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Opinion evolution influenced by informed agents
Physica A xx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
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Opinion evolution influenced by informed agents
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Kangqi Fan a,b,∗ , Witold Pedrycz b,c,d a
School of Mechano-Electronic Engineering, Xidian University, Xi’an 710071, China
b
Department of Electrical and Computer Engineering, University of Alberta, Edmonton T6G 2V4, Canada
c
Systems Research Institute, Polish Academy of Sciences, Warsaw 01447, Poland
d
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
article
info
Article history: Received 31 January 2016 Received in revised form 6 June 2016 Available online xxxx Keywords: Opinion dynamics Informed agents Inner opinions Discrete choices
abstract Guiding public opinions toward a pre-set target by informed agents can be a strategy adopted in some practical applications. The informed agents are common agents who are employed or chosen to spread the pre-set opinion. In this work, we propose a social judgment based opinion (SJBO) dynamics model to explore the opinion evolution under the influence of informed agents. The SJBO model distinguishes between inner opinions and observable choices, and incorporates both the compromise between similar opinions and the repulsion between dissimilar opinions. Three choices (support, opposition, and remaining undecided) are considered in the SJBO model. Using the SJBO model, both the inner opinions and the observable choices can be tracked during the opinion evolution process. The simulation results indicate that if the exchanges of inner opinions among agents are not available, the effect of informed agents is mainly dependent on the characteristics of regular agents, including the assimilation threshold, decay threshold, and initial opinions. Increasing the assimilation threshold and decay threshold can improve the guiding effectiveness of informed agents. Moreover, if the initial opinions of regular agents are close to null, the full and unanimous consensus at the pre-set opinion can be realized, indicating that, to maximize the influence of informed agents, the guidance should be started when regular agents have little knowledge about a subject under consideration. If the regular agents have had clear opinions, the full and unanimous consensus at the preset opinion cannot be achieved. However, the introduction of informed agents can make the majority of agents choose the pre-set opinion. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Opinion evolution and diffusion have been greatly facilitated by the rapid development of modern information technologies. The research on the dynamic evolution of public opinions can improve the understanding of some crucial phenomena arising in the fields of sociology (e.g., the emergence of extremism) [1–4], politics (e.g., political elections) [5–7], and economics (e.g., advertising strategies) [8–10]. As an important research field of sociophysics [11], opinion dynamics provides an effective tool to model and study the evolution and diffusion of opinions among social agents. In terms of the possible values of opinions, the opinion dynamics models can be categorized into three main groups: (1) discrete opinion models; (2) continuous opinion models; (3) continuous opinions and discrete actions (CODA) model [12]. These opinion dynamics models, although are not able to precisely predict the real-world observations in some cases, can help us capture the critical factors that govern the opinion evolution and diffusion.
∗
Corresponding author at: School of Mechano-Electronic Engineering, Xidian University, Xi’an 710071, China. E-mail address: [email protected] (K. Fan).
http://dx.doi.org/10.1016/j.physa.2016.06.110 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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In the continuous opinion models, the opinion of each agent ranges within a continuous opinion space (e.g., [0, 1]). Two significant representatives of continuous opinion models are Deffuant model [13,14] and Hegselmann–Krause (HK) model [15,16], which were all developed using the concept of bounded confidence. This indicates that agents will interact with each other only if the difference in their opinions is smaller than a given threshold. In the Deffuant model [13,14], agents encounter in random pairwise interaction to update their opinions, whereas in the HK model [15,16] each agent moves to the average opinion of all agents whose opinions lie in its coverage of confidence. In the discrete opinion models, each agent carries one of a finite number of different opinions, frequently two opposite binary points of view (e.g., yes or no, support or opposition, accept or reject). Among several different discrete opinion models, we highlight voter model [17,18], Sznajd model [19–21], and majority-rule model [22–24]. In the voter model, each Q3 agent is randomly chosen to adopt the opinion of one of his neighbors, whereas in the Sznajd model two agents with the same opinion persuade their neighbors to accept their opinion. The agents in the majority-rule model update their opinions by following the local majority rule. The CODA model [25] differentiates the inner continuous opinion from the observable binary choices. In this model, the inner opinion is expressed with a certain probability stating that an individual believes one of the two alternatives is the best. The agent in the CODA model observes the choices of his neighbors, and then updates his inner opinion and relevant choice according to a set of Bayesian rules [25–28]. In fact, the choices are a type of opinions as suggested in the discrete opinion models. Since the CODA model involves both the inner continuous opinion and the observable discrete opinion, the latter is named as the choice. This differentiation and nomenclature will also be adopted in this study. In addition to the generic opinion dynamics models, some researchers have attempted to develop the specific models to explore the practical opinion evolution processes on several online social media [29–31]. For example, Xiong and Liu [30] collected people’s comments on three electronic products (i.e., ‘‘iPhone 4’’, ‘‘Blackberry’’, and ‘‘iPad 2’’) from Twitter and proposed an opinion model to explore the evolution processes of people’s opinions on the three electronic products. The aforementioned opinion models have been employed to study the influences of contrarian agents [32], inflexible agents [28,33,34], opinion leaders [35], and extremists [1] on the dynamic evolution of public opinions, and the obtained results have been used to explain some sociological phenomena, such as hung elections [11] and extremism propagation [36,37]. More recently, the continuous opinion models have been extended to capture the opinion evolution trends under the influence of a small portion of informed agents [38,39]. The informed agents are common agents who are employed or chosen to guide the society toward a pre-set opinion. Understanding the opinion evolution influenced by informed agents can help people figure out ways to guide the public opinion toward a desirable aim (such as promote the acceptance of eco-friendly lifestyle) and hinder the spread of undesirable opinion (such as extremist opinions). The simulations showed that only a small number of informed agents were required to successfully change the public opinions. The continuous opinion models are suitable to describe the situations where the exchanges of inner opinions among agents are available. However, in some cases, such as political elections [5] and innovation diffusion [40], people only have a limited number of discrete choices. Moreover, each person is not well acquainted with most of the other people, and then the exchanges of inner opinions with others are hindered. As a result, each person observes others’ choices and then makes his own choice. In these cases, how the informed agents influence the public opinions is still unclear. The discrete opinion models and the CODA model are justified to represent the situations where the inner opinion of each agent is not observable and binary choices are a good description of the problem. In the two categories of models, each agent must declare his choice. However, besides the two opposite choices, people may also keep silent or remain undecided when they are not very sure about their choices [37,41]. In addition, the well-known social judgment theory [42] and the real-world opinion interactions [43] show that the repulsion between distinctly dissimilar opinions may have a significant effect on the opinion evolution. For example, recent studies on Polish Internet fora showed that some online discussions, especially when they were associated with political issues, usually developed into fierce quarrels, provocation, and invectives [43]. However, this opinion interaction mechanism is not included in the aforementioned models. Therefore, a model that incorporates both an agent’s silence and the repulsion between dissimilar opinions is obviously more useful for understanding the opinion evolution in the case that the exchanges among inner opinions are impeded. In this paper, a social judgment based opinion (SJBO) model is proposed to study the opinion evolution under the influence of informed agents. Unlike the discrete opinion model and CODA model where each agent must express his choice, the agent in the SJBO model is allowed to remain undecided or keep silent if his support to either of the two opposite choices is not sufficiently strong. Moreover, both the compromise between similar opinions and the repulsion between dissimilar opinions are included in the SJBO model. The proposed model is simulated in a fully connected network, and the effect of informed agents on the opinion evolution is studied. The results suggest that the informed agents can guide the public opinions toward the pre-set target. But the effectiveness is mainly dependent on the characteristics of regular agents.
