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Opinion clusters in a modified Hegselmann–Krause model with heterogeneous bounded confidences and stubbornness
Physica A 531 (2019) 121791
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Physica A journal homepage: www.elsevier.com/locate/physa
Opinion clusters in a modified Hegselmann–Krause model with heterogeneous bounded confidences and stubbornness Wenchen Han a , Changwei Huang b , Junzhong Yang c ,
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a
College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu, 610068, People’s Republic of China School of Computer, Electronics and Information, Guangxi University, Nanning, 530004, People’s Republic of China c School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China b
highlights • Fewer opinion clusters forms if most open-minded agents are more stubborn. • The effect of increasing the bounded confidence of one subpopulation depends on those of others. • The number of opinion clusters and the size of the largest opinion cluster are almost negatively correlated.
article
info
Article history: Received 20 December 2018 Received in revised form 31 May 2019 Available online 12 June 2019 Keywords: Heterogeneous populations Opinion clusters The size of the largest cluster
a b s t r a c t In opinion dynamics with continuous opinion, bounded confidence is a critical parameter. Agents can interact with each other only when the opinion difference between them is less than the bounded confidence. Larger bounded confidence always leads to fewer opinion clusters. Stubbornness characterizing the insistence of an agent on her own opinion is thought to only affect the transition time. In this work, a modified Hegselmann–Krause model with heterogeneous population is investigated, where agents in different/same subpopulation have different/same bounded confidence and stubbornness. We find that, due to the interaction among subpopulations, increasing the stubbornness of agents in the subpopulation with the largest bounded confidence favors fewer opinion clusters and the expansion of the largest cluster. We also find that increasing the bounded confidence of a subpopulation leads to fewer clusters and a larger largest cluster provided that all the others have large bounded confidence. While one subpopulation is with a small bounded confidence, there exist an optimal bounded confidence of another subpopulation for the smallest number of opinion clusters and that for the largest size of the largest cluster. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Social life is full of social opinions of things or items, e.g., likes or dislikes of tastes of foods, agreements or disagreements on propositions. Almost all social interactions can be shaped by opinions [1,2]. When the evolution of opinions is involved, we are facing opinion dynamics. Opinion dynamics has been a popular topic, in social physics [3– 10], mathematics [11,12], to name but a few. Opinions can be described by discrete variables as well as continuous ones. Accordingly, opinion dynamics models are classified into discrete and continuous opinion dynamics models. The ∗ Corresponding author. E-mail addresses: [email protected] (W. Han), [email protected] (J. Yang). https://doi.org/10.1016/j.physa.2019.121791 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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Sznajd model [13], the voter model [14,15], the Galam majority-rule model [16], the naming game [17,18], and the nonconsensus opinion model [19] as well as its generalized version [20] are examples for discrete models. Deffuant– Weisbuch (DW) model [21] and Hegselmann–Krause (HK) model [22] are two well-known continuous models. In an opinion dynamics, several final configuration can be found, for example consensus where all agents share a same opinion, polarization where only two opinion clusters exist, and fragmentation where several opinion clusters coexist. Especially, the consensus draws much attention in opinion dynamics and several mechanisms to reach consensus are proposed, including social diversity [23], committed minorities [24,25], introducing zealots [26], communication burstiness [27], adaptive convergence rate [28], heterogeneous bounded confidences [29–31], and so on. The bounded confidence is the threshold of opinion difference between two agents, below which these two agents can communicate, and it is usually defined in a continuous opinion dynamics [21,29,32,33]. In a HK model [22,29], opinions are usually treated as real numbers in an interval [0, 1]. Two agents are neighbours only when the opinion difference between them is smaller than the bounded confidence. During the evolution, an agent updates her opinion at each time step by adopting the average opinion of all her neighbors including herself [22,30]. High bounded confidence always leads to consensus and low bounded confidence to a fragmentation. Lorenz studied continuous opinion models where agents are with different bounded confidences and found that agents could achieve consensus even when bounded confidences were below the threshold [34]. When agents are with different bounded confidences, it is found the number of opinion clusters is dominated by the agents with the smallest bounded confidence [35,36]. Besides the bounded confidence, the stubbornness measuring the extent that an agent insists on her own opinion is another feature of agents in the opinion dynamics [22,36–38]. Recently, the effects of heterogeneous stubbornness on opinion dynamics have been taken into consideration, such as heterogeneous stubbornness with public authority [39], heterogeneous stubbornness with interactive radius [40], and heterogeneous stubbornness with social similarity [41], heterogeneous stubbornness in PA/C model with partial antagonism agents and concord agents [42]. In previous works, heterogeneous bounded confidence and heterogeneous stubbornness are studied separately. Considering that stubbornness could control the time scale in the formation of opinion clusters, some questions arise in opinion dynamics with heterogeneous bounded confidence. For example, if we assign agents with high bounded confidence with strong stubbornness to slow down the formation of consensus and assign agents with low bounded confidence with weak stubbornness to speed up the formation of multiple opinion clusters, what happens under the competition between these two processes? To our knowledge, in the previous works, heterogeneous bounded confidence and heterogeneous stubbornness are investigated independently. It will be interesting to study opinion dynamics in the presence of both heterogeneous bounded confidence and heterogeneous stubbornness. Especially, we had like to study the collective effects on opinion dynamics when bounded confidence and stubbornness assigned to an agent are correlated. In this work, we study a modified HK model with a heterogeneous population composed of several subpopulations and focus our attention on the number of opinion clusters and the relative size of the largest opinion cluster in the population. In each subpopulation, agents share a same set of bounded confidence and stubbornness, which is different from that of other subpopulations. We find that opinion clusters are fewer and the largest opinion cluster is larger if agents in the subpopulation with the largest bounded confidence become more stubborn. We also find that, if all other subpopulations are with large bounded confidence, increasing the bounded confidence of one subpopulation favors fewer clusters and a larger largest cluster. If there exits one subpopulation with a small bounded confidence, there exist an optimal bounded confidence of another subpopulation for the smallest number of opinion clusters and that for the smallest size of the largest opinion cluster in the population. 2. Model We consider a population composed of N agents on a complete graph. The state of agent i at discrete time t is described by his opinion xi (t), which is in the range of [0, 1]. The opinion of agent i evolves following the updating rule ∑ { j∈N (t) xj (t) αi xi (t) + (1 − αi ) ∥Ni (t)∥ ∥Ni (t)∥ > 0, i xi (t + 1) = xi (t) ∥Ni (t)∥ = 0,
(1)
αi ∈ [0, 1] represents the stubbornness of agent i, describing the extent that agent i insists on her own opinion. αi = 1 refers to a zealot, sticking to her own opinion and neglecting others, while αi = 0 means a conformist, just following others’ opinions. Bearing this in mind, we say that agent i is more stubborn than agent j if αi > αj . Ni (t) is the set of agents, interacting with agent i, whose opinions satisfy |xj (t) − xi (t)| < σi and j ̸ = i at time t, and ∥Ni (t)∥ is the cardinality of the set Ni (t). The bounded confidence σi depicts the interacting range of agent i and σi ∈ (0, 0.5]. We could name agents with large σ as open-minded ones and agents with small σ as close-minded ones. In this work, ∑ the total population is l=M composed of M subpopulations, where each subpopulation with ρl N agents, where ρl ∈ [0, 1] and l=1 ρl = 1, takes the bounded confidence σl and the stubbornness αl . For convenience, we use Cl to denote the subpopulation with σl and αl . η describes the average of absolute opinion changes in each time t, written as η(t) =
N 1 ∑
N
i=1
|xi (t) − xi (t − 1)|.
