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An opinion formation model with dissipative structure
Physica A 390 (2011) 2504–2510
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Physica A journal homepage: www.elsevier.com/locate/physa
An opinion formation model with dissipative structure Fei Xiong ∗ , Yun Liu, Zhen-jiang Zhang, Xia-meng Si, Fei Ding School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing, 100044, China Key Laboratory of Communication and Information Systems (Beijing Jiaotong University), Beijing Municipal Commission of Education, Beijing, 100044, China
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Article history: Received 1 September 2010 Received in revised form 7 January 2011 Available online 11 March 2011 Keywords: Opinion dynamics Dissipative structure Voter rule Sociopsychology
abstract We propose an opinion formation model which includes both the influence of internal motivation and dissipation. Active agents adopt their neighbors’ opinions according to a simplified set of rules, but they may gradually lose their interests in the discussion and drop out of it. On the other hand, inert agents can become active by initial activation or internal motivation from neighbors, and participate in the discussion. The initial activation is usually due to occurrence of a social event. The internal motivation, opinion update and dissipation procedure take place simultaneously. We apply the voter rule in our model, and carry out analysis and numerical simulations. Results show if nonzero dissipation stays below a threshold value, the system evolves to a balance state where the average concentration of one opinion is equal to that of the other. With dissipation, a number of small-size opinion clusters exist in the end, but the system gains a fast relaxation rate. The final average opinion is closely related to the dissipation intensity and the length of time for initial activation. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Opinion dynamics which aims to investigate the formation of public opinion provides a valuable insight to social interactions and economic decision making [1,2]. Opinion evolution has attracted great attention at present and many researchers are absorbed in it to explain and forecast stable opinion distributions, which may be consensus, polarization or fragmentation [3,4]. A great many opinion models have been presented, considering that a finite number of agents interact with each other and try to persuade others to adopt their own opinions. In the update process these agents, whose points of view are fully mixed in the beginning, change their opinions according to some rules as predefined [5,6]. Statistical physics which is used to build a connection between microscopic and macroscopic behavior is an effective measure to research into the dynamics. Based on the amount of possible states, the main investigation can be divided into two groups: discrete opinion models and continuous opinion models. In the first group these models (such as the voter model [7–9], the majority model [10–12], the Sznajd model [13–15], the Ising model [16], etc.), describe the situations where a limited number of choices are available for agents. The models of the latter group, like the Deffuant [17,18] and the Hegselmann–Krause [19,20] model, use the concept of bounded confidence and have a compromise process. An extension of the voter model was studied on adaptive network topology which can accelerate the evolution to consensus or separate the network into non-interacting groups [21]. A model with discrete actions and continuous opinions was put forward, assuming that agents change their opinions by using a Bayesian description of how likely their neighbors are correct. The model witnesses the appearance of extremists [22,23]. Stark et al. found if the transition rate of an agent into the opposite state decreased with the time
∗ Corresponding address: Beijing Jiaotong University, School of Electronic and Information Engineering, Room 607, South of No. 9 Teaching-Building, No. 3 Shang Yuan Cun, 100044 Hai Dian District, Beijing, China. Tel.: +86 13811992620. E-mail addresses: [email protected] (F. Xiong), [email protected] (Y. Liu), [email protected] (Z.-j. Zhang). 0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.02.040
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since the last state change, the macroscopic ordering progress would be accelerated despite the slowing-down microscopic dynamics [24]. Selective advantages of two competing strategies were studied in finite structured populations, finding out that the probability for both choices to reach a consensus state is independent of topology [25,26]. As a matter of fact, not all individuals attempt to discuss a certain topic, but instead the topic might be ignored sometimes. Using evolutionary game theory, Ding proposed a model where players might bow out of an unpromising interaction in term of their memories of past payoffs. Opinions always tend to converge, and convergence time is affected by memory length [27]. In Ref. [28], the authors presumed after a voter updated its opinion, it would withdraw from the discussion for a stochastic length of waiting period. In their model the average magnetization is not conserved, and the system is driven towards zero magnetization, independently of initial conditions. Voters do not always update their states, and they may become inert. Inspired by their work, we study the influence of neighbors’ behavior on individual activity in the dynamics. The real society is a dissipative structural system, as well as online communities. The open dissipative system remains far from equilibrium [29–31]. Events which occur in real society are regarded as initial activation of interacting systems. If no coherent influence takes place in users’ actions, public attention declines with the time elapsed, and agents may stop discussing the topic. For example, web users who no longer care about a website are unlikely to login the website again. As the number of active agents decreases continually, ordered states will fail to come forth without motivation. On the other hand, actions of agents can provide a good demonstration for their neighbors, mobilizing their neighbors which have lost interests in the topic. When discussions in local areas cause wide public response, the macroscopic dynamics is then changed and the system may converge to a state in which most agents share an identical opinion. In this paper, we propose a model of opinion dynamics which takes into account the impact of activation and deactivation of agents, illuminating a system where only active agents can take part in discussions. The voter rule is applied to the model, and we research into problems of average magnetization and opinion clusters. Analysis and simulation results show there are some differences in absorbing states compared with the standard voter model. The rest paper is structured as follows. Section 2 presents a new model with the motivation and dissipation. In Section 3 the voter rule is applied in our model, and analysis results are given. Section 4 contains simulation results and discussions about the model. We close the paper in Section 5 with concluding remarks. 