimal agreement (at least $0<q_0\le q$ opposite opinions) is sufficient to flip the central agent's opinion, including also the possibility of independent (non conformist) choices. Variants of this model including non-conformist behavior have been previously studied in fully connected networks (mean-field limit). Here we investigate its dynamics in random networks…
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 shear the same opinion to a unimodal phase where almost the same fraction of agent are in opposite states. By the i…
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…
We examine parliamentary presence data of the 2008-2012 and the 2012-2016 legislatures of Lithuanian parliament. We consider 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 anomalous diffusion of the individual agent trajectories and show that the modified model is able to reproduce empirical statistical properties.
We present a novel model for the effect of echo chambers, filter bubbles, and reinforcement on election results. Our model extends the well known voter model with zealots to include reinforcement. We analyze the behaviour of the model, determine the invariant measure, and show that reinforcement may 1) shift the distribution of votes compared to the voter model, and 2) lead to phase transitions. We test whether the model with reinforcement fit better than under its absence in election data from US presidential elections, the Brexit referendum, and parliamentary elections in France, The Netherl…
We study a variant of the voter model with multiple opinions; individuals can imitate each other and also change their opinion randomly in mutation events. We focus on the case of a population with all-to-all interaction. A noise-driven transition between regimes with multi-modal and unimodal stationary distributions is observed. In the former, the population is mostly in consensus states; in the latter opinions are mixed. We derive an effective death-birth process, describing the dynamics from the perspective of one of the opinions, and use it to analytically compute marginals of the stationa…
We consider the herding to non-herding transition caused by idiosyncratic choices or imperfect imitation in the context of the Kirman Model for financial markets, or equivalently the Noisy Voter Model for opinion formation. In these original models this is a finite size transition that disappears for a large number of agents. We show how the introduction of two different mechanisms make this transition robust and well defined. A first mechanism is nonlinear interactions among agents taking into account the nonlinear effect of local majorities. The second one is aging, so that the longer an age…
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 …
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 compa…
We propose a new analytical method to study stochastic, binary-state models on complex networks. Moving beyond the usual mean-field theories, this alternative approach is based on the introduction of an annealed approximation for uncorrelated networks, allowing to deal with the network structure as parametric heterogeneity. As an illustration, we study the noisy voter model, a modification of the original voter model including random changes of state. The proposed method is able to unfold the dependence of the model not only on the mean degree (the mean-field prediction) but also on more compl…
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, 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 a…
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 multi-state voter model in mean-field that incorporates a degree of imperfection or error in the copying process…
