Navigability is a Robust Property

  • Dimitris Achlioptas
  • Paris SiminelakisEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9479)


The Small World phenomenon has inspired researchers across a number of fields. A breakthrough in its understanding was made by Kleinberg who introduced Rank Based Augmentation (RBA): add to each vertex independently an arc to a random destination, selected from a carefully crafted probability distribution. Kleinberg proved that RBA makes many networks navigable, i.e., it allows greedy routing to successfully deliver messages between any two vertices in a polylogarithmic number of steps. Our goal in this work is to prove that navigability is an inherent, robust property of many random networks, requiring no augmentation, coordination, or even independence assumptions. Our framework assigns a cost to each edge and considers the uniform measure over all graphs on n vertices that satisfy a total budget constraint. We show that when the cost function is sufficiently correlated with the underlying geometry of the vertices and for a wide range of budgets, the overwhelming majority of all feasible graphs with the given budget are navigable. We provide a new set of geometric conditions that generalize Kleinberg’s set systems as well as a unified analysis of navigability.


Cost Function Random Graph Product Measure Distance Scale Uniform Measure 
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  1. 1.
    Abraham, I., Gavoille, C., Malkhi, D.: Compact routing for graphs excluding a fixed minor. In: Fraigniaud, P. (ed.) DISC 2005. LNCS, vol. 3724, pp. 442–456. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  2. 2.
    Achlioptas, D., Siminelakis, P.: Symmetric graph properties have independent edges. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 467–478. Springer, Heidelberg (2015) CrossRefGoogle Scholar
  3. 3.
    Alon, N., Spencer, J.: The Probabilistic Method. John Wiley, New York (1992) zbMATHGoogle Scholar
  4. 4.
    Aspnes, J., Diamadi, Z., Shah, G.: Fault-tolerant routing in peer-to-peer systems. In: Proceedings of the 21st Annual ACM Symposium on Principles of Distributed Computing, PODC 2002, pp. 223–232 (2002)Google Scholar
  5. 5.
    Chaintreau, A., Fraigniaud, P., Lebhar, E.: Networks become navigable as nodes move and forget. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 133–144. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  6. 6.
    Clarke, I., Sandberg, O., Wiley, B., Hong, T.W.: Freenet: a distributed anonymous information storage and retrieval system. In: Federrath, H. (ed.) Anonymity 2000. LNCS, vol. 2009, pp. 46–66. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  7. 7.
    Duchon, P., Hanusse, N., Lebhar, E., Schabanel, N.: Could any graph be turned into a small-world? Theor. Comput. Sci. 355(1), 96–103 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Fraigniaud, P.: Greedy routing in tree-decomposed graphs. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 791–802. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  9. 9.
    Fraigniaud, P., Gavoille, C., Kosowski, A., Lebhar, E., Lotker, Z.: Universal augmentation schemes for network navigability. Theor. Comput. Sci. 410(21–23), 1970–1981 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Fraigniaud, P., Giakkoupis, G.: On the searchability of small-world networks with arbitrary underlying structure. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, pp. 389–398 (2010)Google Scholar
  11. 11.
    Fraigniaud, P., Lebhar, E., Lotker, Z.: A lower bound for network navigability. SIAM J. Discrete Math. 24(1), 72–81 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Kleinberg, J.: Complex networks and decentralized search algorithms. In: Proceedings of the International Congress of Mathematicians, ICM 2006, pp. 1019–1044 (2006)Google Scholar
  13. 13.
    Kleinberg, J.M.: The small-world phenomenon: an algorithmic perspective. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, STOC 2000, pp. 163–170 (2000)Google Scholar
  14. 14.
    Kleinberg, J.M.: Small-world phenomena and the dynamics of information. In: Proceedings of Advances in Neural Information Processing Systems 14, NIPS 2001, pp. 431–438 (2001)Google Scholar
  15. 15.
    Lebhar, E., Schabanel, N.: Graph augmentation via metric embedding. In: Baker, T.P., Bui, A., Tixeuil, S. (eds.) OPODIS 2008. LNCS, vol. 5401, pp. 217–225. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  16. 16.
    Liben-Nowell, D., Novak, J., Kumar, R., Raghavan, P., Tomkins, A.: Geographic routing in social networks. Proc. Natl. Acad. Sci. U.S.A. 102(33), 11623–11628 (2005)CrossRefGoogle Scholar
  17. 17.
    Manku, G.S., Naor, M., Wieder, U.: Know thy neighbor’s neighbor: the power of lookahead in randomized P2P networks. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, STOC 2004, pp. 54–63 (2004)Google Scholar
  18. 18.
    Milgram, S.: The small world problem. Psychol. Today 2(1), 60–67 (1967)MathSciNetGoogle Scholar
  19. 19.
    Sandberg, O.: Neighbor selection and hitting probability in small-world graphs. Ann. Appl. Probab. 18(5), 1771–1793 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Slivkins, A.: Distance estimation and object location via rings of neighbors. Distrib. Comput. 19(4), 313–333 (2007)CrossRefzbMATHGoogle Scholar
  21. 21.
    Watts, D.J.: Six Degrees: The Science of a Connected Age. WW Norton, New York (2004)Google Scholar
  22. 22.
    Zeng, J., Hsu, W.-J., Wang, J.: Near optimal routing in a small-world network with augmented local awareness. In: Pan, Y., Chen, D., Guo, M., Cao, J., Dongarra, J. (eds.) ISPA 2005. LNCS, vol. 3758, pp. 503–513. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  23. 23.
    Zhang, H., Goel, A., Govindan, R.: Using the small-world model to improve freenet performance. Comput. Netw. 46(4), 555–574 (2004)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CaliforniaSanta CruzUSA
  2. 2.Department of Electrical EngineeringStanford UniversityStanfordUSA

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