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2. The SJBO model
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2.1. Inner opinions and discrete choices
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In general, people exchange their opinions with others with intent to seek certain mutual understanding and compromise. However, the opinion evolution cannot always lead to a full and unanimous consensus. In practice, the
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Fig. 1. Schematic diagram of inner opinion and discrete choice of an agent.
interaction between dissimilar opinions sometimes leads to a larger divergence and even the extremism [43,44]. To incorporate different opinion dynamics into the model, we refer to the Social Judgment Theory [42]. The basic idea of this theory is that each individual has two attitude thresholds: assimilation threshold and repulsion threshold. If the opinion difference between a persuader and a receiver is smaller than the assimilation threshold, the receiver will shift his opinion toward that of the persuader. On the contrary, if the opinion difference is larger than the repulsion threshold, the receiver will shift his opinion away from that of the persuader. If the opinion difference falls between the two thresholds, the receiver will hold his own opinion. We consider a social community composed of N agents, in which each agent can notice the discrete choices of other agents, but cannot acquire their inner continuous opinions. Depending on his inner opinion, an agent may vote for one of the two opposite choices represented by +1 and −1, or keep silent represented by 0 if it remains undecided, as shown in Fig. 1. Here we introduce the hesitation threshold h to reflect the lack of confidence of an agent regarding which alternative to choose. At time t, if the inner opinion xti of agent i falls within the hesitation range (−h ≤ xti ≤ +h), agent i has not been completely convinced by any of the two sides, and then his choice is neutrality or silence. Otherwise, agent i chooses −1 if xti < −h or +1 if xti > +h. The choice x¯ ti of agent i can be described in the form
−1, x¯ ti = 0, 1,
x t < −h i t x ≤ h
(1)
i
xti
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15
> +h . 16
Since the inner opinion xti of an agent i cannot been observed, his choice x¯ ti could be perceived as his opinion by others. In fact, the choice is an indication of an agent’s opinion. Some highly rational agents may doubt if an agent is completely supportive of the choice that he chooses. But the choice expressed by an agent at least indicates that it supports this choice to a large extent. To simplify the problem, an agent is believed to completely support the choice if he chooses it. 2.2. Opinion update rules for regular agents
xit +1
2
0 ≤ xti − x¯ tj ≤ εR εR < xti − x¯ tj ≤ τR
(2)
where ¯ (−1 or +1) is the opinion of agent j perceived by agent i; a ∈ [0, 0.5] is the assimilation coefficient of agent i being influenced by agent j; r ∈ [0, 1] is the repulsion coefficient of agent i being influenced by agent j; εR and τR are the assimilation threshold and repulsion threshold of agent i, respectively. The assimilation threshold εR in the proposed model represents the uncertain level of agent i, below which agent i tends to move his opinion toward that of the persuader. The repulsion threshold τR reflects the tolerance, beyond which agent i shifts his opinion away from that of the persuader due to the repulsion between their dissimilar opinions. If the persuader’s opinion falls between the two thresholds, agent i allows the presence of a different opinion but keeps his own opinion. However, if agent j remains undecided or keeps silent (x¯ tj = 0) at time t, agent i adjusts his inner opinion xti according to the following rules xit +1 =
xti ,
t x − x¯ t ≤ ρR i j t x − x¯ t > ρR i j
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τR < xti − x¯ tj ≤ 2
xtj
t λ xi ,
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At time t, an agent i is randomly chosen to interact with his neighboring agent j. If agent i is a regular agent and agent j has made his choice between −1 and +1, agent i updates his inner opinion xti according to the Social Judgment Theory, i.e., the following rules
t xi + a x¯ tj − xti , t x, = i t t x¯ − xti x − r j 1 − xti , i
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(3)
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K. Fan, W. Pedrycz / Physica A xx (xxxx) xxx–xxx Table 1 Parameters in the SJBO model.
Opinion assimilation Opinion repulsion Opinion decay Hesitation range
Thresholds
Coefficients
ε τ ρ
a r
λ
h
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where λ is the decay coefficient indicating a drop in the preference for −1 or +1 after observing the silence of agent j; ρR is the decay threshold below which an agent tends to lose his preference for one of the two opposite choices. At present, statistic data or theories like Social Judgment Theory to describe this phenomenon have not been found, and more research is required to quantify the parameters in Eq. (3). However, in fact, Eq. (3) shares the same fundamental as Deffuant model [13], and it can be obtained by letting one of the two opinions in Deffuant model be zero. After updating the inner opinion, agent i declares his choice according to Eq. (1). Since an agent is in the state of hesitation when his inner opinion varies between −h and +h, the influence from both sides can function. Therefore, it is reasonable to assume that the assimilation threshold εR could be larger than one. It should be noted that, if εR = τR = 2 and h = ρR = 0, each agent has only two opposite choices and the persuasion from both sides is acceptable. In this case, the SJBO model described by Eqs. (2) and (3) is similar to the CODA model in the fundamental idea.
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2.3. Opinion update rules for informed agents
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Informed agents are the employed or voluntary individuals who intentionally and invisibly try to pull the public opinions toward a pre-set target. The informed agents differ from the inflexible agents in that the former adjust their opinions with intent to influence other agents’ opinions while the latter keep their opinions always unchanged. For the same reason, the informed agents are also different from the contrarians who always have the opposite opinion to that of the majority of the surrounding agents. Opinion leaders are agents who have relatively high connectivity into the social network and/or larger influence on regular agents’ opinions whereas the informed agents do not have such superiorities. Therefore, it is reasonable to assume that, when an informed agent i encounters agent j, agent i will approach agent j in the opinion with an aim to influence agent j if the difference between their opinions are not too large. Otherwise, the informed agent i will hold his own opinion. The informed agent i is also constantly attracted by the pre-set target so as not to deviate it far away. In addition, the informed agent i should have no preference for the opposite target or choice. In summary, if an informed agent i interacts with agent j who has declared his choice, agent i updates his inner opinion xti via following Eq. (2) with minor adjustments, Q4 as expressed in the following form max (1 − µ) xti + a x¯ tj − xti
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xit +1
=
+ µx T , 0 ,
xti ,
t x − x¯ t ≤ εI j it x − x¯ t > εI i j
(4)
where xT is the pre-set target; µ ∈ [0, 1] is the target coefficient; εI is the assimilation threshold of the informed agent i. Likewise, if agent j keeps silent (x¯ tj = 0), agent i updates his inner opinion xti by following Eq. (3) with minor adjustments, as expressed in the following form xit +1
=
(1 − µ) λxti + µxT , xti ,
t x − x¯ t ≤ ρI j ti x − x¯ t > ρI i j
(5)
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where λ and ρI are the decay coefficient and decay threshold of the informed agent i, respectively. After updating the inner opinion, agent i makes his choice according to Eq. (1). We assume that none of the informed agents is aware of the presence of other informed agents. Therefore, even if agent j is an informed agent, agent i still adjusts his inner opinion according to Eqs. (4) or (5). In practice, the parameters used to characterize an individual vary for different individuals since no two persons are identical in terms of personality. But reasonable assumptions are usually adopted in the theoretical studies to simplify the problem. In this study, the regular agents are supposed to be homogeneous, i.e., they have identical parameters, and all informed agents are also endowed with the same parameters. In addition, the regular agents and the informed agents have the same hesitation threshold h, decay threshold ρ , assimilation coefficient a, repulsion coefficient r, and decay coefficient λ since the informed agents are common agents except that they have a specific aim. The parameters adopted in the SJBO model are listed in Table 1. In this study, the subscript R in the parameters indicates the regular agents, whereas the subscript I represents the informed agents.
