Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution

Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution

H. F. Chau hfchau@hku.hk    C. Y. Wong    F. K. Chow    Chi-Hang Fred Fung Department of Physics and Center of Theoretical and Computational Physics, University of Hong Kong, Pokfulam Road, Hong Kong
July 12, 2019
Abstract

The dynamical origin of opinion polarization in the real world is an interesting topic physical scientists may help to understand. To properly model the dynamics, the theory must be fully compatible with findings by social psychologists on microscopic opinion change. Here we introduce a generic model of opinion formation with homogeneous agents based on the well-known social judgment theory in social psychology by extending a similar model proposed by Jager and Amblard. The agents’ opinions will eventually cluster around extreme and/or moderate opinions forming three phases in a two-dimensional parameter space that describes the microscopic opinion response of the agents. The dynamics of this model can be qualitatively understood by mean-field analysis. More importantly, first-order phase transition in opinion distribution is observed by evolving the system under a slow change in the system parameters, showing that punctuated equilibria in public opinion can occur even in a fully connected social network.

Mean Field Theory, Opinion Dynamicsi, Opinion Polarization, Phase Transition, Punctuated Equilibrium In Social Science, Social Judgment Theory, Sociophysics
pacs:
89.65.Ef, 05.65.+b, 89.75.Fb

I Introduction

Opinion formation and evolution are interesting and important subject of research in social psychology. Many experiments and theories have been conducted and proposed *[See; forexample; ][; inparticularchap.~8.]Eagly:1993; Also see, for example, Griffin, E. (2011), including the elaboration likelihood model, the heuristic-systematic model and the cognitive dissonance theory. In particular, Sherif et al. proposed the well-known social judgment theory (SJT) Eagly and Chaiken (1993); Sherif and Hovland (1961); Sherif et al. (1965); Also see, for example, Griffin, E. (2011) in the 1960’s to explain the microscopic behavior of how individuals evaluate and change their opinions based on interaction with others.

The basic idea of SJT is that attitude change of an individual is a judgmental process. According to SJT, describing the stand of an individual as a point in a continuum of possible opinions is not adequate because the individual’s degree of tolerance is also important in determining his/her response to external stimuli and persuasion Sherif and Hovland (1961); Sherif et al. (1965). In particular, a presented opinion is acceptable (unacceptable) to a person if it is perceived to be sufficiently close to (far from) his/her own stand point. This presented opinion is said to be in his/her latitude of acceptance (rejection). A presented opinion is neither acceptable nor objectable if it is perceived to be neither close to or far from the individual’s own stand point. This opinion is said to be in his/her latitude of noncommitment. Clearly, these three latitudes differ from person to person and they depend on factors such as individual’s ego involvement and the person’s familiarity of the subject of discussion Eagly and Chaiken (1993); Sherif and Hovland (1961); Sherif et al. (1965); Also see, for example, Griffin, E. (2011). When the presented opinion is in one’s latitude of acceptance (rejection) or perhaps also in the nearby latitude of noncommitment, assimilation (contrast) occurs in the sense that the presented opinion is perceived to be closer to (farther from) one’s stand point than it truly is. Moreover, this positively-evaluated (negatively-evaluated) opinion may cause the person to move his stand point towards (away from) it. The greater the difference between the individual’s and the presented opinions, the more the resultant attitude change in general. The phenomenon of moving away from the presented opinion through contrast is called the boomerang effect Eagly and Chaiken (1993); Also see, for example, Griffin, E. (2011); Sherif and Hovland (1961); Sherif et al. (1965). The opinion change due to boomerang effect, however, is generally smaller than the opinion change induced by assimilation. Thus, not every psychological experiment unambiguously shows its existence Also see, for example, Griffin, E. (2011), making it perhaps the most controversial part of the SJT. In fact, some social psychologists do not consider the boomerang effect to be one of the core thesis of SJT and some even cast doubt on its existence Eagly and Chaiken (1993). Here we adopt the view that the boomerang effect is one of the central themes of SJT whose effect, in general, is rather weak in comparison to the opinion change due assimilation. Finally, whenever the presented opinion is in the person’s latitude of noncommitment which is not close to his/her latitudes of acceptance or rejection, then there is little chance for him/her to change his/her mind. Consequently, the most effective method to successfully persuade an individual is to present the opinion near the boundary of his/her latitudes of acceptance and noncommitment Also see, for example, Griffin, E. (2011). And just like most theories in social science, the above findings should be interpreted in statistical sense rather than as definitive rules governing every single persuasion and discussion Eagly and Chaiken (1993); Also see, for example, Griffin, E. (2011); Sherif and Hovland (1961); Sherif et al. (1965). Thoroughly studied and advanced by social psychologists, SJT is one of the most important theories in the field and is strongly supported by many psychological experiments especially concerning the latitudes of acceptance and noncommitment Sherif and Hovland (1961); Sherif et al. (1965); Also see, for example, Griffin, E. (2011); Sakaki (1984); Sarup et al. (1991).

