Opinion evolution and rare events in an open community

Physica A 462 (2016) 1178–1188

Contents lists available at ScienceDirect

Physica A journal homepage: www.elsevier.com/locate/physa

Opinion evolution and rare events in an open community Yusong Ye a , Zhuoqin Yang a , Zili Zhang b,∗ a

School of Mathematics and Systems Science and LMIB, Beihang University, Beijing 100191, China

b

School of Science, China University of Geosciences, Beijing 100083, China

highlights • The model give an deep interpretation of the self-organization and the mechanism of clustering phenomenon in social opinion dynamics.

• If the noise in social opinion large, then the total opinion in macroscopic view (or the clustering phenomenon) will be easily to change to another.

• The potential landscape describe a distinct image of probability density. It’s a useful strategy to get the stability of different meta-stable state. The different meta-stable state means the different clustering opinion.

article

info

Article history: Received 19 August 2015 Received in revised form 27 March 2016 Available online 24 June 2016 Keywords: Stochastic dynamical systems Opinion evolution First-passage time

abstract There are many multi-stable phenomena in society. To explain these multi-stable phenomena, we have studied opinion evolution in an open community. We focus on probability of transition (or the mean transition time) that the system transfer from one state to another. We suggest a bistable model to provide an interpretation of these phenomena. The quasi-potential method that we used is the most important method to calculate the transition time and it can be used to determine the whole probability density. We study the condition of bistability and then discuss rare events in a multistable system. In our model, we find that two parameters, ‘‘temperature’’ and ‘‘persuading intensity,’’ influence the behavior of the system; a suitable ‘‘persuading intensity’’ and low ‘‘temperature’’ make the system more stable. This means that the transition rarely happens. The asymmetric phenomenon caused by ‘‘public-opinion’’ is also discussed. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Multi-stability and rare events are important phenomena in statistical physics. The methods in statistical physics are widely used in sociology. Especially, the study of rare events is a hot topic in biophysics and social physics. There are many multi-stable phenomena in society such as the trends of public opinion (pros and cons) and the stock market (bull market and bear market). We notice that the presence of two states in a society is a bistability phenomenon. The transition from one opinion state to another sometimes rarely occur. However, that will make a major impact, we want to determine the mean transition time. In addition, it is important to estimate the stability of the meta-stable state. Opinion evolution and change in an open community is an interesting field. It is of theoretical interest and has practical applications too. Previous studies mainly focused on determining the dynamic transport of opinion, or community

∗

Corresponding author. E-mail addresses: [email protected] (Z. Yang), [email protected] (Z. Zhang).

http://dx.doi.org/10.1016/j.physa.2016.06.084 0378-4371/© 2016 Elsevier B.V. All rights reserved.