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3. Results and discussion
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In this section, we report on a series of simulations in a fully connected social network to reveal the opinion evolution trends under the influence of informed agents based on the proposed model. The fully connected social network is adopted
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Fig. 2. Temporal evolutions of opinions and choices for 400 agents: (a) inner opinion; (b) percentage of different choices. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 0.8, τR = 1.6, and x0R ∈ [−1, +1], respectively. The assimilation threshold and initial opinions of informed agents are εI = 1.2 and x0I ∈ [0, 1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. The percentage of informed agents is 10%.
because it allows us to focus on the effects of agent’s characteristics or model parameters on the opinion evolution. At the beginning of each simulation, a population of N agents is generated and some of the agents are randomly selected to be the informed agents. Then the regular agents and the informed agents are initialized separately. The pre-set target is defined as the choice of +1, i.e., xT = +1, and the agent whose choice is +1 is also named as the supporter. On the contrary, the agent whose choice is −1 is called the contrarian. At each time step, two agents i and j are randomly selected, with agent i observing the choice of agent j and updating his inner opinion and choice according to the SJBO model. A large number of simulations indicate that the effects of assimilation coefficient a, repulsion coefficient r, decay coefficient λ, and target coefficient µ on the opinion evolution trends are not significant. To simplify the problem, the simulations are conducted with constant coefficients set up as follows: a = r = λ = 0.4, and µ = 0.6. The initial opinions x0I of informed agents are drawn from a uniform distribution between [0, +1] to ensure that none of the informed agents has a preference for the opposite target (−1), whereas the regular agents pick their initial opinions x0R from a uniform distribution between [−1, + 1], unless otherwise stated. Since each agent can only observe the choices of other agents, the informed agents always choose 0 or +1 to avoid the adverse guidance, which is guaranteed by the opinion updating rules described by Eqs. (4) and (5). Figs. 2–4 show the typical temporal evolutions of opinions and choices for a community with N = 400 agents, in which the time step represents the average update per agent. Initially, some agents whose inner opinions fall beyond the hesitation range declare their choices, whereas the majorities keep silent because their support to either choice is not strong enough. After the opinion evolution is started, the silencers or centrists are influenced to gradually express their choices by the agents who have voiced their choices. If the assimilation threshold of regular agents is smaller than one, i.e., εR < 1, the regular agents whose inner opinions are close to zero are immune to the influence from the two opposite choices, and then the opinion evolution leads to three opinion clusters, as shown in Fig. 2. On the contrary, if εR ≥ 1, all the silencers can be influenced to express their choices, causing the opinion bipolarization, as shown in Fig. 3. Under the guidance of informed agents, the population with the pre-set target is larger than that with the opposite choice after the opinion evolution reaches convergence, as shown in Figs. 2 and 3. However, the complete consensus fails to be achieved due to the presence of the agents or contrarians who initially choose the opposite choice because their initial preferences for the opposite choice are strong. These contrarians can tempt some of the silencers to choose the opposite choice, blocking the formation of complete consensus. If the initial opinions of regular agents distribute between [−ρ, +ρ], i.e., [−0.3, +0.3] in this study, all regular agents tend to discard their preferences for the two opposite choices. But the informed agents can guide all regular agents to choose the pre-set target if εR ≥ 1, as shown in Fig. 4. To gain a more fundamental understanding of the opinion evolution influenced by informed agents, a large number of simulations are carried out in a completely connected community with N = 2000 agents. The dynamics evolution runs until it reaches convergence or 300 updates per agent have been completed, and all simulations are averaged over 30 runs. The percentage of agents with the pre-set choice versus the percentage of informed agents is shown in Fig. 5. It can be observed that the supporters increase with the percentage of the informed agents. This observation makes sense because an increasing number of informed agents can improve the guiding effect and then turn more regular agents into supporters. However, even if the informed agents take the overwhelming majority of the population, the contrarians still survive owing to the effect of repulsion between dissimilar opinions. The full and unanimous consensus can be reached only if all agents are informed agents. In general, only a small fraction of agents work as informed agents because recruiting a large number of informed agents is either impractical or prohibitively costly. Therefore, in the following simulations, 10% of the total agents are selected to be the informed agents.
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Fig. 3. Temporal evolutions of opinions and choices for 400 agents: (a) inner opinion; (b) percentage of different choices. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The assimilation threshold and initial opinions of informed agents are εI = 1.2 and x0I ∈ [0, 1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. The percentage of informed agents is 10%.
a
b
Fig. 4. Temporal evolutions of opinions and choices for 400 agents: (a) inner opinion; (b) percentage of different choices. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−0.3, +0.3], respectively. The assimilation threshold and initial opinions of informed agents are εI = 1.2 and x0I ∈ [0, 1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. The percentage of informed agents is 10%.
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The percentage of agents with the pre-set choice versus the assimilation threshold εR of regular agents is shown in Fig. 6. It can be noticed that the characteristics of informed agents, including the assimilation threshold εI and initial opinion distribution x0I , have insignificant effects on the ultimate state of the system. Adjusting the characteristics of informed agents cannot remarkably expand the size of supporters. However, varying the assimilation threshold εR of regular agents from 0.8 to 1.4 increases the percentage of supporters by almost 20%. Moreover, the size of supporters is found to be very sensitive to the variation of εR from 0.8 to 1. This is caused by a phenomenon called freezing effect [45], meaning that each agent is immune to the influence from the choice that it initially tends not to choose. Consequently, three opinion clusters come into being, as shown in Fig. 2. As εR increases from 0.8 to 1, more regular agents can be influenced by the informed agents to choose the pre-set target. If εR ≥ 1, the silencers are influenced by both the supporters and the contrarians, and eventually all silencers are tempted to express their choices, as shown in Fig. 3. But the supporters have a greater influence due to the presence of informed agents, and then the majority of the silencers join the supporters. Fig. 7 shows the percentage of supporters versus the repulsion threshold τR of regular agents. It can be seen that neither the characteristics of informed agents nor the repulsion threshold τR of regular agents can significantly impact the ultimate state of the system. Theoretically, a smaller repulsion threshold τR can push more silencers to supporters or contrarians depending on their initial opinions. Therefore, the amount of silencers can be reduced by decreasing the repulsion threshold. Similar result can also be obtained by increasing the assimilation threshold εR . If εR ≥ 1, the assimilation effect can
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Fig. 5. Percentage of pre-set choice versus that of informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
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b
Fig. 6. Percentage of pre-set choice versus the assimilation threshold εR of regular agents guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The repulsion threshold and initial opinions of regular agents are τR = 1.6 and x0R ∈ [−1, +1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
