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Networks Become Navigable as Nodes Move and Forget

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Automata, Languages and Programming (ICALP 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5125))

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Abstract

We propose a dynamic process for network evolution, aiming at explaining the emergence of the small world phenomenon, i.e., the statistical observation that any pair of individuals are linked by a short chain of acquaintances computable by a simple decentralized routing algorithm, known as greedy routing. Our model is based on the combination of two dynamics: a random walk (spatial) process, and an harmonic forgetting (temporal) process. Both processes reflect natural behaviors of the individuals, viewed as nodes in the network of inter-individual acquaintances. We prove that, in k-dimensional lattices, the combination of these two processes generates long-range links mutually independently distributed as a k-harmonic distribution. We analyze the performances of greedy routing at the stationary regime of our process, and prove that the expected number of steps for routing from any source to any target in any multidimensional lattice is a polylogarithmic function of the distance between the two nodes in the lattice. Up to our knowledge, these results are the first formal proof that navigability in small worlds can emerge from a dynamic process for network evolution. Our dynamica process can find practical applications to the design of spatial gossip and resource location protocols.

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References

  1. Abraham, I., Gavoille, C.: Object Location Using Path Separators. In: 25th ACM Symp. on Principles of Distributed Computing (PODC), pp. 188–197 (2006)

    Google Scholar 

  2. Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Review of modern physics 74, 47–97 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aspnes, J., Diamadi, Z., Shah, G.: Fault-tolerant routing in peer-to-peer systems. In: 21st ACM Symp. on Principles of Distributed Computing (PODC), pp. 223–232 (2002)

    Google Scholar 

  4. 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)

    Chapter  Google Scholar 

  5. Bollobás, B., Chung, F.: The diameter of a cycle plus a random matching. SIAM J. Discrete Math. 1(3), 328–333 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bremaud, P.: Markov Chains, Gibbs Field, Monte Carlo Simulation and Queues. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  7. Chaintreau, A., Fraigniaud, P., Lebhar, E.: Opportunistic spatial gossip over mobile social networks. In: 1st ACM SIGCOMM Workshop on Online Social Net (WOSN) (to appear, 2008)

    Google Scholar 

  8. Chaintreau, A., Fraigniaud, P., Lebhar, E.: Networks Become Navigable as Nodes Move and Forget. Technical Report, arXiv:0803.0248v1 (2008)

    Google Scholar 

  9. Chaintreau, A., Hui, P., Crowcroft, J., Diot, C., Scott, J., Gass, R.: Impact of Human Mobility on Opportunistic Forwarding Algorithms. IEEE Trans. Mob. Comp. 6(6), 606–620 (2007)

    Article  Google Scholar 

  10. Clarke, I., Sandberg, O., Wiley, B., Hong, T.W.: Freenet: A Distributed Anonymous Information Storage and Retrieval System. In: Federrath, H. (ed.) Designing Privacy Enhancing Technologies. LNCS, vol. 2009, pp. 46–66. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  11. Clauset, A., Moore, C.: How Do Networks Become Navigable? Technical Report, arXiv:0309.415v2 (2003)

    Google Scholar 

  12. Demers, A., Greene, D., Hauser, C., Irish, W., Larson, J., Shenker, S., Sturgis, H., Swinehart, D., Terry, D.: Epidemic Algorithms for Replicated Database Maintenance. Operating Systems Review 22(1), 8–32 (1988)

    Article  Google Scholar 

  13. Dodds, P., Muhamad, R., Watts, D.: An experimental study of search in global social networks. Science 301(5634), 827–829 (2003)

    Article  Google Scholar 

  14. Duchon, P., Eggeman, N., Hanusse, N.: Non-Searchability of Random Power Law Graphs. In: Tovar, E., Tsigas, P., Fouchal, H. (eds.) OPODIS 2007. LNCS, vol. 4878. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  15. Duchon, P., Hanusse, N., Lebhar, E., Schabanel, N.: Could any graph be turned into a small-world? Theoretical Computer Science 355(1), 96–103 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Flammini, M., Moscardelli, L., Navarra, A., Perennes, S.: Asymptotically Optimal Solutions for Small World Graphs. In: Fraigniaud, P. (ed.) DISC 2005. LNCS, vol. 3724, pp. 414–428. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  17. Fraigniaud, P.: Greedy routing in tree-decomposed graphs: a new perspective on the small-world phenomenon. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 791–802. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  18. 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: Cunha, J.C., Medeiros, P.D. (eds.) Euro-Par 2005. LNCS, vol. 3648, pp. 1163–1172. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  19. Fraigniaud, P., Gavoille, C., Kosowski, A., Lebhar, E., Lotker, Z.: Universal Augmentation Schemes for Network Navigability: Overcoming the \({\sqrt{n}}\)-Barrier. In: 19th ACM Symp. on Parallelism in Algorithms and Architectures (SPAA), pp. 1–9 (2007)

