Nonlinear voter model with inflexible zealots
Abstract
We study the dynamics of the nonlinear voter model with inflexible zealots in a finite wellmixed population. In this system, each individual supports one of two parties and is either a susceptible voter or an inflexible zealot. At each time step, a susceptible adopts the opinion of a neighbor if this belongs to a group of neighbors all in the same state, whereas inflexible zealots never change their opinion. In the presence of zealots of both parties the model is characterized by a fluctuating stationary state and, below a zealotry density threshold, the distribution of opinions is bimodal. After a characteristic time, most susceptibles become supporters of the party having more zealots and the opinion distribution is asymmetric. When the number of zealots of both parties is the same, the opinion distribution is symmetric and, in the long run, susceptibles endlessly swing from the state where they all support one party to the opposite state. Above the zealotry density threshold, when there is an unequal number of zealots of each type, the probability distribution is singlepeaked and nonGaussian. These properties are investigated analytically and with stochastic simulations. We also study the mean time to reach a consensus when zealots support only one party.
pacs:
89.75.k, 02.50.r, 05.40.a, 89.65.sI Introduction
The voter model (VM) Liggett () is one of the simplest and most influential examples of individualbased systems exhibiting collective behavior. The VM has been used as a paradigm for the dynamics of opinion in socially interacting populations, see e.g. Opinions (); Sociophysics () and references therein. The classical, or linear, VM is closely related to the Ising model Glauber () and describes how consensus results from the interactions between neighboring agents endowed with a discrete set of states (“opinions”). While the VM is one of the rare exactly solvable models in nonequilibrium statistical physics, it relies on oversimplified assumptions such as perfect conformity and lack of selfconfidence of all voters. This is clearly unrealistic as it is recognized that members of a society respond differently to stimuli: Many exhibit conformity while some show independence, and this influences the underlying social dynamics Granovetter (); ConfIndep (); GroupSize (). In order to mimic the dynamics of socially interacting agents with different levels of confidence, this author introduced “zealots” in the VM MM1 (); MM2 (); zealot07 (). Originally zealots were agents favoring one opinion MM1 (); MM2 (). The case of inflexible zealots whose state never changes was then also studied zealot07 (), and the influence of committed and/or independent individuals was considered in various models of opinion and social dynamics MM3 (); otherZealots (). Recently, authors have investigated the effect of zealots in naming and cooperation games, and even in theoretical ecology gamesZealots (); MM3 ().
In recent years, many versions of the VM have been proposed Opinions (). A particularly interesting variant of the VM is the twostate nonlinear voter model (VM) introduced in qVM (). In this model randomly picked neighbors may influence a voter to change its opinion. When , the qVM is closely related to the Sznajd model Sznajd (); Slanina (); genSznajd () and to that of Ref. vacillating (). The properties of the VM have received much attention and there is a debate on the expression of the exit probability in one dimension vacillating (); exitprobq2 (); exitprobq ().
Here, we investigate a generalization of the nonlinear VM, with , in which a wellmixed population consists of inflexible zealots and susceptible voters influenced by their neighbors. As a motivation, this parsimonious model allows to capture three important concepts of social psychology ConfIndep () and sociology Granovetter (): (i) conformity/imitation is an important social mechanism for collective actions; (ii) group pressure is known to influence the degree of conformity, especially when a group size threshold is reached GroupSize (); (iii) the degree of conformity can be radically altered by the presence of some individuals that are capable of resisting group pressure GroupSize (); ConfIndep (). Here, the VM mimics the process of conformity by imitation with groupsize threshold, whereas zealots are independent agents that resist social pressure and can thus prevent to reach unanimity.
In this work, we study the fluctuationdriven dynamics of the twostate VM with zealots in finite wellmixed populations and shed light on the deviations from the mean field description and from the linear case (). We find that below a zealotry density threshold the probability distribution is bimodal instead of Gaussian and, after a characteristic time, most susceptibles become supporters of the party having more zealots. When both parties have the same small number of zealots, susceptibles endlessly swing from the state where they all support one party to the other with a mean switching time that approximately grows exponentially with the population size.
In the next section we introduce the model. Sections III and IV are dedicated to the mean field description and to the model’s stationary probability distribution. In Secs. V and VI we discuss the longtime dynamics and the mean consensus time when there is one type of zealots. We summarize our findings and conclude in Sec. VII.