Y. Ye et al. / Physica A 462 (2016) 1178–1188

1179

clustering. Sudbury and Zanette studied rumor propagation [1,2]. They talked about the SIR (Susceptible–Infective–Removal) model using a complex network. The Ising model is useful when considering the clustering phenomenon. Sznajd suggested a ‘‘vote model’’ to study the community clustering with different opinions [3]. Deffuant, Neau, and Amblard considered a ‘‘relative agreement model’’ to describe everyone’s ‘‘tolerance level’’ and to determine the clustering [4]. Kaizoji, using a ‘‘disagreement function’’, studied the stock market and herding behavior [5]. Vazquez, Krapivsky, and Redner studied opinion formation in a population of leftists, centrists, and rightists, which is controlled by the initial density of centrists [6]. Toscani introduced and discussed certain kinetic models of opinion formation involving both exchange of opinion between individual agents and diffusion of information [7]. The Ising model has also been discussed using a complex network [8–10]. All of the studies cited are focused on either opinion dynamics or social clustering. However, in some sense, the processes of dynamics and clustering are the same. If we can design a model that contains the dynamics of changing opinion and we use the opinion label to divide the different clusters, then we can describe the cluster changing and the transition time. The method of potential landscape is a recent topic of interest in studies of probability distributions and rare events (see Refs. [11–13]). We will use it when discussing our model in this paper. We have studied a typical multi-stable phenomenon: it is based on inner noise or system vibrations. If there is no noise, the model is well defined. Hence, as we have discussed, it is like a clustering phenomenon, but we focus primarily on the dynamics of the clustering or the transition between different clustering states. One of the important phenomena is the different tendencies in community opinion, which may be considered an asymmetric situation. In this paper, we suggest a model to explain the phenomenon in opinion evolution and provide an approximation of stability to describe the rare event (the transition) in different cases of dominating opinion. 2. Model 2.1. Description First, we introduce a model like a molecule collision model with simplicity. Consider a community that contains N sites, which we define as ‘‘persons’’ in that community. The state of a site can be either P or Q . There are two parameters that we use in our model: Define η as the ‘‘stubbornness level’’ in our model; η is a parameter between 0 and 1 to express the strength of ‘‘thermal noise’’. Define γ as the ‘‘persuading intensity’’. It defines the power of influence from people that hold a different view. γ can be any positive real number. The time-discrete dynamics is defined by iterating the following steps: Step 1: choose a site α randomly among the N sites. Step 2: a persuasion is attempted on α with probability (1 − η) (Step 2A) or a noisy change is attempted on α with probability η (Step 2B) Step 2A (Persuading): Calculate the proportion p: the sites in the total number of sites N that are in a different state than α (say, if α is in the P-state, then count sites of Q ). With probability pγ , the transition occurs and the site α changes its state (if α is in P, then change to Q and vice versa). Step 2B (Noisy change): the site α will directly change its state to the other state (if α is Q then it changes to P and vice versa). The two states of a site represent the two opposite opinions. In the election, it can represent tendencies toward two different candidates. Similarly, in public opinion P and Q can represent pro and con, respectively, and in the stock market they can represent the two actions of buying or selling. In our model, if there are X sites in the P-state, then there must be (N − X ) sites in the Q state. The ‘‘persuading’’ (Step 2A) expresses the attachment between the chosen site and the others. In our model, the power of influence from different sites is determined by the parameter γ . In one step, the chosen site α can be influenced by other sites with a different view. If γ is large, it means that the power of influence is small, because with increasing γ the probability pγ decreases. That is the reason it is called ‘‘persuading intensity;’’ the larger the parameter, the harder it is to change the site (in Step 2A). We provide another explanation of the parameter γ : it represents the number of attachments (attaching to other sites that hold the opposite view) that is needed for the site α to change its state in one time step. For the transfer, the states of the attached sites need to be in the other state (different from the chosen or α site); this is the way in which the others can change the opinion of α . In one time step, if we met people that hold views different from us, then we would be likely to consider the other view. The bigger the parameter γ is, the more people we need to meet to change our view, so it represents that the power of influence of others is small. Obviously, the parameter γ need not to be a whole number, it is a kind of extended model of molecule collisions. We could consider γ as an average parameter of all the sites. The noisy change (Step 2B) explains the stochastic phenomenon in real society. We try to employ a simple thermal noise with probability η: the state of the chosen site will change directly to the other state. Thus, η expresses the scale of the noise. Larger η will make larger noise. Hence, we call it ‘‘stubbornness level’’ to describe how stubbornness the person is. Smaller noise always means that people are more stubbornness (or that they just accept the different views from others).

1180

Y. Ye et al. / Physica A 462 (2016) 1178–1188

We can see the chosen site α can change its state in two different ways: One way to change is by the influence from other sites and the other way is by the inherent thermal noise. It is useful to define the factor T as ‘‘temperature’’: T = η/(1 − η). Notice that higher T (or temperature) leads to larger noise, so the factor T is analogous to physical temperature. The factor T can be any positive real number. When T is zero there is no inherent noise (η = 0). The system will be determined only by ‘‘Persuading’’ (Step 2A).

