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The European Physical Journal B

, Volume 84, Issue 4, pp 613–626 | Cite as

Speed of complex network synchronization

  • C. GrabowEmail author
  • S. Grosskinsky
  • M. Timme
Regular Article Focus Section on Frontiers in Network Science: Advances and Applications

Abstract

Synchrony is one of the most common dynamical states emerging on networks. The speed of convergence towards synchrony provides a fundamental collective time scale for synchronizing systems. Here we study the asymptotic synchronization times for directed networks with topologies ranging from completely ordered, grid-like, to completely disordered, random, including intermediate, partially disordered topologies. We extend the approach of master stability functions to quantify synchronization times. We find that the synchronization times strongly and systematically depend on the network topology. In particular, at fixed in-degree, stronger topological randomness induces faster synchronization, whereas at fixed path length, synchronization is slowest for intermediate randomness in the small-world regime. Randomly rewiring real-world neural, social and transport networks confirms this picture.

Keywords

Betweenness Centrality Synchronization Time Average Path Length Synchronous State Global Coupling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Arenas et al., Phys. Rep. 469, 93 (2008)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    M. Barahona, L.M. Pecora, Phys. Rev. Lett. 89, 054101 (2002)CrossRefADSGoogle Scholar
  3. 3.
    M.G. Earl, S.H. Strogatz, Phys. Rev. E 67, 036204 (2003)CrossRefADSGoogle Scholar
  4. 4.
    M. Timme, F. Wolf, T. Geisel, Phys. Rev. Lett. 89, 258701 (2002)CrossRefADSGoogle Scholar
  5. 5.
    J. Travers, S. Milgram, Sociometry 32, 425 (1969)CrossRefGoogle Scholar
  6. 6.
    D. Watts, S.H. Strogatz, Nature 393, 440 (1998)CrossRefADSGoogle Scholar
  7. 7.
    A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization, A universal concept in nonlinear sciences (Cambridge Univ. Press, Cambridge, UK, 2001)Google Scholar
  8. 8.
    S.H. Strogatz, Sync: The emerging science of spontaneous order (Penguin Books, London, UK, 2004)Google Scholar
  9. 9.
    I. Kanter, W. Kinzel, E. Kanter, Europhys. Lett. 57, 141 (2002)CrossRefADSGoogle Scholar
  10. 10.
    Yu. Maistrenko et al., Phys. Rev. Lett. 93, 084102 (2004)CrossRefADSGoogle Scholar
  11. 11.
    T. Netoff et al., J. Neurosci. 24, 8075 (2004)CrossRefGoogle Scholar
  12. 12.
    S.H. Strogatz, Nature 410, 268 (2001)CrossRefADSGoogle Scholar
  13. 13.
    T. Nishikawa et al., Phys. Rev. Lett. 91, 014101 (2003)CrossRefADSGoogle Scholar
  14. 14.
    L.M. Pecora, T. Carroll, Phys. Rev. Lett. 80, 2109 (1998)CrossRefADSGoogle Scholar
  15. 15.
    A. Zumdieck et al., Phys. Rev. Lett. 93, 244103 (2004)CrossRefADSGoogle Scholar
  16. 16.
    R. Zillmer et al., Phys. Rev. E 76, 046102 (2007)CrossRefADSGoogle Scholar
  17. 17.
    S. Jahnke, R.-M. Memmesheimer, M. Timme, Phys. Rev. Lett. 100, 048102 (2008)CrossRefADSGoogle Scholar
  18. 18.
    R. Zillmer, N. Brunel, D. Hansel, Phys. Rev. E 79, 031909 (2009)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    R. Olfati-Saber, Proceedings of the American Control Conference (IEEE, Los Alamitos, CA, USA, 2005), p. 2371Google Scholar
  20. 20.
    N. Uchida, Z.F. Mainen, Nature Neurosci. 11, 1224 (2003)CrossRefGoogle Scholar
  21. 21.
    S. Thorpe, D. Fize, C. Marlot, Nature 381, 520 (1996)CrossRefADSGoogle Scholar
  22. 22.
