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External activation promoting consensus formation in the opinion model with interest decay
Physics Letters A 377 (2013) 362–366
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
External activation promoting consensus formation in the opinion model with interest decay Yun Liu a,∗ , Fei Xiong a , Jiang Zhu b , Ying Zhang b a b
Key Laboratory of Communication and Information Systems, Beijing Municipal Commission of Education, Beijing Jiaotong University, Beijing, 100044, China Carnegie Mellon University, Silicon Valley, Moffett Field, CA 94035, USA
a r t i c l e
i n f o
Article history: Received 31 May 2012 Received in revised form 22 November 2012 Accepted 23 November 2012 Available online 28 November 2012 Communicated by C.R. Doering Keywords: Opinion dynamics Social physics Individual activity Majority model
a b s t r a c t We put forward an opinion model that considers internal decay and external activation or deactivation. Agents may withdraw from the discussion, meanwhile, these inactive agents are likely to be motivated by active neighbors. In addition, external influence from outside circumstances is added to the population. We focus on the majority rule of opinion exchange. Our investigations reveal under the impact of external circumstances, the system evolves to different stable states. One opinion can finally be made dominant when the internal motivation is large sufficiently. However, without external activation, consensus is hardly reached in the system with interest decay. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Complex systems and complex networks have become an absorbing research area. Extensive work related to this field has been conducted, in order to build theoretical models to describe real systems in social life and to investigate the elaborate characteristics of the systems. As a typical representative, opinion dynamics [1,2] researches the formation of public opinion, the convergence of opinion evolution, and the emerging condition of a final state, which may be consensus, polarization, or fragmentation. These models assume that a population of finite sized agents interact with each other, exchange their opinions, and try to persuade other people [3,4]. A certain opinion updating rule is used in a model, and agents adopt other’s opinions following the rule [5,6]. The opinion model with the voter rule is simplified and can be analyzed by the mean-field equation [7–10]. The majority model is intuitive since an agent adopts the majority opinion in its local neighborhood [11–13]. Other opinion models, such as the Deffuant model [14,15], the continuous opinion and discrete action model (CODA) [16,17], Sznajd model [18,19] and its modified version [20] were also put forward, taking their own points of emphasis. To date, extensive explorations have been developed in which personal characters and nonlinear system features are considered [21–28]. Noise was introduced to the voter model on one-
*
Corresponding author at: Beijing Jiaotong University, School of Electronic and Information Engineering, Room 603, South of No. 9 Teaching-Building, No. 3 Shang Yuan Cun, 100044 Hai Dian District, Beijing, China. Tel.: +86 10 51684227. E-mail address: [email protected] (Y. Liu). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.11.044
dimensional chains, and the system with noise could not converge to the completely ordered state in thermodynamic limit [21]. Individuals do not always change their minds, but sometimes they keep their own ideas during interactions. Considering individual memory, the transition to the other opinion decreases with the time during which the agent holds its current opinion [22]. Irrespectively of the slowing down of microscopic dynamics, the macroscopic dynamics is accelerated to reach a consensus state. Moreover, some agents may withdraw from the discussion. Introducing individual latency, agents may stop interacting with other people temporarily for a period of time, after they change their opinions [23]. This waiting period of each agent drives the voter dynamics to the state with equal densities of two opinions. Thus, these nonlinear features are of great importance to the dynamics, and they have a close relationship with relaxation process. Meanwhile, the influence of the underlying network on opinion dynamics has attracted great attention. In [29], collective behavior with the majority rule in complex networks was investigated, and the results showed that degree heterogeneity affects the location of order–disorder transition. The real society, especially online social networks are not isolated and not always active. Users don’t concentrate on the discussion all the time, and they may lose interest and discontinue their interactions. As all agents stop participating in the discussion, the topic will die out. On the other side, active users usually can persuade their inactive neighbors to join in the discussion again [30], and thus, these inactive neighbors have opportunities to get evoked. Meanwhile, in the actual interaction, users are more impressionable to their neighbors holding the same opinion, since
Y. Liu et al. / Physics Letters A 377 (2013) 362–366
people would prefer to communicate with others of similar beliefs rather than argue with opponents [31,32]. Yet, in many real systems, influences come from the outside as well as from the inside. For instance, in an online forum network, users can persuade each other, and meanwhile website administrators also can take some measures outside interactions, to make a topic hot or cold. External effects may be positive or negative. A positive influence from outside can make the system active, while a negative influence makes more agents withdraw from the discussion. Therefore, internal and external factors are always competing for the main role in the evolution. In this Letter, we introduce the impact of outside circumstances in the opinion model with interest decay, and apply the majority interacting rule to investigate the evolution process of system. The rest of the Letter is structured as follows. Section 2 presents a model with the external influence, and the majority rule is used. In Section 3, we show simulation results obtained with the model. We close the Letter in Section 4 with concluding remarks.
