Nonlinear Dynamics

, Volume 93, Issue 2, pp 335–347 | Cite as

Synchronization properties of interconnected network based on the vital node

  • Shasha Feng
  • Li Wang
  • Shiwen Sun
  • Chengyi Xia
Original Paper


In this paper, we perform an intensive study of the synchronization properties of interconnected network and the concepts of vital node, and the simplest and equivalent network is firstly introduced. We strictly derive the eigenvalues of Laplacian matrix and the synchronizability of interconnected network and its simplest and equivalent network through utilizing the master stability function approach. Firstly, we find the synchronizability of interconnected network is identical to its simplest and equivalent network. Secondly, we identify the general factors that determine the synchronizability of interconnected network and further analyze the impact of different factors on the synchronizability. Finally, theoretical analysis and numerical simulations are carried out to indicate the validity and effectiveness of current analysis. The current results are beneficial to understand the dynamical behaviors of complex networked systems.


Synchronization Interconnected network Vital node Simplest and equivalent network Eigenvalue analysis 



This work was partially supported by the National Natural Science Foundation of China under Grant Nos. 61403280, 61374169 and 61773286. LW acknowledge the support from Excellent Young Teachers Program of Tianjin. LW, SWS and CYX acknowledge the support from 131 Innovative Talents Program of Tianjin. CYX also acknowledges the support from the Scientific Research Foundation for the Returned Overseas Chinese Scholars (Ministry of Education).


