Polarization and Consensus in a Voter Model under Time-Fluctuating Influences

Abstract

We study the effect of time-fluctuating social influences on the formation of polarization and consensus in a three-party community consisting of two types of voters (“leftists” and “rightists”) holding extreme opinions, and moderate agents acting as “centrists”. The former are incompatible and do not interact, while centrists hold an intermediate opinion and can interact with extreme voters. When a centrist and a leftist/rightist interact, they can become either both centrists or both leftists/rightists. The population eventually either reaches consensus with one of the three opinions, or a polarization state consisting of a frozen mixture of leftists and rightists. As a main novelty, here agents interact subject to time-fluctuating external influences favouring in turn the spread of leftist and rightist opinions, or the rise of centrism. The fate of the population is determined under various scenarios, and it is shown how the rate of change of external influences can drastically affect the polarization and consensus probabilities, as well as the mean time to reach the final state.

1. Introduction

The relevance of parsimonious individual-based models to describe social phenomena at micro and macro levels has a long history [1,2,3]. In the last few decades, “sociophysics” has grown as a research field that aims at studying collective social behaviour, such as the spread of opinions or the dynamics of cultural diversity, using models and methods from statistical physics [3,4,5,6,7,8,9,10,11,12,13,14]. Typical questions in sociophysics concern the conditions under which consensus, or long-term opinion diversity, emerges in a population of agents (“voters”) whose states (“opinions”) change as they interact.

The voter model (VM) [15], closely related to the Ising model [16], has been commonly used to describe how consensus ensues from the interactions between neighbouring voters. In fact, while the classical two-state VM is arguably the simplest and most popular model of opinion dynamics, it rests on a number of oversimplifying assumptions: voters are endowed with only two possible states, they are blind to any external stimuli, have zero self-confidence and are all identical. In reality, members of social communities respond differently to stimuli, as they can interact in groups [1,17,18,19], and are usually characterised by multiple attributes [20,21,22,23,24]. In light of this, many generalizations of the VM have been proposed: for instance, “zealotry” was introduced in various forms to endow voters with different levels of self-confidence [25,26,27,28,29,30,31,32,33,34,35,36], while group-size influence is notably captured in the nonlinear q-voter model [37] and its variants [38,39,40,41,42,43,44,45,46]. It has also been noted that in many cases only some of the attributes characterising agents are actually compatible for social interactions [20,24,47,48,49,50,51]. It was thus suggested that agents whose opinions are too different would not interact, while voters holding close opinions can interact and attain a global consensus. This motivated the study of multi-state VMs, such as the constrained three-state voter model (3CVM) of Refs. [52,53,54], which is a discrete version of the bounded-compromise model [47,48,49,50,51]. In the 3CVM, incompatible “leftist” and “rightist” voters can only interact with “centrists”, and the final outcome is either consensus with one of the three parties, or a polarized state consisting of mixture of leftists and rightists. In the latter case, the population is stuck in a frozen state of “polarization” in which leftists and rightists hold uncompromising opinions.

In addition to interactions among agents, external stimuli or influences, especially social media and news sources, play an increasingly important role in shaping the social environment that in turn affects the collective social behaviour [1,55,56,57,58,59,60,61]. Yet, with some recent remarkable exceptions [62,63,64,65], the role of social influences is still rarely modelled in sociophysics. Actually, sources of news play both a crucial, yet complex and multifaceted, role in influencing public social behaviour. They are typically characterised by opposing viewpoints, or agenda, that often change over time and can in turn favour consensus or polarization. It is also worth noting that voter-like models subject to a time-periodic external field have recently been studied in the context of economic networks [66,67].

How do the opinions of a social community evolve in a volatile and time-fluctuating environment? Inspired by this question, here, we introduce a generalization of the constrained three-state voter model [52,53,54] in the presence of binary time-fluctuating external influences. In the same vein as in population dynamics [68,69,70,71,72,73,74,75,76,77,78,79,80,81], for the sake of simplicity, we assume that the media influences endlessly switch from favouring the spread of polarization to promoting centrism, and vice versa. The goal of this work is to determine the fate of the population under various scenarios, and in particular to study how the time variation of the external influences affects the probability to reach polarization or a consensus, as well as the mean time for the population to settle in its final state.

The plan of the paper is as follows: The general formulation of the model is introduced in the next Section. Section 3 is dedicated to a thorough study of the polarization and consensus probabilities. Section 4 focuses on the study of the mean exit time. Section 5 is dedicated to a discussion of the results and to the conclusions. Technical details, including useful results in the absence of external influences, and possible generalizations and applications are discussed in three Appendices.