2.2. Mathematical description Suppose there are X sites in the P-state in a system at time t. The total number of sites in the system is N. The fraction of sites in the P-state is x = X /N. The fraction of sites in the Q -state is 1 − x (or (N − X )/N). We have the mean-field equation: dx dt

= (R+ (x) − R− (x)).

(1)

The rate R(x) represents the total flux of x changes in one step. With x as a variable, the flux in (R+ (x)) and flux out (R− (x)) are taken into account. Notice that we take an average over x: it obeys a mean field ordinary differential equation (ODE). However, the ODE cannot give us the whole dynamics of the system because of the stochastic feature. The Langevin form is given as dx dt

= (R+ (x) − R− (x)) + noise.

(2)

The rate functions are R+ (x) = (1 − η)(1 − x)xγ + η(1 − x)

(3)

γ

R (x) = (1 − η)(x)(1 − x) + ηx. −

(4)

Here, the first term is the conversion attempt (or Step 2A); in the case of R+ (x), the conversion occurs with probability (1 − η), and the power of influence from Q is xγ (the probability proportional to xγ ). The first term in R− (x) is understood in a similar manner. The second term is the noise effect (Step 2B), where the changing probability is proportional to η and a site can become a P-site by random conversion from an Q -state (probability proportional to (1 − x)). The second term in R− (x) is interpreted in the same fashion. Notice that the asymmetric situation (that γ can be different in R+ (x) and R− (x)) will be discussed later.

3. Method

3.1. The condition of multi-stability The system that we are interested in is the bistability system. The first question is when it has a bistability feature. Obviously, it depends on two parameters: η and γ . The mean-field ODE equation is dx/dt = (1 − η)[(1 − x)xγ − x(1 − x)γ + T (1 − 2x)].

(5)

The ‘‘critical temperature’’ is the highest temperature at which the system has a bistability phenomenon. It can also be the bifurcation point (in the sense of the mean-field equation): the global attractor transform to two local attractors. Because it is a symmetric system, x = 1/2 is a zero point. Taking relation between γ and the critical temperature Tc : Tc = (γ − 1)

 γ 1

2

.

∂



dx dt

∂x



= 0 and letting x = 1/2, we can formulate the

(6)

Furthermore, we can calculate the maximum value of Tc . When γ ∗ = 2.44, Tc takes the maximum value Tmax ≈ 0.265. It is the highest ‘‘phase transition temperature’’. Remember that γ represents the ‘‘persuading intensity’’. This means that in this situation (γ = γ ∗ ), it is easy to see the bistable phenomenon.

Y. Ye et al. / Physica A 462 (2016) 1178–1188

1181

3.2. The potential landscape To compare with the theoretical results, we use an iterative method to simulate the probability density distribution. The total number of iteration steps is large enough for the probability density of x to be stable. We can derive the probability density function from the frequency of different x-states that appears in our simulation time series. The probability density function is rescaled by the partition function. Arrhenius law tells us that P (x) ∼ A · e−V (x) .

(7)

Hence, we take the form: Vs (x) = − ln(P (x)).

(8)

Here, A is the prefactor of the Arrhenius function (which we will discuss later), P (x) is the probability density function (the result is simulated), and Vs (x) is the ‘‘simulated potential’’ for conveniently comparing the results. The one-dimensional Fokker–Planck equation (FPE) for diffusion of a particle in a potential V (x) is

∂p 1 = −∇x · (bp) + ∇x2 (ap). ∂t 2

(9)

Here, b represents the drift of the diffusion system, a is the variance of the variable x, and V (x) denotes the potential. Consider the probability flux J:

∂p ∂J =− . ∂t ∂x

(10)

Here, the system satisfies the detail balance condition (DBC), which can be verified from the boundary condition. Because it satisfies DBC, J (flux) = 0. We take the form V (x) = −ln(P (x)) from Eq. (8), we then have

∂ V (x) 2b d(ln(a)) =− + ∂x a dx

(11)

or V (x) = −



2b a

dx + ln(a).