2. Opinion model with dissipative structure In most opinion models, agents exchange their opinions about a certain topic based on different rules. Interactions in a closed system will persist all the time until no change comes about, leading to a stable state. A complex network which defines the number of neighbors for each agent, mediates the interactions. Yet in actual society, especially on Internet, conversations about a topic will not continue at all times, and not all agents concentrate on the topic. The attraction of a topic decays with time, and at last the topic may die out. The dynamics is terminated since no one is willing to talk about this. Here we present a dissipative opinion formation model with the following procedures in view of people’s interests in a topic. (i) Initial activation. At first the system is static, and no interaction proceeds between these inactive agents. Then some of the agents are activated by outside circumstance at random. The number of agents activated in the beginning is defined as m. (ii) Opinion update. According to different rules (such as the voter rule, the majority rule, and so on), active agents partake in discussions, and change opinions in accordance with their neighbors. (iii) Internal motivation. Agents are often influenced by neighbors, and they may behave following others. Thus active agents can motivate their inactive like-minded neighbors. If an inert agent holds the same opinion with an active neighbor, the inert agent is driven to be active with a probability λ. People would rather converse with others holding a similar opinion than argue with opponents. (iv) Dissipation. Interests of active agents in the topic decay with time, and they are likely to drop out of discussions. Active agents become dormant with a probability δ . These inactive agents keep silence until the next reactivation. The internal motivation, update and dissipation procedure take place at every time step. The dynamics will go on until all active agents vanish or a stable state is achieved. According to these steps, the evolution of agents’ activity and their opinions are incorporated in the dynamics. Then we will implement the model with the voter rule. 3. Voter rule In the voter model, a system consists of N agents which are positioned at the sites of a certain network. Agents can take either of the two available opinions σ = 0 or σ = 1. In an update event an agent i is randomly selected, together with one of its neighbors. Then agent i adopts the neighbor’s opinion. For the dissipative model where the voter rule is used, only active agents have opportunities to change their opinions. Both the motivation and dissipation procedure are carried out repeatedly with opinion evolution. Now we make an analytical approach of the ordering process. At time t the global density of nodes holding opinion 1 irrespective of their status (active/inactive) is given by ρ1 (t ). Thus we get the ensemble average opinion O(t ) = 0 · (1 − ρ1 (t )) + 1 · ρ1 (t ) = ρ1 (t ). Let ρ1a (t ) and ρ0a (t ) denote the global density of active agents with opinion 1, and opinion 0
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respectively. ρ1a (t ) increases because of two aspects: the internal motivation procedure makes inactive agents with opinion 1 (their density is ρ1 (t ) − ρ1a (t )) become active, while the opinion update procedure makes active agents holding opinion 0 change their opinion to 1. Meanwhile ρ1a (t ) decreases due to two facts: in the dissipation procedure active nodes with opinion 1 lose their activity, and in the opinion update procedure active agents change their opinion from 1 to 0. The probability that an inactive node with opinion 1 becomes active by internal motivation in the time interval [t , t + 1t ] →a is defined as pina (t ). It is obvious that for an inactive agent with opinion 1, the probability that it meets an active neighbor 1 holding the same opinion and becomes active during [t , t + 1t ] is λ1t ρ1a (t ). We assume the underlying networks are homogeneous with the node degree k, so each inactive agent has k opportunities to be motivated by neighbors. In the →a mean-field approach [32,33] the probability pina (t ) is given by 1 →a pina (t ) = 1 − (1 − λ1t ρ1a (t ))k . 1
(1)
Then we conclude ρ (t ) changes as a 1
→a ρ1a (t + 1t ) − ρ1a (t ) = pina (t ) · (ρ1 (t ) − ρ1a (t )) − δρ1a (t )1t + ρ0a (t )ρ1 (t )1t − ρ1a (t ) · (1 − ρ1 (t ))1t . 1
(2)
→a On the right of above equation, the first part pina (t ) · (ρ1 (t ) − ρ1a (t )) is derived from the motivation procedure, while 1 the second part −δρ1a (t )1t is in consideration of dissipation. The third part ρ0a (t )ρ1 (t )1t accounts for the opinion update situation where active agents change their opinion from 0 to 1 following their neighbors holding opinion 1, and the fourth part −ρ1a (t ) · (1 − ρ1 (t ))1t similarly accounts for the opposite situation of opinion update. In the limit 1t → 0, from Eq. (1) we obtain p1ina→a (t ) ≈ λkρ1a (t )1t. Then the transition of ρ1a (t ) is given by ∂t ρ1a (t ) = lim1t →0 (ρ1a (t + 1t )−ρ1a (t ))/1t. Similarly, we can write the corresponding change rate of ρ0a (t ), ρ1 (t ). For simplicity the variables ρ1a (t ), ρ0a (t ),ρ1 (t ) are written as ρ1a , ρ0a ,ρ1 respectively. Therefore we have the following transition equations.
∂t ρ1a = λkρ1a (ρ1 − ρ1a ) − δρ1a + ρ0a ρ1 − ρ1a · (1 − ρ1 ) ∂t ρ0a = λkρ0a (1 − ρ1 − ρ0a ) − δρ0a + ρ1a (1 − ρ1 ) − ρ0a ρ1 ∂t ρ1 = ρ0a ρ1 − ρ1a (1 − ρ1 ).
