Small-World Networks: From Theoretical Bounds to Practical Systems

  • François Bonnet
  • Anne-Marie Kermarrec
  • Michel Raynal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4878)


In small-world networks, each peer is connected to its closest neighbors in the network topology, as well as to additional long-range contact(s), also called shortcut(s). In 2000, Kleinberg provided asymptotic lower bounds on routing performances and showed that greedy routing in an n-peer small-world network performs in \(\Omega(n^\frac{1}{3})\) steps when the distance to shortcuts is chosen uniformly at random, and in Θ(log2n) when the distance to shortcuts is chosen according to a harmonic distribution in a d-dimensional mesh. Yet, we observe through experimental results that peer to peer gossip-based protocols achieving small-world topologies where shortcuts are randomly chosen, perform reasonably well in practice.

Kleinberg results are relevant for extremely large systems while systems considered in practice are usually of smaller size (they are typically made up of less than one million of peers). This paper explores the impact of Kleinberg results in the context of practical systems and small-world networks. More precisely, based on the observation that, despite the fact that the routing complexity of gossip-based small-world overlay networks is not polylogarithmic (as proved by Kleinberg), this type of networks ultimately provide reasonable results in practice. This leads us to think that the asymptotic big O() complexity alone might not always be sufficient to assess the practicality of a system whose size is typically smaller that what the one theory targets. The paper consequently proposes a refined routing complexity measure for small-world networks (namely, a recurrence formula that can be easily computed). Yet, given that Kleinberg proved that the distribution of shortcuts has a strong impact on the routing complexity (when extremely large networks are considered), arises the question of leveraging this result to improve upon current gossip-based protocols. We show that gossip-based protocols (designed for less than one million of peers) can benefit from a good approximation of Kleinberg-like small-world topologies (designed for extremely large networks). Along, are presented simulation results that demonstrate the relevance of the proposed approach.


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  1. 1.
    Barrière, L., Fraigniaud, P., Kranakis, E., Krizanc, D.: Efficient Routing in Networks with Long Range Contacts. In: Welch, J.L. (ed.) DISC 2001. LNCS, vol. 2180, pp. 270–284. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Birman, K.P., Hayden, M., Ozkasap, O., Xiao, Z., Budiu, M., Minsky, V.: Bimodal Multicast. ACM Transactions on Computer Systems 17(2), 41–88 (1999)CrossRefGoogle Scholar
  3. 3.
    Bonnet, F., Kermarrec, A.-M., Raynal, M.: Small-World Networks: Is there a mismatch between theory and practice. In: Research Report IRISA #1849 (2007)Google Scholar
  4. 4.
    Demers, A.J., Greene, D.H., Hauser, C., Irish, W., Larson, J.: Epidemic Algorithms for Replicated Database Maintenance. In: Proc. 6th ACM Symposium on Principles of Distributed Computing (PODC 1987), pp. 1–12 (1987)Google Scholar
  5. 5.
    Eugster, P.T., Guerraoui, G., Handurukande, B., Kermarrec, A.-M., Kouznetsov, P.: Lightweight Probabilistic Broadcast. ACM TOCS 21(1), 341–374 (2003)CrossRefGoogle Scholar
  6. 6.
    Eugster, P.T., Guerraoui, G., Kermarrec, A.-M., Massoulié, L.: Epidemic Information Dissemination in Distributed Systems. IEEE Computer 37(5), 60–67 (2004)Google Scholar
  7. 7.
    Fraigniaud, P., Gauron, P., Latapy, M.: Combining the Use of Clustering and Scale-free Nature of User Exchanges into a Simple and Efficient P2P System. In: Proc. European Conf. on Parallelism (EUROPAR 2005) (2005)Google Scholar
  8. 8.
    Fraigniaud, P., Gavoille, C., Paul, C.: Eclecticism Shrinks even Small Worlds. In: Fraigniaud, P., Gavoille, C., Paul, C. (eds.) Proc. 23th ACM Symposium on Principles of Distributed Computing (PODC 2004), pp. 169–178. ACM Press, New York (2004)CrossRefGoogle Scholar
  9. 9.
    Jelasity, M., Babaoglu, O.: T-Man: Gossip-based Overlay Topology Management. In: Proc. Engineering Self-Organising Applications (ESOA 2005) (2005)Google Scholar
  10. 10.
    Jelasity, M., Guerraoui, G., Kermarrec, A.-M., van Steen, M.: The Peer Sampling Service: Experimental Evaluation of Unstructured Gossip-based Implementations. In: Proc. 5th ACM/IFIP/USENIX Int’l Conference on Middleware. Lecture Notes in Computer Science, pp. 79–98. Springer-Verlag, Heidelberg (2004)Google Scholar
  11. 11.
    Kleinberg, J.: Navigation in a Small World. Nature 845(406) (2000)Google Scholar
  12. 12.
    Kleinberg, J.: The Small-World Phenomenon: an Algorithmic Perspective. In: Proc. 32nd ACM Symposium on Theory of Computing, pp. 163–170. ACM Press, New York (2000)Google Scholar
  13. 13.
    Milgram, S.: The Small-World Problem. Psychology Today 61(2), 60–67 (1967)Google Scholar
  14. 14.
    Naor, M., Wieder, U.: Know The Neighbor’s Neighbor: Better Routing for Skip-Graphs and Small Worlds. In: IPTPS 2004. LNCS, vol. 3279, pp. 269–277. Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Voulgaris, S., Gavidia, D., van Steen, M.: CYCLON: Inexpensive Membership Management for Unstructured P2P Overlays. Journal of Network and Systems Management 13(2), 197–217 (2005)CrossRefGoogle Scholar
  16. 16.
    Voulgaris, S., Rivière, E., Kermarrec, A.-M., van Steen, M.: Sub-2-Sub: Self-Organizing Content-Based Publish and Subscribe for Dynamic and Large Scale Collaborative Networks. In: Proc. 5th Workshop on Peer-to-Peer Systems (2006)Google Scholar
  17. 17.
    Voulgaris, S., van Steen, M.: Epidemic-style Management of Semantic Overlays for Content-Based Searching. In: RSCTC 2000. LNCS (LNAI), vol. 2005, Springer, Heidelberg (2001)Google Scholar
  18. 18.
    Watts, D.J., Strogatz, S.H.: Collective Dynamics of Small-World Networks. Nature 393, 440–442 (1998)CrossRefGoogle Scholar
  19. 19.
    Yao, A.C.C.: On Constructing Minimum Spanning Trees in k-dimensional Space and Related Problems. SIAM Journal of Computing 11, 721–736 (1982)MATHCrossRefGoogle Scholar
  20. 20.

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© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • François Bonnet
    • 1
  • Anne-Marie Kermarrec
    • 1
  • Michel Raynal
    • 1
  1. 1.IRISARennes CedexFrance

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