2. Three-State Constrained Voter Model under Binary Time-Fluctuating Influences

We consider a well-mixed population of N individuals consisting of $N_R$ “rightists”, or R-voters, $N_L$ “leftists”, or L-voters, and $N_C$ “centrists”, or C-voters, with $N=N_L+N_R+N_C$. In the voter model language [7,8,9,13,15], R and L represent extreme opinions, while C-voters hold an intermediate opinion. In the three-state constrained voter model (3CVM) [53,54], an agent selected at random tries to interact with one of its neighbours, that is any other randomly picked voter, at each microscopic update attempt. When the two agents hold the same opinion or if it is a pair of leftist–rightist ($LR$ or $RL$), nothing happens. However, if the pair is a centrist C and an L or an R voter ($LC,CL,RC$ or $CR$), the initial agent adopts the opinion of the neighbour with a probability that depends on the external influences whose effect is encoded into the random variable $\xi \left(t\right)$ that fluctuates with the time t; see below in this Section and Figure 1. Hence, while the interactions between L and R voters and centrists C now change in time with $\xi$, L and R remain incompatible and the system’s final state is, as in the static 3CVM, either a consensus state or a state of polarization consisting of a frozen mixture of leftists and rightists [53,54]; see below in this Section.

The main novel ingredient of this study is the modelling of time-fluctuating social influences by means of the coloured dichotomous (telegraph) noise $\xi \left(t\right)\in \{-1,1\}$ [72,73,74]; see Figure 1. The process $\xi \left(t\right)$ simply encodes all the complex effects of social environment and external influences in the endless random switching between two states: here, $\xi =1$ corresponds to external influences favouring L and R opinions (spread of polarization), whereas $\xi =-1$ favours centrism C (compromise between L and R). Here, the dichotomous noise $\xi$ is always at stationarity, and switches from $\pm 1$ to $\mp 1$ according to

$$\xi ⟶-\xi ,$$

(1)

with a rate $(1-\delta \xi )\nu$, where $\nu$ is the average switching rate [77], and $1<\delta <-1$ denotes the switching asymmetry. Accordingly, the average time spent in state $\xi =\pm 1$ before switching to $-\xi$ is $1/(1\mp \delta )\nu$ (symmetric switching occurs when $\delta =0$), see Figure 1a–c. At stationarity, $\xi =\pm 1$ with probability $(1\pm \delta )/2$ [72,73,74], and its (ensemble-)average is

$$\begin{array}{cc}\hfill \langle \xi \left(t\right)\rangle & =\delta ,\hfill \end{array}$$

(2)

while its autocorrelation is $\langle \xi \left(t\right)\xi \left(t^{\prime }\right)\rangle -\langle \xi \left(t\right)\rangle \langle \xi \left(t^{\prime }\right)\rangle =(1-{\delta }^2)e^{-2\nu |t-t^{\prime }|}$.

The 3CVM switching dynamics is therefore defined by the four reactions: $LC\to LL$ and $RC\to RR$, corresponding to the spread of extreme opinions at rate $(1+b\xi )/2$, and $LC\to CC$ and $RC\to CC$, with centrists replacing L and R voters at rate $(1-b\xi )/2$. Here, $0<b<1$ denotes the social influences bias favouring polarization when $\xi =1$ and centrism in the social environment $\xi =-1$. The 3CVM switching dynamics can thus be schematically described by the following reactions occurring at each time increment:

Figure 2. Illustration of the 3CVM dynamics in the simplex $\ell +r+c=(N_L+N_R+N_C)/N=1$. Circles show the consensus (absorbing) states all-L$(1,0,0)$, all-R$(0,1,0)$, and all-C$(0,0,1)$, and the initial condition is $(x,y,z)$. The thick line indicates polarization state (pol-$LR$) consisting of a frozen mixture of L and R voters made up of a fraction $\ell =N_L/N$ of L-voters coexisting with an incompatible fraction $1-\ell$ of R-voters. Dashed lines are typical trajectories: the dynamics ceases when the line pol-$LR$ is reached (polarization with probability $P_{LR}$), or when there is consensus by reaching one of absorbing states all-L, all-R or all-C (with respective probabilities $P_L,P_R$ and $P_C$). Once a trajectory reaches the line $\ell =0$ or $r=0$, the evolution is restricted on $\ell =0$, $r=0$ until there is a consensus. As $\xi \left(t\right)$ varies, the dynamics favours in turn the spread of L and R ($\xi =1$) or that of C ($\xi =-1$). See text for details.

Figure 2. Illustration of the 3CVM dynamics in the simplex $\ell +r+c=(N_L+N_R+N_C)/N=1$. Circles show the consensus (absorbing) states all-L$(1,0,0)$, all-R$(0,1,0)$, and all-C$(0,0,1)$, and the initial condition is $(x,y,z)$. The thick line indicates polarization state (pol-$LR$) consisting of a frozen mixture of L and R voters made up of a fraction $\ell =N_L/N$ of L-voters coexisting with an incompatible fraction $1-\ell$ of R-voters. Dashed lines are typical trajectories: the dynamics ceases when the line pol-$LR$ is reached (polarization with probability $P_{LR}$), or when there is consensus by reaching one of absorbing states all-L, all-R or all-C (with respective probabilities $P_L,P_R$ and $P_C$). Once a trajectory reaches the line $\ell =0$ or $r=0$, the evolution is restricted on $\ell =0$, $r=0$ until there is a consensus. As $\xi \left(t\right)$ varies, the dynamics favours in turn the spread of L and R ($\xi =1$) or that of C ($\xi =-1$). See text for details.

