Opinion Dynamics on Triad Scale Free Network

Transcription

1 Opinion Dynamics on Triad Scale Free Network Li Qianqian 1 Liu Yijun 1,* Tian Ruya 1,2 Ma Ning 1,2 1 Institute of Policy and Management, Chinese Academy of Sciences, Beijing , China lqqcindy@gmail.com, yijunliu@casipm.ac.cn 2 Graduate University of Chinese Academy of Sciences, Beijing , China tianruya@126.com, maning @163.com Abstract. In this paper, we investigate the opinion dynamics model of social impact theory on triad scale free network with power law degree distribution and tunable clustering coefficient. Based on this opinion dynamic model, we try to observe the clustering coefficient influence on opinion formation by adjusting the triad formation parameter. Simulation result shows that by adjusting triad scale free network parameters, a large clustering coefficient favors development of a consensus. In particular, when in the system with initial opinion proportion of +1, p + =0.5, a consensus seems to be never reached for triad scale free network with any clustering coefficient. Keywords: opinion dynamics, social impact theory, triad scale free network 1 Introduction Recently, there has been a growing interest in study of complex phenomena in social field, whereas statistical physics, mathematics, computer science are very popular research tools. In particular, one of very significant research area is opinion dynamics, which explain how the society reach consensus. The dynamics of opinion formation is a non-linear phenomenon, both personal view interaction and collective behavior emergence play important role in the underlying mechanism. The individual s opinion may likely be affected by its nearest neighbors. In fact, some classic opinion dynamical models were based on personal interaction between their neighbors. For example, Sznajd model [1, 2], Deffuantmodel [3],KHmodel [4] and Galam s Majority Rule model [5]. Most of the time, the topological properties of network govern the dynamical behavior of complex system. Studies on topological structure have been an intriguing issue. A few example include food web [6], actor collaborating network [7], * Liu Yijun: corresponding author

2 paper citation network [8], stock market network [9]. At the end of 20 th century, there have been two milestone progress in network science: 1) Watts and Strogatzs proposed WS small-world network model [10], which explains those systems having highly clustered and small characteristic path length; 2) Barabási and Albert proposed BA scale free network model [8], which described the networks with power-law degree distribution. Since then, some of opinion dynamical models, such as Sznajd model, Deffuant model, KH model et al. have been studied in the context of complex network. Social network often have highly clustering coefficient. If person A knows B and C, then person B and C are more likely to know each other. Moreover, many empirical results discovered the fat-tail property in human behavior [11] The WS model shows a high clustering but without the power-law degree distribution, while the BA model with the scale free nature does not possess the high clustering. Therefore, Holme and Kim proposed a triad scale free network model [12], which has both the perfect power-law degree distribution and highly clustering. We regard this triad scale free network is more suitable for modeling the internet opinion formation net: 1) when there is a hot spot in public opinion, many netizens will post their view. The participating netizens will enlarge the network, which shows the growing property of the network; 2) views of opinion leaders will attract the public attention and influence the attitudes and behavior change of their followers, which can be modeled as the preferential attachment mechanism of BA scale free network, that is, a rich-get-richer phenomenon; 3) the communication between the followers of opinion leaders will increase the highly clustering of network. For example, you can always see the reply 18 floor, etc.. In this paper, we investigate the opinion formation model of social impact theory based on triad scale free network. In this model, we consider both the neighborhood opinion and their influence strength. 2 Triad Scale Free Network Holme and Kim introduced triad formation mechanism to increase clustering coefficient of network. Combining triad formation and BA scale free network could generate both highly clustering and power-law degree distribution.

3 Fig. 1.preferential attachment and triad formation. (a) preferential attachment step: the new added vertex v attaches to vertex u with a probability to its degree; (b) triad formation step: the new added vertex v attaches to w in the neighborhood of one linked to in the previous preferential attachment step. We below describe the model of triad scale free network [12] which established based on BA model [8] but has high clustering. Initial condition: starting with m 0 of vertices; Growth: we add a new vertex v with m( m 0 ) edges that link the new vertex to m different vertices already present in system; Preferential Attachment(PA): we assume that the probability Π that a new vertex v will be connected to vertex u depends on the connectivity k u of that vertex, so that Π v = k v ; j k j Triad Formation(TF): if we already link v to u in the previous PA step, then add one more edge from v to a randomly chosen neighbor of u. If all neighbors of u were already connected to v, do a PA step instead. Briefly speaking, when add a new vertex to the existing network, we first perform one PA step, and then perform a TF step with probability p t or a PA step with probability (1-p t ). When p t = 0, this model reduces to the original BA model.