(2)
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Fig. 1. (a) The number of clusters Nc and the relative size of the largest opinion clusters S against the bounded confidence σ with different stubbornness α in a homogeneous population. The time sequences of opinions (b) when α = 0, σ = 0.1, (c) when α = 0.8, σ = 0.1. (d) Time evolution of η with σ = 0.1.
Similarly, we have the ηl for Cl as
ηl (t) =
1
∥Cl ∥
∑
|xi (t) − xi (t − 1)|,
(3)
i∈Cl
When the system reaches its steady state, where opinions of all agents stop changing (η = 0), we monitor the number of opinion clusters, Nc , where agents within a same cluster share a same opinion. S is the relative size of the largest opinion cluster, the ratio between the number of agents in the largest cluster and the number of agents in the system. We consider the population size N = 1000 and the simulation results are averaged over 100 realizations with uniform random initial condition in opinions. The opinions of agents are updated synchronously. 3. Simulation analysis and results We first consider a population with homogeneous bounded confidence σ and stubbornness α , or interpreted as M = 1, quite similar to the original HK model [22,29,36]. As is known to all, a larger σ leads to fewer opinion clusters, shown in Fig. 1 (a1), and a larger relative size of the largest opinion cluster, shown in Fig. 1 (a2). When the bounded confidence is above the critical one, σc = 0.25, agents reach consensus, Nc = 1 and S = 1. Moreover, the number of opinion clusters Nc and the relative size S is independent of the stubbornness of agents. The time sequences of opinions with different α , presented in Figs. 1(b) and (c), show that increasing α only extends the transient time. This is caused by that agents with larger α will stick to their own opinions rather than be affected by their neighbors’ opinions and that agents interact with the same neighbors under a fixed σ . Fig. 1(d) shows the time evolution of η, the average opinion change in absolute value, with different α . It shows clearly that more stubborn agents (with larger α ) change their opinions slowly. No matter what α is, the gradual decay always follows an initial rise of η. The characteristics of η against time reflects the process of the formation of opinion clusters. The comparison between Fig. 1(b), (c) and (d) suggests that the peak of η occurs roughly at the time when the steady opinion clusters are formed. Before the formation of the steady clusters, agents have to adjust their opinions greatly to form clusters. After the formation of clusters, agents only adjust their opinion in a small pace till they share a same opinion in the same cluster. It is expected that how strong the initial rise is dependent upon how many clusters are formed. Less opinion clusters indicates a weak initial rise of η. While the stubbornness cannot affect the number of opinion clusters in a homogeneous population, we are curious about, in a heterogeneous population, whether the opinion consensus can be promoted when agents in one subpopulation become more stubborn, or vice versa. Considering that increasing the bounded confidence decreases the number of opinion clusters in homogeneous populations, we are curious about whether opinion clusters are always reduced if some agents become more open-minded in a heterogeneous population. 3.1. 2 Subpopulations on complete networks Then we consider a population composed of just 2 subpopulations, ρ1 N agents of C1 with σ1 and α1 , ρ2 N agents of C2 with σ2 and α2 , and ρ2 = 1 − ρ1 . We investigate the dependence of Nc on α1 and the dependence of S on α1 and the results are presented in Fig. 2. We assume σ1 > σ2 without loss of generality. In other words, agents in C1 are more open-minded than those in C2 . In Fig. 2(a) and (b) σ1 = 0.3, where C1 agents will reach consensus, and σ2 = 0.1, where
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Fig. 2. The number of clusters Nc and the relative size of the largest opinion clusters S against α1 . (a) C1 with σ1 = 0.3 and C2 with σ2 = 0.1, α2 = 0.8. (b) C1 with σ1 = 0.3 and C2 with σ2 = 0.1, α2 = 0. (c) C1 with σ1 = 0.1 and C2 with σ2 = 0.04, α2 = 0.8. (d) C1 with σ1 = 0.1 and C2 with σ2 = 0.04, α2 = 0. ρ1 = 0.1 in red and ρ1 = 0.9 in blue.
Fig. 3. The evolutions of opinions with a same initial condition. C1 agents in red and C2 agents in blue. (a) α1 = 0.0, (b) α1 = 0.5, (c) α1 = 0.8, (d) α1 = 0.95. The average opinion change in absolute value of C1 (η1 ) in (e) and that of C2 (η2 ) in (f) with different α1 . Other parameters are fixed. ρ1 = 0.9, σ1 = 0.3, ρ2 =0.1, σ2 = 0.1, and α2 = 0.8.
C2 agents will form 4 clusters. Exchanging the values of α1 and α2 , there always exists a very large opinion cluster (S ≃ 1) when ρ1 is large, e.g. ρ1 = 0.9, while S < 1 when ρ1 is small. But the fact that Nc decreases monotonously with α1 while S increases is true. It is also true for the case where σ1 = 0.1 and σ2 = 0.04, where C1 agents and C2 agents will form 4 and about 10 opinion clusters respectively, as shown in Fig. 2(c) and (d). These facts indicate that fewer opinion clusters and a larger S are favored when the open-minded agents (C1 agents) in a heterogeneous population become more stubborn and that the relation between Nc and α1 and the relation between S and α1 are robust, independent of ρ1 and α2 . These facts can be understood in the following way. In a final configuration in a homogeneous population, C1 agents distribute themselves in fewer opinion clusters than C2 agents. It should be satisfied that the opinion difference between adjacent opinion clusters being larger than σ1 for C1 agents and σ2 for C2 agents. Nevertheless, in a heterogeneous population, how the population evolves to the final configuration strongly depends upon the competition between C1 agents and C2 agents. When the evolution of C1 agents is great faster than C2 agents, for example α1 = 0 and α2 = 0.8 in Fig. 3(a), the route to the final configuration is mainly determined by C2 agents. During this process, the C1 opinion cluster formed in advance will adjust its position when C2 opinion clusters, in C1 agents’ interacting range, are about to form instead of
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Fig. 4. The number of clusters Nc and the relative size of the largest opinion cluster S against σ1 . (a) C1 with α1 = 0 and C2 with σ2 = 0.3, α2 = 0.8. (b) C1 with α1 = 0.8 and C2 with σ2 = 0.3, α2 = 0. (c) C1 with α1 = 0 and C2 with σ2 = 0.04, α2 = 0.8. (d) C1 with α1 = 0.8 and C2 with σ2 = 0.04, α2 = 0. ρ1 = 0.1 in red and ρ1 = 0.9 in blue.