$$\begin{array}{ccc}\hfill LC& ⟶& LL\mathrm{rate}:\frac{1+b\xi }{2},LC⟶CC\mathrm{rate}:\frac{1-b\xi }{2},\hfill \\ \hfill RC& ⟶& RR\mathrm{rate}:\frac{1+b\xi }{2},RC⟶CC\mathrm{rate}:\frac{1-b\xi }{2},\hfill \end{array}$$

where $\xi$ and the rates endlessly fluctuates according to Equation (1). The dynamics of the switching 3CVM is therefore the Markov chain defined by the transition rates (A1) and master Equation (A2) and (A3) [82]; see Appendix A.1. The 3CVM switching dynamics is characterised by three consensus/absorbing states $N_R=N_C=0$ (all-L), $N_L=N_C=0$ (all-R), $N_L=N_R=0$, $N_C>0$ (all-C), and by the polarization state (pol-$LR$) where $N_L+N_R=N$, $N_C=0$; see Figure 2. As in the absence of external influences, the final state of the population is therefore guaranteed to be either one of the consensus/absorbing states or pol-$LR$ [53,54].

It is worth noting that the approach considered here bears some similarities with the two-party model of Refs. [62,63]. However, there are also important differences: first, in the 3CVM, the polarization state corresponds to a frozen mixture (while it is an active state in Refs. [62,63]). Moreover, external influences are here assumed to fluctuate endlessly in time, rather than by establishing a certain number of connections with voters. It is also interesting to notice that the effect of an exogenous time-varying influence has recently been studied in the context of economic networks [66,67]. The ensuing dynamics derived also from an Ising-like (voter-like) model subject to a time-dependent external field [66,67] has been shown to lead to rich dynamics characterized by hysteresis [83], which is not a phenomemon exhibited by the 3CVM switching dynamics. This stems from the fact that Equation (A4) differ from those governing the mean-field dynamics in Refs. [66,67] for not having any non-noisy linear terms, and for the exogenous time dependence being stochastic (via the multiplicative dichotomous noise $\xi \left(t\right)$) rather than periodic.

3. Final State: Polarization and Consensus Probabilities

In its final state, the population is either in the polarized state pol-$LR$, or in one of its three absorbing/consensus states (all-L, all-R or all-C); see Figure 2. Here, $P_{LR}$ denotes the probability to end up in the polarized final state pol-$LR$. The probabilities to reach the absorbing/consensus states all-C, all-L, and all-R are, respectively, denoted by $P_C,P_L$ and $P_R$. The density of voters of each type is $\ell \equiv N_L/N,r\equiv N_R/N$ and $c\equiv N_C/N=1-\ell -r$, and initially the population consists of densities $x,y,z=1-x-y$ of $L,R$ and C voters, respectively.

In the absence of environmental switching, the probabilities of ending in any of the absorbing/consensus or polarization states were found to depend non-trivially on the parameter $s\equiv Nb$ and $(x,y,z)$ [53,54]; see Appendix B. Here, we are interested in finding how $P_{LR},P_C,P_L$ and $P_R$, as well as the final densities $(\ell ,r,c)$ depend on $\nu$ under various scenarios. We consider the same initial density of L and R voters, i.e., $0<x=y=(1-z)/2<1/2$, which suffices for the purposes of this study and simplifies the analysis. (Only in Appendix C, an example with $x\ne y$ is briefly considered.) By symmetry, $x=y$ implies $P_L\left(\nu \right)=P_R\left(\nu \right)$, i.e., the probability of L and R consensus is the same. When it occurs, polarization consists of a fraction $1/2$ of L and R voters; see Figure 2. We thus have: $P_{LR}+P_C+P_L+P_R=P_{LR}+P_C+2P_L=1$, and therefore focus on studying $P_{LR}$ and $P_C$ as functions of $\nu$ for different values of $\delta$ and z, treated as parameters, from which we obtain also the final densities: $(\ell ,r,c)=(\ell ,\ell ,1-2\ell )=((1-P_C)/2,(1-P_C)/2,P_C)$; see Appendix A.2.

3.1. Final State in the Regimes $\nu \to 0$ and $\nu \to \infty$

The polarization and consensus probabilities can be computed analytically in the regimes $\nu \to 0$ and $\nu \to \infty$. For this, we take advantage of the results obtained in the absence of external influences for the polarization probability ${\mathcal{P}}_{LR}$, and the probabilities of $C,L$ and R consensus, respectively, denoted here by ${\mathcal{P}}_C$, ${\mathcal{P}}_L$ and ${\mathcal{P}}_R$. In the absence of external influences, these quantities have been obtained in Refs. [53,54] and are summarized in Appendix B.1.

