Epidemic Self-synchronization in Complex Networks

  • Ingo Scholtes
  • Jean Botev
  • Markus Esch
  • Peter Sturm
Part of the Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering book series (LNICST, volume 5)


In this article we present and evaluate an epidemic algorithm for the synchronization of coupled Kuramoto oscillators in complex network topologies. The algorithm addresses the problem of providing a global, synchronous notion of time in complex, dynamic Peer-to-Peer topologies. For this it requires a periodic coupling of nodes to a single random one-hop-neighbor. The strength of the nodes’ couplings is given as a function of the degrees of both coupling partners. We study the emergence of self-synchronization and the resilience against node failures for different coupling strength functions and network topologies. For Watts/Strogatz networks, we observe critical behavior suggesting that small-world properties of the underlying topology are crucial for self-synchronization to occur. From simulations on networks under the effect of churn, we draw the conclusion that special coupling functions can be used to enhance synchronization resilience in power-law Peer-to-Peer topologies.


Self-Synchronization Networks Coupled Oscillators Kuramoto Model Peer-to-Peer 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Acebrón, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics 77, 137–185 (2005)CrossRefGoogle Scholar
  2. 2.
    Babaoglu, O., Binci, T., Jelasity, M., Montresor, A.: Firefly-inspired heartbeat synchronization in overlay networks. In: SASO 2007: Proceedings of the First International Conference on Self-Adaptive and Self-Organizing Systems, Washington, DC, USA, 2007, pp. 77–86. IEEE Computer Society, Los Alamitos (2007)CrossRefGoogle Scholar
  3. 3.
    Barabasi, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barahona, M., Pecora, L.M.: Synchronization in small-world systems. Phys. Rev. Lett. 89(5) (July 2002)Google Scholar
  5. 5.
    di Bernardo, M., Garofalo, F., Sorrentino, F.: Effects of degree correlation on the synchronizability of networks of nonlinear oscillators. In: Proceedings of the 44th IEEE Conference on Decision and Control (CDC), 2005 and 2005 European Control Conference (ECC), December 2005, pp. 4616–4621 (2005)Google Scholar
  6. 6.
    Ermentrout, B.: Synchronization in a pool of mutually coupled oscillators with random frequencies. Journal of Mathematical Biology 22(1), 1–9 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ermentrout, B.: An adaptive model for synchrony in the firefly pteroptyx malaccae. Journal of Mathematical Biology 29(6), 571–585 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ermentrout, G.B., Kopell, N.: Frequency plateaus in a chain of weakly coupled oscillators, i. SIAM Journal on Mathematical Analysis 15(2), 215–237 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hwang, D.-U., Chavez, M., Amann, A., Boccaletti, S.: Synchronization in complex networks with age ordering. Physical Review Letters 94(13), 138701 (2005)CrossRefGoogle Scholar
  10. 10.
    Jadbabaie, A., Motee, N., Barahona, M.: On the stability of the kuramoto model of coupled nonlinear oscillators. In: Proceedings of the American Control Conference, pp. 4296–4301 (2004)Google Scholar
  11. 11.
    Jelasity, M., Montresor, A., Babaoglu, O.: Gossip-based aggregation in large dynamic networks. ACM Trans. Comput. Syst. 23(3), 219–252 (2005)CrossRefGoogle Scholar
  12. 12.
    Jelasity, M., van Steen, M.: Large-scale newscast computing on the Internet. Tech. Rep. IR-503, Vrije Universiteit Amsterdam, Department of Computer Science, Amsterdam, The Netherlands (October 2002)Google Scholar
  13. 13.
    Kuramoto, Y.: Self-entrainment of a population of coupled nonlinear oscillators. In: International symposium on mathematical problems in theoretical physics, pp. 420–422. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  14. 14.
    Li, X.: Uniform synchronous criticality of diversely random complex networks. Physica A: Statistical Mechanics and its Applications 360, 629–636 (2006)CrossRefGoogle Scholar
  15. 15.
    Li, X.: The role of degree-weighted couplings in the synchronous onset of kuramoto oscillator networks. Physica A: Statistical Mechanics and its Applications 387(26), 6624–6630 (2008)CrossRefGoogle Scholar
  16. 16.
    Lucarelli, D., Wang, I.-J.: Decentralized synchronization protocols with nearest neighbor communication. In: SenSys 2004: Proceedings of the 2nd international conference on Embedded networked sensor systems, pp. 62–68. ACM, New York (2004)Google Scholar
  17. 17.
    Mirollo, R.E., Strogatz, S.H.: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50(6), 1645–1662 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Moreno, Y., Pacheco, A.F.: Synchronization of kuramoto oscillators in scale-free networks. EPL (Europhysics Letters) 68(4), 603–609 (2004)CrossRefGoogle Scholar
  19. 19.
    Motter, A.E., Zhou, C., Kurths, J.: Network synchronization, diffusion, and the paradox of heterogeneity. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 71(1), 016116 (2005)CrossRefGoogle Scholar
  20. 20.
    Peskin, C.S.: Mathematical aspects of heart physiology. Technical report, Courant Institute of Mathematical Sciences (1975)Google Scholar
  21. 21.
    Scholtes, I., Botev, J., Esch, M., Hoehfeld, A., Schloss, H., Zech, B.: Topgen - internet router-level topology generation based on technology constraints. In: Proceedings of the First International Conference on Simulation Tools and Techniques for Communications, Networks and Systems (SIMUTools) (February 2008)Google Scholar
  22. 22.
    Strogatz, S.H.: From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143(1-4), 1–20 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Strogatz, S.H.: Sync: The Emerging Science of Spontaneous Order. Hyperion (2003)Google Scholar
  24. 24.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393 (1998)Google Scholar
  25. 25.
    Wiener, N.: Nonlinear Problems in Random Theory. MIT Press, Cambridge (1958)zbMATHGoogle Scholar
  26. 26.
    Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. Journal of Theoretical Biology 16, 15–42 (1967)CrossRefGoogle Scholar

Copyright information

© ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering 2009

Authors and Affiliations

  • Ingo Scholtes
    • 1
  • Jean Botev
    • 1
  • Markus Esch
    • 2
  • Peter Sturm
    • 1
  1. 1.Systemsoftware and Distributed SystemsUniversity of TrierTrierGermany
  2. 2.Faculty of Sciences, Technology and CommunicationUniversity of LuxembourgLuxembourg-KirchbergLuxembourg

Personalised recommendations