pinning down the locations of C2 opinion clusters. Meanwhile, C1 agents will make some C2 agents, who hold almost the same opinion with these C1 ones, to change their opinions when C1 agents do. On the other hand, when the evolution of C1 agents is great slower than C2 agents, for example α1 close to 1 (α1 = 0.95 shown in Fig. 3(d)), the evolution to the final configuration is determined by C1 agents. Though C2 agents first have formed different opinion clusters of their own, they are slaved by C1 agents due to the interaction which further drives C2 agents to hold opinions closer to others till opinion clusters of C1 agents are formed. Consequently, the number of opinion clusters approaches the one acquired by homogeneous C1 agents. When the evolutions in a homogeneous population of C1 agents and C2 agents become comparable in time scale, C1 agents may be pinned down by the C2 clusters (α = 0.5 in Fig. 3(b)) or slave C2 agents (α = 0.8 in Fig. 3(c)). Fig. 3(a)–(d) shows that there is only one opinion cluster for C1 agents, which is reflected in Fig. 3(f) where η1 against time display an obvious peak. On the other hand, C2 agents forms four opinion clusters at α = 0 and 0.5 and, consequently, the peak in η2 against time seems absent. Moreover, when α1 = 0.8, η1 can remain almost constant for a period when the C1 cluster is moving and η2 remains constant for some C2 agents moving, who holds the opinion very close to C1 ones. Comparing η1 and η2 with different α1 , Fig. 3(e) shows that C1 agents definitely change their opinions slower in a heterogeneous population when they become more stubborn and Fig. 3(f) indicates that the interaction with more stubborn C1 agents prolongs the evolution of C2 agents opinions. The overall effects of the mutual interaction between C1 agents and C2 agents are that more open-minded agents being more stubborn favors a smaller Nc and a larger S. To be mentioned, this correlation is independent of ρ1 , σ1 , σ2 , and α2 . Similarly, we can also infer that increasing the stubbornness of the close-minded agents (C2 agents) will increase the number of opinion clusters and decrease the relative size of the largest opinion cluster. In addition, Figs. 2 and 3 show that both close-minded agents and open-minded agents have significant impacts on Nc and S, which is different from the statement that in a population with homogeneous stubbornness open-minded agents cannot contribute to forging opinion consensus [36]. Now, we investigate how the bounded confidence of agents in one subpopulation influences the number of opinion clusters and the relative size of the largest opinion cluster. Without loss of generality, only the bounded confidence of C1 agents (σ1 ) varies and other parameters are fixed. Here, we first study the correlation of Nc and σ1 and that of S against σ1 in the system with open-minded C2 agents, for example σ2 = 0.3 shown in Fig. 4(a) and (b). In this case, C2 agents always reach consensus if C2 agents do not interact with C1 ones. Increasing σ1 of C1 agents always leads to fewer opinion clusters and a larger size of the largest opinion cluster if C1 agents are alone. Thus increasing σ1 leads to a smaller number of opinion clusters and a larger size of the largest opinion cluster. This dependence is independent of ρ1 , α1 , and α2 are. Moreover, agents reach consensus even when σ1 is blow the critical bounded confidence σc . In the case where C2 agents are close-minded the correlation between the number of opinion clusters Nc and the stubbornness of C1 agents σ1 and the correlation between the relative size of the largest cluster S and σ1 are not monotonous. Bearing a larger σ leading to a smaller Nc and a larger S in mind, it is quite reasonable that larger σ1 leads to a smaller Nc and a larger S when σ1 is small, while it is quite counterintuitive that increasing σ1 even makes Nc larger and S smaller when σ1 is large, with an example σ2 = 0.04 shown in Fig. 4(c) and (d).
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Fig. 5. The time sequences of opinions with a same initial condition. C1 agents in red and C2 agents in blue. (a) σ1 = 0.1, (b) σ1 = 0.24, (c) σ1 = 0.32, (d) σ1 = 0.4. The average opinion change in absolute value of C1 (η1 ) in (e) and that of C2 (η2 ) in (f) with different σ1 . Other parameters are fixed. ρ = 0.9, α1 = 0.8, σ2 = 0.10, and α2 = 0.