2. The model
In this Letter, we introduce the external impact which usually comes from outside circumstances. Meanwhile the modified version of the majority interacting rule [13,23] is applied in our model. In the modified majority rule, an agent and two of its neighbors are selected randomly. If the two neighbors have the same opinion, the agent adopts the opinion of these neighbors. Agents can only hold one of two possible opinions, i.e., σ = 0 or σ = 1. Meanwhile, based on the majority rule, as introduced in Ref. [23], let another variable denote agent’s activity, ϕ = A (Active) or ϕ = I (Inactive). Inactive agents withdraw from the discussion, and only active agents can update their opinions. Agent’s activity may change due to individual interest decay, internal motivation or external impact. We present the opinion model with internal motivation and external impact as described below. At first, there are several active nodes in the system, while other nodes are inactive. The number of initial active agents is given by m, and these agents are assigned randomly. A time step contains N update events for an N sized system. In an update event, an agent (the focal agent) and two of its neighbors are selected. The following four procedures are implemented synchronously. (1) Opinion evolution. If the selected focal agent is active, it updates its opinion according to its neighbors’ opinions following the majority rule. (2) Individual interest decay. If the focal agent is active, it may become inactive with a probability δ . (3) Internal motivation. If the selected focal agent is inactive, it may regain its interest in the discussion through the persuasiveness of its neighbors. For the inactive focal agent, if either of its two selected neighbors is active and holds the same opinion with it, it is evoked to be active with a probability λ. (4) External influence. The external influence acts on some random agents at every time step. We define e (−1 < e < 1) proportion of agents are influenced by the outside circumstances. If e > 0, the outside influence is activation; otherwise, it is deactivation. For the system with individual interest decay, the dynamics stops on the condition that the system achieves a stable state or on the condition that there is no active agent left in the system.
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3. Analytical approach Using the mean-field theory, we make an analytical assessment of the dynamics on the complete graph where each individual links to all other members in the network. The density of agents with opinion 1 and opinion 0 at time t are denoted by ρ1 (t ) and ρ0 (t ), respectively, and they are simplified to ρ1 and ρ0 . It is obvious that ρ1 is equivalent to the average opinion. Similarly, the density of active agents with opinion 1 and opinion 0 are defined as ρ1a (t ) and ρ0a (t ), simplified as ρ1a , ρ0a respectively. From the model, it is easy to see that ρ1a changes because of opinion update, interest decay, internal motivation and external impact. The transition rate of ρ1a during each procedure is listed as follows: (1) Opinion evolution. In the opinion update under majority rule, ρ1a increases as active agents holding opinion 0 change their opinion following two neighbors with opinion 1, but it decreases if active agents with opinion 1 adopt their neighbors’ different opinion. Therefore, for the opinion update, the variation of ρ1a is ρ0a ρ12 − ρ1a ρ02 . (2) Interest decay. With individual interest decay, ρ1a declines by δ ρ1a . (3) Internal motivation. An inactive agent with opinion 1 can be motivated by its two neighbors if one of them is active and holds opinion 1. The probability that neither of the two neighbors is active and holds opinion 1 is (1 − ρ1a )2 , and thus, the increment of ρ1a for internal motivation is λ(ρ1 − ρ1a ) × (1 − (1 − ρ1a )2 ). (4) External influence. Considering the external impact, e proportion of agents are influenced, but only a positive action on these inactive agents or a negative action on these active agents has an effect. The variation of ρ1a for external impact is e (ρ1 − ρ1a ) for e > 0, and e ρ1a for e 0. Similarly, we can write the corresponding rate of change of
ρ0a , ρ1 . Therefore the transition equations of these variables are
listed as follows, taking the condition e 0, for example:
∂t ρ1a = λ(ρ1 − ρ1a ) 1 − (1 − ρ1a )2 − δ ρ1a + e ρ1a + ρ0a ρ12 − ρ1a ρ02 ∂t ρ0a = λ(ρ0 − ρ0a ) 1 − (1 − ρ0a )2 − δ ρ0a + e ρ0a + ρ1a ρ02 − ρ0a ρ12 ∂t ρ1 = ρ0a ρ12 − ρ1a ρ02
(1)
In the final state of the dynamics, i.e. t → ∞, both sides of Eq. (1) tend to be zero. With the auxiliary equation, ρ1 + ρ0 = 1, after some elementary manipulations, the fixed points can be calculated. Analogously to Ref. [23], from the right side of Eq. (1), we gain the coefficient matrix of the first-order Taylor series expansion at the fixed points. Making all the eigenvalues of this matrix have a negative real part, we get the stability of these stationary points. If the stationary point is unstable, it cannot appear in any realization. A tiny fluctuation can make the average opinion deviate from the unstable solution and evolve to the stable solution. Noticing that e 0, if λ < δ − e, the stable stationary point is ρ1a = ρ0a = 0, and the solution refers to the frozen state. In the frozen state, all agents become inactive. If δ − e < λ < 2(δ − e ), the stable solution is ρ1a = ρ0a = 0.5 − 0.5δ/λ + 0.5e /λ, ρ1 = 0.5, and this solution refers to the zero polarization state. When λ > 2(δ − e ), the system becomes active, and the final average opinion depends on the initial value of ρ1 , i.e., ρ1 (0). For ρ1 (0) < 0.5, the stable stationary point is ρ1 ≈ 0.5 − 0.5(1 − 2δ/λ + 2e /λ)0.5 , and for ρ1 (0) > 0.5, the stable solution is ρ1 ≈ 0.5 + 0.5(1 − 2δ/λ + 2e /λ)0.5 . Both of these two
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is zero polarization in which the densities of the two competing opinions become the same. Therefore, under internal actions, the opinions of agents become less ordered, and this outcome coincides with the intrinsic mechanism of some actual systems. When the interest decay rate stays very small, only large external deactivation e < −0.4 can freeze the dynamics. However, small outside activation will change the dynamics and make the system achieve consensus in spite of interest decay. Thus in addition to the internal action, the external impact has a key effect and determines the trend of evolution. A dramatic phenomenon occurs in which large internal motivation doesn’t always mean the system will become more ordered. With larger internal motivation λ = 0.3, more external activation e = 0.02 is needed for total consensus. 4. Simulation results
Fig. 1. Final average opinion derived from the mean-field equations as a function of initial average opinion for (a) λ = 0.3 and δ = 0.1, and (b) δ = 0.4. The original density of active agents is fixed at 0.02.