completely remove the silencers, as shown in Fig. 3. Therefore, when εR = 1.2, no significant difference in the percentage of supporters can be observed for various repulsion threshold τR , as shown in Fig. 7. Fig. 8 shows the percentage of supporters versus the decay threshold ρ . With an increase in the decay threshold, the silencers have a larger influence scope and then can guide more agents to keep silent. However, due to a large assimilation threshold (εR = 1.2), the silencers are influenced by the two competitive groups as well. Under the guidance of informed agents, more silencers choose the pre-set target. Therefore, increasing the decay threshold can enlarge the percentage of supporters, as shown in Fig. 8. Fig. 9 shows the percentage of supporters versus the hesitation threshold h. It can be seen that adjusting the hesitation threshold h from 0.1 to 0.7 could slightly increase the percentage of supporters for various characteristics of informed agents. This minute increase occurs due to the similar cause as discussed in Fig. 8. That is, increasing the hesitation threshold expands the scale of silencers, but the majority of the silencers can be influenced by the supporters to choose the pre-set target when εR = 1.2. However, a visible drop in the percentage of supporters is revealed when εI = 2 and h = 0.7, as shown in Fig. 9(a). To figure out this abnormal drop, we explore the temporal evolutions of opinions and choices for εI = 2 and h = 0.7, as shown in Fig. 10. It can be observed that although the agents have been bipolarized, the informed agents still attempt to approach and guide the contrarians. In the meanwhile, the informed agents are attracted by the pre-set target and the other supporters. Consequently, the opinions of informed agents swing between [0.6, 1], and their choices alternate between 0 and 1, resulting in the abnormal drop in the supporters. If h < 0.6, the opinions of informed agents fall beyond the hesitation
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a
b
Fig. 7. Percentage of pre-set choice versus the repulsion threshold τR of regular agents guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold and initial opinions of regular agents are εR = 1.2 and x0R ∈ [−1, +1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
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Fig. 8. Percentage of pre-set choice versus the decay threshold ρ guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The hesitation threshold is h = 0.5. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
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range, and their choices are always the pre-set choice. In addition, if εI < 2, the informed agents will give up the effort of guiding the contrarians and stick to the pre-set choice. Therefore, if h < 0.6 or εI < 2, no visible drop in the percentage of supporters can be observed. Fig. 11 shows the percentage of supporters versus the initial opinion distribution x0R of regular agents, in which [|x|] indicates the opinion distribution [−x, +x]. It is obvious that the smaller of x0R , the larger of the percentage of supporters. For a small x0R , the regular agents have no strong preferences for either of the two opposite choices, and then the influence of informed agents is highlighted. Particularly, if x0R ≤ 0.2, the regular agents initially have little preferences for the two opposite choices. In this case, the informed agents can guide all regular agents to choose the pre-set choice, leading to the herding behavior [46]. Therefore, to maximize the influence of informed agents, the guidance should be started when the regular agents have no clear opinions, i.e., x0R is close to zero. Moreover, even if the initial opinions of regular agents vary in the range of [−0.8, +0.8], all regular agents, when the dynamics evolution reaches convergence, only hold the opinion of −1 or +1 and the agents with opposite opinions fail to influence each other. As a result, the opinion evolution engenders extreme opinions and inflexible agents, a phenomenon previously revealed by Martins and Galam [28]. The previous studies [38,39] indicated that the informed agents can improve the guiding effect by changing their characteristics if the agents can interact with each other through their inner opinions. However, when the exchanges of inner opinions among agents are hindered, each agent can only observe the choices of other agents. In this case, the informed agents influence the regular agents in a passive way because the direct conversation and persuasion between the two types
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a
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b
Fig. 9. Percentage of pre-set choice versus the hesitation threshold h guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
a
b
Fig. 10. Temporal evolutions of opinions and choices for 400 agents: (a) inner opinion; (b) percentage of different choices. The assimilation threshold, repulsion threshold, initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The assimilation threshold and initial opinions of informed agents are εI = 2 and x0I ∈ [0, 1], respectively. The hesitation threshold is h = 0.7, and the decay threshold is ρ = 0.3. The percentage of informed agents is 10%.
of agents are not available. As a result, the sensitivity of regular agents to the choices of informed agents is mainly dependent on the characteristics of regular agents, including the assimilation threshold, decay threshold, and initial opinions, as shown in Figs. 6, 8 and 11. The simulation results in this work show that, for a comparatively large assimilation threshold, all regular agents can be turned into supporters when their initial opinions are close to zero, as shown in Fig. 11. This indicates that the unanimous consensus at the pre-set opinion can be achieved if the regular agents have no clear opinions on a certain subject. The requirement can be usually satisfied when people have little knowledge about a subject under consideration. This opinion evolution process could be used to characterize and explain the public opinions on a certain product or the rapid propagation of rumor. For example, due to the marketing campaign and the excellent reputation of previous versions of iPhone, the initial comments on iPhone 4 posted on Twitter were almost unanimously positive and commendatory [30]. However, if people have gained some information about a certain subject and have had clear opinions on the subject, unanimous consensus at the pre-set opinion is hardly realized. An interesting finding is that increasing the decay threshold can boost the development of supporters as shown in Fig. 8, indicating that agents tending to keep silent or neutral are susceptible to the informed agents. Nevertheless, the introduction of informed agents makes the people holding the pre-set opinion outnumber the people carrying the opposite opinion. As the public opinions evolve, the majority of people choose the pre-set opinion, as shown in Figs. 6–9. This result can be justified by some practical observations. For example, Xiong and Liu [30] collected and analyzed an abundance of data regarding people’s comments on ‘‘iPad 2’’ from Twitter. They found that initially the
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b
Fig. 11. Percentage of pre-set choice versus the initial opinion distribution x0R of regular agents guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold and repulsion threshold of regular agents are εR = 1.2 and τR = 1.6, respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
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majority of people held positive opinions, and then more people were attracted to post commendatory comments. The SJBO model might also find some applications in the financial risk assessment. Take the stock investment as an example, each investor can observe the choices (purchase, remaining undecided, or sell) of others but cannot acquire their inner opinions (or private information) [47]. Although a substantial number of well informed investors are capable of influencing the stock price [48], a large initial opinion distribution (or high dispersion in earnings forecasts) concerning a stock usually leads to fewer investors and lower returns than otherwise similar stocks [49], as suggested in Fig. 11.
7
4. Conclusions
1 2 3 4 5
27
In this paper, the SJBO dynamics model is proposed to study the opinion evolution influenced by the informed agents. The SJBO model incorporates both the compromise between similar opinions and the repulsion between dissimilar opinions. Moreover, the model differentiates the inner continuous opinions from the observable discrete choices, and each agent in the model update his opinion and choice through observing the choices of other agents. The model can track both the inner opinions and the observable choices during the opinion evolution process. The simulation results suggest that, for a social community in which the direct conversation and persuasion among agents are not available, the informed agents influence the regular agents in a passive way. The sensitivity of regular agents to the choices of informed agents is mainly dependent on the characteristics of regular agents, especially the assimilation threshold and decay threshold. Increasing the assimilation threshold can expand the size of supporters. If the assimilation threshold is large enough, all regular agents will join the supporters when their initial opinions are close to null. This happens when people have little knowledge regarding the subject under consideration. Another interesting finding is that increasing the decay threshold can also boost the development of supporters, implying that agents tending to keep silent or neutral are susceptible to the informed agents. Although the full and unanimous consensus at the pre-set opinion cannot be achieved if the regular agents have had clear opinions, the introduction of informed agents can turn the majority of agents into supporters. At current stage, the SJBO model has not been attempted to map the opinion interactions in a specific real-world social network. To improve the SJBO model, future studies will focus on the following three directions. The first one is on incorporating different network structures, especially the Internet social networks, into the proposed model. The second may concern exploring the coevolution of public opinions and network structure. The third is reducing model parameters and improving the model.