(12)

The drift equation could have been obtained directly from the Langevin equation: [R+ (x) − R− (x)]/N. The variance of x is [R+ (x) + R− (x)]/N 2 . drift = b = (1 − η)[(1 − x)xγ − x(1 − x)γ + T (1 − 2x)]/N γ

γ

var = a = (1 − η)[(1 − x)x + x(1 − x) + T ]/N . 2

(13) (14)

Using Eqs. (13) and (14) and substituting into Eq. (12), we can calculate the quasi-potential V (x). Notice that in most situations, we cannot derive an analytical result. We will use numerical methods to evaluate the integral and get the potential V (x). Take V (x) as: V (x) = −



  1 (1 − x)xγ − x(1 − x)γ + T (1 − 2x) γ γ dx + ln [(1 − x)x + x(1 − x) ] + 1 . 2N (1 − x)xγ + x(1 − x)γ T

(15)

The potential V (x) is composed of two parts: the first part (integral part) is just like the drift, but it needs to be rescaled by the variance of x. It is like a force acting on x. The second part is the ‘‘diffusion of variance’’. Notice that if the variance is a constant, the second part is zero (or a constant). Thus, a changing variance will cause a ‘‘force’’ (analogy to drift) on x. We need to mention that the potential is a result of an integral, so readers need to be aware that the theoretical result of Eq. (15) is not the only possible one. The reason we take this form is for convenience when comparing results. In this form, the simulation result Vs (x) and the theoretical result Vt (x) are exactly the same (we will see this later).

1182

Y. Ye et al. / Physica A 462 (2016) 1178–1188

Fig. 1. Sketched map of the first exit problem. a and c are the two ‘‘valleys’’ and b is the ‘‘hill’’.

3.3. The stability and the average time of transfer We will quantify the average time of transfer between two states. Fig. 1 shows a sketched map of V (x). We define the three states of x: a is a small-x state, b a transition state, and c a large-x state. In analogy with Kramers [14], and using the method introduced in Ref. [15], the mean first-passage time is given by Eq. (16), where V (x) is the potential and θ is the average variance of x (we take the average of Eq. (14) of all x). Eq. (16) is an approximate result. τac is considered the average first-passage time, so if we consider τac , then x must start from the state a, climb the hill, trek over the mountain (state b) and finally go down to state c. We discuss the symmetric situation, which means that τac and τca are the same.

τac ≈

1

θ

c



e−V (x) dx′



x′

′′ eV (x ) dx′′ .

(16)

−∞

a

This double integral can be approximated by a simple expression (see Eq. (19)): when x is near xb , then eV (x) is large, otherwise eV (x) is exponentially smaller. It may therefore be replaced with



exp V (b) − V ′′ (x)

(x − xb )2



2

(17)

i.e., a parabolic approximation. Subsequently, integrate Eq. (17) and we get



 eV (x) ≈

2π V ′′ (xb )

exp[V (xb )].

(18)

For e−V (x) , we use the same approximation method: we take the approximation near state a, and the entire expression Eq. (16) reduces to

τac ≈

1

θ

√

2π

|V ′′ (x

a

)V ′′ (xb )

exp[V (xb ) − V (xa )].

(19)

4. Results Fig. 2 shows dx/dt (drift) with fixed γ and different T . Notice that when dx | = 0, then x0 is a stable point in the mean dt x0 field equation. We can see in Fig. 2 that when T changes, the stable point changes from one to two points. Taking T as a variable parameter, then there are not two stable points when T > Tc . Too much noise will make the system ‘‘plain’’: The persuading influence is too small. We notice that the critical temperature is irrelevant with the scale of the system or the total number of sites N (Eq. (6)). Fig. 3 shows the relation between γ and Tc . If γ < 1 or γ = 1, then there is no Tc , because when η > 0 then T > 0. The sublinear and linear situations do not have the bistability feature. The superlinear situation can have this feature only when γ > 1. Next, we would like to present the potential landscape V (x). The theoretical and simulation results are compared in Fig. 4. Take N = 50 and the two sets of parameters γ = 1.5, T = 0.1 and γ = 2.5, T = 0.15. The theoretical result matches the simulation result quite well.

