Small-World Networks Revisited

  • Thomas Fuhrmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2877)


Small-world networks have received much attention recently. Computer scientists, theoretical physicists, mathematicians, and others use them as basis for their studies. At least partly due to the different mind-sets of these disciplines, these random graph models have not always been correctly applied to questions in, e.g., peer-to-peer computing. This paper tries to shed some light on common misunderstandings in the study of small-world peer-to-peer networks. It shows that, contrary to some recent publications, Gnutella can indeed be described by a model with power-law degree distribution. To further distinguish the proposed model from other random graph models, this paper also applies two mathematical concepts, dimension and curvature, to the study of random graphs. These concepts help to understand the distribution of node distances in small-world networks. It thus becomes clear that the observed deficit in the number of reachable nodes in Gnutella-like networks is quite natural and no sign of any wrong or undesirable effect like, e.g., network partitioning.

Index terms

Random Graphs Small World Networks Peer-to-Peer Computing Gnutella 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, R., Barabasi, A.-L.: Topology of evolving networks: Local events and universality. Physical Review Letters 85(24) (2000)Google Scholar
  2. 2.
    Albert, R., Barabasi, A.-L.: Statistical mechanics of complex networks. Review of Modern Physics 74(47) (2002)Google Scholar
  3. 3.
    Albert, R., Jeong, H., Barabasi, A.-L.: Diameter of the world wide web. Nature 401, 130–131 (1999)CrossRefGoogle Scholar
  4. 4.
    Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, Q., Chang, H., Govindan, R., Jamin, S., Shenker, S.J., Willinger, W.: The origin of power laws in internet topologies revisited. In: Proceedings of the 21st Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2002), vol. 2, pp. 608–617 (2002)Google Scholar
  6. 6.
    Erdös, P., Renyi, A.: On random graphs. Publ. Math. Debrecen 6, 290–297 (1959)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Erdös, P., Renyi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17–61 (1960)zbMATHGoogle Scholar
  8. 8.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. In: SIGCOMM, pp. 251–262 (1999)Google Scholar
  9. 9.
    Mandelbrot, B.B.: The Fractal Geometry of Nature. H. W. Freeman and Company, New York (1977)Google Scholar
  10. 10.
    Milgram, S.: The small world problem. Psychol. Today 2, 60–67 (1967)Google Scholar
  11. 11.
    Ripeanu, M., Foster, I.: Mapping gnutella network: Macroscopic properties of large-scale peer-to-peer systems. In: Proceedings of the First International Workshop on Peer-to-Peer Systems (IPTPS 2002), Cambridge, Massachusetts (March 2002)Google Scholar
  12. 12.
    Ripeanu, M., Foster, I., Iamnitchi, A.: Mapping the gnutella network: Properties of large-scale peer-to-peer systems and implications for system design. IEEE Internet Computing Journal (Special issue on peer-to-peer networking) 6(1) (2002)Google Scholar
  13. 13.
    Schollmeier, R., Hermann, F.: Topology-analysis of pure peer-to-peer networks. In: Proceedings of Kommunikation in Verteilten Systemen, Leipzig, Germany, February 26-28, pp. 359–370 (2003)Google Scholar
  14. 14.
    Schollmeier, R., Schollmeier, G.: Why peer-to-peer does scale: An analysis of P2P traffic patterns. In: Proceedings of the Second International Conference on Peer-to-Peer Computing, Linköping, Sweden, September 5-7, pp. 112–119 (2002)Google Scholar
  15. 15.
    Tutschku, K., deMeer, H.: A measurement study on the dynamics of gnutella overlays. In: Proceedings of Kommunikation in Verteilten Systemen, Leipzig, Germany, February 26-28, pp. 295–306 (2003)Google Scholar
  16. 16.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 440–442 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Thomas Fuhrmann
    • 1
  1. 1.Institut für TelematikUniversität Karlsruhe (TH)Germany

Personalised recommendations