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Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Q1
Opinion evolution influenced by informed agents
Q2
Kangqi Fan a,b,∗ , Witold Pedrycz b,c,d a
School of Mechano-Electronic Engineering, Xidian University, Xi’an 710071, China
b
Department of Electrical and Computer Engineering, University of Alberta, Edmonton T6G 2V4, Canada
c
Systems Research Institute, Polish Academy of Sciences, Warsaw 01447, Poland
d
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
article
info
Article history: Received 31 January 2016 Received in revised form 6 June 2016 Available online xxxx Keywords: Opinion dynamics Informed agents Inner opinions Discrete choices
abstract Guiding public opinions toward a pre-set target by informed agents can be a strategy adopted in some practical applications. The informed agents are common agents who are employed or chosen to spread the pre-set opinion. In this work, we propose a social judgment based opinion (SJBO) dynamics model to explore the opinion evolution under the influence of informed agents. The SJBO model distinguishes between inner opinions and observable choices, and incorporates both the compromise between similar opinions and the repulsion between dissimilar opinions. Three choices (support, opposition, and remaining undecided) are considered in the SJBO model. Using the SJBO model, both the inner opinions and the observable choices can be tracked during the opinion evolution process. The simulation results indicate that if the exchanges of inner opinions among agents are not available, the effect of informed agents is mainly dependent on the characteristics of regular agents, including the assimilation threshold, decay threshold, and initial opinions. Increasing the assimilation threshold and decay threshold can improve the guiding effectiveness of informed agents. Moreover, if the initial opinions of regular agents are close to null, the full and unanimous consensus at the pre-set opinion can be realized, indicating that, to maximize the influence of informed agents, the guidance should be started when regular agents have little knowledge about a subject under consideration. If the regular agents have had clear opinions, the full and unanimous consensus at the preset opinion cannot be achieved. However, the introduction of informed agents can make the majority of agents choose the pre-set opinion. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Opinion evolution and diffusion have been greatly facilitated by the rapid development of modern information technologies. The research on the dynamic evolution of public opinions can improve the understanding of some crucial phenomena arising in the fields of sociology (e.g., the emergence of extremism) [1–4], politics (e.g., political elections) [5–7], and economics (e.g., advertising strategies) [8–10]. As an important research field of sociophysics [11], opinion dynamics provides an effective tool to model and study the evolution and diffusion of opinions among social agents. In terms of the possible values of opinions, the opinion dynamics models can be categorized into three main groups: (1) discrete opinion models; (2) continuous opinion models; (3) continuous opinions and discrete actions (CODA) model [12]. These opinion dynamics models, although are not able to precisely predict the real-world observations in some cases, can help us capture the critical factors that govern the opinion evolution and diffusion.
∗
Corresponding author at: School of Mechano-Electronic Engineering, Xidian University, Xi’an 710071, China. E-mail address: [email protected] (K. Fan).
http://dx.doi.org/10.1016/j.physa.2016.06.110 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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In the continuous opinion models, the opinion of each agent ranges within a continuous opinion space (e.g., [0, 1]). Two significant representatives of continuous opinion models are Deffuant model [13,14] and Hegselmann–Krause (HK) model [15,16], which were all developed using the concept of bounded confidence. This indicates that agents will interact with each other only if the difference in their opinions is smaller than a given threshold. In the Deffuant model [13,14], agents encounter in random pairwise interaction to update their opinions, whereas in the HK model [15,16] each agent moves to the average opinion of all agents whose opinions lie in its coverage of confidence. In the discrete opinion models, each agent carries one of a finite number of different opinions, frequently two opposite binary points of view (e.g., yes or no, support or opposition, accept or reject). Among several different discrete opinion models, we highlight voter model [17,18], Sznajd model [19–21], and majority-rule model [22–24]. In the voter model, each Q3 agent is randomly chosen to adopt the opinion of one of his neighbors, whereas in the Sznajd model two agents with the same opinion persuade their neighbors to accept their opinion. The agents in the majority-rule model update their opinions by following the local majority rule. The CODA model [25] differentiates the inner continuous opinion from the observable binary choices. In this model, the inner opinion is expressed with a certain probability stating that an individual believes one of the two alternatives is the best. The agent in the CODA model observes the choices of his neighbors, and then updates his inner opinion and relevant choice according to a set of Bayesian rules [25–28]. In fact, the choices are a type of opinions as suggested in the discrete opinion models. Since the CODA model involves both the inner continuous opinion and the observable discrete opinion, the latter is named as the choice. This differentiation and nomenclature will also be adopted in this study. In addition to the generic opinion dynamics models, some researchers have attempted to develop the specific models to explore the practical opinion evolution processes on several online social media [29–31]. For example, Xiong and Liu [30] collected people’s comments on three electronic products (i.e., ‘‘iPhone 4’’, ‘‘Blackberry’’, and ‘‘iPad 2’’) from Twitter and proposed an opinion model to explore the evolution processes of people’s opinions on the three electronic products. The aforementioned opinion models have been employed to study the influences of contrarian agents [32], inflexible agents [28,33,34], opinion leaders [35], and extremists [1] on the dynamic evolution of public opinions, and the obtained results have been used to explain some sociological phenomena, such as hung elections [11] and extremism propagation [36,37]. More recently, the continuous opinion models have been extended to capture the opinion evolution trends under the influence of a small portion of informed agents [38,39]. The informed agents are common agents who are employed or chosen to guide the society toward a pre-set opinion. Understanding the opinion evolution influenced by informed agents can help people figure out ways to guide the public opinion toward a desirable aim (such as promote the acceptance of eco-friendly lifestyle) and hinder the spread of undesirable opinion (such as extremist opinions). The simulations showed that only a small number of informed agents were required to successfully change the public opinions. The continuous opinion models are suitable to describe the situations where the exchanges of inner opinions among agents are available. However, in some cases, such as political elections [5] and innovation diffusion [40], people only have a limited number of discrete choices. Moreover, each person is not well acquainted with most of the other people, and then the exchanges of inner opinions with others are hindered. As a result, each person observes others’ choices and then makes his own choice. In these cases, how the informed agents influence the public opinions is still unclear. The discrete opinion models and the CODA model are justified to represent the situations where the inner opinion of each agent is not observable and binary choices are a good description of the problem. In the two categories of models, each agent must declare his choice. However, besides the two opposite choices, people may also keep silent or remain undecided when they are not very sure about their choices [37,41]. In addition, the well-known social judgment theory [42] and the real-world opinion interactions [43] show that the repulsion between distinctly dissimilar opinions may have a significant effect on the opinion evolution. For example, recent studies on Polish Internet fora showed that some online discussions, especially when they were associated with political issues, usually developed into fierce quarrels, provocation, and invectives [43]. However, this opinion interaction mechanism is not included in the aforementioned models. Therefore, a model that incorporates both an agent’s silence and the repulsion between dissimilar opinions is obviously more useful for understanding the opinion evolution in the case that the exchanges among inner opinions are impeded. In this paper, a social judgment based opinion (SJBO) model is proposed to study the opinion evolution under the influence of informed agents. Unlike the discrete opinion model and CODA model where each agent must express his choice, the agent in the SJBO model is allowed to remain undecided or keep silent if his support to either of the two opposite choices is not sufficiently strong. Moreover, both the compromise between similar opinions and the repulsion between dissimilar opinions are included in the SJBO model. The proposed model is simulated in a fully connected network, and the effect of informed agents on the opinion evolution is studied. The results suggest that the informed agents can guide the public opinions toward the pre-set target. But the effectiveness is mainly dependent on the characteristics of regular agents.
