Complex Contagions on Configuration Model Graphs with a Power-Law Degree Distribution

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10123)


In this paper we analyze k-complex contagions (sometimes called bootstrap percolation) on configuration model graphs with a power-law distribution. Our main result is that if the power-law exponent \(\alpha \in (2, 3)\), then with high probability, the single seed of the highest degree node will infect a constant fraction of the graph within time \(O\left( \log ^{\frac{\alpha -2}{3-\alpha }}(n)\right) \). This complements the prior work which shows that for \(\alpha > 3\) boot strap percolation does not spread to a constant fraction of the graph unless a constant fraction of nodes are initially infected. This also establishes a threshold at \(\alpha = 3\).

The case where \(\alpha \in (2, 3)\) is especially interesting because it captures the exponent parameters often observed in social networks (with approximate power-law degree distribution). Thus, such networks will spread complex contagions even lacking any other structures.

We additionally show that our theorem implies that \(\omega (\left( n^{\frac{\alpha -2}{\alpha -1}}\right) \) random seeds will infect a constant fraction of the graph within time \(O\left( \log ^{\frac{\alpha -2}{3-\alpha }}(n)\right) \) with high probability. This complements prior work which shows that \(o\left( n^{\frac{\alpha -2}{\alpha -1}}\right) \) random seeds will have no effect with high probability, and this also establishes a threshold at \(n^{\frac{\alpha -2}{\alpha -1}}\).


Degree Distribution Configuration Model Preferential Attachment Random Seed Full Version 


  1. 1.
    Adamic, L.A., Glance, N.: The political blogosphere, the: divided they blog. In: Proceedings of the 3rd International Workshop on Link Discovery, pp. 36–43. ACM (2005)Google Scholar
  2. 2.
    Adler, J.: Bootstrap percolation. Phys. A: Stat. Theor. Phys. 171(3), 453–470 (1991)CrossRefGoogle Scholar
  3. 3.
    Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002). doi: 10.1103/RevModPhys.74.47. MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Amini, H.: Bootstrap percolation and diffusion in random graphs with given vertex degrees. Electron. J. Comb. 17(1), 1–20 (2010)MathSciNetMATHGoogle Scholar
  5. 5.
    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). doi: 10.1007/978-3-642-35311-6_34 CrossRefGoogle Scholar
  6. 6.
    Backstrom, L., Huttenlocher, D., Kleinberg, J., Lan, X.: Group formation in large social networks: membership, growth, and evolution. In: Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 44–54 (2006)Google Scholar
  7. 7.
    Balogh, J., Pittel, B.: Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30, 257–286 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Banerjee, A., Chandrasekhar, A.G., Duflo, E., Jackson, M.O.: The diffusion of microfinance. Science 341(6144), 1236498 (2013)CrossRefGoogle Scholar
  9. 9.
    Barabási, A., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bollobás, B., McKay, B.D.: The number of matchings in random regular graphs and bipartite graphs. J. Comb. Theory, Series B 41(1), 80–91 (1986)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., Wiener, J.: Graph structure in the web. In: Proceedings of the 9th International World Wide Web Conference on Computer Networks, pp. 309–320 (2000)Google Scholar
  12. 12.
    Centola, D., Macy, M.: Complex contagions and the weakness of long ties1. Am. J. Sociol. 113(3), 702–734 (2007)CrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Coleman, J., Katz, E., Menzel, H.: The diffusion of an innovation among physicians. Sociometry 20(4), 253–270 (1957)CrossRefGoogle Scholar
  15. 15.
    Coleman, J.S., Katz, E., Menzel, H.: Medical Innovation: A Diffusion Study. Bobbs-Merrill Co., Indianapolis (1966)Google Scholar
  16. 16.
    Ebrahimi, R., Gao, J., Ghasemiesfeh, G., Schoenebeck, G.: How complex contagions spread quickly in the preferential attachment model, other time-evolving networks. arXiv preprint arXiv:1404.2668 (2014)
  17. 17.
    Ebrahimi, R., Gao, J., Ghasemiesfeh, G., Schoenebeck, G.: Complex contagions in Kleinberg’s small world model. In: Proceedings of the Conference on Innovations in Theoretical Computer Science, pp. 63–72. ACM (2015)Google Scholar
  18. 18.
    Ghasemiesfeh, G., Ebrahimi, R., Gao, J.: Complex contagion, the weakness of long ties in social networks: revisited. In: Proceedings of the Fourteenth ACM Conference on Electronic Commerce, pp. 507–524, June 2013Google Scholar
  19. 19.
    Granovetter, M.: Threshold models of collective behavior. Am. J. Sociol. 83(6), 1420–1443 (1978)CrossRefGoogle Scholar
  20. 20.
    Janson, S., Łuczak, T., Turova, T., Vallier, T.: Bootstrap percolation on the random graph \(G_{n, p}\). Ann. Appl. Prob. 22(5), 1989–2047 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146 (2003)Google Scholar
  22. 22.
    Macdonald, J.S., Macdonald, L.D.: Chain migration, ethnic neighborhood formation and social networks. Milbank Meml. Fund Q. 42(1), 82–97 (1964)CrossRefGoogle Scholar
  23. 23.
    Mermelstein, R., Cohen, S., Lichtenstein, E., Baer, J.S., Kamarck, T.: Social support and smoking cessation and maintenance. J. Consult. Clin. Psychol. 54(4), 447 (1986)CrossRefGoogle Scholar
  24. 24.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    De Solla Price, D.: Networks of scientific papers. Science 149(3683), 510–515 (1965). doi: 10.1126/science.149.3683.510 CrossRefGoogle Scholar
  26. 26.
    Romero, D.M., Meeder, B., Kleinberg, J.: Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter. In: Proceedings of the 20th International Conference on World Wide Web, pp. 695–704 (2011)Google Scholar
  27. 27.
    Steyvers, M., Tenenbaum, J.B.: The large-scale structure of semantic networks: statistical analyses and a model of semantic growth. Cogn. Sci. 29, 41–78 (2005)CrossRefGoogle Scholar
  28. 28.
    Ugander, J., Backstrom, L., Marlow, C., Kleinberg, J.: Structural diversity in social contagion. Proc. Natl. Acad. Sci. 109(16), 5962–5966 (2012)CrossRefGoogle Scholar
  29. 29.
    Van Der Hofstad, R.: Random graphs and complex networks, p. 11 (2009).
  30. 30.
    Wormald, N.C.: Differential equations for random processes and random graphs. Ann. Appl. Prob. 5(4), 1217–1235 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2016

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA

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