- Email: [email protected]
Opinion dynamics of modified Hegselmann–Krause model in a group-based population with heterogeneous bounded confidence
Physica A xx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Opinion dynamics of modified Hegselmann–Krause model in a group-based population with heterogeneous bounded confidence Q1
Guiyuan Fu a,b , Weidong Zhang a,b,∗ , Zhijun Li c a
Department of Automation, Shanghai Jiaotong University, Shanghai, 200240, PR China
b
Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, 200240, PR China
c
College of Automation Science and Engineering, South China University of Technology, Guangzhou, 510006, PR China
highlights • • • •
Modified HK model is proposed for grouped population. Open-minded agent may even diversify the final opinions. The relative size of the largest cluster varies along concave-parabola-like curve. The tipping point of the curve of relative size occurs at about po = pm .
article
info
Article history: Received 19 December 2013 Received in revised form 6 June 2014 Available online xxxx Keywords: Opinion dynamics Modified Hegselmann–Krause model Group-based population Heterogeneous bounded confidence
abstract Continuous opinion dynamics in a group-based population with heterogeneous bounded confidences is considered in this paper. A slightly modified Hegselmann–Krause model is proposed, and agents are classified into three categories: open-minded-, moderateminded-, and close-minded-agents, while the whole population is divided into three subgroups accordingly. We study how agents of each category and the population size can affect opinion dynamics. It is observed that the number of final opinion clusters is dominated by the close-minded agents; open-minded agents cannot contribute to forming opinion consensus and the existence of open-minded agents may diversify the final opinions instead; for the fixed population size and proportion of close-minded agents, the relative size of the largest final opinion cluster varies along concave-parabola-like curve as the proportion of open-minded agents increases, and there is a tipping point when the number of open-minded agents is almost equal to that of moderate-minded agents; for the fixed proportion of the three categories in the population, as the population size becomes larger, the number of final opinion clusters will reach a plateau. Some of the results are different from the previous studies. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Is gene-modified food harmful to our health and will you eat such food? What kind of fashion style would you like to follow? When you go to the polls, which candidate are you going to support? All such things are determined by our own opinions. It can be even said that almost all social interactions are shaped by our beliefs and opinions [1]. Thus it is of
∗
Corresponding author. E-mail addresses: [email protected] (G. Fu), [email protected] (W. Zhang), [email protected] (Z. Li).
http://dx.doi.org/10.1016/j.physa.2014.10.045 0378-4371/© 2014 Elsevier B.V. All rights reserved.
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high value to study opinion dynamics, and up to now many researchers with different background have proposed various models to analyze the evolution of the opinion dynamics from various aspects, see, for example Refs. [2–12]. One of the most commonly used models is Hegselmann–Krause (HK) model proposed by Ref. [7], which is agent-based and considers the continuous opinion dynamics under bounded confidence, i.e. agents only interact when they are close in opinion to each Q4 other, and where every agent synchronously update their opinion by averaging all the opinions of their neighbors. HK models can be classified into agent-based and density-based models [9]. The stability and convergence of HK model have been studied in Refs. [13–17], however, most of them only consider homogeneous HK model and there is no convincing stability analysis available for heterogeneous HK model. Previous studies have also investigated opinion dynamics based on HK model from various aspects. Ref. [18] studied the consensus threshold of the bounded confidence of homogeneous HK model and indicates that the threshold for complete consensus ϵc can only take one of two possible values ϵc ∼ 0.2 or ∼0.5, which depends on the average degree of the graph when the number of agents approaches infinity. Ref. [19] considered both the bounded confidence and the influence radius of agents and analyzed the influence of the heterogeneity in confidence and influence distribution on the opinion dynamics, where they found that heterogeneity did not always promote consensus and there is an optimal heterogeneity under which the relative size of the largest opinion cluster reaches its peak point. Ref. [20] considered the case where each agent has also several neighbors outside bounded confidence, which are selected according to the similarity-based probability rule. Ref. [21] studied the opinion dynamics of HK model with multi-level confidences, which is similar to Ref. [22], by dividing the agents into three subgroups: close-minded, moderate-minded and open-minded respectively based on social differentiation theory and analyze the influence of the fractions of each subgroup as well as the population size on opinion formation. However, they assume that all the agents in the same subgroup share the same bounded confidence. In this paper, extending our work [23], we consider the continuous opinions dynamics in the space [0 1]. First, slightly modified HK model is proposed based on non-Bayesian learning rule [24], so that every agent can take into account their own opinion independently with different weight rather than the same weight as their neighbors’. The population are classified into three categories: open-minded agents, moderate-minded agents and close-minded agents according to their corresponding confidence levels. The study focus on how these three categories of agents as well as population size affect the final opinion dynamics of the whole population. Unlike the work [21], the confidence levels of the individuals in the same subgroup is uniformly distributed in the corresponding interval. Another important aspect of our work we want to stress here is that the influence of the population size on the number of final opinion clusters is different from the previous study. The rest of this paper is organized as follows. Section 2 reviews HK model briefly and presents modified Hegselmann– Krause (MHK) model; main results are presented in Section 3, devoted to the study of the opinion dynamics according to MHK with respect to different proportions of subgroups and population size by numerous simulations. Finally summary and conclusions are given in Section 4.
