Skip to main content
SpringerLink
Log in

Adaptive Control of the Structural Balance for a Class of Complex Dynamical Networks

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

In social networks, the structural balance is a state of a group of individuals (nodes) with established mutual relationships (connection relationships) between them. It is easy to see that a social network can be described by a complex dynamical network model composed of the nodes subsystem (NS) and the connection relationships subsystem (CS), where the two subsystems are usually coupled with each other. It implies that the dynamic changes of nodes’ states may cause the structural balance in CS. However, few papers have discussed the relationship between the structural balance and the specific dynamic changes of the nodes’ states. This paper proposes a model of complex dynamical networks, and mainly focuses on the dynamic changes of states in NS which can lead to the structural balance in CS. It is proved that if each state in NS is doing a specific dynamic motion via the controller with the parameter adaptive law, then the CS can track a given structural balance matrix via the effective coupling and the structural balance can be achieved. Such a result can be regarded as an explanation of the relationship between the structural balance and the specific dynamic changes of the nodes’ states. Finally, the simulations verify the effectiveness of the proposed method.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Watts D J and Strogatz S H, Collective dynamics of ‘small-world’ networks, Nature, 1998, 393(6684): 440–442.

    MATH  Google Scholar 

  2. Strogatz S H, Exploring complex networks, Nature, 2001, 410: 268–276.

    MATH  Google Scholar 

  3. Wang X F and Chen G R, Complex networks: Small-world, scale-free and beyond, IEEE Circuits and System Magzine, 2003, 3(1): 6–20.

    Google Scholar 

  4. Ji Z J and Yu H S, A new perspective to graphical characterization of multi-agent controllability, IEEE Transactions on Cybernetics, 2017, 47(6): 1471–1483.

    Google Scholar 

  5. Liu X Z and Ji Z J, Controllability of multi-agent systems based on path and cycle graphs, International Journal of Robust and Nonlinear Control, 2018, 28(1): 296–309.

    MathSciNet  MATH  Google Scholar 

  6. Tian L, Ji Z J, Hou T, et al., Bipartite consensus on coopetition networks with time-varying delays, IEEE Access, 2018, 6(1): 10169–10178.

    Google Scholar 

  7. Wang Y H, Fan Y Q, Wang Q Y, et al., Stabilization and synchronization of complex dynamical networks with similar nodes via decentralized controllers, IEEE Transactions on Circuits and Systems-I: Regular Papers, 2012, 59(8): 1786–1795.

    MathSciNet  Google Scholar 

  8. Duan Z S, Wang J Z, and Chen G R, Stability analysis and decentralized control of a class of complex dynamical networks, Automatica, 2008, 44(4): 1028–1035.

    MathSciNet  MATH  Google Scholar 

  9. Zhang L L, Wang Y H, Wang Q R, et al., Synchronization for time-delayed coupling complex dynamical networks with different dimensional nodes via decentralized dynamical compensation controllers, Asian Journal of Control, 2015, 17(2): 664–674.

    MathSciNet  MATH  Google Scholar 

  10. Liu X W, Li P, and Chen T P, Cluster synchronization for delayed complex networks via periodically intermittent pinning control, Neurocomputing, 2015, 162(25): 191–200.

    Google Scholar 

  11. Jin X Z, He T, Xia J W, et al., Adaptive general pinned synchronization of a class of disturbed complex networks, Computation Communication in Nonlinear Science and Numerical Simulation, 2019, 67: 658–669.

    MathSciNet  Google Scholar 

  12. Wang Y W, Bian T, Xiao J W, et al., Global synchronization of complex dynamical networks through digital communication with limited data rate, IEEE Transactions on Neural Networks and Learning Systems, 2015, 26(10): 2487–2499.

    MathSciNet  Google Scholar 

  13. Sandell N R, Varaiya P, Athans M, et al, Survey of decentralized control methods for large scale systems, IEEE Transactions on Automatic Control, 1975, 23(2): 108–128.

    MathSciNet  MATH  Google Scholar 

  14. Bartos M, Vida I, and Jonas P, Synaptic mechanisms of synchronized gamma oscillations in inhibitory interneuron networks, Nature Reviews Neuroscience, 2007, 8(1): 45–56.

    Google Scholar 

  15. Pagilla P R, Siraskar N B, and Dwivedula R V, Decentralized control of web processing lines, IEEE Transactions on Control Systems Technology, 2007, 15(1): 106–117.

    Google Scholar 

  16. Heider F, Attitudes and cognitive organization, Journal of Psychology, 1946, 21(1): 107–112.

    Google Scholar 

  17. Ilany A, Barocas A, Koren L, et al., Structural balance in the social networks of a wild mammal, Animal Behaviour, 2013, 85(6): 1397–1405.

    Google Scholar 

  18. Nakajima S, Sensory preconditioning, the Espinet effect, and Heider’s balance theory: Note on animal reasoning of event relations, Psychological Reports, 2005, 96(3): 1011–1014.