(3)
At the end of the dynamics, these variables will reach stable values, and both sides of Eq. (3) tend to be zero. Therefore from the right side of Eq. (3), after some elementary manipulations it is concluded that the fixed points should satisfy these equations as below:
λkρ1a (ρ1 − ρ1a ) − δρ1a = 0 λkρ0a (1 − ρ1 − ρ0a ) − δρ0a = 0 ρ0a ρ1 − ρ1a (1 − ρ1 ) = 0.
(4)
In particular, for δ = 0, all agents will become active. It is easy to see that the stationary solution is ρ1a = ρ1 , ρ0a = 1 − ρ1 , and the stable value of ρ1 is closely related to initial conditions. Then we consider only this situation δ > 0. From Eq. (4), we get the stationary solutions
ρ1a ρ1a ρ1a ρ1a
= ρ0a = 0 or = ρ0a = 0.5 − δ/λk, ρ1 = 0.5 or = 0, ρ0a = 1 − δ/λk, ρ1 = 0 or = 1 − δ/λk, ρ0a = 0, ρ1 = 1.
(5)
The stationary solution ρ1a = ρ0a = 0 indicates the frozen state, and here ρ1 is determined by initial opinions. Now we analyze stability of these solutions. Take the solution ρ1a = ρ0a = 0.5 − δ/λk, ρ1 = 0.5 for example. To do so, we introduce small deviations around this stationary solution [28], that is, ρ1 = 0.5 + ε1 + o(ε 2 ), ρ1a = 0.5 − δ/λk + ε1a + o(ε 2 ) and ρ0a = 0.5 − δ/λk + ε0a + o(ε2 ). In this case, omitting high-order infinitesimals, the equations of Eq. (3) become
∂ t ε1 1 − 2δ/λk a ∂t ε1 = 0.5λk − 2δ/λk − δ + 1 ∂t ε0a −0.5λk + 2δ/λk + δ − 1
−0.5 −0.5λk + δ − 0.5 0.5
0.5 0.5 −0.5λk + δ − 0.5
ε1 · ε1a . ε0a
(6)
This stationary solution is stable on condition that the real part in each eigenvalue of this set of linearized equations (6) is negative, that is, the dissipation satisfies this limit 0 < δ < 0.5λk. Under this condition, the balance state where the average densities of two competing opinions become the same, is the steady absorbing state of the system. Analogously, stability of other solutions can be examined. The solutions ρ1a = 0, ρ0a = 1 − δ/λk, ρ1 = 0 and ρ1a = 1 − δ/λk, ρ0a = 0, ρ1 = 1 are always unstable whatever values of the parameters are. Thus if δ > 0.5λk, the stable solution is the point ρ1a = ρ0a = 0; in this situation the dynamics is stopped in a short span of time, for all agents become inactive quickly. Since ρ1 is equivalent to the average opinion, Fig. 1 reveals the final average opinion derived from Eq. (3) with different parameters. From the left plot of Fig. 1, if δ = 0.2 < 0.5λk, the solution ρ1a = ρ0a = 0.25, ρ1 = 0.5 is stable despite the initial average opinion. Contrarily for δ = 0.8 > 0.5λk, the system evolves to the frozen state where all agents lose their activity, and the final average opinion equals to the initial average opinion. From the right plot, we observe a clear transition at the point δ = 0.5λk.
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Fig. 1. Final average opinion given by Eq. (3) with different parameters. The initial density of active agents is 0.02, and k = 4 for both plots, so the initial value of ρ1a is 0.02ρ1 (0). In the left plot, we set λ = 0.2. In the right plot, the initial average opinion is 0.3.
4. Simulation results In order to explore the impact of motivation and dissipation procedure on opinion dynamics, we carry out computer simulations for the dissipative opinion model with different parameters. We assume that at first opinions of agents are fully mixed. At every time step all active agents synchronously motivate their inactive neighbors holding the identical opinion, update their opinions in the light of the voter rule, and then may get dormant. Homogeneous and Barabasi–Albert scale-free networks are used as interaction topology, both of which have the identical average degree k = 4. Scale-free networks are built as follows: 5 nodes constitute a fully connected network initially, and every time a new node is added to the network with 2 edges connecting to 2 different nodes already present in the network. The probability that a new node links to a given existing node is directly proportional to the degree of the old node. The network continues to grow until the size of it reaches N. The degree distribution P (k) decays as a power law following P (k) ∼ k−3 . To investigate the dependence of opinion distribution on system parameters and initial conditions, we study the ensemble average opinion of the system. Fig. 2 illustrates the stable average opinion for our model with the voter rule. With strong internal motivation and nonzero dissipation, the conservation of average magnetization is broken, and the system is driven to a balance state. This is expected, for a potential well at ρ1 = 1/2 exists and it is stable in spite of initial opinions. The results of simulations are consistent with analytical solutions. In homogeneous networks, the threshold value of dissipation below which the system evolves to a balance state, is at δ = 0.5λk; however, in scale-free networks the threshold value becomes larger exceeding 0.5λk¯ on account of the inhomogeneous degree distribution. Increasing the connectivity of networks will give rise to a faster relaxation process. In the left plot, when the amount of agents is too small, the finite-size effect causes a deviation from the balance state. From the right plot the motivation and dissipation parameters play a significant role in the dynamics. Without dissipation, in the end an intermediate state between the balance state and initial magnetization is accessible. If the effect of dissipation remains far beyond motivation, the average opinion approaches that of the standard voter model. However, the intrinsic evolution processes are quite different. In the standard voter model, a finite-size system converges to a consensus state in a typical realization due to random fluctuations, while in our model the dynamics will be frozen quickly, if