Y. Ye et al. / Physica A 462 (2016) 1178–1188

1183

Fig. 2. dx/dt when γ = 2.0 for different temperatures. This is the symmetric situation and the temperature varies from 0.1 to 0.25. dx/dt = 0 means that it is the zero point. Thus, obviously with T rising, two stable points appear.

Fig. 3. Relation between γ and Tc . Notice that only γ > 1 makes sense.

The larger V (x) means a small probability. Take J (flux) = 0, which means that V (x) is a quasi-potential. Thus, we can consider V (x) as a potential, similar to the potential energy: A high-energy state means it rarely will appear. Figs. 5 and 6 show the ‘‘two basins’’ transforming to only one basin with changing T . The potential V (x) describes the effective force on x from the combined effect of ‘‘Persuading’’ and ‘‘Noisy change’’. A large positive gradient in V (x) means that the average number of sites that are in the state P will tend to decrease. A potential minimum means that the persuading processes and noisy change balance such that the number of P states typically stays around this minimum. In particular, the valleys and hills of this landscape can be viewed as the metastable states and the barriers between them, respectively. Next, we will discuss how V (x) changes as the parameter γ changes. Fig. 7 shows how Tc changes when γ varies. If T is a constant, then V (x) will first show only one valley (Phase 1), if γ > γa (as Fig. 7 describe), V (x) shows a ‘‘river plain’’, and then there will be two valleys (Phase 2). At first, the hill will be steeper and the valley will be deeper. However, when γ is

1184

Y. Ye et al. / Physica A 462 (2016) 1178–1188

Fig. 4. Comparison between theoretical (solid line) and simulation (squares) results. The two results are the same for the potential. The number of sites is 50. The parameter sets are γ = 1.5, T = 0.1 and γ = 2.5, T = 0.15. The results are from Eq. (15).

Fig. 5. Fix γ = 2.5 with respect to different ‘‘temperature’’ or parameter T . N = 50 and T = 0.15, 0.25, and 0.35. When γ = 2.5, Tc (critical temperature) is 0.2652. When T < 0.26, V (x) has two ‘‘valleys’’ and for T > 0.26, V (x) has only one ‘‘valley’’. The two ‘‘valleys’’ mean that there are two basins of attraction.

Fig. 6. V (x) dependence on T . T varies between 0.15 and 0.35. The landscape changes gradually as T decreases, from a single valley, through an almost equipotential ‘‘river plain’’, to two valleys. This change is associated with stronger ‘‘persuading’’ processes at small T values.

still increasing and near γb , the valley will be shallow. Finally, when γ = γb , V (x) is just like a ‘‘river plain’’ again. When γ > γb the single valley appear (Phase 3).

Y. Ye et al. / Physica A 462 (2016) 1178–1188

1185

Fig. 7. N = 50. Fix T = 0.225 as an example. As γ increases, the potential V (x) will first form a single valley, transform to two valleys, and then back to a single valley again. Phase 1 and Phase 3 corresponds to only one valley. Phase 2 is the two valleys. γa and γb are the ‘‘critical γ ’’, i.e., they are the critical phase transition parameters (for a fixed temperature).

Fig. 8. N = 50 and T = 0.2. V (x) changes gradually as γ varies from 1 to 4.7. Notice that it starts with a single valley, changes to two valleys, at finally changes to a single valley again. The phases are defined as in Fig. 7.