3.1.1. Polarization and Consensus Probabilities in the Regime $\nu \to 0$

When $\nu \to 0$, we can assume that there are no switches before attaining polarization or consensus. In this case, $\xi$ is a quenched random variable, and the kinetics is the superposition, with probability $(1\pm \delta )/2$, of the 3CVM dynamics in the stationary environment $\xi =\pm 1$. As a result, the polarization probability when $\nu \to 0$, $P_{LR}^0$, is the superposition of ${\mathcal{P}}_{LR}(\pm s,z)$, obtained in a static external state $\xi =\pm 1$, with probability $(1\pm \delta )/2$:

$$\begin{array}{c}\hfill P_{LR}^0=\left(\frac{1+\delta }{2}\right){\mathcal{P}}_{LR}(s,z)+\left(\frac{1-\delta }{2}\right){\mathcal{P}}_{LR}(-s,z),\end{array}$$

(3)

which is readily obtained from Equation (A5) or (A7). Similarly, the consensus probabilities when $\nu \to 0$, are obtained from (A5), with Equations (A6) and (A8), and ${\mathcal{P}}_L(s,z)=(1-{\mathcal{P}}_{LR}(s,z)-{\mathcal{P}}_C(s,z))/2$:

$$\begin{array}{c}\hfill P_{C,L}^0=\left(\frac{1+\delta }{2}\right){\mathcal{P}}_{C,L}(s,z)+\left(\frac{1-\delta }{2}\right){\mathcal{P}}_{C,L}(-s,z).\end{array}$$

(4)

With Equations (A9) and (4), we obtain the final density ${\ell }_0=r_0$ of L and R voters when $\nu \to 0$:

$$\begin{array}{c}\hfill {\ell }_0=r_0=\frac{1-P_C^0}{2}.\end{array}$$

(5)

3.1.2. Polarization and Consensus Probabilities in the Regime $\nu \to \infty$

When $\nu \to \infty$, so many switches occur before polarization or consensus that $\xi$ self-averages [71,72,75,76,77,78]. In this case, $\xi$ is an annealed random variable that can be replaced by its average: $\xi \to \langle \xi \rangle =\delta$. The switching dynamics of the 3CVM with $\nu \to \infty$ is thus the same as in Ref. [54], with $s\to s\langle \xi \rangle =s\delta$. In the limit $\nu \to \infty$, the polarization probability, $P_{LR}^{\infty }$, is therefore obtained from Equation (A5) or (A7) according to

$$\begin{array}{c}\hfill P_{LR}^{\infty }={\mathcal{P}}_{LR}(s\delta ,z).\end{array}$$

(6)

Similarly, the consensus probabilities under high $\nu$, $P_{C,L}^{\infty }$, are obtained from Equations (A6) and (A7):

$$\begin{array}{c}\hfill P_{C,L}^{\infty }={\mathcal{P}}_{C,L}(s\delta ,z).\end{array}$$

(7)

With Equations (A9) and (7), the final density ${\ell }_{\infty }=r_{\infty }$ of L and R voters in the regime $\nu \to \infty$:

$$\begin{array}{c}\hfill {\ell }_{\infty }=r_{\infty }=\frac{1-P_C^{\infty }}{2}.\end{array}$$

(8)

Since ${\mathcal{P}}_{LR}\approx 1$ when $s\equiv Nb\gg 1$ and ${\mathcal{P}}_C\approx 1$ when $s<0$ and $\left|s\right|\gg 1$ [54] (see Appendix B.1), the focus here is on the regime where the small influences bias affects a large but finite number of voters, i.e., $b\ll 1$ and $s\gg 1$, with $s\delta =\mathcal{O}\left(1\right)$. This allows us to highlight the effect of the external influences.

3.2. Polarization and Consensus Probabilities When $\delta >0$

When $\delta >0$, most of the time is spent in the external state $\xi =1$, where influences favour polarization (pol-$LR$); see Figure 1 and Figure 2.

In fact, as shown in Figure 3a, when z is not too close to 1, $P_{LR}>P_C$ for all values of $\nu$. In this case, $P_{LR}$ increases with $\nu$ over a large range of values (for $\nu \gtrsim b$), with $P_{LR}\approx P_{LR}^0\approx (1+\delta )/2$ when $\nu \ll b$ and $P_{LR}\approx P_{LR}^{\infty }$ when $\nu \gg b$, whereas $P_C$ is a decreasing function of $\nu$, with $P_C\approx P_C^0\approx (1-\delta )/2$ when $\nu \ll b$ and $P_C\approx P_C^{\infty }\ll 1$ when $\nu \gg b$; see Figure 3a. The fact that $P_{LR}$ and $P_C$ vary little with $\nu$, and are quite close to $P_{LR}^{0,\infty }$ and $P_C^{0,\infty }$, indicates that the analytical predictions for $P_{LR}^{0,\infty }$ and $P_C^{0,\infty }$ not only apply to the limits $\nu \to 0,\infty$, but are also valid approximations of $P_{LR}\left(\nu \right)$ and $P_C\left(\nu \right)$ in the regimes of low and high switching rate (see also Section 4 below). We can also notice in Figure 3a, that $P_{LR}$ exhibits a weak non-monotonic behaviour, with a “dip” for $\nu \sim b$. In this setting, the final fraction of L and R voters, given by $\ell =r=(1-P_C)/2$ (see Appendix A.2), is an increasing function of $\nu$, while the final density of centrists, $c=1-2\ell =P_C$, decreases with $\nu$; see inset in Figure 3a.

When there is a small initial fraction of L and R voters, with z close to 1, centrism can prevail over a large range of values of $\nu$: $P_C>P_{LR}$ for $\nu \gtrsim b$, see Figure 3b. The probability $P_{LR}$ thus decreases with $\nu$, while $P_C$ can have a non-monotonic behaviour as in Figure 3b where it exhibits a “bump” for $\nu \sim b$. In this case, centrists hold the majority opinion over an intermediate range of $\nu$: $P_C>P_{LR}+P_L+P_R=1-P_C$, i.e., $P_C>1/2$ when $\nu =\mathcal{O}\left(b\right)$; see inset in Figure 3b.