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2. Model formulation
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2.1. Introduction of Hegselmann–Krause (HK) model
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We will review the original HK model first. Consider a system of n agents, whose opinions are located in one dimensional Euclidean space R and can be expressed by a real number. Denote the set of n agents as V = {1, 2, . . . , n} and for agent i ∈ V , its opinion at time t is represented by xi (t ) ∈ R. Given the bounded confidence set ϵ = {ϵ1 , ϵ2 , . . . , ϵn }, each agent i only interacts with his neighbors, whose opinions differ from his own not more than the certain confidence level ϵi . The discrete time HK model [9] can be described by Eq. (1): xi ( t + 1 ) =
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|Ni (t )| j∈N (t )
(xj (t ) − xi (t ))
(1)
i
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where Ni (t ) = {j ∈ V | |xi (t ) − xj (t )| ≤ ϵi } is the opinion neighbor set of agent i at time t and |Ni (t )| is the cardinality of Ni (t ). When ϵ1 = ϵ2 = · · · = ϵn , the HK model is homogeneous, otherwise, it is heterogeneous. Q5 Obviously, Eq. (1) inherently indicates that agent i is always self neighbor and it updates its opinion by simply averaging all the opinions of his neighbors with the same weights |N 1(t )| . i
2.2. Modified Hegselmann–Krause (MHK) model As mentioned above, agent i considers his own opinion and his neighbors’ equally, and this may be not realistic in the real world. We thus claim that it would be more reasonable to consider self opinion independently when agent i updates his opinion. To this end, we redefine agent i’s neighbor set as N¯ i (t ) = {j ∈ V ∩ j ̸= i | |xi (t ) − xj (t )| ≤ ϵi }. Based on non-Bayesian updating rule, MHK model is proposed as Eq. (2). xi ( t + 1 ) =
αi xi (t ) + (1 − αi )
xi (t ),
otherwise
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|N¯ i (t )|
j∈N¯ i (t )
xj (t ),
N¯ i (t ) ̸= ∅ (2)
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Fig. 1. Convergence time of MHK equation (2) with different α . The result refers to n = 100 and ϵ = 0.5 and is an average over 50 realizations with the same initial opinion.
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Fig. 2. Time evolution of 100 agent opinions according to MHK model equation (2) with different bounded confidence. ϵi are uniformly distributed in the corresponding interval. (a)–(c) have the same initial opinion, while (d)–(f) have another same initial opinion.
where αi ∈ [0 1] is the weight that agent i assigns to his own opinion, which we can refer to as the measure of self-belief or convergence parameter. Usually, the convergence time of MHK with large α is much longer than that with small α . As is shown in Fig. 1, where we consider the number of agents n = 100 and assume that the homogeneous confidence level ϵ = 0.5. The result is an average over 50 realizations with the same computer configuration and initial opinion profile, and it can be observed that the larger α is, the longer convergence time is. When the bounded confidences are heterogeneous, it displays qualitatively similar trend, thus we omit its result here. Obviously when αi = |N¯ (t1)|+1 , MHK model is equal to HK model. Furthermore, when αi = 1, it means agent i will never
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consider other people’s opinion and we can treat it as stubborn agent. Without loss of generality, we assume α1 = α2 = · · · = αn = α = 0.5 in this paper. Time evolution of 100 agents’ opinion dynamics of MHK equation (2) is shown in Fig. 2, where the initial opinion is uniformly distributed in the opinion space, and confidence levels ϵi are uniformly distributed in the intervals [0.01 0.05], [0.2 0.3], [0.4 0.9], respectively. Since the initial opinion can influence the dynamics of the final opinion (see Ref. [9]), we give results under different initial profiles. As is shown in Fig. 2, the left three (a)–(c) have the same initial opinion, while the right three (d)–(f) have another same initial opinion profile. It can be observed from Fig. 2 that opinion consensus,
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Fig. 3. The number of final opinion clusters with different proportions of close-minded agents pc . The results refer to 100 agents, where po = pm , and are an average over 50 realizations which have the same initial opinion profiles.
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polarization or fragmentation emerges respectively under different confidence levels. Moreover, under different initial profiles, some of the results would remain similar while some would be quite different from each other, for example, Fig. 2(b) and (e), where the confidence levels are uniformly distributed in the interval [0.2 0.3]. This is mainly due to the fact that the mean bounded confidence is near to the consensus threshold. Thus there is transition from consensus to polarization. Meanwhile, if the confidence level is much larger than the consensus threshold, for example ϵi ∈ [0.4 0.9], the population can always reach a consensus, while if ϵi ∈ [0.01 0.05] which is much less than the threshold, the final opinion dynamics will result in fragmentation.
