Transmission of Packets on a Hierarchical Network: Avalanches, Statistics and Explosive Percolation

  • Neelima GupteEmail author
  • Ajay Deep Kachhvah
Part of the Understanding Complex Systems book series (UCS)


We discuss transport on load bearing branching hierarchical networks which can model diverse systems which can serve as models of river networks, computer networks, respiratory networks and granular media. We study avalanche transmissions and directed percolation on these networks, and on the V lattice, i.e., the strongest realization of the lattice. We find that typical realizations of the lattice show multimodal distributions for the avalanche transmissions, and a second order transition for directed percolation. On the other hand, the V lattice shows power-law behavior for avalanche transmissions, and a first order (explosive) transition to percolation. The V lattice is thus the critical case of hierarchical networks. We note that small perturbations to the V lattice destroy the power-law behavior of the distributions, and the first order nature of the percolation. We discuss the implications of our results.


Versus Lattice Test Weight Base Lattice Hierarchical Network Percolation Transition 
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  1. 1.
    J. Duch, A. Arenas, Phys. Rev. Lett. 94, 028701 (2005)CrossRefGoogle Scholar
  2. 2.
    P. Holme, B.J. Kim, C.N. Yoon, S.K. Han, Phys. Rev. E 65, 056109 (2002)CrossRefGoogle Scholar
  3. 3.
    B. Tadić, G.J. Rodgers, Advs. Complex Syst. 05, 445 (2002)CrossRefGoogle Scholar
  4. 4.
    T. Ohira, R. Sawatari, Phys. Rev. E 58, 193 (1998)CrossRefGoogle Scholar
  5. 5.
    L. Zhao, Y.-C. Lai, K. Park, N. Ye, Phys. Rev. E 71, 026125 (2005)CrossRefGoogle Scholar
  6. 6.
    A.D. Kachhvah, N. Gupte, Phys. Rev. E 83, 036107 (2011)CrossRefMathSciNetGoogle Scholar
  7. 7.
    A.D. Kachhvah, N. Gupte, Phys. Rev. E 86, 026104 (2012)CrossRefGoogle Scholar
  8. 8.
    A.E. Scheidegger, Bull. Int. Acc. Sci. Hydrol. 12, 15 (1967)CrossRefGoogle Scholar
  9. 9.
    S.N. Coppersmith, C-h Liu, S. Majumdar, O. Narayan, T.A. Witten. Phys. Rev. E 53, 4673 (1996)CrossRefGoogle Scholar
  10. 10.
    D. Griffeath, in Additive and Cancellative Interacting Particles Systems. Lectures Notes in Mathematics, vol. 724 (Springer, Berlin, 1979)Google Scholar
  11. 11.
    T.M. Liggett, Interacting Particles Systems (Springer, Berlin, 1985)CrossRefGoogle Scholar
  12. 12.
    E. Domany, W. Kinzel, Phys. Rev. Lett. 53, 311 (1984)CrossRefMathSciNetGoogle Scholar
  13. 13.
    B. Suki, A.-L. Barabasi, Z. Hantos, F. Petak, H.E. Stanley, Nature 368, 615 (1994)CrossRefGoogle Scholar
  14. 14.
    B. Suki, J.S. Andrade, M.F. Coughlin, D. Stamenovic, H.E. Stanley, M. Sujeer, S. Zapperi, Ann. Biomed. Eng. 26, 608 (1998)CrossRefGoogle Scholar
  15. 15.
    T.M. Janaki, N. Gupte, Phys. Rev. E 67, 021503 (2003)CrossRefGoogle Scholar
  16. 16.
    H. Takayasu, I. Nishikawa, H. Tasaki, Phys. Rev. A 37, 3110 (1988)CrossRefMathSciNetGoogle Scholar
  17. 17.
    H. Seybold, J.S. Andrade Jr, H.J. Herrmann, Proc. Natl. Acad. Sci. USA 104, 16804 (2007)CrossRefGoogle Scholar
  18. 18.
    D. Reiss, G. Erkeling, K.E. Bauch, H. Hiesinger, Geophys. Res. Lett. 37, L06203 (2010)Google Scholar
  19. 19.
    T. Shinbrot, N.-H. Duong, L. Kwan, M.M. Alvarez, Proc. Natl. Acad. Sci. USA 101, 8542 (2004)CrossRefGoogle Scholar
  20. 20.
    A.D. Kachhvah, N. Gupte, Pramana J. Phys. 77, 873 (2011)CrossRefGoogle Scholar
  21. 21.
    J. Nagler, A. Levina, M. Timme, Nat. Phys. 7, 265 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of PhysicsIndian Institute of Technology, MadrasChennaiIndia
  2. 2.Nonlinear Physics DivisionInstitute for Plasma ResearchGandhinagarIndia

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