Estimating The Size Of Peer-To-Peer Networks Using Lambert's W Function

  • Javier Bustos-Jimenez
  • Nicolas Bersano
  • Satu Elisa Schaeffer
  • Jose Miguel Piquer
  • Alexandru Iosup
  • Augusto Ciuffoletti

In this work, we address the problem of locally estimating the size of a Peerto- Peer (P2P) network using local information. We present a novel approach for estimating the size of a peer-to-peer (P2P) network, fitting the sum of new neighbors discovered at each iteration of a breadth-first search (BFS) with a logarithmic function, and then using Lambert’s W function to solve a root of a ln(n)+b–n = 0, where n is the network size. With rather little computation, we reach an estimation error of at most 10 percent, only allowing the BFS to iterate to the third level.

Keywords

Peer-to-Peer Network size Estimation 

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References

  1. [1]
    Mark Jelasity, Spyros Voulgaris, Rachid Guerraoui, Anne-Marie Kermarrec, and Maarten van Steen. “Gossip-based peer sampling.” In ACM Transactions on Computer Systems, vol. 25, no. 3, 2007.Google Scholar
  2. [2]
    Alexandru Iosup, Pawel Garbacki, and Dick Epema. “Provisioning and Scheduling Resources for World-Wide Data-Sharing Services.” In Proceedings of IEEE e-Science , p. 84, 2006.Google Scholar
  3. [3]
    Arnaud Legout, Nikitas Liogkas, Eddie Kohler, and Lixia Zhang. “Clustering and sharing incentives in BitTorrent systems.” In Proceedings of ACM SIGMETRICS, pp. 301-312, 2007.Google Scholar
  4. [4]
    Keren Horowitz and Dahlia Malkhi. “Estimating network size from local information.” In Information Processing Letters, vol. 88, no. 5, pp. 237-243, 2003.MATHCrossRefGoogle Scholar
  5. [5]
    Alexandru Iosup, Pawel Garbacki, Johan Pouwelse, and Dick Epema. “Correlating Topology and Path Characteristics of Overlay Networks and the Internet.” In Proceedings of CCGrid, p. 10, 2006.Google Scholar
  6. [6]
    Duncan J. Watts and Steven H. Strogatz. “Collective Dynamics of ’Small World’ Networks.” In Nature, vol. 393, no. 6684, pp. 440-442, 1998.CrossRefGoogle Scholar
  7. [7]
    Michalis Faloutsos, Petros Faloutsos, and Christos Faloutsos. “On Power-law Relationships of the Internet Topology.” In Proceedings of ACM SIGCOMM, pp. 251-262, 1999.Google Scholar
  8. [8]
    Albert-Laszlo Barabasi and Reka Albert. “Emergence of scaling in random networks.” In Science, vol. 268, pp. 509-512, 1999.Google Scholar
  9. [9]
    Satu Elisa Schaeffer. “Algorithms for nonuniform networks.” Technical Report HUTTCS-A102, Helsinki University of Technology, Finland, 2006.Google Scholar
  10. [10]
    Stefan Saroiu, P. Krishna Gummadi, and Steven D. Gribble, “Measuring and analyzing the characteristics of napster and gnutella hosts.” In Multimedia Systems, vol. 9, no. 2, pp. 170-184, 2003.CrossRefGoogle Scholar
  11. [11]
    Matei Ripeanu, Adriana Iamnitchi, and Ian T. Foster, “Mapping the gnutella network.” In IEEE Internet Computing, vol. 6, no. 1, pp. 50-57, 2002.CrossRefGoogle Scholar
  12. [12]
    Daniel Stutzbach and Reza Rejaie, “Capturing accurate snapshots of the gnutella net- work.” In Proceedings of IEEE INFOCOM, vol. 4, pp. 2825-2830, 2006.Google Scholar
  13. [13]
    Johan A. Pouwelse, Pawel Garbacki, Dick H. J. Epema, and Henk J. Sips, “The BitTorrent P2P file-sharing system: Measurements and analysis.” In Proceedings of IPTPS, vol. 3640 of LNCS, pp. 205-216, 2005.Google Scholar
  14. [14]
    Wenjie Wang, Hyunseok Chang, Amgad Zeitoun, and Sugih Jamin, “Characterizing guarded hosts in peer-to-peer file sharing systems.” In Proceedings of IEEE GLOBECOM, vol. 3, pp. 1539-1543, 2004.Google Scholar
  15. [15]
    Subhabrata Sen and Jia Wang, “Analyzing peer-to-peer traffic across large networks.” In IEEE/ACM Transactions on Networking, vol. 12, no. 2, pp. 219-232, 2004.CrossRefGoogle Scholar
  16. [16]
    Boris Pittel. “On Spreading a Rumor..” In SIAM Journal on Applied Mathematics, vol. 47, no. 1, pp. 213-223, 1987.MATHCrossRefGoogle Scholar
  17. [17]
    Romualdo Pastor-Satorras and Alessandro Vespignani. “Epidemic spreading in scale-free networks.” In Physical Review Letters, vol. 86, no. 14, pp. 3200-3203, 2001.CrossRefGoogle Scholar
  18. [18]
    Alan Demers, Dan Greene, Carl Hauser, Wes Irish, John Larson, Scott Shenker, Howard Sturgis, Dan Swinehart, and Doug Terry. “Epidemic algorithms for replicated database maintenance.” In Proceedings of ACM PODC, pp. 1-12, 1987.Google Scholar
  19. [19]
    Leonhard Euler. “De serie Lambertina Plurimisque eius insignibus proprietatibus.” Acta Acad. Scient. Petropol. 2, pp. 29-51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350-369, 1921.Google Scholar
  20. [20]
    Robert M. Corless, Gaston H. Gonnet, David E. G. Hare, David J. Jeffrey, and Donald E. Knuth. “On the Lambert W function.” In Advanced Computational Mathematics, vol. 5, pp. 329-359, 1996.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Javier Bustos-Jimenez
    • 1
  • Nicolas Bersano
    • 1
  • Satu Elisa Schaeffer
    • 2
  • Jose Miguel Piquer
    • 3
  • Alexandru Iosup
    • 4
  • Augusto Ciuffoletti
    • 5
  1. 1.Escuela de Ingenieria InformaticaUniversidad Diego PortalesChile
  2. 2.Universidad Autonoma de Nuevo LeonMexico
  3. 3.Departamento de Ciencias de la ComputacionUniversidad de ChileChile
  4. 4.Parallel and Distributed Systems GroupDelft University of TechnologyNetherlands
  5. 5.Department of Computer ScienceUniversity of PisaItaly

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