the dissipation exerts a powerful influence. As shown in Fig. 3, final opinion clusters and relaxation rates are closely related to internal motivation and dissipation parameters. Connected agents sharing the same opinion form an opinion cluster. Small clusters could disappear or merge over time with the emergence of macroscopic-size opinion clusters. For δ = 0, in every realization the system achieves a consensus state with a single macroscopic-size cluster in the end, but the convergence process needs quite a long time. If dissipation is included, at last there are still an extensive number of small-size clusters, as well as several large clusters, and the sizes of large clusters decrease. However, the system tends to stabilize more quickly. The reason for this counterintuitive result is that if dissipation is ignored, more time is needed for incorporation of small clusters. These small clusters have attained group support and are difficult to invade. Moreover, with considerable dissipation, after a slight drop in quantity, opinion clusters level off rapidly. When dissipation stays above zero and below the threshold value, there are always some active agents and opinion updates proceed in the system all along. In a realization of simulation, the consensus state is never reached however long the interaction time is, and the proportions of both opinions have a small fluctuation around 0.5. However, the average opinion and average amount of active agents will become constant after some time steps, and thus we consider the system reaches the final state and stop the simulation. Fig. 4 demonstrates the final distribution of active individuals as a function of node degree. From this chart, it is distinctly realized that the distribution for the number of active nodes versus degree k follows a power law Ak−γ where A is a constant. The power exponents from top to bottom are γ = 2.618, γ = 2.489 and γ = 2.268 respectively. These power exponents fall below the value γ = 3, which is the exponent for degree distribution
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Fig. 2. Final average opinion versus initial average opinion, when m = 10 for both plots. In the left plot, the underlying topology is a homogeneous network, λ = 0.5, and δ = 0.5. In the right plot, a scale-free network is used, and the system size is fixed at 500. Every plot is an average of 100 different simulations.
Fig. 3. Amount of opinion clusters as a function of time, when a scale-free network is used, m = 10 and N = 500. In the beginning opinions are assigned uniformly at random.
Fig. 4. The distribution function of amount of final active nodes versus node degree k, when N = 500 and m = 10. The initial average opinion is 0.5. The results are averaged over 100 different realizations.
of the network. This phenomenon implies agents with a small degree have lower probability of becoming active than high connected agents, for high connected nodes have more opportunities to get in touch with active agents. However, active
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Fig. 5. Total amount of final active nodes versus average degree, when N = 500, m = 10. The initial assignment of opinions is random. We increase the number of edges induced from a new node in every step of network construction.
Fig. 6. Final average opinion versus initial average opinion on scale-free networks, when N = 500, λ = 0.1 and δ = 0.1. The initial activation lasts for a length of time T .
agents whose degrees stay above 10 are still awfully few in number. The network connectivity has a bearing on the final total of active nodes, but this amount will reach a plateau despite the increasing of average degree (Fig. 5). Large dissipation makes more active agents withdraw from the discussion. Now we turn our attention to the significance of extent and duration of initial activation. The initial activation from outside may be transient, or it will persist for a long time in the dynamics. For instance, in an entertainment event, the initial activation takes effect only when the entertainment news is reported, while in a democratic election the outer activation is likely to last throughout the election as a result of frequent speeches of candidates. We suppose the initial activation which always arouses m dormant nodes is performed at every time step of a certain period T . In Fig. 6 we find that in contrast to increasing the acting extent, prolonging acting time of initial activation has more far-reaching outcomes. If dissipation exists no matter how trivial the dissipation is, the final state of the system is independent of m. After activation outside for a long time, the evolution of opinions has been affected. With increasing T , the final average opinion is forced toward the initial average opinion gradually. We also find when T > 200, the system with small dissipation can converge to consensus in a typical run. 5. Conclusions In this paper, we have studied the relation between system dissipation and opinion formation. Agents may lose their interests and keep away from talking about the topic. We put forward a dissipative opinion model in which only active agents can change their opinions. Agents may be motivated by their neighbors, or become inert due to system dissipation. We have analyzed the stability of stationary solutions for our model with the voter rule, and investigated the evolution of average opinion and opinion clusters by simulations. Through the results it is found that in both homogeneous and scale-free networks, when the existent dissipation is small, the average magnetization is conserved no longer, and the average densities of two opinions incline to be equivalent.
