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Opinion dynamics with the increasing peer pressure and prejudice on the signed graph

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Abstract

In this paper, we propose the opinion dynamics model with the increasing peer pressure and the stubborn agents. Cooperation and competition between individuals are considered simultaneously in a social network. Similar to the signed DeGroot model, we adopt a weighted average update rule in our model. We derive conditions under which opinions converge to a fixed opinion distribution. In particular, we find conditions under which opinions reach consensus or polarization (bipartite consensus). Two examples are provided to illustrate the effectiveness of the obtained results.

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References

  1. Liu, F.: Dynamical processes on social networks: modeling, analysis, and control. Ph.D. thesis, Technische Universität München (2019)

  2. Yang, L.X., Li, P., Yang, X., Wu, Y., Tang, Y.Y.: On the competition of two conflicting messages. Nonlinear Dyn. 91(3), 1853–1869 (2018)

    MATH  Google Scholar 

  3. Altafini, C., Ceragioli, F.: Signed bounded confidence models for opinion dynamics. Automatica 93, 114–125 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Tang, Y., Qian, F., Gao, H., Kurths, J.: Synchronization in complex networks and its application—a survey of recent advances and challenges. Annu. Rev. Control 38(2), 184–198 (2014)

    Google Scholar 

  5. Bindel, D., Kleinberg, J., Oren, S.: How bad is forming your own opinion? Games Econ. Behav. 92, 248–265 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Golub, B., Jackson, M.O.: Naive learning in social networks and the wisdom of crowds. Am. Econ. J. Microecon. 2(1), 112–49 (2010)

    Google Scholar 

  7. Xiao, Y., Chen, D., Wei, S., Li, Q., Wang, H., Xu, M.: Rumor propagation dynamic model based on evolutionary game and anti-rumor. Nonlinear Dyn. 95(1), 523–539 (2019)

    Google Scholar 

  8. Friedkin, N.E., Johnsen, E.C.: Social Influence Network Theory: A Sociological Examination of Small Group Dynamics, vol. 33. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  9. Frasca, P., Tarbouriech, S., Zaccarian, L.: Hybrid models of opinion dynamics with opinion-dependent connectivity. Automatica 100, 153–161 (2019)

    MathSciNet  MATH  Google Scholar 

  10. Zhang, W., Han, Q.L., Tang, Y., Liu, Y.: Sampled-data control for a class of linear time-varying systems. Automatica 103, 126–134 (2019)

    MathSciNet  MATH  Google Scholar 

  11. Anderson, B.D., Ye, M.: Recent advances in the modelling and analysis of opinion dynamics on influence networks. Int. J. Autom. Comput. 16(2), 129–149 (2019)

    Google Scholar 

  12. Garofalo, F., Iudice, F.L., Napoletano, E.: Herding as a consensus problem. Nonlinear Dyn. 92(1), 25–32 (2018)

    Google Scholar 

  13. Parsegov, S.E., Proskurnikov, A.V., Tempo, R., Friedkin, N.E.: Novel multidimensional models of opinion dynamics in social networks. IEEE Trans. Autom. Control 62(5), 2270–2285 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Angeli, D., Manfredi, S.: Criteria for asymptotic clustering of opinion dynamics towards bimodal consensus. Automatica 103, 230–238 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Girejko, E., Machado, L., Malinowska, A.B., Martins, N.: Krause’s model of opinion dynamics on isolated time scales. Math. Methods Appl. Sci. 39(18), 5302–5314 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Amelkin, V., Bullo, F., Singh, A.K.: Polar opinion dynamics in social networks. IEEE Trans. Autom. Control 62(11), 5650–5665 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Liu, F., Xue, D., Hirche, S., Buss, M.: Polarizability, consensusability and neutralizability of opinion dynamics on coopetitive networks. IEEE Trans. Autom. Control 64(8), 3339–3346 (2019)

    MathSciNet  MATH  Google Scholar 

  18. Altafini, C.: Consensus problems on networks with antagonistic interactions. IEEE Trans. Autom. Control 58(4), 935–946 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Wu, X., Tang, Y., Cao, J., Mao, X.: Stability analysis for continuous-time switched systems with stochastic switching signals. IEEE Trans. Autom. Control 63(9), 3083–3090 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Liu, Q., Wang, X.: Opinion dynamics with similarity-based random neighbors. Sci. Rep. 3, 2968 (2013)

    Google Scholar 

  21. Etesami, S.R., Basar, T.: Game-theoretic analysis of the Hegselmann–Krause model for opinion dynamics in finite dimensions. IEEE Trans. Autom. Control 60(7), 1886–1897 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Friedkin, N.E.: The problem of social control and coordination of complex systems in sociology: a look at the community cleavage problem. IEEE Control Syst. Mag. 35(3), 40–51 (2015)