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2. The SJBO model
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2.1. Inner opinions and discrete choices
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In general, people exchange their opinions with others with intent to seek certain mutual understanding and compromise. However, the opinion evolution cannot always lead to a full and unanimous consensus. In practice, the
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Fig. 1. Schematic diagram of inner opinion and discrete choice of an agent.
interaction between dissimilar opinions sometimes leads to a larger divergence and even the extremism [43,44]. To incorporate different opinion dynamics into the model, we refer to the Social Judgment Theory [42]. The basic idea of this theory is that each individual has two attitude thresholds: assimilation threshold and repulsion threshold. If the opinion difference between a persuader and a receiver is smaller than the assimilation threshold, the receiver will shift his opinion toward that of the persuader. On the contrary, if the opinion difference is larger than the repulsion threshold, the receiver will shift his opinion away from that of the persuader. If the opinion difference falls between the two thresholds, the receiver will hold his own opinion. We consider a social community composed of N agents, in which each agent can notice the discrete choices of other agents, but cannot acquire their inner continuous opinions. Depending on his inner opinion, an agent may vote for one of the two opposite choices represented by +1 and −1, or keep silent represented by 0 if it remains undecided, as shown in Fig. 1. Here we introduce the hesitation threshold h to reflect the lack of confidence of an agent regarding which alternative to choose. At time t, if the inner opinion xti of agent i falls within the hesitation range (−h ≤ xti ≤ +h), agent i has not been completely convinced by any of the two sides, and then his choice is neutrality or silence. Otherwise, agent i chooses −1 if xti < −h or +1 if xti > +h. The choice x¯ ti of agent i can be described in the form
−1, x¯ ti = 0, 1,
x t < −h i t x ≤ h
(1)
i
xti
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15
> +h . 16
Since the inner opinion xti of an agent i cannot been observed, his choice x¯ ti could be perceived as his opinion by others. In fact, the choice is an indication of an agent’s opinion. Some highly rational agents may doubt if an agent is completely supportive of the choice that he chooses. But the choice expressed by an agent at least indicates that it supports this choice to a large extent. To simplify the problem, an agent is believed to completely support the choice if he chooses it. 2.2. Opinion update rules for regular agents
xit +1
2
0 ≤ xti − x¯ tj ≤ εR εR < xti − x¯ tj ≤ τR
(2)
where ¯ (−1 or +1) is the opinion of agent j perceived by agent i; a ∈ [0, 0.5] is the assimilation coefficient of agent i being influenced by agent j; r ∈ [0, 1] is the repulsion coefficient of agent i being influenced by agent j; εR and τR are the assimilation threshold and repulsion threshold of agent i, respectively. The assimilation threshold εR in the proposed model represents the uncertain level of agent i, below which agent i tends to move his opinion toward that of the persuader. The repulsion threshold τR reflects the tolerance, beyond which agent i shifts his opinion away from that of the persuader due to the repulsion between their dissimilar opinions. If the persuader’s opinion falls between the two thresholds, agent i allows the presence of a different opinion but keeps his own opinion. However, if agent j remains undecided or keeps silent (x¯ tj = 0) at time t, agent i adjusts his inner opinion xti according to the following rules xit +1 =
xti ,
t x − x¯ t ≤ ρR i j t x − x¯ t > ρR i j
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τR < xti − x¯ tj ≤ 2
xtj
t λ xi ,
18
21
At time t, an agent i is randomly chosen to interact with his neighboring agent j. If agent i is a regular agent and agent j has made his choice between −1 and +1, agent i updates his inner opinion xti according to the Social Judgment Theory, i.e., the following rules
t xi + a x¯ tj − xti , t x, = i t t x¯ − xti x − r j 1 − xti , i
17
(3)
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K. Fan, W. Pedrycz / Physica A xx (xxxx) xxx–xxx Table 1 Parameters in the SJBO model.
Opinion assimilation Opinion repulsion Opinion decay Hesitation range
Thresholds
Coefficients
ε τ ρ
a r
λ
h
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where λ is the decay coefficient indicating a drop in the preference for −1 or +1 after observing the silence of agent j; ρR is the decay threshold below which an agent tends to lose his preference for one of the two opposite choices. At present, statistic data or theories like Social Judgment Theory to describe this phenomenon have not been found, and more research is required to quantify the parameters in Eq. (3). However, in fact, Eq. (3) shares the same fundamental as Deffuant model [13], and it can be obtained by letting one of the two opinions in Deffuant model be zero. After updating the inner opinion, agent i declares his choice according to Eq. (1). Since an agent is in the state of hesitation when his inner opinion varies between −h and +h, the influence from both sides can function. Therefore, it is reasonable to assume that the assimilation threshold εR could be larger than one. It should be noted that, if εR = τR = 2 and h = ρR = 0, each agent has only two opposite choices and the persuasion from both sides is acceptable. In this case, the SJBO model described by Eqs. (2) and (3) is similar to the CODA model in the fundamental idea.
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2.3. Opinion update rules for informed agents
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Informed agents are the employed or voluntary individuals who intentionally and invisibly try to pull the public opinions toward a pre-set target. The informed agents differ from the inflexible agents in that the former adjust their opinions with intent to influence other agents’ opinions while the latter keep their opinions always unchanged. For the same reason, the informed agents are also different from the contrarians who always have the opposite opinion to that of the majority of the surrounding agents. Opinion leaders are agents who have relatively high connectivity into the social network and/or larger influence on regular agents’ opinions whereas the informed agents do not have such superiorities. Therefore, it is reasonable to assume that, when an informed agent i encounters agent j, agent i will approach agent j in the opinion with an aim to influence agent j if the difference between their opinions are not too large. Otherwise, the informed agent i will hold his own opinion. The informed agent i is also constantly attracted by the pre-set target so as not to deviate it far away. In addition, the informed agent i should have no preference for the opposite target or choice. In summary, if an informed agent i interacts with agent j who has declared his choice, agent i updates his inner opinion xti via following Eq. (2) with minor adjustments, Q4 as expressed in the following form max (1 − µ) xti + a x¯ tj − xti
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xit +1
=
+ µx T , 0 ,
xti ,
t x − x¯ t ≤ εI j it x − x¯ t > εI i j
(4)
where xT is the pre-set target; µ ∈ [0, 1] is the target coefficient; εI is the assimilation threshold of the informed agent i. Likewise, if agent j keeps silent (x¯ tj = 0), agent i updates his inner opinion xti by following Eq. (3) with minor adjustments, as expressed in the following form xit +1
=
(1 − µ) λxti + µxT , xti ,
t x − x¯ t ≤ ρI j ti x − x¯ t > ρI i j
(5)
40
where λ and ρI are the decay coefficient and decay threshold of the informed agent i, respectively. After updating the inner opinion, agent i makes his choice according to Eq. (1). We assume that none of the informed agents is aware of the presence of other informed agents. Therefore, even if agent j is an informed agent, agent i still adjusts his inner opinion according to Eqs. (4) or (5). In practice, the parameters used to characterize an individual vary for different individuals since no two persons are identical in terms of personality. But reasonable assumptions are usually adopted in the theoretical studies to simplify the problem. In this study, the regular agents are supposed to be homogeneous, i.e., they have identical parameters, and all informed agents are also endowed with the same parameters. In addition, the regular agents and the informed agents have the same hesitation threshold h, decay threshold ρ , assimilation coefficient a, repulsion coefficient r, and decay coefficient λ since the informed agents are common agents except that they have a specific aim. The parameters adopted in the SJBO model are listed in Table 1. In this study, the subscript R in the parameters indicates the regular agents, whereas the subscript I represents the informed agents.