solutions indicate the polarized state. In addition, we can obtain arithmetic solutions from Eq. (1). Fig. 1 shows the final average opinion derived from the meanfield equations. From Fig. 1, the final state of the system with the modified majority rule is divided into three groups, i.e., the frozen state, the zero polarization state and the polarized state. The polarized state means that one opinion predominates. With large interest decay or external deactivation, the dynamics is frozen, and the final average opinion approaches the initial magnetization. If the system is greatly active, a polarized state, or even the total consensus state, is accessible, and we can see a clear phase transition at ρ1 = 0.5. However, it is noticed that, without external activation, the system even with large internal motivation λ = 1 hardly evolves to total consensus. For e = 0 and λ = 0.2, the final average opinion is the same as that in linear voter model, but the essential dynamics is different. In the voter model, consensus is always reached in each run of realization, but in our model the evolution process in each run is stopped quickly since all agents become inactive. When the dynamics is not positive enough, the final state
We conduct Monte Carlo simulations for our opinion model with internal actions and external influence. Our aim is to investigate the opinion evolution and relaxation process of the dynamic system under the influence of outside circumstances. We set the system size N = 1000, and the number of initial active agents m = 10. Simulations are implemented synchronously. At each time step, all active agents update their opinions following two neighbors randomly selected, and then they may lose their activity due to individual interest decay. Meanwhile, all inactive agents may become active by internal motivation. Then, the external influence acts on some random agents at each time step. Simulations are terminated when no active agent exists or when the average opinion becomes constant. Fig. 2 shows the final average opinion as a function of initial average opinion on the complete graph. In Fig. 2, the stable average opinion is closely related to the external influence, internal motivation and interest decay. With adequate external deactivation, the system becomes frozen, and the average opinion remains unchanged. When the system is positive enough, it will converge to a more ordered state, and original polarization is enhanced toward the direction of consensus. Intermediate outside deactivation e = −0.1 leads to zero polarization, making the system more unordered. We realize that it is easy to make an intrinsically inactive system active with small outside activation. On the contrary, great effort is required to cool down an active system. These results are in accordance with the analytical results. However, for e = −0.01, the final average opinion is not strictly limited to the polarized state as the analytical result because of the finite size effect. The difference can be diminished by increasing the number of agents. Now we investigate the opinion dynamics in a complex network. The Barabasi–Albert scale-free network is used as interaction topology, with the average node degree k = 20. The network is built as follows: initially a small network with 20 nodes is fully connected, and every time a new node is added to the network with 10 edges connecting to 10 different old nodes. The degree distribution of this network decays as a power law with the power exponent of −3. We find the final average opinion with different parameters in the scale-free network is in accordance with that on the complete graph. Then we study the detailed evolution process and relaxation time in this network with heterogeneous degree distribution. It is known that the system might achieve a consensus state when the initial average opinion is above or below 0.5. Thus, without loss of generality, we set the initial average opinion at 0.45 to research into the variations of active nodes and relaxation time. Fig. 3 illustrates the time evolution of the total number of active agents and the number of active agents holding opinion 1. The number of active agents rises to a high level quickly, and then it increases slowly until it reaches a plateau. For e = 0.02, there is a transient and small increase in the number of active agents with
Y. Liu et al. / Physics Letters A 377 (2013) 362–366
Fig. 2. Final average opinion versus initial average opinion on the complete graph, N = 1000, λ = 0.3 and δ = 0.1. The plot is an average of 100 different runs.