    Google Scholar 

  19. Marvel S A, Kleinberg J, Kleinberg R D, et al., Continuous-time model of structural balance, Proceedings of the National Academy of Sciences of the United States of America, 2011, 108(5): 1771–1776.

    Google Scholar 

  20. Antal T, Krapivsky P L, and Redner S, Social balance on networks: The dynamics of friendship and enmity, Physica D, 2006, 224(1–2): 130–136.

    MathSciNet  MATH  Google Scholar 

  21. Pattison P and Robins G, Neighborhood-based models for social networks, Sociological Methodology, 2002, 32(1): 301–337.

    Google Scholar 

  22. Hassanibesheli F, Hedayatifar L, Gawroski P, et al., Gain and loss of esteem, direct reciprocity and Heider balance, Physica A, 2017, 468: 334–339.

    MathSciNet  MATH  Google Scholar 

  23. Gawronski P, Krawczyk M J, and Kulakowski K, Emerging communities in networks — A flow of ties, Acta Physica Polonica B, 2015, 46: 911.

    Google Scholar 

  24. Altafini C, Dynamics of opinion forming in structurally balanced social networks, PLoS ONE, 2012, 7(6): e38135.

    Google Scholar 

  25. Antal T, Krapivsky P L, and Redner S, Dynamics of social balance on networks, Physical Review E, 2005, 72: 036121.

    MathSciNet  Google Scholar 

  26. Kulakowski K, Gawronski P, and Gronek P, The Heider balance: A continuous approach, International Journal of Modern Physics C, 2005, 16(5): 707–716.

    MATH  Google Scholar 

  27. Wongkaew S, Caponigro M, Kulakowski K, et al., On the control of the Heider balance model, The European Physical Journal Special Topics, 2015, 224(17–18): 3325–3342.

    Google Scholar 

  28. Su H S, Rong Z H, Chen M Z Q, et al., Decentralized adaptive pinning control for cluster synchronization of complex dynamical networks, IEEE Transactions on Cybernetics, 2013, 43(1): 394–399.

    Google Scholar 

  29. Wang J L, Wu H N, Huang T, et al., Analysis and control of output synchronization in directed and undirected complex dynamical networks, IEEE Transactions on Neural Networks and Learning Systems, 2018, 29(8): 3326–3338.

    MathSciNet  Google Scholar 

  30. Song Q and Cao J D, On pinning synchronization of directed and undirected complex dynamical networks, IEEE Transactions on Circuits and Systems-I: Regular Papers, 2010, 57(3): 672–680.

    MathSciNet  Google Scholar 

  31. Barnett S, Matrix differential equations and Kronecker products, SIAM Journal on Applied Mathematics, 1973, 24(1): 1–5.

    MathSciNet  MATH  Google Scholar 

  32. Traag V A, Van D P, and De L P, Dynamical models explaining social balance and evolution of cooperation, PLoS One, 2013, 8(4): e60063.

    Google Scholar 

  33. Hwang E J, Hyun C H, Kim E, et al., Fuzzy model based adaptive synchronization of uncertain chaotic systems: Robust tracking control approach, Physics Letters A, 2009, 373(22): 1935–1939.

    MATH  Google Scholar 

  34. Marvel S A, Strogatz S H, and Kleinberg J M, Energy landscape of social balance, Physical Review Letters, 2009, 103(19): 198701.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Intelligent Information Processing and Control of Chongqing Municipal Institutions of Higher Education, Chongqing Three Gorges University, Chongqing, 404100, China

    Zilin Gao

  2. School of Computer Science and Engineering, Chongqing Three Gorges University, Chongqing, 404100, China

    Zilin Gao

  3. School of Automation, Guangdong University of Technology, Guangzhou, 510006, China

    Yinhe Wang, Yi Peng & Lizhi Liu

  4. School of Computer Science, Zhongshan Institute of Electronic Science and Technology University of China, Zhongshan, 528402, China

    Haoguang Chen

Authors
  1. Zilin Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yinhe Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lizhi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Haoguang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zilin Gao.

Additional information

This research was supported by the National Science Foundation of China under Grant Nos. 61673120, 61273219, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20134420110003, the Science and Technology Research Program of Chongqing Municipal Education Commission under Grant Nos. KJQN201801215, KJ1710244, KJ1710241, KJQN201801209, the Chongqing Basic and Advanced Technology Research Project under Grant No. cstc2018jcyjAX0202, and the Key Laboratory of Chongqing Municipal Institutions of Higher Education under Grant No. [2017]3.

This paper was recommended for publication by Editor JIA Yingmin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, Z., Wang, Y., Peng, Y. et al. Adaptive Control of the Structural Balance for a Class of Complex Dynamical Networks. J Syst Sci Complex 33, 725–742 (2020). https://doi.org/10.1007/s11424-020-8093-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-020-8093-4

Keywords

Search

Navigation