Recently, physical scientists entered this field by studying the more macroscopic aspects of the problem such as opinion formation and evolution in a social network using simple models and computer simulations Sobkowicz (2009). The variety of models proposed include the use of discrete or continuous opinions, discrete or continuous time, homogeneous or heterogeneous agents, fully connected or more realistic social networks Sobkowicz (2009); Sznajd-Weron and Sznajd (2000); Deffuant et al. (2000); Sousa (2005); Weisbuch et al. (2002); Hegselmann and Krause (2002); Stauffer et al. (2004); Stauffer and Meyer-Ortmanns (2004); Jager and Amblard (2005); Huet et al. (2008); Lorenz (2010); Gandica et al. (2010); Martins et al. (2010); Martins and Kuba (2010); Kurmyshev et al. (2011); Iñiguez et al. (2011); Xie et al. (2011); Li et al. (2011); Sîrbu et al. (2013); Martins and Galam (2013); Crawford et al. (2013). Of particular importance is the continuous opinion agent-based model in a fully connected network introduced by Deffuant et al. (D-W Model) with the feature that players only have latitudes of acceptance and noncommitment so that only the effect of assimilation is considered Deffuant et al. (2000); Weisbuch et al. (2002). The appeal of this model is that it can be simulated efficiently by computers and its dynamics can be qualitatively understood. This model is also consistent with the social psychologists’ finding that opinions can be reasonably well represented and measured as a continuum Hegselmann and Krause (2002); See, for example, Lodge, M. (1981). However, the absence of contrast and boomerang effect imply that D-W Model cannot be used to simulate opinion polarization in the real world in which opinions of the supporters of very different viewpoints become much more extreme.

Various modifications of the D-W Model have been proposed Deffuant et al. (2002); Amblard and Deffuant (2004); Jager and Amblard (2005); Deffuant (2006); Huet et al. (2008); Lorenz (2010); Gandica et al. (2010); Martins et al. (2010); Martins and Kuba (2010); Kurmyshev et al. (2011); Xie et al. (2011); Li et al. (2011); Sîrbu et al. (2013); Martins and Galam (2013); Crawford et al. (2013). To account for opinion polarization, some modified this model by introducing inflexible or contrarian players Deffuant et al. (2002); Amblard and Deffuant (2004); Deffuant (2006); Martins and Kuba (2010); Xie et al. (2011); Li et al. (2011); Martins and Galam (2013), stochastic boomerang effect in the region of assimilation Martins et al. (2010); Sîrbu et al. (2013) and vector-valued opinions Sîrbu et al. (2013). These models are not fully compatible with the SJT as the agents’ response in the latitude of rejection due to contrast are not properly treated. This is not ideal because in order to understand the macroscopic origin of opinion formation and polarization, one should combine the strengths of social psychology and physical science communities by introducing D-W-based models of opinion evolution whose rules are consistent with SJT. In fact, this approach is beginning to gain acceptance among social psychologists Mason et al. (2007). Actually, the only SJT-based models we aware of are the ones proposed by Jager and Amblard (J-A Model) Jager and Amblard (2005) and its recent extension by Crawford et al. Crawford et al. (2013) as well as the model of Huet et al. Huet et al. (2008). Jager and Amblard studied their model only by Monte Carlo simulation with very limited sample and agent sizes Jager and Amblard (2005). The work of Crawford et al. was more extensive, which included a simple analysis on eventual opinion distribution of the agents Crawford et al. (2013). Note that both the models of Jager and Amblard Jager and Amblard (2005) and Crawford et al. Crawford et al. (2013) involved agents with opinions on one issue only. In contrast, the model of Huet et al. Huet et al. (2008) studied the response of agents based on their opinions on two issues by Monte Carlo simulation up to 5000 agents.