Ii The voter model with zealots
We consider a population of voters that can support one of two parties, either A or B, and therefore be in two states. Supporters of party A are in state , and those supporting party B are in state . Among the voters, a fixed number of them are “inflexible zealots” while the others are “susceptibles”. Here, zealots are individuals that never change opinion: they permanently support either party A (Azealots) or party B (Bzealots). Susceptible voters can change their opinion under the pressure of a group of neighbors. The population thus consists of a number of Azealots (pinned in state ) and of Bzealots (pinned in state ), and a total of susceptibles agents, of which are Asusceptibles (nonzealot voters in state ) and are Bsusceptibles (nonzealot voters in state ). The fraction, or density, of susceptibles in the entire population remains constant and is given by . For simplicity we assume that all agents have the same persuasion strength.
At each time step, a susceptible voter consults a group of neighbors (with ) and, if there is consensus in the group, the voter is persuaded to adopt the group’s state with rate qVM (). The dynamics is a generalization of the nonlinear VM qVM () with a finite density of zealots zealot07 (), and consists of the following steps:

Pick a random voter. If this voter is a zealot nothing happens.

If the picked voter is a susceptible, then pick a group of neighbors (for the sake of simplicity repetition is allowed, as in Refs. qVM (); exitprobq ()). If all neighbors are in the same state, the selected voter also adopts that state. Nothing happens in the update if there is no consensus among the neighbors Comment (), or if the voter and its neighbors are already in the same state.

Repeat the above steps ad infinitum or until consensus is reached.
The case corresponds to the classical (linear) voter model Liggett (); MM1 (); MM2 (); zealot07 (), and we therefore focus on .
For the sake of simplicity, we investigate this model on a complete graph (wellmixed population of size ). The state of the population is characterized by the the probability that the number of Asusceptibles at time is . This probability obeys the master equation noise ()
(1)  
The first line accounts for processes in which the number of Asusceptibles after the event equals , while the second term accounts for the complementary loss processes where . Here, represent the rates at which transitions occur and are given by
(2) 
When there are zealots of both types and the system has reflective boundaries at and . When there are only Azealots, and , with being the density of Azealots, then is an absorbing boundary with , while is reflective. The birthanddeath process (1) is here simulated with the Gillespie algorithm Gillespie () upon rescaling time in Eq. (1) as .
A quantity of particular interest is the magnetization that gives the population’s average opinion or, equivalently here, the opinion of a random voter zealot07 (). We have when all susceptibles are state (all Asusceptibles) and when all susceptibles are in state (all Bsusceptibles), with .
To gain an intuitive understanding of the VM dynamics, it is useful to consider the evolution of in typical sample realizations, as those in Fig. 1 where we illustrate the dynamics at low zealotry. In Fig. 1 we notice two distinct regimes and different timescales. In the case of symmetric low zealotry (), the number of susceptibles first approaches either the state (all Bsusceptibles) or (all Asusceptibles). After a characteristic time (see Sec. V.A), all susceptibles suddenly start switching from one state to the other, see Fig. 1 (a). A similar feature has been observed in the Sznajd model () with anticonformity genSznajd (). When , the majority of susceptibles become Asupporters after a typical time (see Sec. V.B). The fluctuations in the number of Asusceptibles then grow endlessly, see Fig. 1 (b). An important aspect of this work is to analyze how demographic fluctuations arising in finite populations alter the mean field predictions. In Section V the phenomena illustrated by Fig. 1 are studied in largebutfinite populations, and we show that these phenomena are beyond the reach of the next section’s mean field analysis.
Iii Mean Field Description
For further reference, it is useful to consider the mean field (MF) limit of an infinitely large population, . In such a setting, demographic fluctuations are negligible and the rates (2) can be written in terms of the density of Asusceptibles, and the densities of zealots of each type: and . The MF dynamics is described by the rate equation obtained by averaging from Eq. (1) (and rescaling time as ) noise ():
(3)  
where the dot denotes the time derivative and .
In the absence of zealotry (, ), Eq. (3) has two stable absorbing fixed points, (all Bsupporters) and (all Asupporters) corresponding to consensus with either A or B party, separated by an unstable fixed point (mixture of A and Bvoters) qVM (). It is worth noting that the dynamics of the VM without zealots ceases when a consensus is reached and this happens in a finite time when the population size is finite qVM (); Sznajd (); Slanina (); genSznajd (); vacillating (). However, in the presence of zealots supporting both parties, the population composition endlessly fluctuates MM2 (); zealot07 (), see, e.g., Fig. 1.