    Google Scholar 

  20. Fraigniaud, P., Lebhar, E., Lotker, Z.: A Doubling Dimension Threshold Θ(loglog n) for Augmented Graph Navigability. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 376–386. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  21. Giakkoupis, G., Hadzilacos, V.: On the complexity of greedy routing in ring-based peer-to-peer networks. In: 26th ACM Symp. on Princ. of Dist. Comp. (PODC) (2007)

    Google Scholar 

  22. Grossglauser, M., Tse, D.: Mobility Increases the Capacity of Ad Hoc Wireless Networks. IEEE/ACM Trans. on Net. 10(4), 477–486 (2002)

    Article  Google Scholar 

  23. Jain, S., Fall, K., Patra, R.: Routing in a delay tolerant network. In: Proc. ACM SIGCOMM (2004)

    Google Scholar 

  24. Kempe, D., Kleinberg, J., Demers, A.: Spatial gossip and resource location protocols. In: 33rd ACM Symposium on Theory of Computing, pp. 163–172 (2001)

    Google Scholar 

  25. Kempe, D., Kleinberg, J.: Protocols and impossibility results for gossip-based communication mechanisms. In: 43st IEEE Symp. on Foundations of Computer Science, pp. 471–480 (2002)

    Google Scholar 

  26. Kleinberg, J.: The small-world phenomenon: an algorithmic perspective. In: 32nd ACM Symp. on Theory of Computing (STOC), pp. 163–170 (2000)

    Google Scholar 

  27. Kleinberg, J.: Complex networks and decentralized search algorithm. In: International Congress of Mathematicians (ICM), Madrid (2006)

    Google Scholar 

  28. Kumar, R., Liben-Nowell, D., Tomkins, A.: Navigating Low-Dimensional and Hierarchical Population Networks. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168. Springer, Heidelberg (2006)

    Google Scholar 

  29. Liben-Nowell, D., Kleinberg, J.: The link prediction problem for social networks. Journal of the American society for information science and technology 58(7), 1019–1031 (2007)

    Article  Google Scholar 

  30. Liben-Nowell, D., Novak, J., Kumar, R., Raghavan, P., Tomkins, A.: Geographic routing in social networks. In: Proc. of the Natl. Academy of Sciences of the USA, vol. 102/3, pp. 11623–11628 (2005)

    Google Scholar 

  31. Milgram, S.: The Small-World Problem. Psychology Today, pp. 60–67 (1967)

    Google Scholar 

  32. Sandberg, O.: Neighbor Selection and Hitting Probability in Small-World Graphs. Annals of Applied Probability (to appear, 2008)

    Google Scholar 

  33. Sandberg, O., Clarke, I.: The evolution of navigable small-world networks. Tech. Report 2007:14, Chalmers University of Technology (2007)

    Google Scholar 

  34. Simon, H.: On a class of skew distribution functions. Biometrika 42(3/4), 425–440 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  35. Slivkins, A.: Distance estimation and object location via rings of neighbors. In: 24th Annual ACM Symp. on Princ. of Dist. Comp. (PODC), pp. 41–50 (2005)

    Google Scholar 

  36. Watts, D., Strogatz, S.: Collective Dynamics of Small-World Networks. Nature 393, 440–442 (1998)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Thomson, Paris, France

    Augustin Chaintreau

  2. CNRS and University Paris Diderot, France

    Pierre Fraigniaud & Emmanuelle Lebhar

Authors
  1. Augustin Chaintreau
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  2. Pierre Fraigniaud
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  3. Emmanuelle Lebhar
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Editor information

Editors and Affiliations

  1. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Luca Aceto

  2. Department of Computer Science, IT-Parken, University of Aarhus, Ã…bogade 34, 8200, Ã…rhusN, Denmark

    Ivan Damgård

  3. Department of Computer Science, University of Liverpool, Ashton Building, L69 3BX, Liverpool, UK

    Leslie Ann Goldberg

  4. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavik, Iceland

    Magnús M. Halldórsson

  5. Department of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Anna Ingólfsdóttir

  6. Université de Bordeaux-1, LaBRI, 351, Cours de la Libération, 33405, Talence cedex, France

    Igor Walukiewicz

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© 2008 Springer-Verlag Berlin Heidelberg

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Chaintreau, A., Fraigniaud, P., Lebhar, E. (2008). Networks Become Navigable as Nodes Move and Forget. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_12

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  • DOI: https://doi.org/10.1007/978-3-540-70575-8_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70574-1

  • Online ISBN: 978-3-540-70575-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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