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3. Simulation analysis and results
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In the real world, there are always some people who are much more willing to accept the new things or much more influenced by others, while some are less likely to. In this sense, we classify the agents into open-minded-, moderateminded-, and close-minded-agent, respectively according to the confidence levels, and divide the whole population into the three subgroups accordingly. By considering the result in Ref. [18], which reveals that the consensus threshold for homogeneous HK model is about ϵc ∼ 0.2 for the complete graph, and the results shown in Fig. 2, we suppose that the confidence levels of close-minded agents are distributed in the interval [0.01 0.05], those of moderate-minded agents in [0.2 0.3] and open-minded agents in [0.4 0.9], respectively. In what follows, we will investigate how these three kinds of agents and the population size can influence the opinion dynamics of MHK (2). We denote the proportion of close-minded-, moderate-minded- and openminded-agent as pc , pm , po , respectively. Suppose that agents for each subgroup are chosen randomly and the bounded confidence ϵi of each agent i is heterogeneous and follows uniform distribution in the corresponding intervals; initial opinions are generated from the opinion space [0 1] with uniform distribution. The simulations are carried out in the complete network and all of the results are an average over 50 realizations.
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3.1. Close-minded agents
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We consider a population of n = 100 agents and assume that the proportion of close-minded agents pc varies from 0% to 30%. Fig. 3 shows the relation between pc and the number of final opinion clusters. By cluster, it is shaped by the agents shared the same opinion, while by the number of final opinion clusters, it means the number of final opinions held by the whole population when they are stable. It is observed that the number of final opinion clusters increases as the percentage of close-minded agents pc becomes larger. This is easy to understand. Because the close-minded agents are less likely to consider other agents’ opinions, more opinions will remain unchanged. As a result, the number of the final opinion clusters becomes larger. But when the number of close-minded agents is large enough, they would come to interact with each other and form clusters eventually, thus it can be also observed from Fig. 3 that the increasing rate becomes smaller and even tends towards zero gradually. It can be said that the number of final opinion dynamics is dominated by close-minded agents.
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3.2. Open-minded agents
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We will investigate how open-minded agents influence the opinion dynamics in this subsection. In order to rule out the influence of initial opinion, we also use the same initial profile as the previous subsection. We assume that the proportion
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Fig. 4. The number of final opinion clusters with different proportions of open-minded agents po . The result refers to n = 100, pc = 10% and pm = 1 − pc − po and is an average over 50 realizations.
Fig. 5. Relative size of the largest cluster with different po . The result refers to n = 100 and is an average over 50 realizations.
of the open-minded agents po varies from 0, i.e. only close-minded and moderate-minded agents are considered, to 1 − pc , i.e. only close-minded and open-minded agents are considered. The number of final opinion clusters with different proportions of open-minded agents po is shown in Fig. 4, where pc = 10%. It can be seen that the number of final opinion clusters increases at the beginning and then remains stable as po increases from 0. It can be indicated that open-minded agent cannot contribute to forging different opinions and instead the existence of open-minded agent may even diversify the final opinions. In order to investigate the influence of open-minded agents on the final opinion dynamics better, the relative size of the largest final opinion clusters as a function of the proportion of open-minded agents po is shown in Fig. 5, with pc = 5%, 10%, 20%, respectively. The relative size is the ratio between the number of agents in the largest final opinion cluster and the whole population size. It is interesting to find that for the same percentage of close-minded agents pc , the relative size of the largest cluster varies along concave-parabola-like curve, as po increases. The phenomenon can be explained by the following fact. Since the initial opinion profile is uniformly distributed in [0 1] and the close-minded agents are chosen randomly, the initial opinion of these close-minded agents would happen to cover both extreme sides of the opinion space, meanwhile, generally, in Eq. (2) the moderate-minded agents drift together towards one side of opinion space and form a large cluster together with the close-minded agents, while the open-minded agents would finally ‘‘sit between the chairs’’ [9] and form a central cluster, i.e. open-minded agents form a cluster between the clusters formed by close-minded agents, although the open-minded agents consider the opinions of both the clusters of close-minded agents synchronously. As a result, the number of agents in the largest cluster when po = 0 is larger than that when pm = 0, as can be seen in Fig. 5. In addition, when there coexist open-minded agents and moderate-minded agents, these moderate-minded agents will also ‘‘sit between the chairs’’, together with open-minded agents. However, it happens that these open-minded agents and moderate-minded agents are not aggregated in the same cluster, though the opinions of the these two kinds of agents
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Fig. 6. Convergence time with different po . The result refers to n = 100, pc = 10% and pm = 1 − pc − po and is an average over 50 realizations.
Fig. 7. The number of final opinion clusters with different proportions of moderate-minded agents pm . The result refers to n = 100, pc = 10% and is an average over 50 realizations.