When σ1 is small, Nc becomes smaller and S becomes larger because of the interaction between C2 agents and C1 agents with larger σ1 , favoring the merging of opinion clusters as presented in Fig. 5(a) and (b). However, the interaction also leads to a larger Nc and a smaller S because of more C2 opinion clusters being left by C1 opinion clusters when σ1 further increases as shown in Fig. 5(c) and (d). It is known that increasing α1 will make the evolution of C1 agents’ opinions slow down directly and the evolution of C2 agents’ opinions slower due to the interaction, shown in Fig. 3(e) and (f). However, it is rarely noticed that increasing the bounded confidence will speed up the evolution of agents’ opinion, manifested by Fig. 5(e). The larger σ1 enlarges the interacting range of C1 agents, which causes that C1 agents can change their opinions faster on average, especially those agents holding extreme opinions, e.g., x ∼ 1 or x ∼ 0. When the C2 agents cannot keep pace with the opinion changing of C1 agents while increasing σ1 , C2 opinion clusters will be left by the C1 opinion clusters, shown in Fig. 5(c) and (d), which causes the increase of Nc and the decrease of S. Meanwhile, the locations of left C2 opinion clusters pin down C1 opinion clusters due to the large σ1 . That the interaction leads to difference outcomes is the result of competition between the effects of the bounded confidence (σ ) and the stubbornness (α ) of agents. 3.2. Several subpopulations on complete networks The results above observed in the 2 subpopulation system are qualitatively the same with that in the system with M subpopulations. In an M subpopulations system, where we simply assume σ1 > σj and j = 2, 3, . . . , M without loss of generality, increasing α1 always make the number of opinion clusters Nc decrease and the relative size of the largest opinion cluster S increase. Fig. 6(a) shows the dependence of Nc on α1 and the dependence S on α1 in 3 subpopulations system where C 1 with σ1 = 0.3, C 2 with σ2 = 0.1, α2 = 0.8, and C3 with ρ3 = 0.1, σ3 = 0.04, α3 = 0 are fixed. Fig. 6(c) shows the same results in 5 subpopulations system where C 1 with σ1 = 0.3, C 2 with σ2 = 0.04, α2 = 0.0, C3 with ρ3 = 0.1, σ3 = 0.1, α3 = 0.8, C4 with ρ4 = 0.1, σ4 = 0.08, α3 = 0.5, and C5 with ρ3 = 0.1, σ3 = 0.3, α3 = 0.3 are fixed. The dependence of Nc on σ1 and the dependence of S on σ1 are not monotonous, when some agents are close-minded, as 3 subpopulations shown in Fig. 6(b) and 5 subpopulations shown in Fig. 6(d). These results show that increasing the stubbornness of the most open-minded agents decreases the number of opinion clusters Nc and increases the relative size of the largest opinion cluster S, which is independent of ρi , αi , and σi with i = 1, 2, . . . , M except for σ1 > σj and j = 2, 3, . . . , M. Moreover, the result of increasing σ1 is due to the overall effect of competition between σ1 and α1 when other parameters are fixed. 3.3. 2 Subpopulations on small-world networks We also investigate a system composed of 2 subpopulation on small-world networks, which is rewired from a ring of degree k with the rewiring probability p = 0.03 [43]. The result is qualitatively the same with those observed in systems on complete networks. That is, increasing α1 will decrease the number of opinion clusters Nc and the relative size of the largest opinion cluster S. And increasing σ1 will make Nc first decrease then increase and S first increase then decrease when C2 are close-minded. Some specific cases with ⟨k⟩ = 20 are shown in Fig. 7.
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Fig. 6. The number of clusters Nc and the relative size of the largest opinion cluster S against α1 in 3 subpopulations systems with σ1 = 0.3 in (a) and against σ1 with α = 0.8 in (b). ρ1 = 0.1, ρ2 = 0.8 for red and ρ1 = 0.8, ρ2 = 0.1 for blue. C2 with σ2 = 0.1, α2 = 0.8 and C3 with ρ3 = 0.1, σ3 = 0.04, α3 = 0 are fixed. In 5 subpopulation systems with σ1 = 0.3 in (c) and against σ1 with α = 0.8 in (d). ρ1 = 0.1, ρ2 = 0.5 for green and ρ1 = 0.5, ρ2 = 0.1 for magenta. C2 with σ2 = 0.04, α2 = 0.0, C3 with ρ3 = 0.1, σ3 = 0.1, α3 = 0.8, C4 with ρ4 = 0.1, σ4 = 0.08, α4 = 0.5, and C5 with ρ5 = 0.1, σ5 = 0.3, α = 0.3 are fixed.
Fig. 7. The number of clusters Nc and the relative size of the largest opinion cluster S on small-world networks with ⟨k⟩ = 20. (a) against α1 with σ1 = 0.3 and σ2 = 0.1, α2 = 0.8. (b) against σ1 with α1 = 0.8 and σ2 = 0.04, α2 = 0.0. ρ1 = 0.1, ρ2 = 0.9 for red and ρ1 = 0.9, ρ2 = 0.1 for blue.
4. Conclusion In this work, we have studied a modified Hegselmann–Krause model with a heterogeneous population composed of M subpopulations, where agents in each subpopulation share a same set of bounded confidence σl and the stubbornness αl . In a homogeneous population, the bounded confidence determines the number of opinion clusters in the steady state and the stubbornness only affects the transient time. There are two interesting findings in our work. The first one is that the stubbornness of agents indeed affects the evolution of opinion systems with a heterogeneous population. Increasing the stubbornness of the most open-minded agents can decrease the number of opinion clusters, which is contrast to the conclusion that open-minded agents cannot contribute to forging opinion consensus when all agents share a same stubbornness in Ref. [36], increase the relative size of the largest opinion cluster in the population as well. This dependence is independent of the population ratio, the stubbornness, and the bounded confidence of other subpopulations. The second point is that increasing the bounded confidence of one subpopulation will not always be the option for forging consensus. When all other agents are open-minded, increasing the bounded confidence of one subpopulation always leads to fewer opinion clusters and a larger size of the largest opinion clusters in the whole population. While, when agents in one subpopulation are very close-minded, there exists an optimal bounded confidence for the smallest number of opinion clusters and that for the biggest relative size of the largest opinion cluster when increasing the bounded confidence of
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another subpopulation. In this case, agents reach consensus, even when the bounded confidences of all agents are below the critical one σc for a homogeneous population, as shown in Ref. [34]. The most close-minded agents mainly affects the number of opinion clusters and the size of the largest opinion cluster, which is consistent with the previous works [35,36]. Our work has studied the opinion dynamics with a heterogeneous population, where agents in each subpopulation with a set of bounded confidence and stubbornness. However, the initial opinion are uniformly distributed in the interval [0, 1]. It may be interesting to investigate the general initial distribution in an opinion system [44] by applying share a drink (SAD) procedure proposed by Häggström [45]. The extreme agents show substantial advantage to other agents [46,47]. It is interesting to find the effects in heterogeneous population with extreme minorities. Moreover, averager–copier–voter models with stubborn agents can show hybrid consensus [48]. In that work, agents obeying different opinion dynamics rather than the same dynamics with different parameters and the stubborn agents are considered with stubbornness α = 1. It will be interesting to investigate a population with different agents following different opinion dynamics. The opinion dynamics with mass media can show order–disorder transition in continuous opinion dynamics [49] and hysteretic phenomena in binary opinion dynamics [50]. Furthermore, the network structures show important effects on the number of opinion clusters in a population with homogeneous bounded confidence [51–55]. It is worth studying heterogeneous population on different networks. Besides this, the multilayer structure will hinders the consensus when the initial opinion configuration is within a bounded range [56]. However, the results of coupled oscillators on multiplex networks show that the multiplexity can promote the synchronizability from some aspects [57,58]. The sum of the bounded confidences on two layers larger than 1 will promise the consensus [59]. The opinion dynamics with heterogeneous population on multilayer networks should attract much attention in the future. It will also be interesting to investigate opinion dynamics involved with evolutionary games [60–63]. Acknowledgments This work is supported by the National Natural Science Foundation of China under grant Nos. 11575036 and 11505016. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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Opinion clusters in a modified Hegselmann–Krause model with heterogeneous bounded confidences and stubbornness Wenchen Han a , Changwei Huang b , Junzhong Yang c ,
∗
a
College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu, 610068, People’s Republic of China School of Computer, Electronics and Information, Guangxi University, Nanning, 530004, People’s Republic of China c School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China b
highlights • Fewer opinion clusters forms if most open-minded agents are more stubborn. • The effect of increasing the bounded confidence of one subpopulation depends on those of others. • The number of opinion clusters and the size of the largest opinion cluster are almost negatively correlated.