Fig. 3. Total number of active agents and active agents with opinion 1 as a function of time in the scale-free network. The initial average opinion is 0.45, N = 1000, λ = 0.3 and δ = 0.2. The results are averaged over 100 different runs.
opinion 1, but a large part of these agents will become extinct. External activation has a significant effect on increasing the number of agents that join in the discussion. However, the number of active agents with opinion 1 does not have a clear change, irrespective of external influence. The quantity levels off after a while, but the mechanisms for activation and deactivation are different. With external activation, most of agents turn to opinion 0 since the initial average opinion is less than 0.5, while, with outside deactivation, many agents become inactive. We know from the model that without internal motivation and external activation, the dynamics will be frozen quickly. In another limit case, without internal decay and external deactivation, the system is very active to achieve total consensus. Now we study the effect of each ingredient in the system. Fig. 4 shows the final average opinion as a function of initial average opinion with different parameters. It is obvious that when λ = 0.3, the system evolves to the zero polarization state with the condition δ = 0.2 and e = 0, or δ = 0 and e = −0.2. Similar results are found for other final states of the system. Therefore, the effect of external deactivation is similar to that of internal decay. However, the effect of external
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Fig. 4. Final average opinion versus initial average opinion with limit cases of parameters in the scale-free network, N = 1000. The plot is an average of 100 different runs.
Fig. 5. Relaxation time versus e in the scale-free network. The average opinion at first is 0.45, N = 1000 and λ = 0.3. The results are averaged over 100 different runs.
activation is very different from that of internal motivation. When δ = 0.3, the system evolves to the frozen state for λ = 0.1 and e = 0, while for λ = 0 and e = 0.1, the final average opinion is the total consensus state. Therefore, although the external activation or internal motivation can make the system active, the influence of external activation is far larger than that of internal motivation. Fig. 5 illustrates the relaxation time with different parameters. Correspondingly to the system state, the relaxation time has a lot to do with the external influence. For the polarized state or frozen state, the relaxation time is small, but for the zero polarization state, a large relaxation time is needed. When the system is extremely active or inactive, it can evolve to the polarized state or frozen state swiftly. Especially with large external deactivation, all agents will withdraw from the discussion, and the dynamics will be frozen immediately. When e > 0.02, the system becomes positive enough to achieve consensus, so the relaxation time declines monotonously with the increase of e. For zero polarization, there are still some active agents in the group at last, but consensus is never reached in every run. Two opinions are competing, and fi-
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Y. Liu et al. / Physics Letters A 377 (2013) 362–366
Fig. 6. Relaxation time versus the system size N in the scale-free network. The average opinion at first is 0.45, λ = 0.3 and δ = 0.1. The results are averaged over 100 different runs.
nally, their densities will become identical. Meanwhile, in Fig. 5, the peak of relaxation time as a function of e changes with the interest decay parameter. Fig. 6 depicts the relaxation time versus the system size. It is clear that the relaxation time increases nearly as a power law with the increase of N. When the system is very inactive, it levels off quickly, and the effect of the system size is not obvious. However, small deactivation slows down the dynamics markedly for a large active system. This suggests that increasing the system size enlarges fluctuations if the consensus state cannot be achieved.
to users, and can be noticed by users easily. If the website administrators want to promote the discussion in a post, they can put the post in the front of the first webpage, so that it can attract more users’ attention, and absorb more participants to interact. On the other side, if the website administrators want to degrade the discussion in a post, they can put the post in the bottom or in an inconspicuous place of previous webpages. Then the post is not easy to find, and becomes less attractive. Users will take much more effort to find the post, and some users are reluctant to do the additional task, so their participation in the discussion decreases. This effect does not come from users themselves spontaneously, but is caused by external deactivation from website administrators. Website administrators even can hide the post from view to freeze the dynamics. No use can discuss about the post, meaning complete external deactivation e = −1. It should be noticed that external influence may not only change users’ activity, but also can change their opinions. However, in some situations especially in online social networks as we just mentioned above, the external influence only has an effect on users’ activity. In the Letter we focus on this kind of influence, and investigate its effect on opinion formation. Furthermore, besides the majority rule, other interacting rule can also be applied in our model, and the analytical approach and simulations are analogous. In future work, the exact action pattern of outside influence in online social networks will be explored. Acknowledgements This work was partially supported by the State Natural Sciences Fund under Grants 61172072, 61271308, the Beijing Natural Science Foundation under Grant 4112045, the Beijing Science and Technology Program under Grant Z121100000312024, the Service Business of Scientists and Engineers Project under Grant 2009GJA00048.