In Figure 3, the predictions (3), (4), (6) and (7) for $P_{LR/C}^0$ and $P_{LR/C}^{\infty }$ are in good agreement with simulation data when $\nu \ll b$ and $\nu \gg b$, respectively. The results reported in Figure 3 show that when $\delta >0$ the main effects of the switching external influences is to tame the bias favouring polarization. $P_{LR}$ and $P_C$ can thus increase or decrease with $\nu$, and even have a local extremum at a nontrivial switching rate. We explain this by solving, at fixed $\delta$, $P_{LR}^{\infty }=P_{LR}^0$ and $P_C^{\infty }=P_C^0$ for $z_{LR}$ and $z_C$ with Equations (3), (4), (6) and (7):

$$\begin{array}{cc}\hfill {\mathcal{P}}_{LR}(s\delta ,z_{LR})& =\left(\frac{1+\delta }{2}\right){\mathcal{P}}_{LR}(s,z_{LR})+\left(\frac{1-\delta }{2}\right){\mathcal{P}}_{LR}(-s,z_{LR}),\hfill \\ \hfill {\mathcal{P}}_C(s\delta ,z_C)& =\left(\frac{1+\delta }{2}\right){\mathcal{P}}_C(s,z_C)+\left(\frac{1-\delta }{2}\right){\mathcal{P}}_C(-s,z_C).\hfill \end{array}$$

(9)

Using (A5) and (A6), these equations are solved for $z_{LR}$ and $z_C$. When $z<z_{LR}$, $P_{LR}^{\infty }>P_{LR}^0$ while $P_{LR}^{\infty }<P_{LR}^0$ when $z>z_{LR}$. Similarly, $P_C^{\infty }>P_C^0$ when $z>z_C$ and $P_C^{\infty }<P_C^0$ for $z<z_C$. In the examples of Figure 3, $z_{LR}\approx 0.708$ and $z_C\approx 0.887$, and therefore $z<z_{LR,C}$ in Figure 3a and $z_{LR}<z<z_C$ in Figure 3b, which agrees with $P_{LR}^{\infty }>P_{LR}^0$ and $P_C^{\infty }<P_C^0$ reported in Figure 3a, and with $P_{LR,C}^{\infty }<P_{LR,C}^0$ in Figure 3b.

The extrema of $P_{LR}$ and $P_C$ in Figure 3 at $\nu \sim b$ can be explained heuristically (and similarly for those in Figure 4, while there are no extrema in Figure 5; see below). In fact, when $\nu \to 0$, the population remains in the initial external state $\xi \left(0\right)$ until polarization or consensus occurs. When $s\gg 1$ and $\nu \ll 1$, polarization occurs with a probability close to 1 if $\xi \left(0\right)=1$ and close to 0 otherwise, and we have $P_{LR}(\nu \ll b)\approx (1+\delta )/2$. We argue that $P_{LR}\left(\nu \right)$ decreases with $\nu$ when this rate is raised from $\nu \to 0$ to $\nu \lesssim b\ll 1$. For the sake of argument, we focus on the switching rate ${\nu }_{*}$ for which there is one external switch before reaching the final state. As discussed in Section 4.2 below, we have ${\nu }_{*}\lesssim b$ (see the inset in Figure 6b), with $\xi (t>1/{\nu }^{*})=-\xi \left(0\right)$, and the population settles in its final state after a time $T\sim \left(\ln N\right)/b$. Hence, if $\xi \left(0\right)=1$, the final state is polarization if pol-$LR$ is reached in a time $t\lesssim 1/{\nu }_{*}$, when still $\xi \left(t\right)=1$. This occurs approximately with a probability $1/\left(T{\nu }_{*}\right)$. On the other hand, if $\xi \left(0\right)=-1$, the final state is polarization if pol-$LR$ is reached after the external switch, when $\xi \left(t\right)=1$, i.e., for $t\gtrsim 1/{\nu }_{*}$, and this occurs with an approximate probability $1-1/\left(T{\nu }_{*}\right)$. Hence, we estimate

$$P_{LR}\left({\nu }_{*}\right)\approx \left(\frac{1+\delta }{2}\right)\frac{1}{T{\nu }_{*}}+\left(\frac{1-\delta }{2}\right)\left(1-\frac{1}{T{\nu }_{*}}\right)=\frac{1}{2}+\delta \left(\frac{1}{T{\nu }_{*}}-\frac{1}{2}\right)<P_{LR}^0,$$

which explains that $P_{LR}\left(\nu \right)$ decreases with $\nu$ when $\nu \lesssim b$. When $\nu$ is increased further above b, $P_{LR}\left(\nu \right)$ increases with $\nu$ and approaches $P_{LR}^{\infty }>P_{LR}^0$. This explains qualitatively the non-monotonic behaviour of $P_{LR}\left(\nu \right)$ in Figure 3a, with a dip at ${\nu }_{*}\approx 0.02$, and $P_{LR}\left({\nu }_{*}\right)\approx 0.565<P_{LR}^0\approx 0.6<P_L^{\infty }\approx 0.876$, while the rough estimate, with $T\approx 63$ (see Figure 6a), gives $P_{LR}\left({\nu }_{*}\right)\approx 0.56$. Similarly, in Figure 3b, $P_C^{\infty }\approx P_C^0<P_C\left({\nu }_{*}\right)$, which results in a “bump” in $P_C\left(\nu \right)$ at some ${\nu }_{*}\lesssim b$.

