双重社会强化谣言传播模型及稳定性分析
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摘要
谣言大规模扩散给社会造成了不良影响甚至严重危害,引发了学术界和政府的高度关注.通过分析群体正负双重社会强化效用对个体传播行为的影响,构建了谣言传播动力学模型,计算了平衡点和基本再生数,并运用Lyapunov函数法、Routh-Hurwitz稳定判据与复合矩阵等方法讨论了平衡点渐进稳定性.理论分析证明,当基本再生数不大于1时,谣言迅速消失,无谣言平衡点全局渐进稳定,否则谣言传播扩散,非零谣言平衡点全局渐进稳定.新浪微博仿真实验发现,在双重社会强化效用影响下,提高个体对谣言独立甄别能力有助于谣言由大规模扩散向迅速消失趋势转移;只有当抑制谣言传播的社会负强化足够大时,才能迅速终止谣言持续传播.
Rumor spreading has bad influence on our society and even leads serious damage. Researchers and governments have paid much attention to rumor spreading.This paper analyzes the effect of positive and negative double social reinforcements on the individual propagation behaviour, and proposes the rumor spreading dynamic model. The two equilibriums and the basic reproduction number are calculated. And then the asymptotical stability of the equilibriums is proved with the methods of Lyapunov functions, Routh-Hurwitz stability criterion and compound matrices. The theoretical analysis shows that if the basic reproduction number is not more than 1,the rumor will disappear quickly and the rumor-free equilibrium is globally asymptotically stable. Otherwise, the rumor will spread and the nonzero rumor equilibrium is globally asymptotically stable. Furthermore, the Sina Weibo numerical simulation results indicate that due to the effect of double social reinforcements, the enhancement of ability to identify the rumor independently is helpful for the tendency of rumor spreading process changing from the large-scale spreading to disappearing immediately. Lastly, it is also confirmed that only when the negative social reinforcement that suppresses rumor spreading is strong enough, the rumor spreading can be controlled.
Rumor spreading has bad influence on our society and even leads serious damage. Researchers and governments have paid much attention to rumor spreading.This paper analyzes the effect of positive and negative double social reinforcements on the individual propagation behaviour, and proposes the rumor spreading dynamic model. The two equilibriums and the basic reproduction number are calculated. And then the asymptotical stability of the equilibriums is proved with the methods of Lyapunov functions, Routh-Hurwitz stability criterion and compound matrices. The theoretical analysis shows that if the basic reproduction number is not more than 1,the rumor will disappear quickly and the rumor-free equilibrium is globally asymptotically stable. Otherwise, the rumor will spread and the nonzero rumor equilibrium is globally asymptotically stable. Furthermore, the Sina Weibo numerical simulation results indicate that due to the effect of double social reinforcements, the enhancement of ability to identify the rumor independently is helpful for the tendency of rumor spreading process changing from the large-scale spreading to disappearing immediately. Lastly, it is also confirmed that only when the negative social reinforcement that suppresses rumor spreading is strong enough, the rumor spreading can be controlled.
引文
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[28]Zhang J,Liu B,Tang J,et al.Social influence locality for modeling retweeting behaviors.International Joint Conference on Artificial Intelligence.AAAI Press,2013,2761-2767.
[2]Wang W,Tang M,Zhang H F,et al.Epidemic spreading on complex networks with general degree and weight distributions.Physical Review E,2014,90(4):042803.
[3]Zhang Y M,Tang C S,Li W G.Cooperative and competitive dynamics model for information propagation in online social networks.Journal of Applied Mathematics,2014,2014(2),Article ID 610382,12 pages.
[4]Moreno Y,Nekovee M,Pacheco A F.Dynamics of rumor spreading in complex networks.Physical Review E,2004,69(6):066130.
[5]Daley D J,Kendall D G.Stochastic rumours.IMA Journal of Applied Mathematics,1965,1(1):42—55.
[6]Maki D P,Thompson M.Mathematical Models and Applications.New Jersey:Prentice-Hall,1973.
[7]Sudbury A.The proportion of the population never hearing a rumour.Journal of Applied Probability,1985,22(2):443-446.
