Cascades and Myopic Routing in Nonhomogeneous Kleinberg’s Small World Model

  • Jie Gao
  • Grant Schoenebeck
  • Fang-Yi YuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10660)


Kleinberg’s small world model [20] simulates social networks with both strong and weak ties. In his original paper, Kleinberg showed how the distribution of weak-ties, parameterized by \(\gamma \), influences the efficacy of myopic routing on the network. Recent work on social influence by k-complex contagion models discovered that the distribution of weak-ties also impacts the spreading rate in a crucial manner on Kleinberg’s small world model [15]. In both cases the parameter of \(\gamma = 2\) proves special: when \(\gamma \) is anything but 2 the properties no longer hold.

In this work, we propose a natural generalization of Kleinberg’s small world model to allow node heterogeneity: instead of a single global parameter \(\gamma \), each node has a personalized parameter \(\gamma \) chosen independently from a distribution \(\mathcal {D}\). In contrast to the original model, we show that this model enables myopic routing and k-complex contagions on a large range of the parameter space, improving the robustness of the model. Moreover, we show that our generalization is supported by real-world data. Analysis of four different social networks shows that the nodes do not show homogeneity in terms of the variance of the lengths of edges incident to the same node.


  1. 1.
    Adler, J.: Bootstrap percolation. Phys. A: Stat. Theor. Phys. 171(3), 453–470 (1991)CrossRefGoogle Scholar
  2. 2.
    Amini, H.: Bootstrap percolation and diffusion in random graphs with given vertex degrees. Electr. J. Comb. 17(1), R25 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Amini, H., Fountoulakis, N.: What i tell you three times is true: bootstrap percolation in small worlds. In: Goldberg, P.W. (ed.) WINE 2012. LNCS, vol. 7695, pp. 462–474. Springer, Heidelberg (2012). CrossRefGoogle Scholar
  4. 4.
    Balogh, J., Pittel, B.: Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30, 257–286 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boguna, M., Krioukov, D., Claffy, K.C.: Navigability of complex networks. Nat. Phys. 5, 74–80 (2009)CrossRefGoogle Scholar
  6. 6.
    Bollobás, B., Chung, F.R.K.: The diameter of a cycle plus a random matching. SIAM J. Discret. Math. 1(3), 328–333 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Burt, R.S.: Structural Holes: The Social Structure of Competition. Cambridge University Press, Cambridge (1992)Google Scholar
  8. 8.
    Burt, R.S.: Structural Holes: The social structure of competition. Harvard University Press, Cambridge (1995)Google Scholar
  9. 9.
    Chalupa, J., Leath, P.L., Reich, G.R.: Bootstrap percolation on a Bethe lattice. J. Phys. C: Solid State Phys. 12(1), L31 (1979)CrossRefGoogle Scholar
  10. 10.
    Dodds, P.S., Muhamad, R., Watts, D.J.: An experimental study of search in global social networks. Science 301, 827 (2003)CrossRefGoogle Scholar
  11. 11.
    Watts, D., Strogatz, S.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 409–410 (1998)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ebrahimi, R., Gao, J., Ghasemiesfeh, G., Schoenebeck, G.: How complex contagions in preferential attachment models and other time-evolving networks. IEEE Trans. Netw. Sci. Eng. PP(99), 1 (2017). ISSN 2327–4697CrossRefGoogle Scholar
  13. 13.
    Ebrahimi, R., Gao, J., Ghasemiesfeh, G., Schoenebeck, G.: Complex contagions in Kleinberg’s small world model. In: Proceedings of the 6th Innovations in Theoretical Computer Science (ITCS 2015), pp. 63–72. January 2015Google Scholar
  14. 14.
    Gao, J., Ghasemiesfeh, G., Schoenebeck, G., Yu, F.-Y.: General threshold model for social cascades: analysis and simulations. In: Proceedings of the 2016 ACM Conference on Economics and Computation, pp. 617–634. ACM (2016)Google Scholar
  15. 15.
    Ghasemiesfeh, G., Ebrahimi, R., Gao, J.: Complex contagion and the weakness of long ties in social networks: revisited. In: Proceedings of the fourteenth ACM conference on Electronic Commerce, pp. 507–524. ACM (2013)Google Scholar
  16. 16.
    Granovetter, M.: Threshold models of collective behavior. Am. J. Sociol. 83(6), 1420–1443 (1978)CrossRefGoogle Scholar
  17. 17.
    Jackson, M.O.: Social and Economic Networks. Princeton University Press, Princeton (2008). ISBN 0691134405, 9780691134406zbMATHGoogle Scholar
  18. 18.
    Janson, S., Luczak, T., Turova, T., Vallier, T.: Bootstrap percolation on the random graph \({G}_{n, p}\). Ann. Appl. Probab. 22(5), 1989–2047 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jeong, H., Mason, S.P., Barabasi, A.-L., Oltvai, Z.N.: Lethality and centrality in protein networks. Nature 411, 41–42 (2001)CrossRefGoogle Scholar
  20. 20.
    Kleinberg, J., The small-world phenomenon: an algorithm perspective. In: Proceedings of the 32-nd Annual ACM Symposium on Theory of Computing, pp. 163–170 (2000)Google Scholar
  21. 21.
    Krioukov, D., Papadopoulos, F., Boguna, M., Vahdat, A.: Greedy forwarding in scale-free networks embedded in hyperbolic metric spaces. In: ACM SIGMETRICS Workshop on Mathematical Performance Modeling and Analysis (MAMA) June 2009Google Scholar
  22. 22.
    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, pp. 480–491. Springer, Heidelberg (2006). ISBN 3-540-38875-3CrossRefGoogle Scholar
  23. 23.
    Milgram, S.: The small world problem. Phychol. Today 1, 61–67 (1967)Google Scholar
  24. 24.
    Newman, M.E.J., Moore, C., Watts, D.J.: Mean-field solution of the small-world network model. Phys. Rev. Lett. 84, 3201–3204 (2000)CrossRefGoogle Scholar
  25. 25.
    Schoenebeck, G., Yu, F.-Y.: Complex contagions on configuration model graphs with a power-law degree distribution. In: Cai, Y., Vetta, A. (eds.) WINE 2016. LNCS, vol. 10123, pp. 459–472. Springer, Heidelberg (2016). CrossRefGoogle Scholar
  26. 26.
    Travers, J., Milgram, S.: An experimental study of the small world problem. Sociometry 32, 425 (1969)CrossRefGoogle Scholar
  27. 27.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)CrossRefzbMATHGoogle Scholar
  28. 28.
    Williams, R.J., Berlow, E.L., Dunne, J.A., Barabasi, A.L., Martinez, N.D.: Two degrees of separation in complex food webs. Proc. Nat. Acad. Sci. 99(20), 12913–12916 (2002)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computer ScienceStony Brook UniversityStony BrookUSA
  2. 2.Department of EECSUniversity of MichiganAnn ArborUSA

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