While these works Jager and Amblard (2005); Crawford et al. (2013); Huet et al. (2008) point to the right direction, we argue in Sec. II that the microscopic rules adopted in their models are questionable. Here we first proposed a minimalist SJT-based model of opinion formation by extending the J-A Model in Ref. Jager and Amblard (2005). This minimalist model is free of the questionable assumption implicitly used in Refs. Jager and Amblard (2005); Huet et al. (2008); Crawford et al. (2013). Then in Secs. III and IV, we report that our minimalist model is simple enough to be studied both semi-analytically and numerically, and at the same time refined enough to show opinion polarization even in the case of homogeneous agents. By studying the agents’ dynamics in Sec. IV, we can understand the process of opinion clustering. In particular, using a simple mean-field analysis, we find that the most important parameters to determine the formation of extreme opinion clusters as well as the coexistence of both extreme and moderate opinion clusters are the values of two parameters and to be defined in Sec. II which determine the widths of the regions for assimilation and boomerang effect to occur. Our analysis also shows that other factors such as network topology, agent’s heterogeneity, and the detailed response dynamics due to assimilation and boomerang effect chiefly affect the opinion formation timescales. More importantly, we find in Sec. V that first-order phase transition in opinion clustering can occur occasionally when the widths of the assimilation and boomerang effect regions change very slowly. This shows that punctuated equilibrium in opinion distribution — the observation that opinion distribution change often comes in a short burst between a long period of stasis, a notion first pointed out by Gould and Eldredge Gould and Eldredge (1977) in evolution biology — can occur even in a fully connected network, repudiating one of the criticisms Tilcsik and Marquis (2013) to the punctuated equilibrium theory in social science Baumgartner and Jones (1993). Finally, we give a brief outlook in Sec. VI.

Ii The model

Just like the D-W Model Deffuant et al. (2000); Weisbuch et al. (2002) and the J-A Model Jager and Amblard (2005), we consider a fixed connected network of agents each with a randomly and uniformly assigned initial opinion in a bounded interval, say, . We call the opinions and extreme while those in between moderate. At each time step, we randomly pick two neighboring agents, say, and , in the network and simulate their opinion changes after they meet and discuss by the following rules:

  • The assimilation rule: If , then and are simultaneously updated as

    (1a)
    (1b)

    where is the convergence parameter.

  • The boomerang effect rule: If , then and are simultaneously updated as

    (2a)
    (2b)

    where is the divergence parameter, and

    (3)

    is the normalization function which maps extreme opinions back to the range .

  • The neutral rule: The values of and do not change otherwise.

This serial opinion updating is repeated until the system is equilibrated.

Clearly, our model is well-defined if and is compatible with the SJT with and reflecting the widths of the assimilation and boomerang effect regions, respectively. Furthermore, the rules are symmetric about . More importantly, our model is highly flexible. Adapting it to model heterogeneous agents (in which each has different values of and ), different opinion change dynamics (by modifying Eqs. (1)–(2) — something that we are going to do in Sec. IV below), and network topology are easy.