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are close to each other in the end. Consequently, as po increases, the number of agents in the largest clusters decreases in the beginning and then increases again. The tipping point occurs at about pm = po . Thus it can be seen in Fig. 5 that the minimum point with larger pc is left-drift. Fig. 6 demonstrates the convergence time as a function of po when pc = 10%, which increases dramatically at the beginning and then decreases gradually as po increases from 0 to 1 − pc . By convergence, in this paper, means that all of the agents’ opinions remain unchanged over several time steps. Thus it can be conjectured that open-minded agents cannot always accelerate the convergence, especially when the number of open-minded agents is small, they even slow down the convergent rate, however, when the proportion of open-minded agents is large enough, they can speed up the convergent rate, compared to the population without open-minded agents. 3.3. Moderate-minded agents The influence of moderate-minded agents on the opinion dynamics will be investigated in this subsection. The same initial opinion profile as that in the previous subsections is employed, and the proportion of moderate-minded agents pm is likewise assumed to vary from 0, i.e. without moderate-minded agents in the population, to 1 − pc , i.e. without open-minded agents in the population. Fig. 7 demonstrates the relation between the number of final opinion clusters and the proportion of moderate-minded agents pm , where pc = 10%. It is shown that for the most part of value range of pm , the number of final opinion clusters remains stable. However, when moderate-minded agents account for the majority of the population, i.e. pm is very large and the proportion of open-minded agents is very small or even approaches zero, the number of final opinion clusters becomes much smaller. This is because the existence of open-minded agents may diversify the final opinions. It thus can be concluded that moderate-minded agents cannot diversify final opinions.
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Fig. 8. The number of final opinion clusters with different population size. The result refers to pc = 0.1, po = 0.2, pm = 0.7 and is an average over 50 realizations.
3.4. Population size Inspired by Ref. [21], we will investigate the impact of population size on the opinion dynamics. The population size n is assumed to increase from 100 to 2500, and the proportion of each subgroup is set as pc = 0.1, pm = 0.7, po = 0.2, respectively. The average number of final opinion clusters with different population size is shown in Fig. 8, which indicates that the average number of final opinion clusters increases until it approaches the stable level, as the population size n increases. This can be explained by the fact that the number of final opinion clusters is dominated by close-minded agents. For the fixed proportion of close-minded agents pc , when the population size n is small, i.e. the number of close-minded agents is relatively small, the number of final opinion clusters will increase as population size n increases, however, when the population size is larger than about 500 in this case, i.e. when the number of close-minded agents is large enough, the number of final opinion clusters will remain stable as n continues to increase, since these close-minded agents would come to interact with each other and form clusters eventually. The result is quite different from that in the work of Ref. [21], which claims that under the same proportion of closeminded agents pc , the number of final opinion clusters is proportional to the total number of agents, which can be approximated by a linear function. 4. Discussion and conclusions In this paper, we have proposed a slightly modified HK model, where each agent considers self opinion independently, and classified the whole population into three categories: open-minded-, moderate-minded-, and close-minded-agents according to their confidence levels. The bounded confidence of the agents in the same subgroup is assumed to follow uniform distribution within the corresponding intervals. The influences of the agents of the three categories and the population size on opinion dynamics in the population level have been investigated. It has been found from the simulation results that the number of final opinion clusters is dominated by close-minded agents. The open-minded agents cannot contribute to forging opinion consensus. The relative size of the largest cluster varies along concave-parabola-like curve with increasing proportion of open-minded agents po , and there is a minimum point at about pm = po , namely the amount of open-minded agent is equal to that of moderate-minded agents. This may be explained by the fact that, on the one hand, open-minded agents will rather finally ‘‘sit between chairs’’ and form central clusters, while the moderate-minded agents will drift together towards one side of opinion space and form a large cluster together with the close-minded agents; on another hand, when the open-minded agents and moderate-minded agents coexist, these moderate-minded agent will also ‘‘sit between the chairs’’, but they will not finally converge into one cluster together with the open-minded agents. In addition, if the proportion of open-minded agent is large enough, open-minded agents can speed up the convergent rate, compared to the population without open-minded agents, but they cannot always accelerate convergence, and when the number of open-minded agents is small, they even increase convergence time. By contrast, moderate-minded agent cannot diversify the final opinion. For the fixed proportion pc , po , pm , as the population size n increases, the average number of final opinion clusters increases too, but will reach a plateau, if n is large enough. This is not the same as is stated in Ref. [21]. It is indicated from the presented results that very open-minded agents do not help to forge opinion consensus in the mixed population, and rather diversify the final opinions. This might give insights to take strategies in the group debate and uncover the dynamics process in the real world. Further work will be directed towards the theoretical analysis from
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mathematical perspective and the influence of network topology and the non-uniform distribution of the initial opinion profiles.
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Acknowledgments
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This work was partly supported by the National Science Foundation of China (61025016, 61034008, 61221003), Program of Shanghai Subject Chief Scientist (14XD1402400), and SJTU M&E Joint Research Foundation (YG2013MS04). The authors are also thankful to the anonymous referees for their helpful comments.