article
info
Article history: Received 20 December 2018 Received in revised form 31 May 2019 Available online 12 June 2019 Keywords: Heterogeneous populations Opinion clusters The size of the largest cluster
a b s t r a c t In opinion dynamics with continuous opinion, bounded confidence is a critical parameter. Agents can interact with each other only when the opinion difference between them is less than the bounded confidence. Larger bounded confidence always leads to fewer opinion clusters. Stubbornness characterizing the insistence of an agent on her own opinion is thought to only affect the transition time. In this work, a modified Hegselmann–Krause model with heterogeneous population is investigated, where agents in different/same subpopulation have different/same bounded confidence and stubbornness. We find that, due to the interaction among subpopulations, increasing the stubbornness of agents in the subpopulation with the largest bounded confidence favors fewer opinion clusters and the expansion of the largest cluster. We also find that increasing the bounded confidence of a subpopulation leads to fewer clusters and a larger largest cluster provided that all the others have large bounded confidence. While one subpopulation is with a small bounded confidence, there exist an optimal bounded confidence of another subpopulation for the smallest number of opinion clusters and that for the largest size of the largest cluster. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Social life is full of social opinions of things or items, e.g., likes or dislikes of tastes of foods, agreements or disagreements on propositions. Almost all social interactions can be shaped by opinions [1,2]. When the evolution of opinions is involved, we are facing opinion dynamics. Opinion dynamics has been a popular topic, in social physics [3– 10], mathematics [11,12], to name but a few. Opinions can be described by discrete variables as well as continuous ones. Accordingly, opinion dynamics models are classified into discrete and continuous opinion dynamics models. The ∗ Corresponding author. E-mail addresses: [email protected] (W. Han), [email protected] (J. Yang). https://doi.org/10.1016/j.physa.2019.121791 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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Sznajd model [13], the voter model [14,15], the Galam majority-rule model [16], the naming game [17,18], and the nonconsensus opinion model [19] as well as its generalized version [20] are examples for discrete models. Deffuant– Weisbuch (DW) model [21] and Hegselmann–Krause (HK) model [22] are two well-known continuous models. In an opinion dynamics, several final configuration can be found, for example consensus where all agents share a same opinion, polarization where only two opinion clusters exist, and fragmentation where several opinion clusters coexist. Especially, the consensus draws much attention in opinion dynamics and several mechanisms to reach consensus are proposed, including social diversity [23], committed minorities [24,25], introducing zealots [26], communication burstiness [27], adaptive convergence rate [28], heterogeneous bounded confidences [29–31], and so on. The bounded confidence is the threshold of opinion difference between two agents, below which these two agents can communicate, and it is usually defined in a continuous opinion dynamics [21,29,32,33]. In a HK model [22,29], opinions are usually treated as real numbers in an interval [0, 1]. Two agents are neighbours only when the opinion difference between them is smaller than the bounded confidence. During the evolution, an agent updates her opinion at each time step by adopting the average opinion of all her neighbors including herself [22,30]. High bounded confidence always leads to consensus and low bounded confidence to a fragmentation. Lorenz studied continuous opinion models where agents are with different bounded confidences and found that agents could achieve consensus even when bounded confidences were below the threshold [34]. When agents are with different bounded confidences, it is found the number of opinion clusters is dominated by the agents with the smallest bounded confidence [35,36]. Besides the bounded confidence, the stubbornness measuring the extent that an agent insists on her own opinion is another feature of agents in the opinion dynamics [22,36–38]. Recently, the effects of heterogeneous stubbornness on opinion dynamics have been taken into consideration, such as heterogeneous stubbornness with public authority [39], heterogeneous stubbornness with interactive radius [40], and heterogeneous stubbornness with social similarity [41], heterogeneous stubbornness in PA/C model with partial antagonism agents and concord agents [42]. In previous works, heterogeneous bounded confidence and heterogeneous stubbornness are studied separately. Considering that stubbornness could control the time scale in the formation of opinion clusters, some questions arise in opinion dynamics with heterogeneous bounded confidence. For example, if we assign agents with high bounded confidence with strong stubbornness to slow down the formation of consensus and assign agents with low bounded confidence with weak stubbornness to speed up the formation of multiple opinion clusters, what happens under the competition between these two processes? To our knowledge, in the previous works, heterogeneous bounded confidence and heterogeneous stubbornness are investigated independently. It will be interesting to study opinion dynamics in the presence of both heterogeneous bounded confidence and heterogeneous stubbornness. Especially, we had like to study the collective effects on opinion dynamics when bounded confidence and stubbornness assigned to an agent are correlated. In this work, we study a modified HK model with a heterogeneous population composed of several subpopulations and focus our attention on the number of opinion clusters and the relative size of the largest opinion cluster in the population. In each subpopulation, agents share a same set of bounded confidence and stubbornness, which is different from that of other subpopulations. We find that opinion clusters are fewer and the largest opinion cluster is larger if agents in the subpopulation with the largest bounded confidence become more stubborn. We also find that, if all other subpopulations are with large bounded confidence, increasing the bounded confidence of one subpopulation favors fewer clusters and a larger largest cluster. If there exits one subpopulation with a small bounded confidence, there exist an optimal bounded confidence of another subpopulation for the smallest number of opinion clusters and that for the smallest size of the largest opinion cluster in the population. 2. Model We consider a population composed of N agents on a complete graph. The state of agent i at discrete time t is described by his opinion xi (t), which is in the range of [0, 1]. The opinion of agent i evolves following the updating rule ∑ { j∈N (t) xj (t) αi xi (t) + (1 − αi ) ∥Ni (t)∥ ∥Ni (t)∥ > 0, i xi (t + 1) = xi (t) ∥Ni (t)∥ = 0,
(1)
αi ∈ [0, 1] represents the stubbornness of agent i, describing the extent that agent i insists on her own opinion. αi = 1 refers to a zealot, sticking to her own opinion and neglecting others, while αi = 0 means a conformist, just following others’ opinions. Bearing this in mind, we say that agent i is more stubborn than agent j if αi > αj . Ni (t) is the set of agents, interacting with agent i, whose opinions satisfy |xj (t) − xi (t)| < σi and j ̸ = i at time t, and ∥Ni (t)∥ is the cardinality of the set Ni (t). The bounded confidence σi depicts the interacting range of agent i and σi ∈ (0, 0.5]. We could name agents with large σ as open-minded ones and agents with small σ as close-minded ones. In this work, ∑ the total population is l=M composed of M subpopulations, where each subpopulation with ρl N agents, where ρl ∈ [0, 1] and l=1 ρl = 1, takes the bounded confidence σl and the stubbornness αl . For convenience, we use Cl to denote the subpopulation with σl and αl . η describes the average of absolute opinion changes in each time t, written as η(t) =
N 1 ∑
N
i=1
|xi (t) − xi (t − 1)|.
