Component Evolution in General Random Intersection Graphs

  • Milan Bradonjić
  • Aric Hagberg
  • Nicolas W. Hengartner
  • Allon G. Percus
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6516)


Random intersection graphs (RIGs) are an important random structure with algorithmic applications in social networks, epidemic networks, blog readership, and wireless sensor networks. RIGs can be interpreted as a model for large randomly formed non-metric data sets. We analyze the component evolution in general RIGs, giving conditions on the existence and uniqueness of the giant component. Our techniques generalize existing methods for analysis of component evolution: we analyze survival and extinction properties of a dependent, inhomogeneous Galton-Watson branching process on general RIGs. Our analysis relies on bounding the branching processes and inherits the fundamental concepts of the study of component evolution in Erdős-Rényi graphs. The major challenge comes from the underlying structure of RIGs, which involves both a set of nodes and a set of attributes, with different probabilities associated with each attribute.


General random intersection graphs random graphs branching processes giant component stochastic processes in relation with random discrete structures 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47–97 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.H.: The probabilistic method, 2nd edn. John Wiley & Sons, Inc., New York (2000)CrossRefMATHGoogle Scholar
  3. 3.
    Barabási, A.L., Albert, R.: Emergence of Scaling in Random Networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Behrisch, M.: Component evolution in random intersection graphs. Electr. J. Comb. 14 (2007)Google Scholar
  5. 5.
    Bernstein, S.N.: On a modification of chebyshevs inequality and of the error formula of laplace. Ann. Sci. Inst. Sav. Ukraine, Sect. Math. 4(25) (1924)Google Scholar
  6. 6.
    Bloznelis, M., Jaworski, J., Rybarczyk, K.: Component evolution in a secure wireless sensor network. Netw. 53(1), 19–26 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chung, F., Lu, L.: The average distances in random graphs with given expected degrees. Proceedings of the National Academy of Sciences of the United States of America 99(25), 15879–15882 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Deijfen, M., Kets, W.: Random intersection graphs with tunable degree distribution and clustering. Probab. Eng. Inf. Sci. 23(4), 661–674 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Erdős, P., Goodman, A.W., Pósa, L.: The representation of a graph by set intersections. Canad. J. Math. 18, 106–112 (1966)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Godehardt, E., Jerzy Jaworski, K.R.: Random intersection graphs and classification. In: Advances in Data Analysis, vol. 45, pp. 67–74 (2007)Google Scholar
  11. 11.
    Eubank, S., Guclu, H., Anil Kumar, V.S., Marathe, M.V., Srinivasan, A., Toroczkai, Z., Wang, N.: Modelling disease outbreaks in realistic urban social networks. Nature 429(6988), 180–184 (2004)CrossRefGoogle Scholar
  12. 12.
    Fill, J.A., Scheinerman, E.R., Singer-Cohen, K.B.: Random intersection graphs when m = ω(n): An equivalence theorem relating the evolution of the g(n, m, p) and g(n,p) models. Random Struct. Algorithms 16(2), 156–176 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Godehardt, E., Jaworski, J.: Two models of random intersection graphs and their applications. Electronic Notes in Discrete Mathematics 10, 129–132 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Guillaume, J.-L., Latapy, M.: Bipartite graphs as models of complex networks. Physica A: Statistical and Theoretical Physics 371(2), 795–813 (2006)CrossRefGoogle Scholar
  15. 15.
    Jaworski, J., Stark, D.: The vertex degree distribution of passive random intersection graph models. Comb. Probab. Comput. 17(4), 549–558 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Karoński, M., Scheinerman, E., Singer-Cohen, K.: On random intersection graphs:the subgraph problem. Combinatorics, Probability and Computing 8 (1999)Google Scholar
  17. 17.
    Lagerås, A.N., Lindholm, M.: A note on the component structure in random intersection graphs. Electronic Journal of Combinatorics 15(1) (2008)Google Scholar
  18. 18.
    Newman, M.E.J.: Scientific collaboration networks. I. Network construction and fundamental results. Phys. Rev. E 64(1), 016131 (2001)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Newman, M.E.J., Park, J.: Why social networks are different from other types of networks. Phys. Rev. E 68(3), 036122 (2003)CrossRefGoogle Scholar
  20. 20.
    Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64(2), 026118 (2001)CrossRefGoogle Scholar
  21. 21.
    Nikoletseas, S., Raptopoulos, C., Spirakis, P.: Large independent sets in general random intersection graphs. Theor. Comput. Sci. 406, 215–224 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Nikoletseas, S.E., Raptopoulos, C., Spirakis, P.G.: The existence and efficient construction of large independent sets in general random intersection graphs. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1029–1040. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    Nikoletseas, S.E., Raptopoulos, C., Spirakis, P.G.: Expander properties and the cover time of random intersection graphs. Theor. Comput. Sci. 410(50), 5261–5272 (2009)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    van der Hofstad, R.: Random graphs and complex networks. Lecture notes in preparation,
  25. 25.
    Rybarczyk, K.: Equivalence of the random intersection graph and G(n,p) (2009) (submitted),
  26. 26.
    Singer-Cohen, K.: Random intersection graphs. PhD thesis, Johns Hopkins University (1995)Google Scholar
  27. 27.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of Small-World networks. Nature 393(6684), 440–442 (1998)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Milan Bradonjić
    • 1
  • Aric Hagberg
    • 2
  • Nicolas W. Hengartner
    • 3
  • Allon G. Percus
    • 4
  1. 1.Theoretical Division and Center for Nonlinear StudiesLos Alamos National LaboratoryLos AlamosUSA
  2. 2.Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA
  3. 3.Information Sciences GroupLos Alamos National LaboratoryLos AlamosUSA
  4. 4.School of Mathematical SciencesClaremont Graduate UniversityClaremontUSA

Personalised recommendations