3.3. Polarization and Consensus Probabilities When $\delta <0$

When $\delta <0$, most time is spent in the external state $\xi =-1$, where influences favour centrist consensus. Hence, when $s=Nb\gg 1$ and z is not too close to 0, centrism prevails over the other opinions: $P_C>P_{LR}\gg P_{L,R}$ for all values of $\nu$; see Figure 4a. In this case, we find: $P_C\approx P_C^0\approx (1+|\delta \left|\right)/2>P_{LR}\approx P_{LR}^0\approx (1-|\delta \left|\right)/2$ when $\nu \ll 1$, while $P_C\approx P_C^{\infty }\approx 1$ and $P_{LR}\approx P_{L,R}\approx P_{LR}^{\infty }\approx 0$ when $\nu \gg 1$. In Figure 4a, $P_C$ increases with $\nu$ from $(1+|\delta \left|\right)/2$ to 1, while, $P_{LR}$ decreases from $(1-|\delta \left|\right)/2$ to 0 as $\nu$ is raised. This results in a final state consisting of a majority of C voters, whose final density c increases from $(1+|\delta \left|\right)/2$ to 1, and a minority of L and R voters, whose final density $\ell =r$ decreases with $\nu$ from $(1-|\delta \left|\right)/4$ to 0; see the inset in Figure 4a. However, while all-C is the most likely final state, there is a finite probability of polarization at low and intermediate switching rate.

When $s\gg 1$ and the initial population consists mainly of L and R voters ($z\ll 1$), polarization is the most likely final state, with $P_{LR}>P_C>P_{L,R}$ for $\nu \gg b$; see Figure 4b. Centrists thus generally hold the minority opinion when $\nu \gg b$, while C is the majority opinion ($P_C>1/2$) only under low switching rate; see inset in Figure 4b. In the limiting regimes $\nu \ll b$ and $\nu \gg b$, $P_{LR}$ and $P_C$, respectively, approach the values of $P_{LR}^{0,\infty },P_C^{0,\infty }$. Here, again Equation (9) can be used to determine the initial density $z_{LR}$, such that $P_{LR}^{\infty }>P_{LR}^0$ if $z<z_{LR}$ and $P_{LR}^{\infty }\le P_{LR}^0$ otherwise, and $z_C$ such that $P_C^{\infty }>P_C^0$ if $z>z_C$ and $P_C^{\infty }\le P_C^0$ otherwise. In the example of Figure 4, $z_{LR}\approx z_C\approx 0.112$. As for $\delta >0$, in the regime of intermediate switching, as for $\delta >0$, $P_{LR}$ and $P_{L,R}$ can exhibit bumps and $P_C$ a dip, see Figure 4b where $P_{LR},P_{L,R}$ and $P_C$ have modest extrema around $\nu \sim b$.

3.4. Polarization and Consensus Probabilities under Symmetric Switching ($\delta =0$)

When $\delta =0$, social switching is symmetric, and the same average amount of time is spent in $\xi =\pm 1$. External influences thus favour centrism ($\xi =-1$) and polarization ($\xi =1$) in turn (see Figure 1c) and all realizations start in either of the external states $\xi =\pm 1$ with the same probability $1/2$.

In the regime where $\nu \ll b$, the final state is reached with a probability close to $1/2$ from either external state $\xi =\pm 1$. When $\xi =1$, polarization is almost certain, whereas there is centrist consensus with probability 1 when $\xi =-1$. Hence, $P_{LR}\approx P_C\approx P_{LR}^0=P_C^0=1/2$ and $P_L=P_R\approx 0$ when $\nu \ll b$. Under low switching rate, $\nu <b$, a small number switches can occur prior reaching the final state, but switching being symmetric, the net effect mostly cancels out and $P_{LR}=P_C\approx P_{LR}^0=P_C^0=1/2$ when $s\gg 1$, as in Figure 5a,b.

When $\nu \gg b$, there are many switches before reaching the final state (see Section 4.2 below). This results in the self-averaging of $\xi$, yielding $\xi \to \langle \xi \rangle =0$. Hence, when $\nu \gg b$ polarization and centrist consensus occur with probabilities:

$$P_{LR}\approx P_{LR}^{\infty }=1-\frac{1-{(1-z)}^2}{\sqrt{1+{(1-z)}^2}}\mathrm{and}{P}_C\approx P_C^{\infty }=z,$$

where we have used Equations (6) and (7) with (A7).