Our model is very different from that of Huet et al. Huet et al. (2008) since theirs is based on the repeated interaction of randomly picked pairs of agents whose responses are based on their opinion differences on two issues. Also, the most important difference between our model in the above form and the J-A Model model Jager and Amblard (2005) as well as its extension by Crawford et al. Crawford et al. (2013) is that we use different convergence and divergence parameters and ; while they set both to the same value. Their choice is not very natural since would then imply the magnitude of opinion change due to boomerang effect must be greater than or equal to that due to assimilation, whose validity is not without doubt Eagly and Chaiken (1993). Note that both groups used Monte Carlo simulations to study their models Jager and Amblard (2005); Crawford et al. (2013). In fact, Jager and Amblard did not perform any analytical or semi-analytical study and Crawford et al. only carried out a basic mean-field analysis which focused mainly on the asymptotic behavior rather than the detailed opinion dynamics of the agents. In contrast, our detailed mean-field analysis in Sec. IV below shows that the dynamics of this type of models are so general that asymptotic behavior is very robust against any change in the assimilation and boomerang effect rules as well as the network topology provided that the average connectivity of the network is not too low.

Iii Simulation results

We first present our findings for agents in a fully connected network with and . These values are chosen to reflect the reality that agents generally have to interact several times before becoming extremists or sharing almost identical opinions. Besides, this choice makes sure that the magnitude of opinion change due to boomerang effect need not be greater than due to assimilation, which is consistent with findings of psychological experiments Also see, for example, Griffin, E. (2011). Since the network is fully connected, the dynamics of opinion distribution can be written as a master equation in the mean-field approximation. The master equation approach is computationally more efficient and generally more suitable to study the steady state opinion distribution than Monte Carlo method for a fully connected network Ben-Naim et al. (2003). The results reported below are found by both Monte Carlo simulations and numerically solving the master equation. Both methods give similar results.

By numerically solving the master equation, Fig. 1 shows that the equilibrated opinion distribution for different values of and can be divided into three regions. In region A (), the system evolves to clusters of moderate opinions similar to that of the D-W Model. In region B (), the system equilibrates to two clusters of extreme opinions plus one or more clusters of moderate opinions similar to the J-A Model. And in region C (), the system evolves to two clusters of extreme opinions only. Again, this is similar to the results of the J-A Model. These findings are consistent with our Monte Carlo simulations except for the small region C’ in which . We shall discuss this difference when we talk about the agents’ dynamics below.

Fig. 1 also shows that the fraction of agents in an opinion cluster upon equilibration vary greatly for different values of and . In fact, Ben-Naim et al. found that equilibrated opinion clusters of vastly different sizes can be present in the D-W Model. They called an opinion cluster with () fraction of agents a major (minor) cluster Ben-Naim et al. (2003). Here, we clarify what an opinion cluster means in this paper. In our subsequent theoretical analysis, it refers to a connected subgraph of the network such that each agent in this subgraph has the same opinion. In addition, the ratio of agents in this subgraph to is non-zero in the large limit. Whereas in our Monte Carlo program, an opinion cluster is a connected subgraph of the network with at least fraction of the agents such that opinion difference between any two agents in the subgraph is less than after equilibration. On the other hand, in our master equation program, consecutive discretized opinion bins each with fraction of opinion greater than a threshold of upon equilibration is considered to be a cluster. In other words, unless otherwise stated, we do not consider minor opinion clusters in our simulations.

Figure 1: [Color online] (a) Fraction of extreme opinion agents and (b) number of moderate (major) opinion clusters in our model in a complete network found by numerically solving the master equation with the opinions divided into  bins for and .

Finally, we remark that we have tried several parameters pairs in our simulations and they all exhibit similar dynamics. In fact, our mean-field analysis in Sec. IV shows why this is the case.

Iv Understanding our simulation results

Consider the following mean-field analysis. Let be the opinion of one of the agents chosen to interact at time . Since the initial opinions are randomly and uniformly assigned, the net rate for to increase after the interaction equals

(4)

for . In addition, , . There are three cases to consider.