  1. 1.
    Xing, Z.C., Liu, Q., D́Asiri, A.M., Sun, X.P.: Closely interconnected network of molybdenum phosphide nanoparticles: a highly efficient electrocatalyst for generating hydrogen from water. Adv. Mater. 26(32), 5702–5707 (2014)CrossRefGoogle Scholar
  2. 2.
    Wang, H.J., Li, Q., Agostino, G., Havlin, S., Stanley, H.E., Mieghem, P.V.: Effect of the interconnected network structure on the epidemic threshold. Phys. Rev. E 88, 022801 (2013)CrossRefGoogle Scholar
  3. 3.
    Boccaletti, S., Bianconi, G., Criado, R., del Genio, C.I., Gómez-Gardeñes, J., Romance, M., Sendiña-Nadal, I., Wang, Z., Zanin, M.: The structure and dynamics of multilayer networks. Phys. Rep. 544(1), 1–122 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. J. Complex Netw. 2, 203–271 (2014)CrossRefGoogle Scholar
  5. 5.
    Li, J., Wang, J., Sun, S.W., Xia, C.Y.: Cascading crashes induced by the individual heterogeneity in complex networks. Appl. Math. Comput. 323, 182–192 (2018)MathSciNetGoogle Scholar
  6. 6.
    Carvalho, R., Buzna, L., Bono, F., Gutiérrez, E., Just, M., Arrowsmith, D.: Robustness of trans-European gas networks. Phys. Rev. E 80, 016106 (2009)CrossRefGoogle Scholar
  7. 7.
    Scardovi, L., Arcak, M., Sontag, E.D.: Synchronization of interconnected systems with applications to biochemical networks: an input-output approach. IEEE Control Syst. Soc. 55(6), 1367–1379 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Louzada, V.H.P., Araújo, N.A.M., Andrade, J.S., Herrmann, H.J.: Breathing synchronization in interconnected networks. Sci. Rep. 3, 3289 (2013)CrossRefGoogle Scholar
  9. 9.
    Feng, J.W., Yang, P., Zhao, Y.: Cluster synchronization for nonlinearly time-varying delayed coupling complex networks with stochastic perturbation via periodically intermittent pinning control. Appl. Math. Comput. 291, 52–68 (2016)MathSciNetGoogle Scholar
  10. 10.
    Saumell-Mendiola, A., Serrano, M.Á., Boguñá, M.: Epidemic spreading on interconnected networks. Phys. Rev. E 86, 026106 (2012)CrossRefGoogle Scholar
  11. 11.
    Wang, Y., Xiao, G.: Epidemics spreading in interconnected complex networks. Phys. Lett. A 376(42–43), 2689–2696 (2012)CrossRefzbMATHGoogle Scholar
  12. 12.
    Dickison, M., Havlin, S., Stanley, H.E.: Epidemics on interconnected networks. Phys. Rev. E 85, 066109 (2012)CrossRefGoogle Scholar
  13. 13.
    Li, C., Wang, L., Sun, S.W., Xia, C.Y.: Identication of inuential spreaders based on classied neighbors in real-world complex networks. Appl. Math. Comput. 320, 512–523 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zhao, Z., Zhang, P., Yang, H.J.: Cascading failures in interconnected networks with dynamical redistribution of loads. Phys. A Stat. Mech. Appl. 433(1), 204–210 (2015)CrossRefGoogle Scholar
  15. 15.
    Yi, C.Q., Bao, Y.Y., Jiang, J.C., Xue, Y.B.: Modeling cascading failures with the crisis of trust in social networks. Phys. A Stat. Mech. Appl. 436(15), 256–271 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang, C.J., Wang, L., Wang, J., Sun, S.W., Xia, C.Y.: Inferring the reputation enhances the cooperation in the public goods game on interdependent lattices. Appl. Math. Comput. 293, 18–29 (2017)MathSciNetGoogle Scholar
  17. 17.
    Chen, M.H., Wang, L., Sun, S.W., Wang, J., Xia, C.Y.: Evolution of cooperation in the spatial public goods game with adaptive reputation assortment. Phys. Lett. A 380, 40–47 (2016)CrossRefzbMATHGoogle Scholar
  18. 18.
    Chen, M.H., Wang, L., Wang, J., Sun, S.W., Xia, C.Y.: Impact of individual response strategy on the spatial public goods game within mobile agents. Appl. Math. Comput. 251, 192–202 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Chen, F., Ren, W.: A connection between dynamic region-following formation control and distributed average tracking. IEEE Trans. Cybern. 99, 1–13 (2017)Google Scholar
  20. 20.
    Chen, F., Ren, W., Lin, Z.L.: Multi-leader multi-follower coordination with cohesion, dispersion, and containment control via proximity graphs. Sci. China Inf. Sci. 60, 110204 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zeng, X.L., Hui, Q.: Global synchronization of biological network systems with time-varying delays. IFAC Proc. Vol. 45(12), 75–80 (2012)CrossRefGoogle Scholar
  22. 22.
    Dos, F.A., Silva, S., Lopes, S.R., Viana, R.L.: Synchronization of biological clock cells with a coupling mediated by the local concentration of a diffusing substance. Commun. Nonlinear Sci. Numer. Simul. 35, 37–52 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zeng, X.L., Hui, Q., Haddad, W.M., Hayakawa, T., Bailey, J.M.: Synchronization of biological neural network systems with stochastic perturbations and time delays. J. Frankl. Inst. 351(3), 1205–1225 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Enjieu Kadji, H.G., Chabi Orou, J.B., Woafo, P.: Synchronization dynamics in a ring of four mutually coupled biological systems. Commun. Nonlinear Sci. Numer. Simul. 13(7), 1361–1372 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Marek, M., Stuchl, I.: Synchronization in two interacting oscillatory systems. Biophys. Chem. 3(3), 241–248 (1975)CrossRefGoogle Scholar
  26. 26.
    Feng, S.S., Wang, L., Li, Y.J., Sun, S.W., Xia, C.Y.: A nonlinear merging protocol for consensus in multi-agent systems on singed and weighted graphs. Physica A 490, 653–663 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Park, Y.J., Rim, J.H., Yim, J., Lee, S.G., Kim, J.H.: Effects of two types of medical contrast media on routine chemistry results by three automated chemistry analyzers. Clin. Biochem. 50(12), 719–725 (2017)CrossRefGoogle Scholar
  28. 28.
    Yan, J., Bloom, M., Bae, S.C., Luijten, E., Granick, S.: Linking synchronization to self-assembly using magnetic Janus colloids. Nature; London 491(7425), 578–81 (2012)CrossRefGoogle Scholar
  29. 29.
    Goedgebuer, J.P., Larger, L., Porte, H.: Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode. Phys. Rev. Lett. 80, 2249 (1998)CrossRefGoogle Scholar
  30. 30.
    Wang, L., Sun, S.W., Xia, C.Y.: Finite-time stability of multi-agent system in disturbed environment. Nonlinear Dyn. 67, 2009–2016 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Argyris, A., Syvridis, D., Larger, L., Annovazzi-Lodi, V.: Chaos-based communications at high bit rates using commercial fibre-optic links. Nature; London 438(7066), 343–346 (2005)CrossRefGoogle Scholar
  32. 32.
    York, R.A.: Nonlinear analysis of phase relationships in quasi-optical oscillator arrays. IEEE Trans. Microw. Theory Tech. 41(10), 1799–1809 (1993)CrossRefGoogle Scholar
  33. 33.
    Cattadori, I.M., Haydon, D.T., Hudson, P.J.: Parasites and climate synchronize red grouse populations. Nature; London 433(7027), 737–741 (2005)CrossRefGoogle Scholar
  34. 34.
    Zhang, X.Y., Boccaletti, S., Guan, S.G., Liu, Z.H.: Explosive synchronization in adaptive and multilayer networks. Phys. Rev. Lett. 114, 038701 (2015)CrossRefGoogle Scholar
  35. 35.
    Wider, N., Garas, A., Scholtes, I., Schweitzer, F.: An ensemble perspective on multi-layer networks. Interconnected, Networks; pp. 37–59 (2016)Google Scholar
  36. 36.
    Aguirre, J., Sevilla-Escoboza, R., Gutiérrez, R., Papo, D., Buldú, J.M.: Synchronization of interconnected networks: the role of connector nodes. Phys. Rev. Lett. 112, 268701 (2014)CrossRefzbMATHGoogle Scholar
  37. 37.
    Xu, M.M., Lu, J.A., Zhou, J.: Synchronizability and eigenvalues of two-layer star networks. Acta Phys. Sinica 65(2), 028902 (2016)Google Scholar
  38. 38.
    Balch, T., Arkin, R.C.: Behavior-based formation control for multirobot teams. IEEE Trans. Robot. Autom. 14(6), 926–939 (1998)CrossRefGoogle Scholar
  39. 39.
    The Algebra Group of Teaching and Research Section of Algebra and Gemotry, Mathematics Department, Beijing University 2003 Advanced Algebra, 3rd edn. p 4382. Higher Education Press, Beijing (in Chinese)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Shasha Feng
    • 1
  • Li Wang
    • 1
    • 2
  • Shiwen Sun
    • 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 System (Ministry of Education)Tianjin University of TechnologyTianjinChina

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