In the presence of zealotry, the interior fixed points of Eq. (3) satisfy , which leads to
(4) 
Depending on the values of and , this equation has either three physical roots, or a single physical solution.
iii.1 The symmetric case
When the density of zealots of both types is identical, and with , Eq. (3) becomes
that is characterized by a fixed point . When is sufficiently low, Eq. (3) has two further fixed points: and . The analysis for arbitrary is unwieldy, but insight can be gained by focusing on and , for which
We readily verify that are both stable when , with and . When , the fixed points are unphysical and is stable. This picture holds for arbitrary finite value of : are stable and the MF dynamics is characterized by bistability below a critical zealotry density , while is unstable when and stable when , see Fig. 2(a,b). By determining when Eq. (4) has three physical roots, we have found the critical zealotry density
(5) 
while since in the linear VM Eq. (3) has always one single stable fixed point zealot07 (). Hence, the value of increases with , while the values of and get closer to the values (all Bsusceptibles) and (all Asusceptibles) as increases with kept fixed.
In this MF picture, the population’s average opinion given by the magnetization undergoes a supercritical pitchfork bifurcation at Strogatz (): At , the critical value separates an ordered phase (), where a majority of susceptibles supports one party, from a disordered phase () in which each party is supported by half of the susceptibles, see Fig. 2 (a,b). The stationary MF magnetization thus depends on the initial condition: when , if and if , while the magnetization vanishes when (or if ). Using Eqs. (4) and (5), it can be directly checked that just below the critical zealotry density, i.e. for , the stationary magnetization is characterized by the scaling relationship .
iii.2 The asymmetric case
When the number of Azealots exceeds that of Bzealots, with , it is convenient to use the parametrization
(6) 
where quantifies the zealotry asymmetry. With Eq. (6), we still have with and Eq. (3) becomes
This rate equation is also characterized by bistability at low zealotry, with two stable fixed points separated by an unstable fixed point , and by the sole stable fixed point at higher zealotry, see Fig. 2(c,d). By determining when Eq. (3) has three physical fixed points, we have determined the critical density of zealotry , see Fig. 3: At fixed and , the fixed points are stable when while only is stable when . We have found that decreases with (at fixed ) and increases with (at fixed ).
In this MF picture, the opinion of a random individual is given by the magnetization . The critical zealotry density separates a bistable phase () from a phase where most susceptibles support the party having more zealots, see Fig. 2 (c,d). Hence, the stationary MF magnetization at low zealotry () depends on the initial condition and is if and if . When the stationary MF magnetization is .
iii.3 The absorbing case
When there are only Azealots, and , Eq. (3) becomes
and has an absorbing fixed point . Below a critical zealotry density , this rate equation admits two other fixed points: , that is stable, and that is unstable and separates and , see Fig. 2(e,f). When , the absorbing state is the only fixed point. For and , we explicitly find
(7) 
and
(8) 
From these expressions, and more generally by determining when Eq. (3) has three physical fixed points, we have found the critical zealotry density in the absorbing case:
(9) 
We thus distinguish two regimes:
(i) When both are stable and the dynamics crucially depends on the initial density of Asusceptibles: If , the final state is the consensus with party A; whereas the steady state consists of a vast majority of Bparty voters when . In Sec. VI, we show that random fluctuations drastically alter this picture: In a finite population, is a metastable state when and , and we shall see that the Aconsensus is reached after a very long transient that scales exponentially with the population size.
(ii) When , as well as when and , the absorbing state is rapidly reached.
Iv Stationary probability distribution
In this section, we compute the stationary probability distribution (SPD) of the VM with zealotry when there is no absorbing state, and show that it shape generally differs from the Gaussianlike distribution obtained in the linear VM with zealots zealot07 ().
The SPD obeys the following stationary master equation, obtained from Eq. (1):
The exact SPD is uniquely obtained by iterating the detailed balance relation noise (), yielding
(10)  
where the normalization gives and .
Since , the stationary magnetization distribution has the same shape as , with
In large populations, a useful approximation of (10) is obtained by writing with , and by using EulerMacLaurin formula , where we have neglected higher order terms Arfken ().
When , it is useful to work in the continuum limit with the rates , as in Sec. III. By introducing
(12) 
we have to leading order in . Hence, the leading contribution to the SPD when is
(13) 
The local extrema of satisfy , see (12), and thus coincide with the fixed points of Eq. (3). As a consequence, in large populations is either characterized by a single peak at when , or has two peaks at the metastable states when . In this case, there is bistability and the amplitudes of the peaks at are in the ratio ()
(14) 
The integrals in Eqs. (13) and (14) can be computed, but their expressions are unenlightening. Here, we infer the properties of and from those of .
iv.1 Stationary probability distribution in the symmetric case
In the symmetric case, , Eq. (12) becomes
and has the symmetry . We distinguish the cases of low and high zealotry density:
(i) When , the fixed points and of Eq. (3) are also the roots of . Hence, when , is a symmetric bimodal SPD characterized by two peaks at . As a consequence, is an even function.