(2)
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Fig. 1. (a) The number of clusters Nc and the relative size of the largest opinion clusters S against the bounded confidence σ with different stubbornness α in a homogeneous population. The time sequences of opinions (b) when α = 0, σ = 0.1, (c) when α = 0.8, σ = 0.1. (d) Time evolution of η with σ = 0.1.
Similarly, we have the ηl for Cl as
ηl (t) =
1
∥Cl ∥
∑
|xi (t) − xi (t − 1)|,
(3)
i∈Cl
When the system reaches its steady state, where opinions of all agents stop changing (η = 0), we monitor the number of opinion clusters, Nc , where agents within a same cluster share a same opinion. S is the relative size of the largest opinion cluster, the ratio between the number of agents in the largest cluster and the number of agents in the system. We consider the population size N = 1000 and the simulation results are averaged over 100 realizations with uniform random initial condition in opinions. The opinions of agents are updated synchronously. 3. Simulation analysis and results We first consider a population with homogeneous bounded confidence σ and stubbornness α , or interpreted as M = 1, quite similar to the original HK model [22,29,36]. As is known to all, a larger σ leads to fewer opinion clusters, shown in Fig. 1 (a1), and a larger relative size of the largest opinion cluster, shown in Fig. 1 (a2). When the bounded confidence is above the critical one, σc = 0.25, agents reach consensus, Nc = 1 and S = 1. Moreover, the number of opinion clusters Nc and the relative size S is independent of the stubbornness of agents. The time sequences of opinions with different α , presented in Figs. 1(b) and (c), show that increasing α only extends the transient time. This is caused by that agents with larger α will stick to their own opinions rather than be affected by their neighbors’ opinions and that agents interact with the same neighbors under a fixed σ . Fig. 1(d) shows the time evolution of η, the average opinion change in absolute value, with different α . It shows clearly that more stubborn agents (with larger α ) change their opinions slowly. No matter what α is, the gradual decay always follows an initial rise of η. The characteristics of η against time reflects the process of the formation of opinion clusters. The comparison between Fig. 1(b), (c) and (d) suggests that the peak of η occurs roughly at the time when the steady opinion clusters are formed. Before the formation of the steady clusters, agents have to adjust their opinions greatly to form clusters. After the formation of clusters, agents only adjust their opinion in a small pace till they share a same opinion in the same cluster. It is expected that how strong the initial rise is dependent upon how many clusters are formed. Less opinion clusters indicates a weak initial rise of η. While the stubbornness cannot affect the number of opinion clusters in a homogeneous population, we are curious about, in a heterogeneous population, whether the opinion consensus can be promoted when agents in one subpopulation become more stubborn, or vice versa. Considering that increasing the bounded confidence decreases the number of opinion clusters in homogeneous populations, we are curious about whether opinion clusters are always reduced if some agents become more open-minded in a heterogeneous population. 3.1. 2 Subpopulations on complete networks Then we consider a population composed of just 2 subpopulations, ρ1 N agents of C1 with σ1 and α1 , ρ2 N agents of C2 with σ2 and α2 , and ρ2 = 1 − ρ1 . We investigate the dependence of Nc on α1 and the dependence of S on α1 and the results are presented in Fig. 2. We assume σ1 > σ2 without loss of generality. In other words, agents in C1 are more open-minded than those in C2 . In Fig. 2(a) and (b) σ1 = 0.3, where C1 agents will reach consensus, and σ2 = 0.1, where
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Fig. 2. The number of clusters Nc and the relative size of the largest opinion clusters S against α1 . (a) C1 with σ1 = 0.3 and C2 with σ2 = 0.1, α2 = 0.8. (b) C1 with σ1 = 0.3 and C2 with σ2 = 0.1, α2 = 0. (c) C1 with σ1 = 0.1 and C2 with σ2 = 0.04, α2 = 0.8. (d) C1 with σ1 = 0.1 and C2 with σ2 = 0.04, α2 = 0. ρ1 = 0.1 in red and ρ1 = 0.9 in blue.
Fig. 3. The evolutions of opinions with a same initial condition. C1 agents in red and C2 agents in blue. (a) α1 = 0.0, (b) α1 = 0.5, (c) α1 = 0.8, (d) α1 = 0.95. The average opinion change in absolute value of C1 (η1 ) in (e) and that of C2 (η2 ) in (f) with different α1 . Other parameters are fixed. ρ1 = 0.9, σ1 = 0.3, ρ2 =0.1, σ2 = 0.1, and α2 = 0.8.
C2 agents will form 4 clusters. Exchanging the values of α1 and α2 , there always exists a very large opinion cluster (S ≃ 1) when ρ1 is large, e.g. ρ1 = 0.9, while S < 1 when ρ1 is small. But the fact that Nc decreases monotonously with α1 while S increases is true. It is also true for the case where σ1 = 0.1 and σ2 = 0.04, where C1 agents and C2 agents will form 4 and about 10 opinion clusters respectively, as shown in Fig. 2(c) and (d). These facts indicate that fewer opinion clusters and a larger S are favored when the open-minded agents (C1 agents) in a heterogeneous population become more stubborn and that the relation between Nc and α1 and the relation between S and α1 are robust, independent of ρ1 and α2 . These facts can be understood in the following way. In a final configuration in a homogeneous population, C1 agents distribute themselves in fewer opinion clusters than C2 agents. It should be satisfied that the opinion difference between adjacent opinion clusters being larger than σ1 for C1 agents and σ2 for C2 agents. Nevertheless, in a heterogeneous population, how the population evolves to the final configuration strongly depends upon the competition between C1 agents and C2 agents. When the evolution of C1 agents is great faster than C2 agents, for example α1 = 0 and α2 = 0.8 in Fig. 3(a), the route to the final configuration is mainly determined by C2 agents. During this process, the C1 opinion cluster formed in advance will adjust its position when C2 opinion clusters, in C1 agents’ interacting range, are about to form instead of
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Fig. 4. The number of clusters Nc and the relative size of the largest opinion cluster S against σ1 . (a) C1 with α1 = 0 and C2 with σ2 = 0.3, α2 = 0.8. (b) C1 with α1 = 0.8 and C2 with σ2 = 0.3, α2 = 0. (c) C1 with α1 = 0 and C2 with σ2 = 0.04, α2 = 0.8. (d) C1 with α1 = 0.8 and C2 with σ2 = 0.04, α2 = 0. ρ1 = 0.1 in red and ρ1 = 0.9 in blue.