Case (1): , namely, most of the region C. We only need to analyze the situation for opinions as our model is symmetric about . Eq. (4) implies for and . Thus, initially opinion tends to move towards ; and is an unstable equilibrium point. Whereas those with initial opinion may change to an opinion in the range after its first interaction due to the assimilation rule. Hence, opinions pile up around in the large limit shortly after . Note that is close to a linear function, increasing from to . So the number of agents with opinions around almost stays constant shortly after . Provided that , the assimilation rule has no effect between these piled up opinions in and those near . In this case, the net rate for opinion to increase at time is greater than . More importantly, this positive feedback mechanism quickly kicks opinions out of . Finally, the assimilation rule among opinions in and the boomerang effect rule between opinions in and assure that only the extreme opinions and will be present in the long run. Fig. 2 as well as Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”a and Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”a in the Supplemental Material SM () show that this is indeed the observed dynamics in region C.

The situation is more complex when due to the competing dynamics of the assimilation and boomerang effect rules between opinions in and opinions near . Depending on the details of the dynamics, the master equation method finds that the assimilation rule may win resulting in a single moderate peak around ; whereas the Monte Carlo method shows that this moderate peak may then be repelled to one of the extreme ends by a handful of remaining extreme opinion agents at the other end via the boomerang effect rule. (See Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”b and Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”b as well as the discussions in the Supplemental Material SM () on why the results of the two methods differ.)

Note that one point is certain — extreme and moderate opinion clusters cannot coexist for .

Case (2): , that is, most of the region B. Here Eq. (4) in the region becomes for , and for . So we have a pile up of opinions in the interval and a migration of opinions from to in the large limit shortly after . The same positive feedback mechanism acting on region C then leads to the formation of the two extremist clusters provided that . Note that the opinion interval is in unstable equilibrium initially because local opinion clustering by the assimilation rule can grow. Besides, the depletion of opinions in due to migration will in effect pull the opinions slightly less than to a lower value. These are precisely the effects governing the dynamics of the D-W Model. Thus, we end up with two extreme clusters plus several moderate ones as shown in Fig. 1. Moreover, the distance between two successive moderate (major) opinion clusters are separated by in case of a fully connected network Deffuant et al. (2000); Weisbuch et al. (2002); Ben-Naim et al. (2003) so that there are about of them. (See Fig. 3 as well as Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”c–d and Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”c–d in the Supplemental Material SM ().)

There are two exceptions to this rule. Just like case (1), if , it is possible for opinions in to merge with opinions near forming a single moderate cluster due to the assimilation rule. (See Fig. 3 as well as Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”e and Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”e in the Supplemental Material SM (). Unlike region C’, both master equation and Monte Carlo approaches give the same conclusion here.) Another situation is when so that the region , where , is very small. Depending on the details of the dynamics, the proportion of agents in may not be high enough to keep them in place before the region is depleted. If this happens, the system will evolve to two extreme opinion peaks at and ; and this is what we find in Fig. 1 as well as Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”f and Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”f in the Supplemental Material SM ().

Figure 2: Master equation solution of the opinion distribution at different time in region C with , . All other parameters are the same as those used in Fig. 1.
Figure 3: Opinion dynamics in region B with , . All other parameters are the same as those used in Fig. 2.

Case (3): , namely, region A and part of region B. Here, for , and for . Hence, there is an initial migration of opinions from to opinions around . Similar analysis in case (2) shows that at least one moderate opinion cluster will form. (See Fig. 4 as well as Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”g and Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”g in the Supplemental Material SM ().) Nevertheless, there is a subtlety. If is close to and is sufficiently large, it is still possible for a small portion of agents to become extremists before they have time to join a moderate opinion cluster. The boundary between regions A and B, however, depends on the detailed dynamics of the system. Nevertheless, it is not possible to have extreme opinion peaks only in this case. (See Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”h and Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”h in the Supplemental Material SM ().)

Figure 4: Opinion dynamics in region A with , . All other parameters are the same as those used in Fig. 2.
Figure 5: [Color online] Phase diagram for our model in the B-A network under different assimilation and boomerang effect rules described in the main text with and obtained by averaging the results over 1000 independent Monte Carlo runs each with 1000 agents. Region C (extreme opinions only region) is colored cyan. The portion of extreme opinion agents in region B is represented by the intensity of purple color.