We can determine when these probabilities increase with $\nu$ by solving Equation (9). When $s\gg 1$, we find $z_{LR}=(4-\sqrt{18-2\sqrt{22}})/4\approx 0.3621$ and $z_C=1/2$. When $s\gg 1$ $P_{LR}$ and $P_C$ hence increase with $\nu$ when $z<z_{LR}$ and $z>1/2$, respectively; see Figure 5. In the case $z<1/2$, $P_C$ is a decreasing function of $\nu$ and C is generally the minority opinion ($P_C<1/2$); see the inset in Figure 5b. As shown in Figure 5a, when $z=1/2$ then $P_C^{\infty }=P_C^0=1/2$, and we find that the probability of centrist consensus remains essentially constant: $P_C\left(\nu \right)\approx 1/2$. When $s\gg 1,\delta =0$ and $z=2x=2y=1/2$, the final densities of voters are $c=P_C\approx 1/2$ and $\ell =r=(1-P_C)/2\approx 1/4$; see the inset in Figure 5a.

4. Mean Exit Time

The mean exit time (MET), here denoted by $T\left(\nu \right)$, is the average time to reach one of three absorbing/consensus states (all-L, all-R, all-C) or the polarization line (pol-$LR$). The MET hence complements the information provided by the polarization and consensus probabilities, and is therefore of great interest. Here, we study how the MET varies with $\nu$ for different given values of $\delta$ and z.

4.1. Mean Exit Time in the Regimes $\nu \to 0$ and $\nu \to \infty$

The MET is obtained analytically in the regimes $\nu \to 0$ and $\nu \to \infty$ in terms of $\mathcal{T}(s,z)$, its counterpart in the absence of external influences, as discussed in Appendix B.2.

4.1.1. MET in the Regime $\nu \to 0$

When $\nu \to 0$, there are no switches before reaching the final state. The MET, $T^0$, can thus be obtained by averaging $\mathcal{T}(s,z)$ over the stationary distribution of $\xi$. Since $\xi =\pm 1$ with probability $(1\pm \delta )/2$, in this regime the MET reads:

$$T^0(s,\delta ,z)=\left(\frac{1+\delta }{2}\right)\mathcal{T}(s,z)+\left(\frac{1-\delta }{2}\right)\mathcal{T}(-s,z).$$

(10)

$T^0(s,\delta ,z)$ and its scaling are thus readily obtained from Equations (A10) and (A11). From the symmetry $\mathcal{T}(-s,z)=\mathcal{T}(s,1-z)$, we have $T^0(s,\delta ,1/2)=T^0(s,-\delta ,1/2)=\mathcal{T}(s,1/2)$ for $z=1/2$ (the unwieldy expression of $\mathcal{T}(s,1/2)$ is given in Ref. [54]). From Figure 6, we find that the predictions of Equation (10) are in good agreement with simulation results when $\nu \ll 1$ in all scenarios $(\delta >0$, $\delta <0$ and $\delta =0$).

4.1.2. MET in the Regime $\nu \to \infty$

When $\nu \to \infty$, many external influences switches occur before the population reaches its final state. This leads to the self-averaging of $\xi$, which can therefore be replaced by its average: $\xi \to \langle \xi \rangle =\delta$. Hence, under high switching rate, the influences bias is rescaled according to $b\to b\delta$, yielding $s\to s\delta$, at fixed N. When $\nu \to \infty$ the MET, $T^{\infty }$, therefore satisfies Equation (A10) with s substituted by $s\delta$, and thus

$$T^{\infty }(s,\delta ,z)=\mathcal{T}(s\delta ,z).$$

(11)

In this regime, the MET follows readily from the solution of (A10), and its scaling is obtained from Equation (A11). Using the symmetry of $\mathcal{T}$, we have $T^{\infty }(s,\delta ,z)=T^{\infty }(s,-\delta ,1-z)$, which boils down to $T^{\infty }(s,\delta ,1/2)=T^{\infty }(s,-\delta ,1/2)$ when $z=1/2$. Furthermore, when $\delta \to 0$, we obtain $T^{\infty }(s,0,z)=\mathcal{T}(0,z)=-2N\left[z\ln z+(1-z)\ln (1-z)\right]$, which is independent of the influences bias b. In Figure 6, the predictions of Equation (11) are in close to perfect agreement with the simulation results for $\nu \gg 1$ in all scenarios ($\delta >0$, $\delta <0$ and $\delta =0$).

4.2. Mean Exit Time as Function of the Switching Rate

We now consider the dependence of the MET under arbitrary $\nu$.

When the switching rate is low, $\nu \ll b$, it is unlikely that there are any switches before reaching the final states, and thus $T\left(\nu \right)\approx T^0(s,\delta ,z)$; see Equation (10). When $b\ll 1$, $1\ll s\ll N$, and z is not too close to 0 and 1, then $T\left(\nu \right)\sim \left(\ln N\right)/b$ (see Equation (A11)), and the average number of switches in this regime prior to reaching the final state is $\nu T\left(\nu \right)\sim \left(\ln N\right)\nu /b$; see Figure 6a,b. Hence, when $\nu \sim b$ there are on average $\mathcal{O}\left(\ln N\right)$ switches before reaching the final state.

Under high $\nu$, the system experiences a large number of switches causing the self-average of $\xi$ (with $\xi \left(t\right)\to \delta$). In this regime, $T\left(\nu \right)\approx T^{\infty }(s,\delta ,z)$; see Equation (11), with $T\left(\nu \right)\sim N$. In this case, the average number of switches before reaching the final state is $\nu T\left(\nu \right)\sim \nu N$.