4 Fig. 2.Example of a scale free network. The number of Nodes is 50 with triad formation probability p t = 0.6 and m 0 = 5, m = 2. So that the new added vertex is linked twice. In order to preserve the clarity of the network, the size of the network has been to kept small. This plot has been realized with the ORA software [13]. 3 Opinion dynamical model In complex network, we use node to represent individual(agent), edge linked one node to another represent information propagation relationship(we consider undirected network only). We set up a triad scale free network according given parameters. Once the network has been completely constructed. We establish social impact opinion formation model the network. This opinion formation model was based on psychology theory of social impact [14] which describes how individuals feel the presence of their peers and how they in turn influence other individuals. Ho lyst, Kacperski and Schweitzer developed a opinion formation model based on social impact theory [15]. We modify the impact function I i of the opinion dynamical model. The initial condition is a population of N agents. Agent i is characterized by an opinion σ i = ±1 and by clustering coefficient c i [0,1]. σ i = 1 represents individual i supports a viewpoint, whereas σ i = 1 represents individual i opposes a viewpoint. The total impact I i that an individual experiences from its neighbors(social environment also) is I i = σ i c i N i j=1 c j σ j (1) Where N i :the number of i-th agent, c i : the clustering coefficient of i-th agent. I i > 0 represents individual i gets supportiveness from its neighbors, whereas I i < 0 represents the neighbors of individual i have opposite opinion with i.

5 Opinions of individuals may change asynchronously(asynchronous dynamics) in discrete time steps according to the rule e I i /T σ i (t + 1) = σ i(t), I i 0,with probability e I i /T +e I i /T (2) e σ i (t), I i < 0,with probability I i /T e I i /T +e I i /T Where the parameter T is interpreted as a social temperature describing a degree of randomness (disturbing factors) in the behavior of individuals. 4 Numerical simulations At first, we establish a triad scale free network and update opinion of individuals asynchronously, i.e., we choose an agent randomly, calculate its social environment I i and update its opinion according to the above rule(see Equation(2)). (a) (b)

6 Fig. 3. relationship between opinion dynamics and triad formation probability p t. (a) T=2, N=500, m=3, p + =0.6 of opinion +1; (b) T=2, N=500, m=3, p + =0.4 of opinion +1; (c) T=2, N=500, m=3, p + =0.5, of opinion +1. The results are obtained by averaging over 100 independent realizations. In order to describe the evolution process of the model, we employ a parameter, average opinion: σ = N i=1 σ i N When N + = N = N/2, σ =0. In Fig.3, we present the average opinion dynamic process for different triad formation probability. From the Fig.(3)(a) we can find when the initial opinion proportion of +1, p + >0.5, the average opinion dynamic result on a high clustering coefficient scale free network is larger than the result on a low clustering coefficient scale free network. From the Fig.(3)(b) we can find when the initial opinion proportion of +1, p + <0.5, the average opinion dynamic result on a high clustering coefficient scale free network is smaller than the result on a low clustering coefficient scale free network. Especially, from the Fig.(3)(c), we find that when the initial opinion proportion of +1, p + =0.5, the opinion dynamic result is approximately to 0 on a triad scale free network with any clustering coefficient. (c) (3) 5 Conclusion In the present work we have constructed a triad scale free network in order to approach the real network topology of online information propagation. The decision updating is governed by a social environment. By simulating this opinion formation model on a triad scale free network, we observe that in scale free network, a large clustering coefficient scale free network favors the development

7 of dominant opinion when the initial opinion proportion +1, p However, when p + = 0.5, non-consensus can be observed. Acknowledgments. The authors gratefully acknowledge the support of National Natural Science Foundation of China (Grant No ) and Projects of Young Scientist Funds of Institute of Policy and Management, Chinese Academy of Sciences (Y200571Q01). References 1. Sznajd-Weron, K., Sznajd, J.: Opinion evolution in closed community. Int J Mod Phys C 11, (2000) 2. Sznajd-Weron, K.: Sznajd model and its applications. Arxiv preprint physics/ (2005) 3. Deffuant, G., Neau, D., Amblard, F., Weisbuch, G.: Mixing beliefs among interacting agents. Adv Complex Syst 3,87-98 (2000) 4. Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence: models, analysis and simulation. Journal of Artificial Societies and Social Simulation 5,(2002) 5. Galam, S., Zucker, J.D.: From individual choice to group decision-making. Physica A 287, (2000) 6. Camacho, J., Guimerà, R., Nunes Amaral, L.A.: Robust patterns in food web structure. Phys Rev Lett 88, (2002) 7. Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys Rev E 64, (2001) 8. Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, (1999) 9. Bonanno, G., Caldarelli, G., Lillo, F., Mantegna, R.N.: Topology of correlation-based minimal spanning trees in real and model markets. Phys Rev E 68,(2003) 10. Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, (1998) 11. Han, X.P., Wang, B.H., Zhou, T.: Researches of human dynamics. Complex Systems and Complexity Science 7,(2010) 12. Holme, P., Kim, B.J.: Growing scale-free networks with tunable clustering. Phys Rev E 65,(2002) 13. Carley, K.M., Reminga, J.: Ora: Organization risk analyzer. DTIC Document (2004) 14. Latane, B.: The psychology of social impact. American psychologist 36,343 (1981) 15. LYST, J.A.H.O., Kacperski, K., Schweitzer, F.: Social impact models of opinion dynamics. Annual reviews of computational physics 9, (2002)