To summarize, we have argued that and are the most important conditions for the formation of extreme and moderate opinion clusters, respectively. More importantly, the mean-field argument we have used does not depend on the precise form of the assimilation and boomerang effect rules. In fact, the key conditions used in our mean-field analysis are:

  • Agent’s opinions can be described by a real number in .

  • All agents have the same and .

  • The assimilation (boomerang effect) rule makes the opinions of the two agents closer (farther) whereas the opinions are unchanged by the neutral rule.

  • The criteria for applying the assimilation and boomerang effect rules are based only on the distance between two opinions .

  • The three rules governing the microscopic opinion change are symmetric about . The last two conditions ensures that there is no prior bias toward one of the extreme opinions.

In other words, the appeal of the above analysis is that conclusions can be drawn that are insensitive to factors such as network topology and the precise form of the agent dynamics as long as the average network connectivity is not too low and the agent dynamics is consistent with the SJT. Indeed, Fig. 5 shows a similar phase diagram of our model in the Barabási-Albert (B-A) scale-free network Barabási and Albert (1999) even when the assimilation and boomerang effect rules for and in Eqs. (1) and (2) are changed to

(5a)
(5b)

and

(6a)
(6b)

respectively, where and are fixed positive parameters. Clearly, these modified assimilation and boomerang effect rules are very different from those used in the D-W Model and the J-A Model. More importantly, unlike our original rules in Eqs. (1) and (2), the modified assimilation and boomerang effect rules are chosen so that agent’s response is continuous across different latitudes, thereby demonstrating that the phase diagram is not sensitive to discontinuity in response across different latitudes. Note further that the number of moderate clusters in this case can be more than , which is consistent with the behavior of the D-W Model model in the B-A network Stauffer et al. (2004); Stauffer and Meyer-Ortmanns (2004). (See Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”a–h in the Supplemental Material SM ().)

The major shortcoming in our mean-field analysis is that we cannot predict the height of each opinion cluster and the most likely location of each of them. Actually, by choosing and near the boundaries between regions A, B and C, some of the equilibrated opinion clusters may contain less than 1% of the population.

V Opinion dynamics of slowly driving and

We go on to study the situation that the thresholds and in Eqs. (1) and (2) change to reflect the change in the level of opinion tolerance in the society. While one’s opinion may change by interacting with another agent once, it probably takes a much longer time for and to change since it reflects a fundamental change in the way the agents evaluate and response to the opinions of others. Here we consider the idealized situation that and change gradually in a timescale much longer than the opinion equilibration time of the system in a way analogous to the study of quasi-static equilibrium processes in thermodynamics.

From the above analysis, we only need to consider the evolution of equilibrated opinions, which consists of extreme and/or moderate clusters, upon a small change in and . Note that upon equilibration, the opinion difference between two adjacent agents belonging to two different opinion clusters must either be (i) outside both the regions of assimilation and boomerang effect or (ii) in the boomerang effect region and the two agents hold extreme opinions of and . Consequently, the opinion distribution will not change as one perturbs and unless increases above or decreases below the opinion difference between two adjacent clusters. Some opinion clusters will merge in either cases. Thus, the opinion distribution stays constant most of the time and then suddenly change by opinion merging in a first-order phase transition. More importantly, a single moderate opinion cluster or two extreme opinion clusters are the only two stable fixed points of the system due to slowly random drifting of and .

Actually, opinion sudden changes in social issues after a long period of stasis, known as punctuated equilibria in social theory, is commonly observed Baumgartner and Jones (1993). Our analysis here shows that they may occur even in a fully connected social network, therefore repudiating the criticism by Tilcsik and Marquis Tilcsik and Marquis (2013). In fact, punctuated equilibrium in our model originates from the separation of timescales between agents’ interactions and the change in and similar to the case of first-order phase transition in a quasi-statically evolving thermal system.

