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

  • 41 Accesses


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.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13


  1. 1.

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

  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)

  3. 3.

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

  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)

  5. 5.

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

  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)

  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)

  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)

  9. 9.

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

  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)

  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)

  12. 12.

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

  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)

  14. 14.

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

  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)

  16. 16.

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

  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)

  18. 18.

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

  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)

  20. 20.

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

  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)

  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)

  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)

  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)

  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)

  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)

  27. 27.

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

  28. 28.

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

  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)

  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)

  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)

  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)

  33. 33.

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

  34. 34.

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

  35. 35.

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

  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)

  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)

  38. 38.

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

  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)

  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)

  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)

  42. 42.

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

  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).

  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)

  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)

  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)

  47. 47.

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

  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)

  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)

  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)

  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)

  52. 52.

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

  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)

  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)

  55. 55.

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

  56. 56.

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

  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)

  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)

  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)

Download references


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.

Author information

Correspondence to Guang He.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


  • Opinion dynamics
  • Peer pressure
  • Prejudice
  • Bipartite consensus
  • Signed graph