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Increasing the length of time for initial activation will make the stable average opinion approach the initial state. Intensity of internal motivation and dissipation has noteworthy influences on opinion clusters and final amount of active nodes. Furthermore, other update rules of spin systems can also be applied in our model, such as the majority rule [10–12], the Sznajd update rule [13–15] and the modified version of Sznajd rule [34]. In future work we will research into the nonlinear relationship between the action of agents and outside circumstances. Actual social data will be used to verify availability of our model. Acknowledgements This work was partially supported by the State Natural Sciences Fund under Grant 60972012, the Beijing Natural Science Foundation under Grant 4102047, the Major Program for Research on Philosophy & Humanity Social Sciences of the Ministry of Education of China under Grant 08WL1101, the Academic Discipline and Postgraduate Education Project of Beijing Municipal Commission of Education, the Service Business of Scientists and Engineers Project under Grant 2009GJA00048, and the Fundamental Research Funds for the Central Universities under Grant 2009YJS007, 2011YJS005. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
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Physica A journal homepage: www.elsevier.com/locate/physa
An opinion formation model with dissipative structure Fei Xiong ∗ , Yun Liu, Zhen-jiang Zhang, Xia-meng Si, Fei Ding School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing, 100044, China Key Laboratory of Communication and Information Systems (Beijing Jiaotong University), Beijing Municipal Commission of Education, Beijing, 100044, China
article
info
Article history: Received 1 September 2010 Received in revised form 7 January 2011 Available online 11 March 2011 Keywords: Opinion dynamics Dissipative structure Voter rule Sociopsychology
abstract We propose an opinion formation model which includes both the influence of internal motivation and dissipation. Active agents adopt their neighbors’ opinions according to a simplified set of rules, but they may gradually lose their interests in the discussion and drop out of it. On the other hand, inert agents can become active by initial activation or internal motivation from neighbors, and participate in the discussion. The initial activation is usually due to occurrence of a social event. The internal motivation, opinion update and dissipation procedure take place simultaneously. We apply the voter rule in our model, and carry out analysis and numerical simulations. Results show if nonzero dissipation stays below a threshold value, the system evolves to a balance state where the average concentration of one opinion is equal to that of the other. With dissipation, a number of small-size opinion clusters exist in the end, but the system gains a fast relaxation rate. The final average opinion is closely related to the dissipation intensity and the length of time for initial activation. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Opinion dynamics which aims to investigate the formation of public opinion provides a valuable insight to social interactions and economic decision making [1,2]. Opinion evolution has attracted great attention at present and many researchers are absorbed in it to explain and forecast stable opinion distributions, which may be consensus, polarization or fragmentation [3,4]. A great many opinion models have been presented, considering that a finite number of agents interact with each other and try to persuade others to adopt their own opinions. In the update process these agents, whose points of view are fully mixed in the beginning, change their opinions according to some rules as predefined [5,6]. Statistical physics which is used to build a connection between microscopic and macroscopic behavior is an effective measure to research into the dynamics. Based on the amount of possible states, the main investigation can be divided into two groups: discrete opinion models and continuous opinion models. In the first group these models (such as the voter model [7–9], the majority model [10–12], the Sznajd model [13–15], the Ising model [16], etc.), describe the situations where a limited number of choices are available for agents. The models of the latter group, like the Deffuant [17,18] and the Hegselmann–Krause [19,20] model, use the concept of bounded confidence and have a compromise process. An extension of the voter model was studied on adaptive network topology which can accelerate the evolution to consensus or separate the network into non-interacting groups [21]. A model with discrete actions and continuous opinions was put forward, assuming that agents change their opinions by using a Bayesian description of how likely their neighbors are correct. The model witnesses the appearance of extremists [22,23]. Stark et al. found if the transition rate of an agent into the opposite state decreased with the time
∗ Corresponding address: Beijing Jiaotong University, School of Electronic and Information Engineering, Room 607, South of No. 9 Teaching-Building, No. 3 Shang Yuan Cun, 100044 Hai Dian District, Beijing, China. Tel.: +86 13811992620. E-mail addresses: [email protected] (F. Xiong), [email protected] (Y. Liu), [email protected] (Z.-j. Zhang). 0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.02.040
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since the last state change, the macroscopic ordering progress would be accelerated despite the slowing-down microscopic dynamics [24]. Selective advantages of two competing strategies were studied in finite structured populations, finding out that the probability for both choices to reach a consensus state is independent of topology [25,26]. As a matter of fact, not all individuals attempt to discuss a certain topic, but instead the topic might be ignored sometimes. Using evolutionary game theory, Ding proposed a model where players might bow out of an unpromising interaction in term of their memories of past payoffs. Opinions always tend to converge, and convergence time is affected by memory length [27]. In Ref. [28], the authors presumed after a voter updated its opinion, it would withdraw from the discussion for a stochastic length of waiting period. In their model the average magnetization is not conserved, and the system is driven towards zero magnetization, independently of initial conditions. Voters do not always update their states, and they may become inert. Inspired by their work, we study the influence of neighbors’ behavior on individual activity in the dynamics. The real society is a dissipative structural system, as well as online communities. The open dissipative system remains far from equilibrium [29–31]. Events which occur in real society are regarded as initial activation of interacting