[8]张亚明,唐朝生,李伟钢.在线社交网络谣言传播兴趣衰减与社会强化机制研究.情报学报,2015,34(8):833-844.(Zhang Y M,Tang C S,Li W G.Research on internet attenuation and social reinforcement mechanism for rumor spreading in online social networks.Journal of the China Society for Scientific and Technical Information,2015,34(8):833-844.)
[9]Wang W,Tang M,Shu P,et al.Dynamics of social contagions with heterogeneous adoption thresholds:crossover phenomena in phase transition.New Journal of Physics,2016,18(1):013029.
[10]House T.Modelling behavioural contagion.Journal of the Royal Society Interface,2011,8(59):909-912.
[11]Romero D M,Meeder B,Kleinberg J.Differences in the mechanics of information diffusion across topics:idioms,political hashtags,and complex contagion on twitter.Proceedings of the20th International Conference on World Wide Web.ACM,2011,695-704.
[12]Zhao L,Cui H,Qiu X,et al.SIR rumor spreading model in the new media age.Physica A:Statistical Mechanics and Its Applications,2013,392(4):995-1003.
[13]Wang Y,Yang X,Han Y,et al.Rumor spreading model with trust mechanism in complex social networks.Communications in Theoretical Physics,2013,59(4):510-516.
[14]Afassinou K.Analysis of the impact of education rate on the rumor spreading mechanism.Physica A:Statistical Mechanics and Its Applications,2014,414:43-52.
[15]Huang J,Jin X.Preventing rumor spreading on small-world networks.Journal of Systems Science and Complexity,2011,24(3):449-456.
[16]Allport G W,Postman L.An analysis of rumor.Public Opinion Quatrerly,1947,10(4):501-517.
[17]Xia L L,Jiang G P,Song B,et al.Rumor spreading model considering hesitating mechanism in complex social networks.Physica A:Statistical Mechanics and Its Applications,2015,437:295-303.
[18]Wang W,Shu P,Zhu Y X,et al.Dynamics of social contagions with limited contact capacity.Chaos:An Interdisciplinary Journal of Nonlinear Science,2015,25(10):103102.
[19]Wang W,Tang M,Zhang H F,et al.Dynamics of social contagions with memory of nonredundant information.Physical Review E,2015,92(1):012820.
[20]张江波,洪奕光.有限信域观点动力学建模和动态分析.系统科学与数学,2016,36(3):319-335.(Zhang J B,Hong Y G.Modeling and dynamic analysis of bounded confidence opinion dynamics.Journal of Systems Science and Mathematical Sciences,2016,36(3):319-335.)
[21]Karrer B,Newman M E J.Message passing approach for general epidemic models.Physical Review E,2010,82(1):016101.
[22]Centola D.The spread of behavior in an online social network experiment.Science,2010,329(5996):1194-1197.
[23]Zheng M H,Lu L Y,Zhao M.Spreading in online social networks:The role of social reinforcement.Physical Review E,2013,88(1):012818.
[24]阚佳倩,谢家荣,张海峰.社会强化效应及连边权重对网络信息传播的影响分析.电子科技大学学报,2014,43(1):21-25.(Kan J Q,Xie J R,Zhang H F.Impacts of social reinforcement and edge weight on the spreading of information in networks.Journal of University of Elect,ronic Science and Technology of China,2014,43(1):21-25.)
[25]Korobeinikov A,Wake G C.Lyapunov functions and global stability for SIR,SIRS,and SIS epidemiological models.Applied Mathematics Letters,2002,15(8):955-960.
[26]Muldowney J S.Compound matrices and ordinary differential equations.Rocky Mountain Journal of Mathematics,1990,20(4):857-872.
[27]Li M Y,Muldowney J S.A geometric approach to global-stability problems.SIAM Journal on Mathematical Analysis,1996,27(4):1070-1083.
[28]Zhang J,Liu B,Tang J,et al.Social influence locality for modeling retweeting behaviors.International Joint Conference on Artificial Intelligence.AAAI Press,2013,2761-2767.