Quasi-Interval Bipartite Consensus Problems on Discrete-Time Signed Networks

  • Jianqiang Liang
  • Deyuan MengEmail author
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 528)


In this paper, discrete-time signed networks with cooperative and antagonistic interactions are considered in the presence of time-varying topologies. A separation approach is proposed such that the cooperations and antagonisms can be clearly distinguished. It is shown that given the repeated joint strong connectivity, signed networks can achieve bipartite consensus (respectively, stability) if and only if the repeated joint structural balance (respectively, unbalance) are ensured. Furthermore, when only joint spanning tree condition is satisfied, quasi-interval bipartite consensus (respectively, stability) holds if and only if the repeated joint structural balance (respectively, unbalance) can be guaranteed for the signed digraphs formed by only these joint root nodes. The simulation tests are included to verify the effectiveness of our obtained results.


Discrete-timedynamics Signed networks Time-varying topology Repeated joint structural balance 



This work was supported in part by the National Natural Science Foundation of China (61873013, 61473010, 61473015, 61520106010), in part by the Beijing Natural Science Foundation (4162036), and in part by the Fundamental Research Funds for the Central Universities.


  1. 1.
    L. Jure, H. Daniel, K. Jon, Signed networks in social media, in Proceedings of the 28th International Conference on Human Factors in Computing Systems, Atlanta, GA, USA, April 2010, pp. 1361–1370Google Scholar
  2. 2.
    A. Daron, O. Asuman, Opinion dynamics and learning in social networks. Dyn. Games Appl. 1(1), 3–49 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A.V. Proskurnikov, A. Matveev, M. Cao, Opinion dynamics in social networks with hostile camps: consensus vs. polarization. IEEE Trans. Autom. Control 61(6), 1524–1536 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M.E. Valcher, P. Misra, On the consensus and bipartite consensus in high-order multi-agent dynamical systems with antagonistic interactions. Syst. Control Lett. 66(66), 94–103 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Hu, W. Zheng, Bipartite consensus for multi-agent systems on directed signed networks, in Proceeding of the 52nd IEEE Conference on Decision and Control (Florence, Italy, 2013), pp. 3452–3456Google Scholar
  6. 6.
    H. Zhang, J. Chen, Bipartite consensus of multiagent systems over signed graphs: state feedback and output feedback control approaches. Int. J. Robust Nonlinear Control 27(1), 3–14 (2017)CrossRefGoogle Scholar
  7. 7.
    W. Ren, R.W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control 50(5), 655–661 (2005)CrossRefGoogle Scholar
  8. 8.
    R. Olfati-Saber, J.A. Fax, R.M. Murray, Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007)CrossRefGoogle Scholar
  9. 9.
    W. Ren, R.W. Beard, Distributed consensus in multi-vehicle cooperative control. Commun. Control Eng. 27(2), 71–82 (2008)zbMATHGoogle Scholar
  10. 10.
    W. Xia, M. Cao, K.H. Johansson, Structural balance and opinion separation in trust-mistrust social networks. IEEE Trans. Control Netw. Syst. 3(1), 46–56 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    C. Altafini, Consensus problems on networks with antagonistic interactions. IEEE Trans. Autom. Control 58(4), 935–946 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    D. Meng, M. Du, Y. Jia, Interval bipartite consensus of networked agents associated with signed digraphs. IEEE Trans. Autom. Control 61(12), 3755–3770 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Z. Meng, G. Shi, K.H. Johansson, M. Cao, Y. Hong, Behaviors of networks with antagonistic interactions and switching topologies. Automatica 73, 110–116 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Liu, X. Chen, T. Basar, M.A. Belabbas, Exponential convergence of the discrete and continuous-time Altafini’s models. IEEE Trans. Autom. Control 62(12), 6168–6182 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    J.M. Hendrickx, A lifting approach to models of opinion dynamics with antagonisms, in Proceedings of the IEEE Conference on Decision and Control (Los Angeles, CA, USA, 2014), pp. 2118–2123Google Scholar
  16. 16.
    D. Meng, Convergence analysis of directed signed networks via an M-matrix approach. Int. J. Control,
  17. 17.
    D. Cartwright, F. Harary, Structural balance: a generalization of Heider’s theory. Psychol. Rev. 63(5), 9–25 (1977)Google Scholar
  18. 18.
    R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, 1985)Google Scholar
  19. 19.
    D. Meng, Z. Meng, Y. Hong, A state transition matrix-based approach to separation of cooperations and antagonisms in opinion dynamics. arXiv:1705.04430

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.The Seventh Research DivisionBeihang University (BUAA)BeijingChina
  2. 2.School of Automation Science and Electrical EngineeringBeihang University (BUAA)BeijingChina

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