We could see the transfer more clearly in Fig. 8. It employs a fix temperature T = 0.2 and γ = 1.0 ∼ 4.7. We see that V (x) varies as γ changes. A ‘‘hillock’’ is bulging from the ‘‘river plain’’ on the ‘‘surface’’ in Fig. 8. The ‘‘hillock’’ represents Phase 2 (with two ‘‘valleys’’), which was shown in Fig. 7. It is quite clear now how V (x) varies as γ and T changes. Notice that we have mentioned that T represents the scale of noise and γ the ‘‘persuading intensity’’. When T increases, it is easy to explain why V (x) loses its bistability: The large noise will let x more easily ‘‘climb’’ the hill, so when T is increasing up to a certain temperature(Tc ), there is no hill and we cannot see the bistability phenomenon anymore. However, when we fix T and let γ increase, the result is different. Notice that the change is from one valley to two valleys and back to one valley again. When γ increases, a single valley becomes shallow and splits into two valleys (the transition from phase 1 to phase 2 is described in Fig. 7). γ is a parameter of ‘‘linear degree’’. As we have mentioned in Fig. 3, when γ = 1, it is totally linear. The bistability needs a larger γ in a specific range. A larger γ will make the bistable phenomenon more obvious. When γ is sufficiently large, and if γ is still increasing, the two stable attractors will lose their stability. Too large γ will depress the influence of the ‘‘Persuading’’ step, because pγ decreases. Finally, when γ > γb , there is only one stable state. Next, we will use this analogy to show the transition time for stochastic switching between the two states. Fig. 9 shows the simulation and theoretical result. The two results agree well. In Fig. 9, N = 50, γ = 2.5 and 1.5, T varies from T = 0.08 to 0.2 and from 0.03 to 0.15. When T goes large, τ decreases. The two Tc are 0.176 (γ = 1.5) and 0.265

1186

Y. Ye et al. / Physica A 462 (2016) 1178–1188

Fig. 9. Stability of the macroscopic state. Here, γ = 2.5 and γ = 1.5. The first passage time, or the average lifetime of a stable state (a or c described in Fig. 1) τ is shown. The squares are the simulation result and the solid line is the theoretical result.

(γ = 2.5), T must be smaller than Tc in the two situations. The results show that low T gives a deeper ‘‘valley’’, which the transition time longer. Note that Eq. (19) is an approximation result. The parabolic approximation needs the difference between Va (Vc ) and Vb to be large. Thus, when T increases and approaches Tc , the theoretical result (Eq. (19)) is not accurate. 5. Discussion 5.1. Symmetric situation This study went deeper than mean field research in the sense that it allows one to explore the probability distribution. Unfortunately, using this method in a higher-dimension situation is hard because one cannot easily find a potential energy surface: with more than one dimensions, we cannot use the FPE to derive the potential directly. The most important thing is to find the stability (average transition time τ ). Thus, to find the most-probable transition path (in the higher-dimensional case), we can use the string-method [16] or the MAM (or gMAM) method [17,18] to find the minimal energy path and estimate the stability. The relative stability is important, it is easier than to estimate an accurate transition time τ . Potential landscapes present an appealing way to discuss multi-stability of opinion states in social systems. As the parameter changes, there is a switch-like transfer that controls the characteristic of the system. Thus, with the technique of potential landscape, the evolution of the whole landscape can be visualized. The WKB (Wentzel–Kramers–Brillouin) approximation is a useful strategy [19]. We expect that our model can be extended easily to explain other similar phenomena. We have studied the onedimensional case, so the first question concerns the two- or higher-dimensional cases. For example, if there are three different opinion (not two as in our model), what is the probability distribution in such cases. Another question is the dynamics of opinion in complex network. In our case, we choose a homogeneous network. The opinion changing in a complex network is very different from our case. The multi-stability is an interesting phenomenon not just in social system but also in other dynamical system such as chemical kinetics, biophysics, and even in economic system. The model can be widely use in these fields to study the multistability phenomenon. 5.2. Asymmetric situation So far, we have discussed the symmetric situation. The parameter γ represents ‘‘persuading intensity’’ or the power of influence from others. If γ is different in R+ and R− , the two different opinions have different powers of influence; they reflect the public opinion. First we express the rate functions as R+ (x) = (1 − η)(1 − x)xγ1 + η(1 − x)

(20)

R− (x) = (1 − η)x(1 − x)γ2 + ηx.