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Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Opinion dynamics of modified Hegselmann–Krause model in a group-based population with heterogeneous bounded confidence Q1
Guiyuan Fu a,b , Weidong Zhang a,b,∗ , Zhijun Li c a
Department of Automation, Shanghai Jiaotong University, Shanghai, 200240, PR China
b
Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, 200240, PR China
c
College of Automation Science and Engineering, South China University of Technology, Guangzhou, 510006, PR China
highlights • • • •
Modified HK model is proposed for grouped population. Open-minded agent may even diversify the final opinions. The relative size of the largest cluster varies along concave-parabola-like curve. The tipping point of the curve of relative size occurs at about po = pm .
article
info
Article history: Received 19 December 2013 Received in revised form 6 June 2014 Available online xxxx Keywords: Opinion dynamics Modified Hegselmann–Krause model Group-based population Heterogeneous bounded confidence
abstract Continuous opinion dynamics in a group-based population with heterogeneous bounded confidences is considered in this paper. A slightly modified Hegselmann–Krause model is proposed, and agents are classified into three categories: open-minded-, moderateminded-, and close-minded-agents, while the whole population is divided into three subgroups accordingly. We study how agents of each category and the population size can affect opinion dynamics. It is observed that the number of final opinion clusters is dominated by the close-minded agents; open-minded agents cannot contribute to forming opinion consensus and the existence of open-minded agents may diversify the final opinions instead; for the fixed population size and proportion of close-minded agents, the relative size of the largest final opinion cluster varies along concave-parabola-like curve as the proportion of open-minded agents increases, and there is a tipping point when the number of open-minded agents is almost equal to that of moderate-minded agents; for the fixed proportion of the three categories in the population, as the population size becomes larger, the number of final opinion clusters will reach a plateau. Some of the results are different from the previous studies. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Is gene-modified food harmful to our health and will you eat such food? What kind of fashion style would you like to follow? When you go to the polls, which candidate are you going to support? All such things are determined by our own opinions. It can be even said that almost all social interactions are shaped by our beliefs and opinions [1]. Thus it is of
∗
Corresponding author. E-mail addresses: [email protected] (G. Fu), [email protected] (W. Zhang), [email protected] (Z. Li).
http://dx.doi.org/10.1016/j.physa.2014.10.045 0378-4371/© 2014 Elsevier B.V. All rights reserved.
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high value to study opinion dynamics, and up to now many researchers with different background have proposed various models to analyze the evolution of the opinion dynamics from various aspects, see, for example Refs. [2–12]. One of the most commonly used models is Hegselmann–Krause (HK) model proposed by Ref. [7], which is agent-based and considers the continuous opinion dynamics under bounded confidence, i.e. agents only interact when they are close in opinion to each Q4 other, and where every agent synchronously update their opinion by averaging all the opinions of their neighbors. HK models can be classified into agent-based and density-based models [9]. The stability and convergence of HK model have been studied in Refs. [13–17], however, most of them only consider homogeneous HK model and there is no convincing stability analysis available for heterogeneous HK model. Previous studies have also investigated opinion dynamics based on HK model from various aspects. Ref. [18] studied the consensus threshold of the bounded confidence of homogeneous HK model and indicates that the threshold for complete consensus ϵc can only take one of two possible values ϵc ∼ 0.2 or ∼0.5, which depends on the average degree of the graph when the number of agents approaches infinity. Ref. [19] considered both the bounded confidence and the influence radius of agents and analyzed the influence of the heterogeneity in confidence and influence distribution on the opinion dynamics, where they found that heterogeneity did not always promote consensus and there is an optimal heterogeneity under which the relative size of the largest opinion cluster reaches its peak point. Ref. [20] considered the case where each agent has also several neighbors outside bounded confidence, which are selected according to the similarity-based probability rule. Ref. [21] studied the opinion dynamics of HK model with multi-level confidences, which is similar to Ref. [22], by dividing the agents into three subgroups: close-minded, moderate-minded and open-minded respectively based on social differentiation theory and analyze the influence of the fractions of each subgroup as well as the population size on opinion formation. However, they assume that all the agents in the same subgroup share the same bounded confidence. In this paper, extending our work [23], we consider the continuous opinions dynamics in the space [0 1]. First, slightly modified HK model is proposed based on non-Bayesian learning rule [24], so that every agent can take into account their own opinion independently with different weight rather than the same weight as their neighbors’. The population are classified into three categories: open-minded agents, moderate-minded agents and close-minded agents according to their corresponding confidence levels. The study focus on how these three categories of agents as well as population size affect the final opinion dynamics of the whole population. Unlike the work [21], the confidence levels of the individuals in the same subgroup is uniformly distributed in the corresponding interval. Another important aspect of our work we want to stress here is that the influence of the population size on the number of final opinion clusters is different from the previous study. The rest of this paper is organized as follows. Section 2 reviews HK model briefly and presents modified Hegselmann– Krause (MHK) model; main results are presented in Section 3, devoted to the study of the opinion dynamics according to MHK with respect to different proportions of subgroups and population size by numerous simulations. Finally summary and conclusions are given in Section 4.