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3. Results and discussion
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In this section, we report on a series of simulations in a fully connected social network to reveal the opinion evolution trends under the influence of informed agents based on the proposed model. The fully connected social network is adopted
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Fig. 2. Temporal evolutions of opinions and choices for 400 agents: (a) inner opinion; (b) percentage of different choices. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 0.8, τR = 1.6, and x0R ∈ [−1, +1], respectively. The assimilation threshold and initial opinions of informed agents are εI = 1.2 and x0I ∈ [0, 1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. The percentage of informed agents is 10%.
because it allows us to focus on the effects of agent’s characteristics or model parameters on the opinion evolution. At the beginning of each simulation, a population of N agents is generated and some of the agents are randomly selected to be the informed agents. Then the regular agents and the informed agents are initialized separately. The pre-set target is defined as the choice of +1, i.e., xT = +1, and the agent whose choice is +1 is also named as the supporter. On the contrary, the agent whose choice is −1 is called the contrarian. At each time step, two agents i and j are randomly selected, with agent i observing the choice of agent j and updating his inner opinion and choice according to the SJBO model. A large number of simulations indicate that the effects of assimilation coefficient a, repulsion coefficient r, decay coefficient λ, and target coefficient µ on the opinion evolution trends are not significant. To simplify the problem, the simulations are conducted with constant coefficients set up as follows: a = r = λ = 0.4, and µ = 0.6. The initial opinions x0I of informed agents are drawn from a uniform distribution between [0, +1] to ensure that none of the informed agents has a preference for the opposite target (−1), whereas the regular agents pick their initial opinions x0R from a uniform distribution between [−1, + 1], unless otherwise stated. Since each agent can only observe the choices of other agents, the informed agents always choose 0 or +1 to avoid the adverse guidance, which is guaranteed by the opinion updating rules described by Eqs. (4) and (5). Figs. 2–4 show the typical temporal evolutions of opinions and choices for a community with N = 400 agents, in which the time step represents the average update per agent. Initially, some agents whose inner opinions fall beyond the hesitation range declare their choices, whereas the majorities keep silent because their support to either choice is not strong enough. After the opinion evolution is started, the silencers or centrists are influenced to gradually express their choices by the agents who have voiced their choices. If the assimilation threshold of regular agents is smaller than one, i.e., εR < 1, the regular agents whose inner opinions are close to zero are immune to the influence from the two opposite choices, and then the opinion evolution leads to three opinion clusters, as shown in Fig. 2. On the contrary, if εR ≥ 1, all the silencers can be influenced to express their choices, causing the opinion bipolarization, as shown in Fig. 3. Under the guidance of informed agents, the population with the pre-set target is larger than that with the opposite choice after the opinion evolution reaches convergence, as shown in Figs. 2 and 3. However, the complete consensus fails to be achieved due to the presence of the agents or contrarians who initially choose the opposite choice because their initial preferences for the opposite choice are strong. These contrarians can tempt some of the silencers to choose the opposite choice, blocking the formation of complete consensus. If the initial opinions of regular agents distribute between [−ρ, +ρ], i.e., [−0.3, +0.3] in this study, all regular agents tend to discard their preferences for the two opposite choices. But the informed agents can guide all regular agents to choose the pre-set target if εR ≥ 1, as shown in Fig. 4. To gain a more fundamental understanding of the opinion evolution influenced by informed agents, a large number of simulations are carried out in a completely connected community with N = 2000 agents. The dynamics evolution runs until it reaches convergence or 300 updates per agent have been completed, and all simulations are averaged over 30 runs. The percentage of agents with the pre-set choice versus the percentage of informed agents is shown in Fig. 5. It can be observed that the supporters increase with the percentage of the informed agents. This observation makes sense because an increasing number of informed agents can improve the guiding effect and then turn more regular agents into supporters. However, even if the informed agents take the overwhelming majority of the population, the contrarians still survive owing to the effect of repulsion between dissimilar opinions. The full and unanimous consensus can be reached only if all agents are informed agents. In general, only a small fraction of agents work as informed agents because recruiting a large number of informed agents is either impractical or prohibitively costly. Therefore, in the following simulations, 10% of the total agents are selected to be the informed agents.
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Fig. 3. Temporal evolutions of opinions and choices for 400 agents: (a) inner opinion; (b) percentage of different choices. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The assimilation threshold and initial opinions of informed agents are εI = 1.2 and x0I ∈ [0, 1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. The percentage of informed agents is 10%.
a
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Fig. 4. Temporal evolutions of opinions and choices for 400 agents: (a) inner opinion; (b) percentage of different choices. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−0.3, +0.3], respectively. The assimilation threshold and initial opinions of informed agents are εI = 1.2 and x0I ∈ [0, 1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. The percentage of informed agents is 10%.
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The percentage of agents with the pre-set choice versus the assimilation threshold εR of regular agents is shown in Fig. 6. It can be noticed that the characteristics of informed agents, including the assimilation threshold εI and initial opinion distribution x0I , have insignificant effects on the ultimate state of the system. Adjusting the characteristics of informed agents cannot remarkably expand the size of supporters. However, varying the assimilation threshold εR of regular agents from 0.8 to 1.4 increases the percentage of supporters by almost 20%. Moreover, the size of supporters is found to be very sensitive to the variation of εR from 0.8 to 1. This is caused by a phenomenon called freezing effect [45], meaning that each agent is immune to the influence from the choice that it initially tends not to choose. Consequently, three opinion clusters come into being, as shown in Fig. 2. As εR increases from 0.8 to 1, more regular agents can be influenced by the informed agents to choose the pre-set target. If εR ≥ 1, the silencers are influenced by both the supporters and the contrarians, and eventually all silencers are tempted to express their choices, as shown in Fig. 3. But the supporters have a greater influence due to the presence of informed agents, and then the majority of the silencers join the supporters. Fig. 7 shows the percentage of supporters versus the repulsion threshold τR of regular agents. It can be seen that neither the characteristics of informed agents nor the repulsion threshold τR of regular agents can significantly impact the ultimate state of the system. Theoretically, a smaller repulsion threshold τR can push more silencers to supporters or contrarians depending on their initial opinions. Therefore, the amount of silencers can be reduced by decreasing the repulsion threshold. Similar result can also be obtained by increasing the assimilation threshold εR . If εR ≥ 1, the assimilation effect can
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Fig. 5. Percentage of pre-set choice versus that of informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
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Fig. 6. Percentage of pre-set choice versus the assimilation threshold εR of regular agents guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The repulsion threshold and initial opinions of regular agents are τR = 1.6 and x0R ∈ [−1, +1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
completely remove the silencers, as shown in Fig. 3. Therefore, when εR = 1.2, no significant difference in the percentage of supporters can be observed for various repulsion threshold τR , as shown in Fig. 7. Fig. 8 shows the percentage of supporters versus the decay threshold ρ . With an increase in the decay threshold, the silencers have a larger influence scope and then can guide more agents to keep silent. However, due to a large assimilation threshold (εR = 1.2), the silencers are influenced by the two competitive groups as well. Under the guidance of informed agents, more silencers choose the pre-set target. Therefore, increasing the decay threshold can enlarge the percentage of supporters, as shown in Fig. 8. Fig. 9 shows the percentage of supporters versus the hesitation threshold h. It can be seen that adjusting the hesitation threshold h from 0.1 to 0.7 could slightly increase the percentage of supporters for various characteristics of informed agents. This minute increase occurs due to the similar cause as discussed in Fig. 8. That is, increasing the hesitation threshold expands the scale of silencers, but the majority of the silencers can be influenced by the supporters to choose the pre-set target when εR = 1.2. However, a visible drop in the percentage of supporters is revealed when εI = 2 and h = 0.7, as shown in Fig. 9(a). To figure out this abnormal drop, we explore the temporal evolutions of opinions and choices for εI = 2 and h = 0.7, as shown in Fig. 10. It can be observed that although the agents have been bipolarized, the informed agents still attempt to approach and guide the contrarians. In the meanwhile, the informed agents are attracted by the pre-set target and the other supporters. Consequently, the opinions of informed agents swing between [0.6, 1], and their choices alternate between 0 and 1, resulting in the abnormal drop in the supporters. If h < 0.6, the opinions of informed agents fall beyond the hesitation
