Structural Controllability of Optimized Networks with Onion-Like Topologies

  • Manli Li
  • Shiwen SunEmail author
  • Yafang Wu
  • Chengyi Xia
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 528)


Recently, an optimization method has been proposed to increase the ability of complex networks to resist intentional attacks on hub nodes. The finally optimized networks exhibit a novel type of “onion-like” structure. At the same time, structural controllability of complex networks also has been a hot research topic in recent years. Thus, structural controllability of “onion-like” networks deserves sufficient discussion. In this study, we explored the relationship between the attack robustness and structural controllability of scale-free networks before and after optimization. After implementing large quantity of numerical simulations, it has been found that the optimized scale-free networks have both increased robustness and enhanced structural controllability. Current research results can shed some light on the deep understanding of structural complexity and dynamical properties of real-world networked systems.


Structural controllability Network attack Scale-free networks Network optimization Onion-like topologies 



SWS and CYX acknowledge the support from Middle-aged and Young Innovative Talents Training Project of the Higher Education Institutions of Tianjin.


  1. 1.
    R. Albert, A.L. Barabási, Statistical mechanics of complex networks. Rev. Modern Phys. 74, 47–97 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    M.E.J. Newman, The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Boccalettia, V. Latorab, Y. Morenod, Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    D.J. Watts, S.H. Strogztz, Collective dynamics of small world networks. Nature 393, 440–442 (1998)CrossRefGoogle Scholar
  5. 5.
    A.L. Barabási, R. Albert, Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    X. Li, X.F. Wang, G.R. Chen, Pinning a complex dynamical network to its equilibrium. IEEE Trans. Circuit Syst. I: Regul. Pap. 51(10), 2074–2087 (2004)Google Scholar
  7. 7.
    G.R. Chen, Pinning control and synchronization on complex dynamical networks. Int. J. Control Autom. 12(2), 221–230 (2014)Google Scholar
  8. 8.
    X.F. Wang, X. Li, G.R. Chen, Network Science: An Introduction (Higher Education Press, Beijing, China, 2012)Google Scholar
  9. 9.
    C.T. Lin, Structural controllability. IEEE Trans. Autom. Control 19(3), 201–208 (1974)Google Scholar
  10. 10.
    Y.Y. Liu, J.J. Slotine, A.L. Barabási, Controllability of complex networks. Nature 473, 167–173 (2011)CrossRefGoogle Scholar
  11. 11.
    Z.Z. Yuan, Z. Chen, Z.R. Di, W.X. Wang, Y.C. Lai, Exact controllability of complex networks. Nat. Commun. 4, 2447 (2013)CrossRefGoogle Scholar
  12. 12.
    J.E. Hopcroft, R.M. Karp, An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)Google Scholar
  13. 13.
    Y.Y. Liu, J.J. Slotine, A.L. Barabási, Control centrality and hierarchical structure in complex networks. PLoS ONE 7, e44459 (2012)CrossRefGoogle Scholar
  14. 14.
    Y. Pan, X. Li, Structural controllability and controlling centrality of temporal networks. PLoS ONE 9(4), e94998 (2014)CrossRefGoogle Scholar
  15. 15.
    J. Ruths, D. Ruths, Control profiles of complex networks. Science 343, 1373 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    G. Yan, J. Ren, Y.C. Lai, C.H. Lai, B. Li, Controlling complex networks: how much energy is needed. Phys. Rev. Lett. 108(21), 218703 (2012)CrossRefGoogle Scholar
  17. 17.
    W.X. Wang, N. Xuan, Y.C. Lai, C. Grebogi, Optimizing controllability of complex networks by minimum structural perturbations. Phys. Rev. E 85, 026115 (2012)CrossRefGoogle Scholar
  18. 18.
    Y.D. Xiao, S.Y. Lao, L.L. Hou, L. Bai, Edge orientation for optimizing controllability of complex networks. Phys. Rev. E 90, 042804 (2014)Google Scholar
  19. 19.
    M. Pósfai, Y.Y. Liu, J.-J. Slotine, A.L. Barabási, Effect of correlations on network controllability. Sci. Rep. 3, 1067 (2013)CrossRefGoogle Scholar
  20. 20.
    G. Menichetti, L. DallÁsta, G. Bianconi, Network controllability is determined by the density of low in-degree and out-degree nodes. Phys. Rev. Lett. 113, 078701 (2014)CrossRefGoogle Scholar
  21. 21.
    S.W. Sun, Y.L. Ma, Y.F. Wu, L. Wang, C.Y. Xia, Towards structural controllability of local-world networks. Phys. Lett. A 380(22–23), 1912–1917 (2016)CrossRefGoogle Scholar
  22. 22.
    R. Albert, H. Jeong, A.L. Barabási, The Internets Achilles Heel: error and attack tolerance of complex networks. Nature 406, 378–382 (2000)CrossRefGoogle Scholar
  23. 23.
    R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, Resilience of the internet to random breakdowns. Phys. Rev. Lett. 85(21), 4626–4628 (2000)Google Scholar
  24. 24.
    R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, Breakdown of the internet under intentional attack. Phys. Rev. Lett. 86(16), 3682–3685 (2001)Google Scholar
  25. 25.
    D.S. Callaway, M.E.J. Newmann, S.H. Strogatz, D.J. Watts, Network robustness and fragility: percolation on random graphs. Phys. Rev. Lett. 85(25), 5468–5471 (2000)CrossRefGoogle Scholar
  26. 26.
    C.M. Schneider, A.A. Moreira, J.S. Andrade, S.Havlin, H.J. Herrmann, Mitigation of malicious attacks on networks. Proc. Natl. Acad. Sci. (USA) 108(10), 3838–3841 (2011)Google Scholar
  27. 27.
    Z.X. Wu, P. Holme, Onion structure and network robustness. Phys. Rev. E 84, 026106 (2011)CrossRefGoogle Scholar
  28. 28.
    T. Tanizawa, S. Havlin, H.E. Stanley, Robustness of onionlike correlated networks against targeted attacks. Phys. Rev. E 85, 046109 (2012)CrossRefGoogle Scholar
  29. 29.
    S.W. Sun, R.Q. Li, L. Wang, C.Y. Xia, Reduced synchronizability of dynamical scale-free networks with onion-like topologies. Appl. Math. Comput. 252, 249–256 (2015)Google Scholar
  30. 30.
    S.W. Sun, Y.L. Ma, R.Q. Li, L. Wang, C.Y. Xia, Tabu search enhances network robustness under targeted attacks. Phys. A 446, 82–91 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Manli Li
    • 1
    • 2
  • Shiwen Sun
    • 1
    • 2
    Email author
  • Yafang Wu
    • 1
    • 2
  • Chengyi Xia
    • 1
    • 2
  1. 1.Tianjin Key Laboratory of Intelligence Computing and Novel Software TechnologyTianjin University of TechnologyTianjinChina
  2. 2.Key Laboratory of Computer Vision and SystemTianjin University of TechnologyTianjinChina

Personalised recommendations