(21)

When γ1 ̸= γ2 , the different opinions will have different ‘‘persuading intensity’’ forces on a person. We have defined x as the number of people in state P. The larger γ is, the harder it is to persuade. Thus, if, e.g., γ1 > γ2 , the person in state Q will have more ‘‘persuading intensity’’ or the persuading (switch from P to Q ) will be easier. If γ1 < γ2 , the person in state Q will have less ‘‘persuading intensity’’ or persuasion (switch from P to Q ) will be harder. We have mentioned two x states: the large-x state and small-x state in Fig. 1. This differentiation is useful for discussing. We consider the asymmetric situation because of the public opinion application. If the public opinion is leaning toward P, then the person in state P will be more powerful and the persuading will be easier, which means that γ1 < γ2 . If the public

Y. Ye et al. / Physica A 462 (2016) 1178–1188

1187

opinion is leaning to Q , then the person in state Q will resist more to agree with the suggestions from the person in state P, so γ1 > γ2 . A small γ1 will make x concentrate more on the large-x state, which means that V (x) will have a deep valley for the large-x state (and vice versa). The asymmetric case is the same and we can estimate the transition rate as discussed in Section 3.3. Using Eq. (19), we get

τac = τca



V ′′ (xc ) eVb −Va V ′′ (xa ) eVb −Vc

 =

V ′′ (xc ) V ′′ (xa )

exp(Vc − Va ).

(22)

In a symmetric situation, Va and Vc (described in Fig. 1) are the same, so we get τac /τca = 1. The difference between Va and Vc represents the relative stability of the two metastable states a and c. If Va < Vc , xa is more stable than xc . The potential V (x) can be derived in the same manner as discussed, using Eqs. (20) and (21): V (x) = −



  1 (1 − x)xγ1 − x(1 − x)γ2 + T (1 − 2x) γ2 γ1 dx + ln [(1 − x)x + x(1 − x) ] + 1 . 2N (1 − x)xγ1 + x(1 − x)γ2 T

(23)

6. Conclusion A stochastic dynamical system can describe and explain some complex phenomena such as social, economics, and biophysics self-organized phenomena. This paper has explored the multi-stable phenomenon and rare events in a social system. Our approach was to find a way to explain the self-organized bistability phenomenon in society. We used a simple iterative process to describe the behavior and people’s decisions in every time step. By introducing a simple model for such a dynamic system, we demonstrated that the model could give a way to explain the bistability. The mechanism is closely related to the two parameters γ and T . If the model is used in other problems or situations, the two parameters should be treated with caution. We first defined two influential parameters in our model: the ‘‘persuading intensity’’ γ and ‘‘stubbornness level’’ T . ‘‘Persuading’’ and ‘‘stubbornness’’ in a social system are the two powerful mechanisms that maintain a dynamic bistable system; it needs γ > 1 and T < Tc . First, we discussed the ‘‘bistable condition’’. The temperature T and parameter γ can influence the features of the system. As we have analyzed, if γ is too large or too small, we cannot get bistability, unless γ is near a suitable value (as Fig. 7 shows). The lower the temperature T is, the more stubbornness the people are. Higher temperature means that people are more likely to change their views. Hence, if the temperature is high enough, people’s views will be inconsistent; therefore, there will be no multi-stable phenomenon. The movements are small, dominated by single jump-process (as we described model), so the movement can be fully determined by the Langevin or Fokker–Planck equation. The first escape time can also be calculated. The theoretical results fit well as shown in Figs. 4 and 9. Finally, we studied an asymmetric landscape caused by different γ in R(x). If there are no social opinion tendency, γ1 = γ2 . The ‘‘persuading intensity’’ of the two opinions is the same. Different γ values indicate the influence of public opinion because γ represents the ‘‘persuading intensity’’. A large influence means a small γ , which means that persuasion is easier or there is a large ‘‘persuading intensity’’. We used Eq. (19) to estimate the average first exit time. This is useful in one dimension, but this theoretical result cannot easily be extended to higher dimensions. For comparison of stability, Eq. (22) can be used. Our analysis opened the understanding of changes and evolution in social-opinion system’s potential landscapes. The whole landscape can be visualized, which makes it an appealing method to study the system. It is also an effective method to calculate and estimate the stability in the multi-stability system. The model we show may be significant in explaining some critical behaviors in social-opinion changes. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 11372017 and 11472009). We especially thank Zhuoqin Yang, Zili Zhang, Tiejun Li, and Fangting Li. References [1] [2] [3] [4] [5] [6] [7] [8]