    C. Grabow et al., Europhys. Lett. 90, 48002 (2010)CrossRefADSGoogle Scholar
  23. 23.
    M. Timme, F. Wolf, T. Geisel, Phys. Rev. Lett. 92, 074101 (2004)CrossRefADSGoogle Scholar
  24. 24.
    M. Timme, T. Geisel, F. Wolf, Chaos 16, 015108 (2006)CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    M. Timme, Europhys. Lett. 76, 367 (2006)CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    G.X. Qi et al., Phys. Rev. E 77, 056205 (2008)CrossRefADSGoogle Scholar
  27. 27.
    G.X. Qi et al., Europhys. Lett. 82, 38003 (2008)CrossRefADSGoogle Scholar
  28. 28.
    U. Ernst, K. Pawelzik, T. Geisel, Phys. Rev. Lett. 74, 1570 (1995)CrossRefADSGoogle Scholar
  29. 29.
    U. Ernst, K. Pawelzik, T. Geisel, Phys. Rev. E 57, 2150 (1998)CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    M. Denker et al., Phys. Rev. Lett. 92, 074103 (2004)CrossRefADSGoogle Scholar
  31. 31.
    R. Tönjes, N. Masuda, H. Kori, Chaos 20, 033108 (2010)CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    G. Fagiolo, Phys. Rev. E 76, 026107 (2007)CrossRefADSGoogle Scholar
  33. 33.
    J. Acebron et al., Rev. Mod. Phys. 77, 137 (2005)CrossRefADSGoogle Scholar
  34. 34.
    J.A. Almendral, A. Diaz-Guilera, New J. Phys. 9, 1211 (2007)Google Scholar
  35. 35.
    K.S. Fink et al., Phys. Rev. E 61, 5080 (2000)CrossRefADSGoogle Scholar
  36. 36.
    L. Huang et al., Phys. Rev. E 80, 36204 (2009)CrossRefADSGoogle Scholar
  37. 37.
    D.U. Hwang et al., Phys. Rev. Lett. 94, 138701 (2005)CrossRefADSGoogle Scholar
  38. 38.
    L.M. Pecora et al., Chaos 7, 520 (1997)CrossRefzbMATHADSMathSciNetGoogle Scholar
  39. 39.
    E. Ott, Chaos in Dynamical Systems (Cambridge University Press, New York, 1993)Google Scholar
  40. 40.
    R. Mirollo, S.H. Strogatz, SIAM J. Appl. Math. 50, 366 (1990)Google Scholar
  41. 41.
    M. Timme, F. Wolf, T. Geisel, Chaos 13, 377 (2003)CrossRefzbMATHADSMathSciNetGoogle Scholar
  42. 42.
    M. Timme, F. Wolf, Nonlinearity 21, 1579 (2008)CrossRefzbMATHADSMathSciNetGoogle Scholar
  43. 43.
    S. Lee, P. Kim, H. Jeong, Phys. Rev. E 73, 016102 (2006)CrossRefADSGoogle Scholar
  44. 44.
    K. Goh et al., Phys. Rev. Lett. 96, 018701 (2006)CrossRefADSGoogle Scholar
  45. 45.
    A. Motter, C. Zhou, J. Kurths, Phys. Rev. E 71, 016116 (2005)CrossRefADSGoogle Scholar
  46. 46.
    A. Arenas, A. Diaz-Guilera, C.J. Perez-Vicente, Phys. Rev. Lett. 96, 114102 (2006)CrossRefADSGoogle Scholar
  47. 47.
    C. Zhou, J. Kurths, Chaos 16, 015104 (2006)CrossRefADSMathSciNetGoogle Scholar
  48. 48.
    L.C. Freeman, Sociometry 40, 35 (1977)CrossRefGoogle Scholar
  49. 49.
    V. Colizza, R. Pastor-Satorras, A. Vespignani, Nature 3, 276 (2007)Google Scholar
  50. 50.
    M.A.J. Van Duijn et al., J. Math. Sociol. 27, (2003)Google Scholar
  51. 51.
    T.B. Achacoso, W.S. Yamamoto, AY’s Neuroanatomy of C. Elegans for Computation (CRC Press, Boca Raton, FL, 1992)Google Scholar
  52. 52.
    R. Cross, A. Parker, The Hidden Power of Social Networks (Harvard Business School Press, Boston, MA, 2001)Google Scholar
  53. 53.
    D. Brockmann, L. Hufnagel, T. Geisel, Nature 439, 462 (2006)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Network Dynamics GroupMax Planck Institute for Dynamics and Self-OrganizationGöttingenGermany
  2. 2.Centre for Complexity Science and Mathematics InstituteUniversity of WarwickCoventryUK
  3. 3.Bernstein Center for Computational Neuroscience (BCCN) GöttingenGöttingenGermany
  4. 4.Faculty of PhysicsUniversity GöttingenGöttingenGermany

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