5. Conclusions References We investigated the effect of outside circumstance on the opinion evolution in the system with individual interest decay. The interior dynamics includes opinion evolution, individual interest decay and neighboring motivation. Agents may lose their activity, but they can be motivated by the persuasion of like-minded neighbors. In addition, the external influence was used as an input to the system, either to activate or deactivate the system. The modified version of the majority rule was applied in the model, and analysis and simulations were conducted. We also researched the final average magnetization and relaxation time. The results indicate that the final average opinion depends significantly on the external influence and internal actions. Small external activation drives the initially inactive system to total consensus quickly, but large external deactivation is required to freeze the active dynamics. Without outside activation, if the internal motivation is not adequately strong, the internal actions will make an isolated system less ordered. In that situation, great fluctuations exist in the dynamics, and the numbers of two opinions approach each other, leading to a large relaxation time. In addition, small deactivation makes the relaxation process much slower in a large, active system. The external influence exists obviously in Internet communities and online social networks. For example, in online forums, after a post is created, users can read the post and publish replies to discuss about it. Users usually read posts from the top to the bottom of the webpage. After users read the posts in the first webpage, they may turn to previous webpages to read more posts. Therefore, these posts in the top of the first webpage are more attractive
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Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
External activation promoting consensus formation in the opinion model with interest decay Yun Liu a,∗ , Fei Xiong a , Jiang Zhu b , Ying Zhang b a b
Key Laboratory of Communication and Information Systems, Beijing Municipal Commission of Education, Beijing Jiaotong University, Beijing, 100044, China Carnegie Mellon University, Silicon Valley, Moffett Field, CA 94035, USA
a r t i c l e
i n f o
Article history: Received 31 May 2012 Received in revised form 22 November 2012 Accepted 23 November 2012 Available online 28 November 2012 Communicated by C.R. Doering Keywords: Opinion dynamics Social physics Individual activity Majority model
a b s t r a c t We put forward an opinion model that considers internal decay and external activation or deactivation. Agents may withdraw from the discussion, meanwhile, these inactive agents are likely to be motivated by active neighbors. In addition, external influence from outside circumstances is added to the population. We focus on the majority rule of opinion exchange. Our investigations reveal under the impact of external circumstances, the system evolves to different stable states. One opinion can finally be made dominant when the internal motivation is large sufficiently. However, without external activation, consensus is hardly reached in the system with interest decay. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Complex systems and complex networks have become an absorbing research area. Extensive work related to this field has been conducted, in order to build theoretical models to describe real systems in social life and to investigate the elaborate characteristics of the systems. As a typical representative, opinion dynamics [1,2] researches the formation of public opinion, the convergence of opinion evolution, and the emerging condition of a final state, which may be consensus, polarization, or fragmentation. These models assume that a population of finite sized agents interact with each other, exchange their opinions, and try to persuade other people [3,4]. A certain opinion updating rule is used in a model, and agents adopt other’s opinions following the rule [5,6]. The opinion model with the voter rule is simplified and can be analyzed by the mean-field equation [7–10]. The majority model is intuitive since an agent adopts the majority opinion in its local neighborhood [11–13]. Other opinion models, such as the Deffuant model [14,15], the continuous opinion and discrete action model (CODA) [16,17], Sznajd model [18,19] and its modified version [20] were also put forward, taking their own points of emphasis. To date, extensive explorations have been developed in which personal characters and nonlinear system features are considered [21–28]. Noise was introduced to the voter model on one-
*
Corresponding author at: Beijing Jiaotong University, School of Electronic and Information Engineering, Room 603, South of No. 9 Teaching-Building, No. 3 Shang Yuan Cun, 100044 Hai Dian District, Beijing, China. Tel.: +86 10 51684227. E-mail address: [email protected] (Y. Liu). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.11.044
dimensional chains, and the system with noise could not converge to the completely ordered state in thermodynamic limit [21]. Individuals do not always change their minds, but sometimes they keep their own ideas during interactions. Considering individual memory, the transition to the other opinion decreases with the time during which the agent holds its current opinion [22]. Irrespectively of the slowing down of microscopic dynamics, the macroscopic dynamics is accelerated to reach a consensus state. Moreover, some agents may withdraw from the discussion. Introducing individual latency, agents may stop interacting with other people temporarily for a period of time, after they change their opinions [23]. This waiting period of each agent drives the voter dynamics to the state with equal densities of two opinions. Thus, these nonlinear features are of great importance to the dynamics, and they have a close relationship with relaxation process. Meanwhile, the influence of the underlying network on opinion dynamics has attracted great attention. In [29], collective behavior with the majority rule in complex networks was investigated, and the results showed that degree heterogeneity affects the location of order–disorder transition. The real society, especially online social networks are not isolated and not always active. Users don’t concentrate on the discussion all the time, and they may lose interest and discontinue their interactions. As all agents stop participating in the discussion, the topic will die out. On the other side, active users usually can persuade their inactive neighbors to join in the discussion again [30], and thus, these inactive neighbors have opportunities to get evoked. Meanwhile, in the actual interaction, users are more impressionable to their neighbors holding the same opinion, since
Y. Liu et al. / Physics Letters A 377 (2013) 362–366
people would prefer to communicate with others of similar beliefs rather than argue with opponents [31,32]. Yet, in many real systems, influences come from the outside as well as from the inside. For instance, in an online forum network, users can persuade each other, and meanwhile website administrators also can take some measures outside interactions, to make a topic hot or cold. External effects may be positive or negative. A positive influence from outside can make the system active, while a negative influence makes more agents withdraw from the discussion. Therefore, internal and external factors are always competing for the main role in the evolution. In this Letter, we introduce the impact of outside circumstances in the opinion model with interest decay, and apply the majority interacting rule to investigate the evolution process of system. The rest of the Letter is structured as follows. Section 2 presents a model with the external influence, and the majority rule is used. In Section 3, we show simulation results obtained with the model. We close the Letter in Section 4 with concluding remarks.