In Figure 4 (a), we show the exact SPD for characterized by two peaks of same intensity at and a local minimum at . We remark that when is increased, the SPD vanishes dramatically away from the peaks. In fact, since is close to zero or negative on , see Fig. 4 (a, upper inset), vanishes exponentially with and when is increased. Fig. 4(a) shows that the SPD steepens and its peaks are more pronounced when and are increased and is kept fixed. We have also obtained the (quasi)SPD from stochastic simulations, see Fig. 4 (a, lower inset), by averaging over realizations after simulation steps. While unavoidably more noisy, the simulation results reproduce the predictions of Eq. (10). Fig. 4(b) shows how scales with for different population sizes, and we notice that the main influence of raising is to concentrate the probability density around the peaks whose location are essentially unaffected by (when ). In Fig. 4, we also notice that the symmetric peaks are clearly identifiable when and , but almost coincide with and for . This is because approach the values when is increased.
(ii) When , the only physical root of is , as in the classical voter model zealot07 (). Hence, has a single maximum at when . The resulting symmetric Gaussianlike distribution centered at when , see Fig. 5, is very similar to the SPD obtained in the classical voter model with zealots zealot07 (). Fig. 5(inset) illustrates that the probability density steepens around when the population size is increased.
iv.2 Stationary probability distribution in the asymmetric case
In the asymmetric case, with zealot densities and , Eq. (12) is
and has either three or one physical roots:
(i) At fixed and , when , the fixed points and of Eq. (3) are the physical roots of . Since when , the SPD is again a bimodal distribution peaked at . However, has a greater basin of attraction than and . As a consequence, the SPD is asymmetric, with the peak at being much stronger than the one at . The ratio of the peaks is given by (14), which shows that the asymmetry of grows exponentially with and increases with , see Fig. 6(a). While an asymmetry in the zealotry in the linear VM does not significantly affect the form of the SPD zealot07 (), we here find that in the VM even a small bias in the zealotry drastically changes the shape of the SPD and leads to marked dominance of the party with more zealots.
In Fig. 6 (a), we report the exact SPD for and illustrate its asymmetric bimodal nature, with marked peaks of different intensities at . We notice that the asymmetry in the peaks intensity, given by (14), is stronger when we increase and . As in the symmetric case, the SPD decays dramatically away from the peaks and vanishes with and when is increased. In Fig. 6 (a, inset) we show that the SPD remains bimodal when the population size is increased, and the main influence of raising is to concentrate the probability density near its peak at (when ).
(ii) At fixed and , when , the only real root of is . This lies closer to than to , and hence is an asymmetric function with a single maximum at . Therefore, in large populations is an asymmetric leftskewed SPD with a single peak at , as shown in Fig. 5(b) where we see that the SPD broadens when is increased and that it steepens when is increased. As above, the probability density steepens around when is increased.
V Fluctuationdriven dynamics at low zealotry
We now study how a small nonzero density of zealots of both parties () affects the VM longtime dynamics. We show that, after a typically long transient, all susceptibles voters switch allegiance from the state (all Bsusceptibles) to state (all Asusceptibles) in a typical switching time. In the symmetric case, there is “swingstate dynamics” with all susceptibles endlessly swinging allegiance. In the asymmetric case where party A has more zealots than party B, the dynamics is characterized by various timescales and by growing fluctuations around the metastable state . Below, we show that the longtime VM dynamics is driven by fluctuations and characterized by a mean switching time that scales (approximately) exponentially with the system size in largebutfinite populations.
v.1 Swingstate dynamics and switching time in the case of symmetric zealotry
As illustrated in Fig. 1(a), the longtime dynamics in the symmetric case is characterized by the continuous swinging from states to and vice versa. When , all susceptibles thus continuously switch allegiance in the long run. In that regime, the magnetization is thus characterized by abrupt jumps from to , see Fig. 7, while the stationary ensembleaveraged magnetization , since is even and each agent is as likely to be in one or the opposite state. A similar phenomenon has been found in the Sznajd model () with anticonformity genSznajd ().
This swingstate phenomenon is not captured by the mean field description of Sec. III and is here characterized by the mean time to switch for the first time from state to . The scaling of on allows us to rationalize the data of Fig. 7 where the switching time is found to dramatically increase with the population size. Clearly, the symmetry implies that the mean switching, or swinging, time is identical to the mean time to switch from to .
