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

  • Guang HeEmail author
  • Wenbing Zhang
  • Jing Liu
  • Haoyue Ruan
Original paper

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.

Keywords

Opinion dynamics Peer pressure Prejudice Bipartite consensus Signed graph 

Notes

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.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Liu, F.: Dynamical processes on social networks: modeling, analysis, and control. Ph.D. thesis, Technische Universität München (2019)Google Scholar
  2. 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)zbMATHCrossRefGoogle Scholar
  3. 3.
    Altafini, C., Ceragioli, F.: Signed bounded confidence models for opinion dynamics. Automatica 93, 114–125 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Bindel, D., Kleinberg, J., Oren, S.: How bad is forming your own opinion? Games Econ. Behav. 92, 248–265 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)zbMATHCrossRefGoogle Scholar
  9. 9.
    Frasca, P., Tarbouriech, S., Zaccarian, L.: Hybrid models of opinion dynamics with opinion-dependent connectivity. Automatica 100, 153–161 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 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)CrossRefGoogle Scholar
  12. 12.
    Garofalo, F., Iudice, F.L., Napoletano, E.: Herding as a consensus problem. Nonlinear Dyn. 92(1), 25–32 (2018)CrossRefGoogle Scholar
  13. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Angeli, D., Manfredi, S.: Criteria for asymptotic clustering of opinion dynamics towards bimodal consensus. Automatica 103, 230–238 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Amelkin, V., Bullo, F., Singh, A.K.: Polar opinion dynamics in social networks. IEEE Trans. Autom. Control 62(11), 5650–5665 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Altafini, C.: Consensus problems on networks with antagonistic interactions. IEEE Trans. Autom. Control 58(4), 935–946 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Liu, Q., Wang, X.: Opinion dynamics with similarity-based random neighbors. Sci. Rep. 3, 2968 (2013)CrossRefGoogle Scholar
  21. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 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)MathSciNetCrossRefGoogle Scholar
  23. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 27.
    Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56(4), 577–621 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Mirtabatabaei, A., Bullo, F.: Opinion dynamics in heterogeneous networks: convergence conjectures and theorems. SIAM J. Control Optim. 50(5), 2763–2785 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 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)Google Scholar
  30. 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)Google Scholar
  31. 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)CrossRefGoogle Scholar
  32. 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. 33.
    DeGroot, M.H.: Reaching a consensus. J. Am. Stat. Assoc. 69(345), 118–121 (1974)zbMATHCrossRefGoogle Scholar
  34. 34.
    Ghaderi, J., Srikant, R.: Opinion dynamics in social networks with stubborn agents: equilibrium and convergence rate. Automatica 50(12), 3209–3215 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Milanese, M., Tempo, R., Vicino, A.: Optimal error predictors for economic models. Int. J. Syst. Sci. 19(7), 1189–1200 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 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. 38.
    Tian, Y., Wang, L.: Opinion dynamics in social networks with stubborn agents: an issue-based perspective. Automatica 96, 213–223 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 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)CrossRefGoogle Scholar
  40. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Flache, A., Macy, M.W.: Small worlds and cultural polarization. J. Math. Sociol. 35(1–3), 146–176 (2011)MathSciNetCrossRefGoogle Scholar
  43. 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 CrossRefGoogle Scholar
  44. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Ye, M.: Opinion Dynamics and the Evolution of Social Power in Social Networks. Springer, Berlin (2019)CrossRefGoogle Scholar
  48. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Altafini, C., Lini, G.: Predictable dynamics of opinion forming for networks with antagonistic interactions. IEEE Trans. Autom. Control 60(2), 342–357 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 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)CrossRefGoogle Scholar
  52. 52.
    Noutsos, D.: On Perron–Frobenius property of matrices having some negative entries. Linear Algebra Appl. 412(2–3), 132–153 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 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)CrossRefGoogle Scholar
  54. 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)CrossRefGoogle Scholar
  55. 55.
    Chen, X.: Culture, peer interaction, and socioemotional development. Child. Dev. Perspect. 6(1), 27–34 (2012)CrossRefGoogle Scholar
  56. 56.
    Hou, Y., Li, J., Pan, Y.: On the laplacian eigenvalues of signed graphs. Linear Multilinear Algebra 51(1), 21–30 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 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)Google Scholar
  58. 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. 59.
    Agaev, R., Chebotarev, P.: On the spectra of nonsymmetric Laplacian matrices. Linear Algebra Appl. 399, 157–168 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 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)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2020

Authors and Affiliations

  1. 1.Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education and School of Mathematics and PhysicsAnhui Polytechnic UniversityWuhuPeople’s Republic of China
  2. 2.Department of MathematicsYangzhou UniversityYangzhouPeople’s Republic of China
  3. 3.College of Information Science and TechnologyDonghua UniversityShanghaiPeople’s Republic of China
  4. 4.School of Mathematics and PhysicsAnhui Polytechnic UniversityWuhuPeople’s Republic of China

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