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2. Model formulation
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2.1. Introduction of Hegselmann–Krause (HK) model
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We will review the original HK model first. Consider a system of n agents, whose opinions are located in one dimensional Euclidean space R and can be expressed by a real number. Denote the set of n agents as V = {1, 2, . . . , n} and for agent i ∈ V , its opinion at time t is represented by xi (t ) ∈ R. Given the bounded confidence set ϵ = {ϵ1 , ϵ2 , . . . , ϵn }, each agent i only interacts with his neighbors, whose opinions differ from his own not more than the certain confidence level ϵi . The discrete time HK model [9] can be described by Eq. (1): xi ( t + 1 ) =
1
|Ni (t )| j∈N (t )
(xj (t ) − xi (t ))
(1)
i
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where Ni (t ) = {j ∈ V | |xi (t ) − xj (t )| ≤ ϵi } is the opinion neighbor set of agent i at time t and |Ni (t )| is the cardinality of Ni (t ). When ϵ1 = ϵ2 = · · · = ϵn , the HK model is homogeneous, otherwise, it is heterogeneous. Q5 Obviously, Eq. (1) inherently indicates that agent i is always self neighbor and it updates its opinion by simply averaging all the opinions of his neighbors with the same weights |N 1(t )| . i
2.2. Modified Hegselmann–Krause (MHK) model As mentioned above, agent i considers his own opinion and his neighbors’ equally, and this may be not realistic in the real world. We thus claim that it would be more reasonable to consider self opinion independently when agent i updates his opinion. To this end, we redefine agent i’s neighbor set as N¯ i (t ) = {j ∈ V ∩ j ̸= i | |xi (t ) − xj (t )| ≤ ϵi }. Based on non-Bayesian updating rule, MHK model is proposed as Eq. (2). xi ( t + 1 ) =
αi xi (t ) + (1 − αi )
xi (t ),
otherwise
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j∈N¯ i (t )
xj (t ),
N¯ i (t ) ̸= ∅ (2)
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Fig. 1. Convergence time of MHK equation (2) with different α . The result refers to n = 100 and ϵ = 0.5 and is an average over 50 realizations with the same initial opinion.
a
d
b
e
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Fig. 2. Time evolution of 100 agent opinions according to MHK model equation (2) with different bounded confidence. ϵi are uniformly distributed in the corresponding interval. (a)–(c) have the same initial opinion, while (d)–(f) have another same initial opinion.
where αi ∈ [0 1] is the weight that agent i assigns to his own opinion, which we can refer to as the measure of self-belief or convergence parameter. Usually, the convergence time of MHK with large α is much longer than that with small α . As is shown in Fig. 1, where we consider the number of agents n = 100 and assume that the homogeneous confidence level ϵ = 0.5. The result is an average over 50 realizations with the same computer configuration and initial opinion profile, and it can be observed that the larger α is, the longer convergence time is. When the bounded confidences are heterogeneous, it displays qualitatively similar trend, thus we omit its result here. Obviously when αi = |N¯ (t1)|+1 , MHK model is equal to HK model. Furthermore, when αi = 1, it means agent i will never
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i
consider other people’s opinion and we can treat it as stubborn agent. Without loss of generality, we assume α1 = α2 = · · · = αn = α = 0.5 in this paper. Time evolution of 100 agents’ opinion dynamics of MHK equation (2) is shown in Fig. 2, where the initial opinion is uniformly distributed in the opinion space, and confidence levels ϵi are uniformly distributed in the intervals [0.01 0.05], [0.2 0.3], [0.4 0.9], respectively. Since the initial opinion can influence the dynamics of the final opinion (see Ref. [9]), we give results under different initial profiles. As is shown in Fig. 2, the left three (a)–(c) have the same initial opinion, while the right three (d)–(f) have another same initial opinion profile. It can be observed from Fig. 2 that opinion consensus,
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Fig. 3. The number of final opinion clusters with different proportions of close-minded agents pc . The results refer to 100 agents, where po = pm , and are an average over 50 realizations which have the same initial opinion profiles.
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polarization or fragmentation emerges respectively under different confidence levels. Moreover, under different initial profiles, some of the results would remain similar while some would be quite different from each other, for example, Fig. 2(b) and (e), where the confidence levels are uniformly distributed in the interval [0.2 0.3]. This is mainly due to the fact that the mean bounded confidence is near to the consensus threshold. Thus there is transition from consensus to polarization. Meanwhile, if the confidence level is much larger than the consensus threshold, for example ϵi ∈ [0.4 0.9], the population can always reach a consensus, while if ϵi ∈ [0.01 0.05] which is much less than the threshold, the final opinion dynamics will result in fragmentation.