    MathSciNet  Google Scholar 

  23. Dandekar, P., Goel, A., Lee, D.T.: Biased assimilation, homophily, and the dynamics of polarization. Proc. Natl. Acad. Sci. 110(15), 5791–5796 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Blondel, V.D., Hendrickx, J.M., Tsitsiklis, J.N.: Continuous-time average-preserving opinion dynamics with opinion-dependent communications. SIAM J. Control Optim. 48(8), 5214–5240 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Blondel, V.D., Hendrickx, J.M., Tsitsiklis, J.N.: On Krause’s multi-agent consensus model with state-dependent connectivity. IEEE Trans. Autom. Control 54(11), 2586–2597 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Tang, Y., Zhang, D., Ho, D.W., Qian, F.: Tracking control of a class of cyber-physical systems via a flexray communication network. IEEE Trans. Cybern. 49(4), 1186–1199 (2018)

    Google Scholar 

  27. Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56(4), 577–621 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Mirtabatabaei, A., Bullo, F.: Opinion dynamics in heterogeneous networks: convergence conjectures and theorems. SIAM J. Control Optim. 50(5), 2763–2785 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Nedić, A., Touri, B.: Multi-dimensional Hegselmann–Krause dynamics. In: 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), pp. 68–73 (2012)

  30. Bhawalkar, K., Gollapudi, S., Munagala, K.: Coevolutionary opinion formation games. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp. 41–50 (2013)

  31. Chae, D.H., Clouston, S., Hatzenbuehler, M.L., Kramer, M.R., Cooper, H.L., Wilson, S.M., Stephens-Davidowitz, S.I., Gold, R.S., Link, B.G.: Association between an internet-based measure of area racism and black mortality. PLoS ONE 10(4), e0122,963 (2015)

    Google Scholar 

  32. Stephens-Davidowitz, S., Pabon, A.: Everybody Lies: Big Data, New Data, and What the Internet Can Tell Us About Who We Really Are. HarperCollins, New York (2017)

    Google Scholar 

  33. DeGroot, M.H.: Reaching a consensus. J. Am. Stat. Assoc. 69(345), 118–121 (1974)

    MATH  Google Scholar 

  34. Ghaderi, J., Srikant, R.: Opinion dynamics in social networks with stubborn agents: equilibrium and convergence rate. Automatica 50(12), 3209–3215 (2014)

    MathSciNet  MATH  Google Scholar 

  35. Milanese, M., Tempo, R., Vicino, A.: Optimal error predictors for economic models. Int. J. Syst. Sci. 19(7), 1189–1200 (1988)

    MathSciNet  MATH  Google Scholar 

  36. Ravazzi, C., Frasca, P., Tempo, R., Ishii, H.: Ergodic randomized algorithms and dynamics over networks. IEEE Trans. Control Netw. Syst. 2(1), 78–87 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Joshi, A., Dencker, J.C., Franz, G., Martocchio, J.J.: Unpacking generational identities in organizations. Acad. Manag. Rev. 35(3), 392–414 (2010)

    Google Scholar 

  38. Tian, Y., Wang, L.: Opinion dynamics in social networks with stubborn agents: an issue-based perspective. Automatica 96, 213–223 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Semonsen, J., Griffin, C., Squicciarini, A., Rajtmajer, S.: Opinion dynamics in the presence of increasing agreement pressure. IEEE Trans. Cybern. 49(4), 1270–1278 (2018)

    Google Scholar 

  40. Ye, M., Liu, J., Anderson, B.D., Yu, C., Basar, T.: Evolution of social power in social networks with dynamic topology. IEEE Trans. Autom. Control 63(11), 3793–3808 (2018)

    MathSciNet  MATH  Google Scholar 

  41. Proskurnikov, A.V., Matveev, A.S., Cao, M.: Opinion dynamics in social networks with hostile camps: consensus vs. polarization. IEEE Trans. Autom. Control 61(6), 1524–1536 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Flache, A., Macy, M.W.: Small worlds and cultural polarization. J. Math. Sociol. 35(1–3), 146–176 (2011)

    MathSciNet  Google Scholar 

  43. Xue, D., Hirche, S., Cao, M.: Opinion behavior analysis in social networks under the influence of coopetitive media. IEEE Trans. Netw. Sci. Eng. (2019). https://doi.org/10.1109/TNSE.2019.2894565