We can therefore revisit and characterise the three switching regimes:

• (i) when $\nu \ll b$, the system is in the regime of “low switching rate” and the final state can possibly be reached without experiencing any switches;
• (ii) when $\nu \sim b$, the population is in a regime of “intermediate switching rate”, where there are typically $\mathcal{O}\left(\ln N\right)$ switches before reaching the final state;
• (iii) when $\nu \gg b$, the system is in the regime of “high switching rate”, where the external noise self-averages, and the number of switches before reaching the final state is $\mathcal{O}\left(\nu N\right)$ and hence grows linearly with $\nu N$, where $\nu N\gg Nb=s\gg 1$; see inset in Figure 6b.

When the switching rate is raised across the above regimes, from $\nu \approx 0$ to $\nu \gg 1$, the MET changes its scaling behaviour. In regime (i), the MET scales as $T\sim \left(\ln N\right)/b$ and therefore $T/N\ll 1$ when $s\gg 1$, while in regime (iii) the MET scales as $T\sim N$, and therefore $T/N=f\left(z\right)$ is a scaling function of the initial fraction z. In the intermediate regime (ii), when $\nu \sim b$, the MET increases steeply with $\nu$ and interpolates between $T\left(\nu \right)/N\ll 1$ and $T\left(\nu \right)/N\approx f\left(z\right)$ as $\nu$ sweeps from regimes (ii) into (iii). This picture is confirmed by the results shown in Figure 6a,b.

Figure 6a,b, illustrates that the initial fraction z has only a marginal effect on the MET in the limiting regimes $\nu \ll b$ and $\nu \gg b$, which is well captured by Equations (10) and (11). Figure 6b shows that, as predicted by Equation (11), the MET at fixed $N,b,z$ is maximum when $\delta =0$, with $T/N\approx T^{\infty }/N=-2\left[z\ln z+(1-z)\ln (1-z)\right]$; see Appendix B.2. The inset in Figure 6a illustrates that the MET scales linearly with N in the regime (iii), while N has a marginal effect on $T\left(\nu \right)$ in the regimes (i) and (ii): When $\nu \gg b$, we find $T\left(\nu \right)/N=\mathcal{O}\left(1\right)$ as in Ref. [54]. When $\nu \ll b$, we find that $T\left(\nu \right)/N$ decreases with N in agreement with $T\left(\nu \right)/N\sim \left(\ln N\right)/\left(Nb\right)$. The inset of Figure 6b confirms that when $\nu \gg b$, the average number of switches exhibits the same linear scaling with $\nu N$ for different parameter sets.

Here, time-varying external influences are therefore responsible for a drastic change in the MET scaling: the MET scales as $\left(\ln N\right)/b$ under low switching rate, while it grows and scales linearly with N under high switching rate. When $s=Nb\gg 1$, reaching the final state thus takes much longer under $\nu \gg 1$ than $\nu \ll 1$.

5. Conclusions

Motivated by the evolution of opinions in a volatile social environment shaped by time-fluctuating external influences, arising, e.g., from news or social media, we have introduced a constrained three-state voter model subject to randomly binary time-fluctuating (switching) external influences. The voters of this population can hold either incompatible leftist or rightist opinions, or behave as centrists. The changing external influences (social environment) is modelled by a dichotomous random process that favours in turn the rise of polarization or the spread of centrism. The fate of this population is either to reach a consensus with leftists, rightists, or centrists, or to achieve polarization, which consists of a frozen state comprised of only non-interacting leftists and rightists. By combining analytical and computational means, we have investigated the effect of the time-fluctuating external influences on the population’s final state under various scenarios. In particular, we have studied how the rate of switching, as well as the switching asymmetry and initial population composition, affect the fate of this population.

Focusing on the interesting case of a small influences bias affecting a finite, yet large, number of voters, we have shown that the consensus and polarization probabilities can vary greatly with the rate of change of the external influences: these probabilities can either increase or decrease with the rate of external variations, and can also exhibit extrema. Remarkably, when there is a large initial majority of voters holding the opinions opposed by the external influences, the majority can resist the influences: the opinions supported by the majority of agents and opposed by the influences are the most likely to prevail over a range of parameters characterising the external variations. When this occurs, the population settles in a final state in which a majority of voters holds the opinions opposed by the external influences, see Figure 3b and Figure 4b. The study has also shown that time-switching influences are responsible for a drastic change in the scaling of the mean time to reach final state: the mean exit time is generally much bigger under high switching rate, when it scales linearly with population size, than under low switching rate.

The goal of this paper is to study how time-varying external influences may affect the social dynamics of an idealized population. It can be anticipated that the model analysed here is too simple to realistically capture the various complex effects of social and news media, and other time-varying external stimuli, on opinion dynamics. It is however natural to ask how this model can be generalized to become more realistic, and which kind of practical information it could then possibly provide. These points are addressed in Appendix C, where a potential application is briefly discussed. In fact, while the direct applications of the current model are admittedly limited, it is expected to still be useful since it sheds light on nontrivial effects that exogenous time-fluctuating influences can have on opinion dynamics. Hence, this work can be envisaged as a step towards more realistic modelling approaches to this challenging interdisciplinary problem.