systems. If no coherent influence takes place in users’ actions, public attention declines with the time elapsed, and agents may stop discussing the topic. For example, web users who no longer care about a website are unlikely to login the website again. As the number of active agents decreases continually, ordered states will fail to come forth without motivation. On the other hand, actions of agents can provide a good demonstration for their neighbors, mobilizing their neighbors which have lost interests in the topic. When discussions in local areas cause wide public response, the macroscopic dynamics is then changed and the system may converge to a state in which most agents share an identical opinion. In this paper, we propose a model of opinion dynamics which takes into account the impact of activation and deactivation of agents, illuminating a system where only active agents can take part in discussions. The voter rule is applied to the model, and we research into problems of average magnetization and opinion clusters. Analysis and simulation results show there are some differences in absorbing states compared with the standard voter model. The rest paper is structured as follows. Section 2 presents a new model with the motivation and dissipation. In Section 3 the voter rule is applied in our model, and analysis results are given. Section 4 contains simulation results and discussions about the model. We close the paper in Section 5 with concluding remarks. 2. Opinion model with dissipative structure In most opinion models, agents exchange their opinions about a certain topic based on different rules. Interactions in a closed system will persist all the time until no change comes about, leading to a stable state. A complex network which defines the number of neighbors for each agent, mediates the interactions. Yet in actual society, especially on Internet, conversations about a topic will not continue at all times, and not all agents concentrate on the topic. The attraction of a topic decays with time, and at last the topic may die out. The dynamics is terminated since no one is willing to talk about this. Here we present a dissipative opinion formation model with the following procedures in view of people’s interests in a topic. (i) Initial activation. At first the system is static, and no interaction proceeds between these inactive agents. Then some of the agents are activated by outside circumstance at random. The number of agents activated in the beginning is defined as m. (ii) Opinion update. According to different rules (such as the voter rule, the majority rule, and so on), active agents partake in discussions, and change opinions in accordance with their neighbors. (iii) Internal motivation. Agents are often influenced by neighbors, and they may behave following others. Thus active agents can motivate their inactive like-minded neighbors. If an inert agent holds the same opinion with an active neighbor, the inert agent is driven to be active with a probability λ. People would rather converse with others holding a similar opinion than argue with opponents. (iv) Dissipation. Interests of active agents in the topic decay with time, and they are likely to drop out of discussions. Active agents become dormant with a probability δ . These inactive agents keep silence until the next reactivation. The internal motivation, update and dissipation procedure take place at every time step. The dynamics will go on until all active agents vanish or a stable state is achieved. According to these steps, the evolution of agents’ activity and their opinions are incorporated in the dynamics. Then we will implement the model with the voter rule. 3. Voter rule In the voter model, a system consists of N agents which are positioned at the sites of a certain network. Agents can take either of the two available opinions σ = 0 or σ = 1. In an update event an agent i is randomly selected, together with one of its neighbors. Then agent i adopts the neighbor’s opinion. For the dissipative model where the voter rule is used, only active agents have opportunities to change their opinions. Both the motivation and dissipation procedure are carried out repeatedly with opinion evolution. Now we make an analytical approach of the ordering process. At time t the global density of nodes holding opinion 1 irrespective of their status (active/inactive) is given by ρ1 (t ). Thus we get the ensemble average opinion O(t ) = 0 · (1 − ρ1 (t )) + 1 · ρ1 (t ) = ρ1 (t ). Let ρ1a (t ) and ρ0a (t ) denote the global density of active agents with opinion 1, and opinion 0
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respectively. ρ1a (t ) increases because of two aspects: the internal motivation procedure makes inactive agents with opinion 1 (their density is ρ1 (t ) − ρ1a (t )) become active, while the opinion update procedure makes active agents holding opinion 0 change their opinion to 1. Meanwhile ρ1a (t ) decreases due to two facts: in the dissipation procedure active nodes with opinion 1 lose their activity, and in the opinion update procedure active agents change their opinion from 1 to 0. The probability that an inactive node with opinion 1 becomes active by internal motivation in the time interval [t , t + 1t ] →a is defined as pina (t ). It is obvious that for an inactive agent with opinion 1, the probability that it meets an active neighbor 1 holding the same opinion and becomes active during [t , t + 1t ] is λ1t ρ1a (t ). We assume the underlying networks are homogeneous with the node degree k, so each inactive agent has k opportunities to be motivated by neighbors. In the →a mean-field approach [32,33] the probability pina (t ) is given by 1 →a pina (t ) = 1 − (1 − λ1t ρ1a (t ))k . 1
(1)
Then we conclude ρ (t ) changes as a 1
→a ρ1a (t + 1t ) − ρ1a (t ) = pina (t ) · (ρ1 (t ) − ρ1a (t )) − δρ1a (t )1t + ρ0a (t )ρ1 (t )1t − ρ1a (t ) · (1 − ρ1 (t ))1t . 1
(2)
→a On the right of above equation, the first part pina (t ) · (ρ1 (t ) − ρ1a (t )) is derived from the motivation procedure, while 1 the second part −δρ1a (t )1t is in consideration of dissipation. The third part ρ0a (t )ρ1 (t )1t accounts for the opinion update situation where active agents change their opinion from 0 to 1 following their neighbors holding opinion 1, and the fourth part −ρ1a (t ) · (1 − ρ1 (t ))1t similarly accounts for the opposite situation of opinion update. In the limit 1t → 0, from Eq. (1) we obtain p1ina→a (t ) ≈ λkρ1a (t )1t. Then the transition of ρ1a (t ) is given by ∂t ρ1a (t ) = lim1t →0 (ρ1a (t + 1t )−ρ1a (t ))/1t. Similarly, we can write the corresponding change rate of ρ0a (t ), ρ1 (t ). For simplicity the variables ρ1a (t ), ρ0a (t ),ρ1 (t ) are written as ρ1a , ρ0a ,ρ1 respectively. Therefore we have the following transition equations.
∂t ρ1a = λkρ1a (ρ1 − ρ1a ) − δρ1a + ρ0a ρ1 − ρ1a · (1 − ρ1 ) ∂t ρ0a = λkρ0a (1 − ρ1 − ρ0a ) − δρ0a + ρ1a (1 − ρ1 ) − ρ0a ρ1 ∂t ρ1 = ρ0a ρ1 − ρ1a (1 − ρ1 ).