    Article  Google Scholar 

  44. Xia, W., Cao, M., Johansson, K.H.: Structural balance and opinion separation in trust–mistrust social networks. IEEE Trans. Control Netw. Syst. 3(1), 46–56 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Meng, Z., Shi, G., Johansson, K.H., Cao, M., Hong, Y.: Behaviors of networks with antagonistic interactions and switching topologies. Automatica 73, 110–116 (2016)

    MathSciNet  MATH  Google Scholar 

  46. Yang, W., Wang, Y.W., Xiao, J.W., Liu, Z.W.: Coordination of networked delayed singularly perturbed systems with antagonistic interactions and switching topologies. Nonlinear Dyn. 89(1), 741–754 (2017)

    MathSciNet  MATH  Google Scholar 

  47. Ye, M.: Opinion Dynamics and the Evolution of Social Power in Social Networks. Springer, Berlin (2019)

    Google Scholar 

  48. Valcher, M.E., Misra, P.: On the consensus and bipartite consensus in high-order multi-agent dynamical systems with antagonistic interactions. Syst. Control Lett. 66, 94–103 (2014)

    MathSciNet  MATH  Google Scholar 

  49. Altafini, C., Lini, G.: Predictable dynamics of opinion forming for networks with antagonistic interactions. IEEE Trans. Autom. Control 60(2), 342–357 (2014)

    MathSciNet  MATH  Google Scholar 

  50. Jiang, Y., Zhang, H., Chen, J.: Sign-consensus over cooperative–antagonistic networks with switching topologies. Int. J. Robust Nonlinear Control 28(18), 6146–6162 (2018)

    MathSciNet  MATH  Google Scholar 

  51. Jiang, Y., Zhang, H., Chen, J.: Sign-consensus of linear multi-agent systems over signed directed graphs. IEEE Trans. Ind. Electron. 64(6), 5075–5083 (2016)

    Google Scholar 

  52. Noutsos, D.: On Perron–Frobenius property of matrices having some negative entries. Linear Algebra Appl. 412(2–3), 132–153 (2006)

    MathSciNet  MATH  Google Scholar 

  53. Bapna, R., Umyarov, A.: Do your online friends make you pay a randomized field experiment on peer influence in online social networks. Manag. Sci. 61(8), 1902–1920 (2015)

    Google Scholar 

  54. Harakeh, Z., Vollebergh, W.A.: The impact of active and passive peer influence on young adult smoking: an experimental study. Drug Alcohol Depend. 121(3), 220–223 (2012)

    Google Scholar 

  55. Chen, X.: Culture, peer interaction, and socioemotional development. Child. Dev. Perspect. 6(1), 27–34 (2012)

    Google Scholar 

  56. Hou, Y., Li, J., Pan, Y.: On the laplacian eigenvalues of signed graphs. Linear Multilinear Algebra 51(1), 21–30 (2003)

    MathSciNet  MATH  Google Scholar 

  57. Xu, Z., Liu, J., Basar, T.: On a modified Degroot–Friedkin model of opinion dynamics. In: 2015 American Control Conference (ACC), pp. 1047–1052 (2015)

  58. Tian, Y., Jia, P., Mirtabatabaei, A., Wang, L., Friedkin, N.E., Bullo, F.: Social power evolution in influence networks with stubborn individuals (2019). arXiv:1901.08727

  59. Agaev, R., Chebotarev, P.: On the spectra of nonsymmetric Laplacian matrices. Linear Algebra Appl. 399, 157–168 (2005)

    MathSciNet  MATH  Google Scholar 

  60. Ren, W., Beard, R.W.: Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control 50(5), 655–661 (2005)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under grant 61703003, 61873294, 61873230, in part by the Research Fund for Distinguished Young Scholars of Anhui Province under grant 1908085J04, in part by the National Natural Science Foundation of Anhui under grant 1708085QA16, in part by the Top Talent Project of Department of Anhui Education under grant gxgwfx2018038, in part by the Top Talent Project of Anhui Polytechnic University under grant 2017BJRC012, 2018JQ01, 2016BJRC009.

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Authors and Affiliations

  1. Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education and School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000, Anhui, People’s Republic of China

    Guang He

  2. Department of Mathematics, Yangzhou University, Yangzhou, 225002, Jiangsu, People’s Republic of China

    Wenbing Zhang

  3. College of Information Science and Technology, Donghua University, Shanghai, 201620, People’s Republic of China

    Jing Liu

  4. School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000, Anhui, People’s Republic of China

    Haoyue Ruan

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  1. Guang He
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Correspondence to Guang He.

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He, G., Zhang, W., Liu, J. et al. Opinion dynamics with the increasing peer pressure and prejudice on the signed graph. Nonlinear Dyn 99, 3421–3433 (2020). https://doi.org/10.1007/s11071-020-05473-1

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