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3. Simulation analysis and results
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In the real world, there are always some people who are much more willing to accept the new things or much more influenced by others, while some are less likely to. In this sense, we classify the agents into open-minded-, moderateminded-, and close-minded-agent, respectively according to the confidence levels, and divide the whole population into the three subgroups accordingly. By considering the result in Ref. [18], which reveals that the consensus threshold for homogeneous HK model is about ϵc ∼ 0.2 for the complete graph, and the results shown in Fig. 2, we suppose that the confidence levels of close-minded agents are distributed in the interval [0.01 0.05], those of moderate-minded agents in [0.2 0.3] and open-minded agents in [0.4 0.9], respectively. In what follows, we will investigate how these three kinds of agents and the population size can influence the opinion dynamics of MHK (2). We denote the proportion of close-minded-, moderate-minded- and openminded-agent as pc , pm , po , respectively. Suppose that agents for each subgroup are chosen randomly and the bounded confidence ϵi of each agent i is heterogeneous and follows uniform distribution in the corresponding intervals; initial opinions are generated from the opinion space [0 1] with uniform distribution. The simulations are carried out in the complete network and all of the results are an average over 50 realizations.
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3.1. Close-minded agents
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We consider a population of n = 100 agents and assume that the proportion of close-minded agents pc varies from 0% to 30%. Fig. 3 shows the relation between pc and the number of final opinion clusters. By cluster, it is shaped by the agents shared the same opinion, while by the number of final opinion clusters, it means the number of final opinions held by the whole population when they are stable. It is observed that the number of final opinion clusters increases as the percentage of close-minded agents pc becomes larger. This is easy to understand. Because the close-minded agents are less likely to consider other agents’ opinions, more opinions will remain unchanged. As a result, the number of the final opinion clusters becomes larger. But when the number of close-minded agents is large enough, they would come to interact with each other and form clusters eventually, thus it can be also observed from Fig. 3 that the increasing rate becomes smaller and even tends towards zero gradually. It can be said that the number of final opinion dynamics is dominated by close-minded agents.
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3.2. Open-minded agents
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We will investigate how open-minded agents influence the opinion dynamics in this subsection. In order to rule out the influence of initial opinion, we also use the same initial profile as the previous subsection. We assume that the proportion
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Fig. 4. The number of final opinion clusters with different proportions of open-minded agents po . The result refers to n = 100, pc = 10% and pm = 1 − pc − po and is an average over 50 realizations.
Fig. 5. Relative size of the largest cluster with different po . The result refers to n = 100 and is an average over 50 realizations.
of the open-minded agents po varies from 0, i.e. only close-minded and moderate-minded agents are considered, to 1 − pc , i.e. only close-minded and open-minded agents are considered. The number of final opinion clusters with different proportions of open-minded agents po is shown in Fig. 4, where pc = 10%. It can be seen that the number of final opinion clusters increases at the beginning and then remains stable as po increases from 0. It can be indicated that open-minded agent cannot contribute to forging different opinions and instead the existence of open-minded agent may even diversify the final opinions. In order to investigate the influence of open-minded agents on the final opinion dynamics better, the relative size of the largest final opinion clusters as a function of the proportion of open-minded agents po is shown in Fig. 5, with pc = 5%, 10%, 20%, respectively. The relative size is the ratio between the number of agents in the largest final opinion cluster and the whole population size. It is interesting to find that for the same percentage of close-minded agents pc , the relative size of the largest cluster varies along concave-parabola-like curve, as po increases. The phenomenon can be explained by the following fact. Since the initial opinion profile is uniformly distributed in [0 1] and the close-minded agents are chosen randomly, the initial opinion of these close-minded agents would happen to cover both extreme sides of the opinion space, meanwhile, generally, in Eq. (2) the moderate-minded agents drift together towards one side of opinion space and form a large cluster together with the close-minded agents, while the open-minded agents would finally ‘‘sit between the chairs’’ [9] and form a central cluster, i.e. open-minded agents form a cluster between the clusters formed by close-minded agents, although the open-minded agents consider the opinions of both the clusters of close-minded agents synchronously. As a result, the number of agents in the largest cluster when po = 0 is larger than that when pm = 0, as can be seen in Fig. 5. In addition, when there coexist open-minded agents and moderate-minded agents, these moderate-minded agents will also ‘‘sit between the chairs’’, together with open-minded agents. However, it happens that these open-minded agents and moderate-minded agents are not aggregated in the same cluster, though the opinions of the these two kinds of agents
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Fig. 6. Convergence time with different po . The result refers to n = 100, pc = 10% and pm = 1 − pc − po and is an average over 50 realizations.
Fig. 7. The number of final opinion clusters with different proportions of moderate-minded agents pm . The result refers to n = 100, pc = 10% and is an average over 50 realizations.