Vi Outlook

In summary, we have proposed an agent-based and SJT-compatible model by extending the works of Jager and Amblard Jager and Amblard (2005) and Crawford et al. Crawford et al. (2013). Our model can serve as a blueprint to study opinion formation dynamics. In our model, formation of extreme and/or moderate opinion clusters as well as punctuated equilibria are observed even in the case of homogeneous agents in a complete network. Besides, we identify the most important conditions for forming extreme and moderate opinion clusters by mean-field analysis. Further works should be done, including the addition of noise to the agent’s response and a more detailed model of how and change, to make our model more realistic. Our next goal is to model opinion cluster splitting.

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Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”

Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution” and Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution” depict the dynamics of our model in a fully connected network in various regions of the parameter space found by numerically solving the master equation and in typical runs of Monte Carlo simulation using agents, respectively. These two approaches give similar results except in region C’ (as shown in Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”b and Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”b) where are slightly less than . In this exceptional case, the master equation approach gives a single moderate peak at in the steady state; while our Monte Carlo simulation shows that this moderate peak can be meta-stable. More precisely, after a long time, the moderate peak at sometimes move towards one of the extreme ends giving eventually a major extreme peak plus a very small minor extreme peak at the other end. In fact, by finite-size scaling analysis, our Monte Carlo simulation suggests that all the steady states in region C’ are made up of one major and one minor extreme peaks in the large limit.

This discrepancy may be caused by the followings. For our numerical solution to the master equation, numerical truncation error and a long decay time of the meta-stable state may lead to a wrong conclusion. More importantly, as an approach that deals with the evolution of opinion distribution, the master equation approach fails to capture the dynamics of certain opinion distributions in our model. For example, consider the opinion distribution in which agents are of opinion almost surely and at the same time with a measure zero number of agents with opinion . (One may think of the system configuration in which there is only one agent with and all the remaining agents has . Then we take the limit .) For and , this configuration will evolve to the steady state with one major opinion peak at and the opinions of those agents with are unchanged. It takes a long time for the system to evolve to this state though. Surely, this is what we will find by running the Monte Carlo simulation for a sufficiently long time. However, since the opinion distribution in this example is equal to that of the Dirac delta function at , the master equation approach will wrongly predicts that the system is already in steady state. However, we are not sure if this is the cause of the discrepancy because we do not know if a uniformly distributed initial opinion will almost surely evolve to a meta-stable state like the one in the above example.

In addition, our Monte Carlo simulation is not without trouble in region C’ because the convergence time is too long to be computationally feasible to perform finite-size scaling when are very close to . So we cannot rule out the existence of a single stable moderate peak at in the large limit when and are very close to .

Finally, Video Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution” shows the dynamics of our model in the B-A network with using different assimilation and boomerang effect rules. It demonstrates that it has very similar dynamics as in the case of the complete network although the number of moderate peaks may differ.

{video}

Due to file size limit, all videos are not uploaded. Please contact the first author at hfchau@hku.hk if you want a copy. Videos showing the dynamics of our model in a fully connected network in different regions of the parameter space found by numerically solving the master equation with the opinions divided into  bins. In particular, Video Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”a shows the typical dynamics in region C using parameters ; Video Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”b shows the dynamics in region C’ using parameters ; Video Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”c shows the dynamics in region B using parameters which results in three moderate opinion clusters plus two extreme opinion clusters; Video Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”d shows the dynamics in region B using parameters which results in two moderate opinion clusters plus two extreme opinion clusters; Video Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”e shows the dynamics in region A with using parameters which results in a single moderate opinion cluster; Video Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”f shows the dynamics in region C with using parameters ; Video Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”g shows the dynamics in region A using parameters ; Video Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”h shows the dynamics in region B with using parameters .

{video}

The same as Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”a–h except that these are typical runs of Monte Carlo simulations for a network of  agents.

{video}

The same as Videos Supplemental Material For “Social Judgment Theory Based Model On Opinion Formation, Polarization And Evolution”a–h but for the case of the B-A network using different assimilation and contrast rules as mentioned in the main text.

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