pinning down the locations of C2 opinion clusters. Meanwhile, C1 agents will make some C2 agents, who hold almost the same opinion with these C1 ones, to change their opinions when C1 agents do. On the other hand, when the evolution of C1 agents is great slower than C2 agents, for example α1 close to 1 (α1 = 0.95 shown in Fig. 3(d)), the evolution to the final configuration is determined by C1 agents. Though C2 agents first have formed different opinion clusters of their own, they are slaved by C1 agents due to the interaction which further drives C2 agents to hold opinions closer to others till opinion clusters of C1 agents are formed. Consequently, the number of opinion clusters approaches the one acquired by homogeneous C1 agents. When the evolutions in a homogeneous population of C1 agents and C2 agents become comparable in time scale, C1 agents may be pinned down by the C2 clusters (α = 0.5 in Fig. 3(b)) or slave C2 agents (α = 0.8 in Fig. 3(c)). Fig. 3(a)–(d) shows that there is only one opinion cluster for C1 agents, which is reflected in Fig. 3(f) where η1 against time display an obvious peak. On the other hand, C2 agents forms four opinion clusters at α = 0 and 0.5 and, consequently, the peak in η2 against time seems absent. Moreover, when α1 = 0.8, η1 can remain almost constant for a period when the C1 cluster is moving and η2 remains constant for some C2 agents moving, who holds the opinion very close to C1 ones. Comparing η1 and η2 with different α1 , Fig. 3(e) shows that C1 agents definitely change their opinions slower in a heterogeneous population when they become more stubborn and Fig. 3(f) indicates that the interaction with more stubborn C1 agents prolongs the evolution of C2 agents opinions. The overall effects of the mutual interaction between C1 agents and C2 agents are that more open-minded agents being more stubborn favors a smaller Nc and a larger S. To be mentioned, this correlation is independent of ρ1 , σ1 , σ2 , and α2 . Similarly, we can also infer that increasing the stubbornness of the close-minded agents (C2 agents) will increase the number of opinion clusters and decrease the relative size of the largest opinion cluster. In addition, Figs. 2 and 3 show that both close-minded agents and open-minded agents have significant impacts on Nc and S, which is different from the statement that in a population with homogeneous stubbornness open-minded agents cannot contribute to forging opinion consensus [36]. Now, we investigate how the bounded confidence of agents in one subpopulation influences the number of opinion clusters and the relative size of the largest opinion cluster. Without loss of generality, only the bounded confidence of C1 agents (σ1 ) varies and other parameters are fixed. Here, we first study the correlation of Nc and σ1 and that of S against σ1 in the system with open-minded C2 agents, for example σ2 = 0.3 shown in Fig. 4(a) and (b). In this case, C2 agents always reach consensus if C2 agents do not interact with C1 ones. Increasing σ1 of C1 agents always leads to fewer opinion clusters and a larger size of the largest opinion cluster if C1 agents are alone. Thus increasing σ1 leads to a smaller number of opinion clusters and a larger size of the largest opinion cluster. This dependence is independent of ρ1 , α1 , and α2 are. Moreover, agents reach consensus even when σ1 is blow the critical bounded confidence σc . In the case where C2 agents are close-minded the correlation between the number of opinion clusters Nc and the stubbornness of C1 agents σ1 and the correlation between the relative size of the largest cluster S and σ1 are not monotonous. Bearing a larger σ leading to a smaller Nc and a larger S in mind, it is quite reasonable that larger σ1 leads to a smaller Nc and a larger S when σ1 is small, while it is quite counterintuitive that increasing σ1 even makes Nc larger and S smaller when σ1 is large, with an example σ2 = 0.04 shown in Fig. 4(c) and (d).
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Fig. 5. The time sequences of opinions with a same initial condition. C1 agents in red and C2 agents in blue. (a) σ1 = 0.1, (b) σ1 = 0.24, (c) σ1 = 0.32, (d) σ1 = 0.4. The average opinion change in absolute value of C1 (η1 ) in (e) and that of C2 (η2 ) in (f) with different σ1 . Other parameters are fixed. ρ = 0.9, α1 = 0.8, σ2 = 0.10, and α2 = 0.