Finding the mean switching time can be formulated as a firstpassage time problem and, when , can be computed using the framework of the backward FokkerPlanck equation (bFPE) noise (). In this context, the model’s bFPE infinitesimal generator is
(15) 
The mean time to be absorbed at (all Asusceptibles), starting from the initial state , with a reflective boundary at (all Bsusceptibles), obeys
(16) 
with and (reflective and absorbing boundaries) noise (); KramersReview (). To obtain the mean switching time we solve Eq. (16) with using standard methods noise (), and obtain
(17) 
where . As with other fluctuationdriven phenomena associated with metastable states, see e.g. Kramers (); KramersReview (); 3SVMZ (); WKB (); otherWKB () and below, this result predicts that the mean switching time grows (approximately) exponentially with the population size . This explains the difference of various orders of magnitude in the switching time observed in Figs. 7(a) and 7(b).
The predictions of (17) are reported in Fig. 8 for various values of . These are in good agreement with the results of numerical simulations (averaged over 1000 samples, each run for simulation steps). When is lowered well below , the peaks of the SPD approach and . In this case, increases and switching allegiance takes very long. At fixed , we find that increases with . Interestingly, we also find that can exhibit a nonmonotonic dependence on just below when is kept fixed, as shown in Fig. 8.
v.2 Timescale separation and growing fluctuations in the asymmetric case
In the asymmetric case , the party A has more zealot supporters than party B. In this situation, when the SPD has a marked peaked near , see Fig. 6(a). As shown in Fig. 1(b), the long time dynamics is characterized by a large majority of susceptibles becoming A supporters independently of the initial state. The magnetization thus fluctuates around its MF value before reaching when all susceptibles are supporters of party A, see Fig. 9(a). The population composition then endlessly fluctuates, with a majority of susceptibles supporting party A. In this case, with Eq. (IV), the stationary ensembleaveraged magnetization is positive.
The VM dynamics is thus characterized by various regimes not captured by the mean field description. For concreteness, we consider that the initial density of Asusceptibles is , as in Fig. 9, and distinguish four time scales:
(i) After a mean time of order , the system quickly relaxes toward the metastable state where a random voter has the MF opinion , see Fig. 9(a).
(ii) After a mean time , almost all realizations suddenly approach the metastable state where , see Figs. 1(a) and 9(b). The mean transition time , as well as the average relaxation times, can be estimated using Kramers’ classical escape rate theory Kramers (). The latter gives the mean transition time between the two local minima of the doublewell potential in which an overdamped Brownian particle is moving subject to a zeromean deltacorrelated Gaussian white noise force . Here, we consider a potential such that , and the noise correlations . The bFPE generator of this Brownian particle is (15) with a constant diffusive term evaluated at . Kramer’s formula hence gives Kramers (); KramersReview ():
where , and denotes the mean relaxation time from state to .
(iii) The system then fluctuates around before reaching the state (all Asusceptibles) where , see Fig. 9(a) after a mean time . In the realm of the bFPE, the mean time for all susceptibles to become A supporters for the first time is
(18) 
When is close to the state , the main contribution to is given by the mean transition time that is independent of , as illustrated by Fig. 9(b). This is well approximated by Kramer’s formula, yielding
(19) 
showing that the mean switching time scales exponentially with the population size.
(iv) The amplitude of the fluctuations around , where grows endlessly in time, see Fig. 1(b), and the system eventually returns to the state (all Bsusceptibles). Yet, this occurs after an enormous amount of time, of order , that is generally not physically observable when .
The predictions (18) and its approximation (19) are reported in Fig. 9(b), where they are in good agreement with the results of stochastic simulations. These results confirm that grows approximately exponentially with when . In Fig. 9(b), we also see that increases with , and with when and are fixed. As illustrated in Fig. 9(a), contrary to the case of symmetric zealotry, there is no “swingstate dynamics”: After a mean time the population persists near where most susceptibles are A supporters and the magnetization is , and there is virtually no switching back to state . Hence, a small bias in the zealotry, combined with fluctuations and nonlinearity, can greatly affect the voters’ opinion in the VM.
Vi Mean consensus time in the presence of one type of zealots
When there are only Azealots, with and , an Aparty consensus is always reached. Yet, the dynamics leading to the corresponding absorbing state depends nontrivially on the zealotry density and on the initial density of Asusceptibles.
Here, the fluctuationdriven dynamics is characterized by the mean consensus time (MCT). As illustrated in Fig. 10, the MCT can change by several order of magnitudes when and change over a small range: (i) Below the critical zealotry density , the MCT grows exponentially with the population size when , see Fig. 10(b); (ii) Otherwise the MCT grows logarithmically with , see Fig. 10(inset). These phenomena are analyzed as follows: