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Consensus in Networks under Transmission Delays and the Normalized Laplacian

  • Fatihcan M. Atay
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 423)

Abstract

We study the consensus problem on directed and weighted networks in the presence of time delays. We focus on information transmission delays, as opposed to information processing delays, so that each node of the network compares its current state to the past states of its neighbors. The connection structure of the network is described by a normalized Laplacian matrix. We show that consensus is achieved if and only if the underlying graph contains a spanning tree. Furthermore, this statement holds independently of the value of the delay, in contrast to the case of processing delays. We also calculate the consensus value and show that, unlike the case of processing delays, the consensus value is determined not just by the initial states of the nodes at time zero, but also on their past history over an interval of time.

Keywords

Span Tree Transmission Delay Functional Differential Equation Negative Real Part Vertex Degree 
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.
    Atay, F.M.: Oscillator death in coupled functional differential equations near Hopf bifurcation. J. Differential Equations 221(1), 190–209 (2006), doi:10.1016/j.jde.2005.01.007MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Atay, F.M.: Consensus in networks with transmission delays (forthcoming)Google Scholar
  3. 3.
    Atay, F.M., Bıyıkoğlu, T., Jost, J.: Synchronization of networks with prescribed degree distributions. IEEE Trans. Circuits and Systems I 53(1), 92–98 (2006), doi:10.1109/TCSI.2005.854604MathSciNetCrossRefGoogle Scholar
  4. 4.
    Atay, F.M., Jost, J., Wende, A.: Delays, connection topology, and synchronization of coupled chaotic maps. Phys. Rev. Lett. 92(14), 144101 (2004), doi:10.1103/PhysRevLett.92.144101CrossRefGoogle Scholar
  5. 5.
    Atay, F.M., Karabacak, Ö.: Stability of coupled map networks with delays. SIAM J. Applied Dyn. Syst. 5(3), 508–527 (2006), doi:10.1137/060652531MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bauer, F.: Spectral graph theory of directed graphs (forthcoming)Google Scholar
  7. 7.
    Bauer, F., Atay, F.M., Jost, J.: Synchronization in discrete-time networks with general pairwise coupling. Nonlinearity 22(1), 2333–2351 (2009), doi: 10.1088/0951-7715/22/10/001MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)Google Scholar
  9. 9.
    Hale, J.K.: Theory of Functional Differential Equations. Springer (1977)Google Scholar
  10. 10.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
  11. 11.
    Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra and its Applications 197-198, 143–176 (1994), doi:10.1016/0024-3795(94)90486-3MathSciNetCrossRefGoogle Scholar
  12. 12.
    Michiels, W., Morărescu, C.I., Niculescu, S.I.: Consensus problems with distributed delays, with application to traffic flow models. SIAM Journal on Control and Optimization 48(1), 77–101 (2009), doi:10.1137/060671425MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Moreau, L.: Stability of continuous-time distributed consensus algorithms. In: Proc. 43rd IEEE Conference on Decision and Control, pp. 3998–4003 (2004)Google Scholar
  14. 14.
    Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control 49(9), 1520–1533 (2004), doi:10.1109/TAC.2004.834113MathSciNetCrossRefGoogle Scholar
  15. 15.
    Seuret, A., Dimarogonas, D.V., Johansson, K.H.: Consensus under communication delays. In: Proc. 47th IEEE Conference on Decision and Control, CDC 2008, Cancun, Mexique (2008)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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