(3)
At the end of the dynamics, these variables will reach stable values, and both sides of Eq. (3) tend to be zero. Therefore from the right side of Eq. (3), after some elementary manipulations it is concluded that the fixed points should satisfy these equations as below:
λkρ1a (ρ1 − ρ1a ) − δρ1a = 0 λkρ0a (1 − ρ1 − ρ0a ) − δρ0a = 0 ρ0a ρ1 − ρ1a (1 − ρ1 ) = 0.
(4)
In particular, for δ = 0, all agents will become active. It is easy to see that the stationary solution is ρ1a = ρ1 , ρ0a = 1 − ρ1 , and the stable value of ρ1 is closely related to initial conditions. Then we consider only this situation δ > 0. From Eq. (4), we get the stationary solutions
ρ1a ρ1a ρ1a ρ1a
= ρ0a = 0 or = ρ0a = 0.5 − δ/λk, ρ1 = 0.5 or = 0, ρ0a = 1 − δ/λk, ρ1 = 0 or = 1 − δ/λk, ρ0a = 0, ρ1 = 1.
(5)
The stationary solution ρ1a = ρ0a = 0 indicates the frozen state, and here ρ1 is determined by initial opinions. Now we analyze stability of these solutions. Take the solution ρ1a = ρ0a = 0.5 − δ/λk, ρ1 = 0.5 for example. To do so, we introduce small deviations around this stationary solution [28], that is, ρ1 = 0.5 + ε1 + o(ε 2 ), ρ1a = 0.5 − δ/λk + ε1a + o(ε 2 ) and ρ0a = 0.5 − δ/λk + ε0a + o(ε2 ). In this case, omitting high-order infinitesimals, the equations of Eq. (3) become
∂ t ε1 1 − 2δ/λk a ∂t ε1 = 0.5λk − 2δ/λk − δ + 1 ∂t ε0a −0.5λk + 2δ/λk + δ − 1
−0.5 −0.5λk + δ − 0.5 0.5
0.5 0.5 −0.5λk + δ − 0.5
ε1 · ε1a . ε0a
(6)
This stationary solution is stable on condition that the real part in each eigenvalue of this set of linearized equations (6) is negative, that is, the dissipation satisfies this limit 0 < δ < 0.5λk. Under this condition, the balance state where the average densities of two competing opinions become the same, is the steady absorbing state of the system. Analogously, stability of other solutions can be examined. The solutions ρ1a = 0, ρ0a = 1 − δ/λk, ρ1 = 0 and ρ1a = 1 − δ/λk, ρ0a = 0, ρ1 = 1 are always unstable whatever values of the parameters are. Thus if δ > 0.5λk, the stable solution is the point ρ1a = ρ0a = 0; in this situation the dynamics is stopped in a short span of time, for all agents become inactive quickly. Since ρ1 is equivalent to the average opinion, Fig. 1 reveals the final average opinion derived from Eq. (3) with different parameters. From the left plot of Fig. 1, if δ = 0.2 < 0.5λk, the solution ρ1a = ρ0a = 0.25, ρ1 = 0.5 is stable despite the initial average opinion. Contrarily for δ = 0.8 > 0.5λk, the system evolves to the frozen state where all agents lose their activity, and the final average opinion equals to the initial average opinion. From the right plot, we observe a clear transition at the point δ = 0.5λk.
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Fig. 1. Final average opinion given by Eq. (3) with different parameters. The initial density of active agents is 0.02, and k = 4 for both plots, so the initial value of ρ1a is 0.02ρ1 (0). In the left plot, we set λ = 0.2. In the right plot, the initial average opinion is 0.3.
4. Simulation results In order to explore the impact of motivation and dissipation procedure on opinion dynamics, we carry out computer simulations for the dissipative opinion model with different parameters. We assume that at first opinions of agents are fully mixed. At every time step all active agents synchronously motivate their inactive neighbors holding the identical opinion, update their opinions in the light of the voter rule, and then may get dormant. Homogeneous and Barabasi–Albert scale-free networks are used as interaction topology, both of which have the identical average degree k = 4. Scale-free networks are built as follows: 5 nodes constitute a fully connected network initially, and every time a new node is added to the network with 2 edges connecting to 2 different nodes already present in the network. The probability that a new node links to a given existing node is directly proportional to the degree of the old node. The network continues to grow until the size of it reaches N. The degree distribution P (k) decays as a power law following P (k) ∼ k−3 . To investigate the dependence of opinion distribution on system parameters and initial conditions, we study the ensemble average opinion of the system. Fig. 2 illustrates the stable average opinion for our model with the voter rule. With strong internal motivation and nonzero dissipation, the conservation of average magnetization is broken, and the system is driven to a balance state. This is expected, for a potential well at ρ1 = 1/2 exists and it is stable in spite of initial opinions. The results of simulations are consistent with analytical solutions. In homogeneous networks, the threshold value of dissipation below which the system evolves to a balance state, is at δ = 0.5λk; however, in scale-free networks the threshold value becomes larger exceeding 0.5λk¯ on account of the inhomogeneous degree distribution. Increasing the connectivity of networks will give rise to a faster relaxation process. In the left plot, when the amount of agents is too small, the finite-size effect causes a deviation from the balance state. From the right plot the motivation and dissipation parameters play a significant role in the dynamics. Without dissipation, in the end an intermediate state between the balance state and initial magnetization is accessible. If the effect of dissipation remains far beyond motivation, the average opinion approaches that of the standard voter model. However, the intrinsic evolution processes are quite different. In the standard voter model, a finite-size system converges to a consensus state in a typical realization due to random fluctuations, while in our model the dynamics will be frozen quickly, if the dissipation exerts a powerful influence. As shown in Fig. 3, final opinion clusters and relaxation rates are closely related to internal motivation and dissipation parameters. Connected agents sharing the same opinion form an opinion cluster. Small clusters could disappear or merge over time with the emergence of macroscopic-size opinion clusters. For δ = 0, in every realization the system achieves a consensus state with a single macroscopic-size cluster in the end, but the convergence process needs quite a long time. If dissipation is included, at last there are still an extensive number of small-size clusters, as well as several large clusters, and the sizes of large