A. Sudbury, The proportion of the population never hearing a rumour, J. Appl. Probab. (1985) 443–446. D.H. Zanette, Dynamics of rumor propagation on small-world networks, Phys. Rev. E 65 (4) (2002) 041908. K. Sznajd-Weron, J. Sznajd, Opinion evolution in closed community, Internat. J. Modern Phys. C 11 (06) (2000) 1157–1165. G. Deffuant, D. Neau, F. Amblard, G. Weisbuch, Mixing beliefs among interacting agents, Adv. Complex Syst. 3 (01n04) (2000) 87–98. T. Kaizoji, Speculative bubbles and crashes in stock markets: an interacting-agent model of speculative activity, Physica A 287 (3) (2000) 493–506. F. Vazquez, P.L. Krapivsky, S. Redner, Constrained opinion dynamics: Freezing and slow evolution, J. Phys. A: Math. Gen. 36 (3) (2003) L61. G. Toscani, et al., Kinetic models of opinion formation, Commun. Math. Sci. 4 (3) (2006) 481–496. P.L. Krapivsky, S. Redner, A statistical physics perspective on web growth, Comput. Netw. 39 (3) (2002) 261–276.

1188 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

Y. Ye et al. / Physica A 462 (2016) 1178–1188 C.P. Herrero, Ising model in small-world networks, Phys. Rev. E 65 (6) (2002) 066110. S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Ising model on networks with an arbitrary distribution of connections, Phys. Rev. E 66 (1) (2002) 016104. C. Lv, X. Li, F. Li, T. Li, Constructing the energy landscape for genetic switching system driven by intrinsic noise, PLoS One 9 (2) (2014) e88167. C. Lv, X. Li, F. Li, T. Li, Energy landscape reveals that the budding yeast cell cycle is a robust and adaptive multi-stage process, PLoS Comput. Biol. 11 (3) (2015) e1004156. W. Zhang, T. Li, P. Zhang, Numerical study for the nucleation of one-dimensional stochastic Cahn-Hilliard dynamics, Commun. Math. Sci. 10 (4) (2012) 1105–1132. H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (4) (1940) 284–304. N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, Vol. 1, Elsevier, 1992. E. Weinan, W. Ren, E. Vanden-Eijnden, String method for the study of rare events, Phys. Rev. B 66 (5) (2002) 052301. W. Ren, E. Vanden-Eijnden, et al., Minimum action method for the study of rare events, Comm. Pure Appl. Math. 57 (5) (2004) 637–656. E. Vanden-Eijnden, M. Heymann, The geometric minimum action method for computing minimum energy paths, J. Chem. Phys. 128 (6) (2008) 061103. B.C. Garrett, D.G. Truhlar, Wkb approximation for the reaction-path Hamiltonian: Application to variational transition state theory, vibrationally adiabatic excited-state barrier heights, and resonance calculations, J. Chem. Phys. 81 (1) (1984) 309–317.