2. The model
In this Letter, we introduce the external impact which usually comes from outside circumstances. Meanwhile the modified version of the majority interacting rule [13,23] is applied in our model. In the modified majority rule, an agent and two of its neighbors are selected randomly. If the two neighbors have the same opinion, the agent adopts the opinion of these neighbors. Agents can only hold one of two possible opinions, i.e., σ = 0 or σ = 1. Meanwhile, based on the majority rule, as introduced in Ref. [23], let another variable denote agent’s activity, ϕ = A (Active) or ϕ = I (Inactive). Inactive agents withdraw from the discussion, and only active agents can update their opinions. Agent’s activity may change due to individual interest decay, internal motivation or external impact. We present the opinion model with internal motivation and external impact as described below. At first, there are several active nodes in the system, while other nodes are inactive. The number of initial active agents is given by m, and these agents are assigned randomly. A time step contains N update events for an N sized system. In an update event, an agent (the focal agent) and two of its neighbors are selected. The following four procedures are implemented synchronously. (1) Opinion evolution. If the selected focal agent is active, it updates its opinion according to its neighbors’ opinions following the majority rule. (2) Individual interest decay. If the focal agent is active, it may become inactive with a probability δ . (3) Internal motivation. If the selected focal agent is inactive, it may regain its interest in the discussion through the persuasiveness of its neighbors. For the inactive focal agent, if either of its two selected neighbors is active and holds the same opinion with it, it is evoked to be active with a probability λ. (4) External influence. The external influence acts on some random agents at every time step. We define e (−1 < e < 1) proportion of agents are influenced by the outside circumstances. If e > 0, the outside influence is activation; otherwise, it is deactivation. For the system with individual interest decay, the dynamics stops on the condition that the system achieves a stable state or on the condition that there is no active agent left in the system.
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3. Analytical approach Using the mean-field theory, we make an analytical assessment of the dynamics on the complete graph where each individual links to all other members in the network. The density of agents with opinion 1 and opinion 0 at time t are denoted by ρ1 (t ) and ρ0 (t ), respectively, and they are simplified to ρ1 and ρ0 . It is obvious that ρ1 is equivalent to the average opinion. Similarly, the density of active agents with opinion 1 and opinion 0 are defined as ρ1a (t ) and ρ0a (t ), simplified as ρ1a , ρ0a respectively. From the model, it is easy to see that ρ1a changes because of opinion update, interest decay, internal motivation and external impact. The transition rate of ρ1a during each procedure is listed as follows: (1) Opinion evolution. In the opinion update under majority rule, ρ1a increases as active agents holding opinion 0 change their opinion following two neighbors with opinion 1, but it decreases if active agents with opinion 1 adopt their neighbors’ different opinion. Therefore, for the opinion update, the variation of ρ1a is ρ0a ρ12 − ρ1a ρ02 . (2) Interest decay. With individual interest decay, ρ1a declines by δ ρ1a . (3) Internal motivation. An inactive agent with opinion 1 can be motivated by its two neighbors if one of them is active and holds opinion 1. The probability that neither of the two neighbors is active and holds opinion 1 is (1 − ρ1a )2 , and thus, the increment of ρ1a for internal motivation is λ(ρ1 − ρ1a ) × (1 − (1 − ρ1a )2 ). (4) External influence. Considering the external impact, e proportion of agents are influenced, but only a positive action on these inactive agents or a negative action on these active agents has an effect. The variation of ρ1a for external impact is e (ρ1 − ρ1a ) for e > 0, and e ρ1a for e 0. Similarly, we can write the corresponding rate of change of
ρ0a , ρ1 . Therefore the transition equations of these variables are
listed as follows, taking the condition e 0, for example:
∂t ρ1a = λ(ρ1 − ρ1a ) 1 − (1 − ρ1a )2 − δ ρ1a + e ρ1a + ρ0a ρ12 − ρ1a ρ02 ∂t ρ0a = λ(ρ0 − ρ0a ) 1 − (1 − ρ0a )2 − δ ρ0a + e ρ0a + ρ1a ρ02 − ρ0a ρ12 ∂t ρ1 = ρ0a ρ12 − ρ1a ρ02
(1)
In the final state of the dynamics, i.e. t → ∞, both sides of Eq. (1) tend to be zero. With the auxiliary equation, ρ1 + ρ0 = 1, after some elementary manipulations, the fixed points can be calculated. Analogously to Ref. [23], from the right side of Eq. (1), we gain the coefficient matrix of the first-order Taylor series expansion at the fixed points. Making all the eigenvalues of this matrix have a negative real part, we get the stability of these stationary points. If the stationary point is unstable, it cannot appear in any realization. A tiny fluctuation can make the average opinion deviate from the unstable solution and evolve to the stable solution. Noticing that e 0, if λ < δ − e, the stable stationary point is ρ1a = ρ0a = 0, and the solution refers to the frozen state. In the frozen state, all agents become inactive. If δ − e < λ < 2(δ − e ), the stable solution is ρ1a = ρ0a = 0.5 − 0.5δ/λ + 0.5e /λ, ρ1 = 0.5, and this solution refers to the zero polarization state. When λ > 2(δ − e ), the system becomes active, and the final average opinion depends on the initial value of ρ1 , i.e., ρ1 (0). For ρ1 (0) < 0.5, the stable stationary point is ρ1 ≈ 0.5 − 0.5(1 − 2δ/λ + 2e /λ)0.5 , and for ρ1 (0) > 0.5, the stable solution is ρ1 ≈ 0.5 + 0.5(1 − 2δ/λ + 2e /λ)0.5 . Both of these two
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is zero polarization in which the densities of the two competing opinions become the same. Therefore, under internal actions, the opinions of agents become less ordered, and this outcome coincides with the intrinsic mechanism of some actual systems. When the interest decay rate stays very small, only large external deactivation e < −0.4 can freeze the dynamics. However, small outside activation will change the dynamics and make the system achieve consensus in spite of interest decay. Thus in addition to the internal action, the external impact has a key effect and determines the trend of evolution. A dramatic phenomenon occurs in which large internal motivation doesn’t always mean the system will become more ordered. With larger internal motivation λ = 0.3, more external activation e = 0.02 is needed for total consensus. 4. Simulation results
Fig. 1. Final average opinion derived from the mean-field equations as a function of initial average opinion for (a) λ = 0.3 and δ = 0.1, and (b) δ = 0.4. The original density of active agents is fixed at 0.02.