clusters decrease. However, the system tends to stabilize more quickly. The reason for this counterintuitive result is that if dissipation is ignored, more time is needed for incorporation of small clusters. These small clusters have attained group support and are difficult to invade. Moreover, with considerable dissipation, after a slight drop in quantity, opinion clusters level off rapidly. When dissipation stays above zero and below the threshold value, there are always some active agents and opinion updates proceed in the system all along. In a realization of simulation, the consensus state is never reached however long the interaction time is, and the proportions of both opinions have a small fluctuation around 0.5. However, the average opinion and average amount of active agents will become constant after some time steps, and thus we consider the system reaches the final state and stop the simulation. Fig. 4 demonstrates the final distribution of active individuals as a function of node degree. From this chart, it is distinctly realized that the distribution for the number of active nodes versus degree k follows a power law Ak−γ where A is a constant. The power exponents from top to bottom are γ = 2.618, γ = 2.489 and γ = 2.268 respectively. These power exponents fall below the value γ = 3, which is the exponent for degree distribution
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Fig. 2. Final average opinion versus initial average opinion, when m = 10 for both plots. In the left plot, the underlying topology is a homogeneous network, λ = 0.5, and δ = 0.5. In the right plot, a scale-free network is used, and the system size is fixed at 500. Every plot is an average of 100 different simulations.
Fig. 3. Amount of opinion clusters as a function of time, when a scale-free network is used, m = 10 and N = 500. In the beginning opinions are assigned uniformly at random.
Fig. 4. The distribution function of amount of final active nodes versus node degree k, when N = 500 and m = 10. The initial average opinion is 0.5. The results are averaged over 100 different realizations.
of the network. This phenomenon implies agents with a small degree have lower probability of becoming active than high connected agents, for high connected nodes have more opportunities to get in touch with active agents. However, active
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Fig. 5. Total amount of final active nodes versus average degree, when N = 500, m = 10. The initial assignment of opinions is random. We increase the number of edges induced from a new node in every step of network construction.
Fig. 6. Final average opinion versus initial average opinion on scale-free networks, when N = 500, λ = 0.1 and δ = 0.1. The initial activation lasts for a length of time T .
agents whose degrees stay above 10 are still awfully few in number. The network connectivity has a bearing on the final total of active nodes, but this amount will reach a plateau despite the increasing of average degree (Fig. 5). Large dissipation makes more active agents withdraw from the discussion. Now we turn our attention to the significance of extent and duration of initial activation. The initial activation from outside may be transient, or it will persist for a long time in the dynamics. For instance, in an entertainment event, the initial activation takes effect only when the entertainment news is reported, while in a democratic election the outer activation is likely to last throughout the election as a result of frequent speeches of candidates. We suppose the initial activation which always arouses m dormant nodes is performed at every time step of a certain period T . In Fig. 6 we find that in contrast to increasing the acting extent, prolonging acting time of initial activation has more far-reaching outcomes. If dissipation exists no matter how trivial the dissipation is, the final state of the system is independent of m. After activation outside for a long time, the evolution of opinions has been affected. With increasing T , the final average opinion is forced toward the initial average opinion gradually. We also find when T > 200, the system with small dissipation can converge to consensus in a typical run. 5. Conclusions In this paper, we have studied the relation between system dissipation and opinion formation. Agents may lose their interests and keep away from talking about the topic. We put forward a dissipative opinion model in which only active agents can change their opinions. Agents may be motivated by their neighbors, or become inert due to system dissipation. We have analyzed the stability of stationary solutions for our model with the voter rule, and investigated the evolution of average opinion and opinion clusters by simulations. Through the results it is found that in both homogeneous and scale-free networks, when the existent dissipation is small, the average magnetization is conserved no longer, and the average densities of two opinions incline to be equivalent.
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Increasing the length of time for initial activation will make the stable average opinion approach the initial state. Intensity of internal motivation and dissipation has noteworthy influences on opinion clusters and final amount of active nodes. Furthermore, other update rules of spin systems can also be applied in our model, such as the majority rule [10–12], the Sznajd update rule [13–15] and the modified version of Sznajd rule [34]. In future work we will research into the nonlinear relationship between the action of agents and outside circumstances. Actual social data will be used to verify availability of our model. Acknowledgements This work was partially supported by the State Natural Sciences Fund under Grant 60972012, the Beijing Natural Science Foundation under Grant 4102047, the Major Program for Research on Philosophy & Humanity Social Sciences of the Ministry of Education of China under Grant 08WL1101, the Academic Discipline and Postgraduate Education Project of Beijing Municipal Commission of Education, the Service Business of Scientists and Engineers Project under Grant 2009GJA00048, and the Fundamental Research Funds for the Central Universities under Grant 2009YJS007, 2011YJS005. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
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