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Fig. 7. Percentage of pre-set choice versus the repulsion threshold τR of regular agents guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold and initial opinions of regular agents are εR = 1.2 and x0R ∈ [−1, +1], respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
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Fig. 8. Percentage of pre-set choice versus the decay threshold ρ guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The hesitation threshold is h = 0.5. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
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range, and their choices are always the pre-set choice. In addition, if εI < 2, the informed agents will give up the effort of guiding the contrarians and stick to the pre-set choice. Therefore, if h < 0.6 or εI < 2, no visible drop in the percentage of supporters can be observed. Fig. 11 shows the percentage of supporters versus the initial opinion distribution x0R of regular agents, in which [|x|] indicates the opinion distribution [−x, +x]. It is obvious that the smaller of x0R , the larger of the percentage of supporters. For a small x0R , the regular agents have no strong preferences for either of the two opposite choices, and then the influence of informed agents is highlighted. Particularly, if x0R ≤ 0.2, the regular agents initially have little preferences for the two opposite choices. In this case, the informed agents can guide all regular agents to choose the pre-set choice, leading to the herding behavior [46]. Therefore, to maximize the influence of informed agents, the guidance should be started when the regular agents have no clear opinions, i.e., x0R is close to zero. Moreover, even if the initial opinions of regular agents vary in the range of [−0.8, +0.8], all regular agents, when the dynamics evolution reaches convergence, only hold the opinion of −1 or +1 and the agents with opposite opinions fail to influence each other. As a result, the opinion evolution engenders extreme opinions and inflexible agents, a phenomenon previously revealed by Martins and Galam [28]. The previous studies [38,39] indicated that the informed agents can improve the guiding effect by changing their characteristics if the agents can interact with each other through their inner opinions. However, when the exchanges of inner opinions among agents are hindered, each agent can only observe the choices of other agents. In this case, the informed agents influence the regular agents in a passive way because the direct conversation and persuasion between the two types
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Fig. 9. Percentage of pre-set choice versus the hesitation threshold h guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold, repulsion threshold, and initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
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Fig. 10. Temporal evolutions of opinions and choices for 400 agents: (a) inner opinion; (b) percentage of different choices. The assimilation threshold, repulsion threshold, initial opinions of regular agents are εR = 1.2, τR = 1.6, and x0R ∈ [−1, +1], respectively. The assimilation threshold and initial opinions of informed agents are εI = 2 and x0I ∈ [0, 1], respectively. The hesitation threshold is h = 0.7, and the decay threshold is ρ = 0.3. The percentage of informed agents is 10%.
of agents are not available. As a result, the sensitivity of regular agents to the choices of informed agents is mainly dependent on the characteristics of regular agents, including the assimilation threshold, decay threshold, and initial opinions, as shown in Figs. 6, 8 and 11. The simulation results in this work show that, for a comparatively large assimilation threshold, all regular agents can be turned into supporters when their initial opinions are close to zero, as shown in Fig. 11. This indicates that the unanimous consensus at the pre-set opinion can be achieved if the regular agents have no clear opinions on a certain subject. The requirement can be usually satisfied when people have little knowledge about a subject under consideration. This opinion evolution process could be used to characterize and explain the public opinions on a certain product or the rapid propagation of rumor. For example, due to the marketing campaign and the excellent reputation of previous versions of iPhone, the initial comments on iPhone 4 posted on Twitter were almost unanimously positive and commendatory [30]. However, if people have gained some information about a certain subject and have had clear opinions on the subject, unanimous consensus at the pre-set opinion is hardly realized. An interesting finding is that increasing the decay threshold can boost the development of supporters as shown in Fig. 8, indicating that agents tending to keep silent or neutral are susceptible to the informed agents. Nevertheless, the introduction of informed agents makes the people holding the pre-set opinion outnumber the people carrying the opposite opinion. As the public opinions evolve, the majority of people choose the pre-set opinion, as shown in Figs. 6–9. This result can be justified by some practical observations. For example, Xiong and Liu [30] collected and analyzed an abundance of data regarding people’s comments on ‘‘iPad 2’’ from Twitter. They found that initially the
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Fig. 11. Percentage of pre-set choice versus the initial opinion distribution x0R of regular agents guided by informed agents with diverse characteristics: (a) effect of assimilation threshold εI of informed agents; (b) effect of initial opinion distribution x0I of informed agents. The assimilation threshold and repulsion threshold of regular agents are εR = 1.2 and τR = 1.6, respectively. The hesitation threshold is h = 0.5, and the decay threshold is ρ = 0.3. In (a), the initial opinion distribution of informed agents is x0I ∈ [0, 1]. In (b), the assimilation threshold of informed agents is εI = 1.2.
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majority of people held positive opinions, and then more people were attracted to post commendatory comments. The SJBO model might also find some applications in the financial risk assessment. Take the stock investment as an example, each investor can observe the choices (purchase, remaining undecided, or sell) of others but cannot acquire their inner opinions (or private information) [47]. Although a substantial number of well informed investors are capable of influencing the stock price [48], a large initial opinion distribution (or high dispersion in earnings forecasts) concerning a stock usually leads to fewer investors and lower returns than otherwise similar stocks [49], as suggested in Fig. 11.
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4. Conclusions
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In this paper, the SJBO dynamics model is proposed to study the opinion evolution influenced by the informed agents. The SJBO model incorporates both the compromise between similar opinions and the repulsion between dissimilar opinions. Moreover, the model differentiates the inner continuous opinions from the observable discrete choices, and each agent in the model update his opinion and choice through observing the choices of other agents. The model can track both the inner opinions and the observable choices during the opinion evolution process. The simulation results suggest that, for a social community in which the direct conversation and persuasion among agents are not available, the informed agents influence the regular agents in a passive way. The sensitivity of regular agents to the choices of informed agents is mainly dependent on the characteristics of regular agents, especially the assimilation threshold and decay threshold. Increasing the assimilation threshold can expand the size of supporters. If the assimilation threshold is large enough, all regular agents will join the supporters when their initial opinions are close to null. This happens when people have little knowledge regarding the subject under consideration. Another interesting finding is that increasing the decay threshold can also boost the development of supporters, implying that agents tending to keep silent or neutral are susceptible to the informed agents. Although the full and unanimous consensus at the pre-set opinion cannot be achieved if the regular agents have had clear opinions, the introduction of informed agents can turn the majority of agents into supporters. At current stage, the SJBO model has not been attempted to map the opinion interactions in a specific real-world social network. To improve the SJBO model, future studies will focus on the following three directions. The first one is on incorporating different network structures, especially the Internet social networks, into the proposed model. The second may concern exploring the coevolution of public opinions and network structure. The third is reducing model parameters and improving the model.
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