solutions indicate the polarized state. In addition, we can obtain arithmetic solutions from Eq. (1). Fig. 1 shows the final average opinion derived from the meanfield equations. From Fig. 1, the final state of the system with the modified majority rule is divided into three groups, i.e., the frozen state, the zero polarization state and the polarized state. The polarized state means that one opinion predominates. With large interest decay or external deactivation, the dynamics is frozen, and the final average opinion approaches the initial magnetization. If the system is greatly active, a polarized state, or even the total consensus state, is accessible, and we can see a clear phase transition at ρ1 = 0.5. However, it is noticed that, without external activation, the system even with large internal motivation λ = 1 hardly evolves to total consensus. For e = 0 and λ = 0.2, the final average opinion is the same as that in linear voter model, but the essential dynamics is different. In the voter model, consensus is always reached in each run of realization, but in our model the evolution process in each run is stopped quickly since all agents become inactive. When the dynamics is not positive enough, the final state
We conduct Monte Carlo simulations for our opinion model with internal actions and external influence. Our aim is to investigate the opinion evolution and relaxation process of the dynamic system under the influence of outside circumstances. We set the system size N = 1000, and the number of initial active agents m = 10. Simulations are implemented synchronously. At each time step, all active agents update their opinions following two neighbors randomly selected, and then they may lose their activity due to individual interest decay. Meanwhile, all inactive agents may become active by internal motivation. Then, the external influence acts on some random agents at each time step. Simulations are terminated when no active agent exists or when the average opinion becomes constant. Fig. 2 shows the final average opinion as a function of initial average opinion on the complete graph. In Fig. 2, the stable average opinion is closely related to the external influence, internal motivation and interest decay. With adequate external deactivation, the system becomes frozen, and the average opinion remains unchanged. When the system is positive enough, it will converge to a more ordered state, and original polarization is enhanced toward the direction of consensus. Intermediate outside deactivation e = −0.1 leads to zero polarization, making the system more unordered. We realize that it is easy to make an intrinsically inactive system active with small outside activation. On the contrary, great effort is required to cool down an active system. These results are in accordance with the analytical results. However, for e = −0.01, the final average opinion is not strictly limited to the polarized state as the analytical result because of the finite size effect. The difference can be diminished by increasing the number of agents. Now we investigate the opinion dynamics in a complex network. The Barabasi–Albert scale-free network is used as interaction topology, with the average node degree k = 20. The network is built as follows: initially a small network with 20 nodes is fully connected, and every time a new node is added to the network with 10 edges connecting to 10 different old nodes. The degree distribution of this network decays as a power law with the power exponent of −3. We find the final average opinion with different parameters in the scale-free network is in accordance with that on the complete graph. Then we study the detailed evolution process and relaxation time in this network with heterogeneous degree distribution. It is known that the system might achieve a consensus state when the initial average opinion is above or below 0.5. Thus, without loss of generality, we set the initial average opinion at 0.45 to research into the variations of active nodes and relaxation time. Fig. 3 illustrates the time evolution of the total number of active agents and the number of active agents holding opinion 1. The number of active agents rises to a high level quickly, and then it increases slowly until it reaches a plateau. For e = 0.02, there is a transient and small increase in the number of active agents with
Y. Liu et al. / Physics Letters A 377 (2013) 362–366
Fig. 2. Final average opinion versus initial average opinion on the complete graph, N = 1000, λ = 0.3 and δ = 0.1. The plot is an average of 100 different runs.