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are close to each other in the end. Consequently, as po increases, the number of agents in the largest clusters decreases in the beginning and then increases again. The tipping point occurs at about pm = po . Thus it can be seen in Fig. 5 that the minimum point with larger pc is left-drift. Fig. 6 demonstrates the convergence time as a function of po when pc = 10%, which increases dramatically at the beginning and then decreases gradually as po increases from 0 to 1 − pc . By convergence, in this paper, means that all of the agents’ opinions remain unchanged over several time steps. Thus it can be conjectured that open-minded agents cannot always accelerate the convergence, especially when the number of open-minded agents is small, they even slow down the convergent rate, however, when the proportion of open-minded agents is large enough, they can speed up the convergent rate, compared to the population without open-minded agents. 3.3. Moderate-minded agents The influence of moderate-minded agents on the opinion dynamics will be investigated in this subsection. The same initial opinion profile as that in the previous subsections is employed, and the proportion of moderate-minded agents pm is likewise assumed to vary from 0, i.e. without moderate-minded agents in the population, to 1 − pc , i.e. without open-minded agents in the population. Fig. 7 demonstrates the relation between the number of final opinion clusters and the proportion of moderate-minded agents pm , where pc = 10%. It is shown that for the most part of value range of pm , the number of final opinion clusters remains stable. However, when moderate-minded agents account for the majority of the population, i.e. pm is very large and the proportion of open-minded agents is very small or even approaches zero, the number of final opinion clusters becomes much smaller. This is because the existence of open-minded agents may diversify the final opinions. It thus can be concluded that moderate-minded agents cannot diversify final opinions.
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Fig. 8. The number of final opinion clusters with different population size. The result refers to pc = 0.1, po = 0.2, pm = 0.7 and is an average over 50 realizations.
3.4. Population size Inspired by Ref. [21], we will investigate the impact of population size on the opinion dynamics. The population size n is assumed to increase from 100 to 2500, and the proportion of each subgroup is set as pc = 0.1, pm = 0.7, po = 0.2, respectively. The average number of final opinion clusters with different population size is shown in Fig. 8, which indicates that the average number of final opinion clusters increases until it approaches the stable level, as the population size n increases. This can be explained by the fact that the number of final opinion clusters is dominated by close-minded agents. For the fixed proportion of close-minded agents pc , when the population size n is small, i.e. the number of close-minded agents is relatively small, the number of final opinion clusters will increase as population size n increases, however, when the population size is larger than about 500 in this case, i.e. when the number of close-minded agents is large enough, the number of final opinion clusters will remain stable as n continues to increase, since these close-minded agents would come to interact with each other and form clusters eventually. The result is quite different from that in the work of Ref. [21], which claims that under the same proportion of closeminded agents pc , the number of final opinion clusters is proportional to the total number of agents, which can be approximated by a linear function. 4. Discussion and conclusions In this paper, we have proposed a slightly modified HK model, where each agent considers self opinion independently, and classified the whole population into three categories: open-minded-, moderate-minded-, and close-minded-agents according to their confidence levels. The bounded confidence of the agents in the same subgroup is assumed to follow uniform distribution within the corresponding intervals. The influences of the agents of the three categories and the population size on opinion dynamics in the population level have been investigated. It has been found from the simulation results that the number of final opinion clusters is dominated by close-minded agents. The open-minded agents cannot contribute to forging opinion consensus. The relative size of the largest cluster varies along concave-parabola-like curve with increasing proportion of open-minded agents po , and there is a minimum point at about pm = po , namely the amount of open-minded agent is equal to that of moderate-minded agents. This may be explained by the fact that, on the one hand, open-minded agents will rather finally ‘‘sit between chairs’’ and form central clusters, while the moderate-minded agents will drift together towards one side of opinion space and form a large cluster together with the close-minded agents; on another hand, when the open-minded agents and moderate-minded agents coexist, these moderate-minded agent will also ‘‘sit between the chairs’’, but they will not finally converge into one cluster together with the open-minded agents. In addition, if the proportion of open-minded agent is large enough, open-minded agents can speed up the convergent rate, compared to the population without open-minded agents, but they cannot always accelerate convergence, and when the number of open-minded agents is small, they even increase convergence time. By contrast, moderate-minded agent cannot diversify the final opinion. For the fixed proportion pc , po , pm , as the population size n increases, the average number of final opinion clusters increases too, but will reach a plateau, if n is large enough. This is not the same as is stated in Ref. [21]. It is indicated from the presented results that very open-minded agents do not help to forge opinion consensus in the mixed population, and rather diversify the final opinions. This might give insights to take strategies in the group debate and uncover the dynamics process in the real world. Further work will be directed towards the theoretical analysis from
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mathematical perspective and the influence of network topology and the non-uniform distribution of the initial opinion profiles.
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Acknowledgments
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This work was partly supported by the National Science Foundation of China (61025016, 61034008, 61221003), Program of Shanghai Subject Chief Scientist (14XD1402400), and SJTU M&E Joint Research Foundation (YG2013MS04). The authors are also thankful to the anonymous referees for their helpful comments.
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