When σ1 is small, Nc becomes smaller and S becomes larger because of the interaction between C2 agents and C1 agents with larger σ1 , favoring the merging of opinion clusters as presented in Fig. 5(a) and (b). However, the interaction also leads to a larger Nc and a smaller S because of more C2 opinion clusters being left by C1 opinion clusters when σ1 further increases as shown in Fig. 5(c) and (d). It is known that increasing α1 will make the evolution of C1 agents’ opinions slow down directly and the evolution of C2 agents’ opinions slower due to the interaction, shown in Fig. 3(e) and (f). However, it is rarely noticed that increasing the bounded confidence will speed up the evolution of agents’ opinion, manifested by Fig. 5(e). The larger σ1 enlarges the interacting range of C1 agents, which causes that C1 agents can change their opinions faster on average, especially those agents holding extreme opinions, e.g., x ∼ 1 or x ∼ 0. When the C2 agents cannot keep pace with the opinion changing of C1 agents while increasing σ1 , C2 opinion clusters will be left by the C1 opinion clusters, shown in Fig. 5(c) and (d), which causes the increase of Nc and the decrease of S. Meanwhile, the locations of left C2 opinion clusters pin down C1 opinion clusters due to the large σ1 . That the interaction leads to difference outcomes is the result of competition between the effects of the bounded confidence (σ ) and the stubbornness (α ) of agents. 3.2. Several subpopulations on complete networks The results above observed in the 2 subpopulation system are qualitatively the same with that in the system with M subpopulations. In an M subpopulations system, where we simply assume σ1 > σj and j = 2, 3, . . . , M without loss of generality, increasing α1 always make the number of opinion clusters Nc decrease and the relative size of the largest opinion cluster S increase. Fig. 6(a) shows the dependence of Nc on α1 and the dependence S on α1 in 3 subpopulations system where C 1 with σ1 = 0.3, C 2 with σ2 = 0.1, α2 = 0.8, and C3 with ρ3 = 0.1, σ3 = 0.04, α3 = 0 are fixed. Fig. 6(c) shows the same results in 5 subpopulations system where C 1 with σ1 = 0.3, C 2 with σ2 = 0.04, α2 = 0.0, C3 with ρ3 = 0.1, σ3 = 0.1, α3 = 0.8, C4 with ρ4 = 0.1, σ4 = 0.08, α3 = 0.5, and C5 with ρ3 = 0.1, σ3 = 0.3, α3 = 0.3 are fixed. The dependence of Nc on σ1 and the dependence of S on σ1 are not monotonous, when some agents are close-minded, as 3 subpopulations shown in Fig. 6(b) and 5 subpopulations shown in Fig. 6(d). These results show that increasing the stubbornness of the most open-minded agents decreases the number of opinion clusters Nc and increases the relative size of the largest opinion cluster S, which is independent of ρi , αi , and σi with i = 1, 2, . . . , M except for σ1 > σj and j = 2, 3, . . . , M. Moreover, the result of increasing σ1 is due to the overall effect of competition between σ1 and α1 when other parameters are fixed. 3.3. 2 Subpopulations on small-world networks We also investigate a system composed of 2 subpopulation on small-world networks, which is rewired from a ring of degree k with the rewiring probability p = 0.03 [43]. The result is qualitatively the same with those observed in systems on complete networks. That is, increasing α1 will decrease the number of opinion clusters Nc and the relative size of the largest opinion cluster S. And increasing σ1 will make Nc first decrease then increase and S first increase then decrease when C2 are close-minded. Some specific cases with ⟨k⟩ = 20 are shown in Fig. 7.
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Fig. 6. The number of clusters Nc and the relative size of the largest opinion cluster S against α1 in 3 subpopulations systems with σ1 = 0.3 in (a) and against σ1 with α = 0.8 in (b). ρ1 = 0.1, ρ2 = 0.8 for red and ρ1 = 0.8, ρ2 = 0.1 for blue. C2 with σ2 = 0.1, α2 = 0.8 and C3 with ρ3 = 0.1, σ3 = 0.04, α3 = 0 are fixed. In 5 subpopulation systems with σ1 = 0.3 in (c) and against σ1 with α = 0.8 in (d). ρ1 = 0.1, ρ2 = 0.5 for green and ρ1 = 0.5, ρ2 = 0.1 for magenta. C2 with σ2 = 0.04, α2 = 0.0, C3 with ρ3 = 0.1, σ3 = 0.1, α3 = 0.8, C4 with ρ4 = 0.1, σ4 = 0.08, α4 = 0.5, and C5 with ρ5 = 0.1, σ5 = 0.3, α = 0.3 are fixed.
Fig. 7. The number of clusters Nc and the relative size of the largest opinion cluster S on small-world networks with ⟨k⟩ = 20. (a) against α1 with σ1 = 0.3 and σ2 = 0.1, α2 = 0.8. (b) against σ1 with α1 = 0.8 and σ2 = 0.04, α2 = 0.0. ρ1 = 0.1, ρ2 = 0.9 for red and ρ1 = 0.9, ρ2 = 0.1 for blue.
4. Conclusion In this work, we have studied a modified Hegselmann–Krause model with a heterogeneous population composed of M subpopulations, where agents in each subpopulation share a same set of bounded confidence σl and the stubbornness αl . In a homogeneous population, the bounded confidence determines the number of opinion clusters in the steady state and the stubbornness only affects the transient time. There are two interesting findings in our work. The first one is that the stubbornness of agents indeed affects the evolution of opinion systems with a heterogeneous population. Increasing the stubbornness of the most open-minded agents can decrease the number of opinion clusters, which is contrast to the conclusion that open-minded agents cannot contribute to forging opinion consensus when all agents share a same stubbornness in Ref. [36], increase the relative size of the largest opinion cluster in the population as well. This dependence is independent of the population ratio, the stubbornness, and the bounded confidence of other subpopulations. The second point is that increasing the bounded confidence of one subpopulation will not always be the option for forging consensus. When all other agents are open-minded, increasing the bounded confidence of one subpopulation always leads to fewer opinion clusters and a larger size of the largest opinion clusters in the whole population. While, when agents in one subpopulation are very close-minded, there exists an optimal bounded confidence for the smallest number of opinion clusters and that for the biggest relative size of the largest opinion cluster when increasing the bounded confidence of
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another subpopulation. In this case, agents reach consensus, even when the bounded confidences of all agents are below the critical one σc for a homogeneous population, as shown in Ref. [34]. The most close-minded agents mainly affects the number of opinion clusters and the size of the largest opinion cluster, which is consistent with the previous works [35,36]. Our work has studied the opinion dynamics with a heterogeneous population, where agents in each subpopulation with a set of bounded confidence and stubbornness. However, the initial opinion are uniformly distributed in the interval [0, 1]. It may be interesting to investigate the general initial distribution in an opinion system [44] by applying share a drink (SAD) procedure proposed by Häggström [45]. The extreme agents show substantial advantage to other agents [46,47]. It is interesting to find the effects in heterogeneous population with extreme minorities. Moreover, averager–copier–voter models with stubborn agents can show hybrid consensus [48]. In that work, agents obeying different opinion dynamics rather than the same dynamics with different parameters and the stubborn agents are considered with stubbornness α = 1. It will be interesting to investigate a population with different agents following different opinion dynamics. The opinion dynamics with mass media can show order–disorder transition in continuous opinion dynamics [49] and hysteretic phenomena in binary opinion dynamics [50]. Furthermore, the network structures show important effects on the number of opinion clusters in a population with homogeneous bounded confidence [51–55]. It is worth studying heterogeneous population on different networks. Besides this, the multilayer structure will hinders the consensus when the initial opinion configuration is within a bounded range [56]. However, the results of coupled oscillators on multiplex networks show that the multiplexity can promote the synchronizability from some aspects [57,58]. The sum of the bounded confidences on two layers larger than 1 will promise the consensus [59]. The opinion dynamics with heterogeneous population on multilayer networks should attract much attention in the future. It will also be interesting to investigate opinion dynamics involved with evolutionary games [60–63]. Acknowledgments This work is supported by the National Natural Science Foundation of China under grant Nos. 11575036 and 11505016. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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