Fig. 3. Total number of active agents and active agents with opinion 1 as a function of time in the scale-free network. The initial average opinion is 0.45, N = 1000, λ = 0.3 and δ = 0.2. The results are averaged over 100 different runs.
opinion 1, but a large part of these agents will become extinct. External activation has a significant effect on increasing the number of agents that join in the discussion. However, the number of active agents with opinion 1 does not have a clear change, irrespective of external influence. The quantity levels off after a while, but the mechanisms for activation and deactivation are different. With external activation, most of agents turn to opinion 0 since the initial average opinion is less than 0.5, while, with outside deactivation, many agents become inactive. We know from the model that without internal motivation and external activation, the dynamics will be frozen quickly. In another limit case, without internal decay and external deactivation, the system is very active to achieve total consensus. Now we study the effect of each ingredient in the system. Fig. 4 shows the final average opinion as a function of initial average opinion with different parameters. It is obvious that when λ = 0.3, the system evolves to the zero polarization state with the condition δ = 0.2 and e = 0, or δ = 0 and e = −0.2. Similar results are found for other final states of the system. Therefore, the effect of external deactivation is similar to that of internal decay. However, the effect of external
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Fig. 4. Final average opinion versus initial average opinion with limit cases of parameters in the scale-free network, N = 1000. The plot is an average of 100 different runs.
Fig. 5. Relaxation time versus e in the scale-free network. The average opinion at first is 0.45, N = 1000 and λ = 0.3. The results are averaged over 100 different runs.
activation is very different from that of internal motivation. When δ = 0.3, the system evolves to the frozen state for λ = 0.1 and e = 0, while for λ = 0 and e = 0.1, the final average opinion is the total consensus state. Therefore, although the external activation or internal motivation can make the system active, the influence of external activation is far larger than that of internal motivation. Fig. 5 illustrates the relaxation time with different parameters. Correspondingly to the system state, the relaxation time has a lot to do with the external influence. For the polarized state or frozen state, the relaxation time is small, but for the zero polarization state, a large relaxation time is needed. When the system is extremely active or inactive, it can evolve to the polarized state or frozen state swiftly. Especially with large external deactivation, all agents will withdraw from the discussion, and the dynamics will be frozen immediately. When e > 0.02, the system becomes positive enough to achieve consensus, so the relaxation time declines monotonously with the increase of e. For zero polarization, there are still some active agents in the group at last, but consensus is never reached in every run. Two opinions are competing, and fi-
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Fig. 6. Relaxation time versus the system size N in the scale-free network. The average opinion at first is 0.45, λ = 0.3 and δ = 0.1. The results are averaged over 100 different runs.
nally, their densities will become identical. Meanwhile, in Fig. 5, the peak of relaxation time as a function of e changes with the interest decay parameter. Fig. 6 depicts the relaxation time versus the system size. It is clear that the relaxation time increases nearly as a power law with the increase of N. When the system is very inactive, it levels off quickly, and the effect of the system size is not obvious. However, small deactivation slows down the dynamics markedly for a large active system. This suggests that increasing the system size enlarges fluctuations if the consensus state cannot be achieved.
to users, and can be noticed by users easily. If the website administrators want to promote the discussion in a post, they can put the post in the front of the first webpage, so that it can attract more users’ attention, and absorb more participants to interact. On the other side, if the website administrators want to degrade the discussion in a post, they can put the post in the bottom or in an inconspicuous place of previous webpages. Then the post is not easy to find, and becomes less attractive. Users will take much more effort to find the post, and some users are reluctant to do the additional task, so their participation in the discussion decreases. This effect does not come from users themselves spontaneously, but is caused by external deactivation from website administrators. Website administrators even can hide the post from view to freeze the dynamics. No use can discuss about the post, meaning complete external deactivation e = −1. It should be noticed that external influence may not only change users’ activity, but also can change their opinions. However, in some situations especially in online social networks as we just mentioned above, the external influence only has an effect on users’ activity. In the Letter we focus on this kind of influence, and investigate its effect on opinion formation. Furthermore, besides the majority rule, other interacting rule can also be applied in our model, and the analytical approach and simulations are analogous. In future work, the exact action pattern of outside influence in online social networks will be explored. Acknowledgements This work was partially supported by the State Natural Sciences Fund under Grants 61172072, 61271308, the Beijing Natural Science Foundation under Grant 4112045, the Beijing Science and Technology Program under Grant Z121100000312024, the Service Business of Scientists and Engineers Project under Grant 2009GJA00048.
5. Conclusions References We investigated the effect of outside circumstance on the opinion evolution in the system with individual interest decay. The interior dynamics includes opinion evolution, individual interest decay and neighboring motivation. Agents may lose their activity, but they can be motivated by the persuasion of like-minded neighbors. In addition, the external influence was used as an input to the system, either to activate or deactivate the system. The modified version of the majority rule was applied in the model, and analysis and simulations were conducted. We also researched the final average magnetization and relaxation time. The results indicate that the final average opinion depends significantly on the external influence and internal actions. Small external activation drives the initially inactive system to total consensus quickly, but large external deactivation is required to freeze the active dynamics. Without outside activation, if the internal motivation is not adequately strong, the internal actions will make an isolated system less ordered. In that situation, great fluctuations exist in the dynamics, and the numbers of two opinions approach each other, leading to a large relaxation time. In addition, small deactivation makes the relaxation process much slower in a large, active system. The external influence exists obviously in Internet communities and online social networks. For example, in online forums, after a post is created, users can read the post and publish replies to discuss about it. Users usually read posts from the top to the bottom of the webpage. After users read the posts in the first webpage, they may turn to previous webpages to read more posts. Therefore, these posts in the top of the first webpage are more attractive
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