Giant Component and Connectivity in Geographical Threshold Graphs

  • Milan Bradonjić
  • Aric Hagberg
  • Allon G. Percus
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4863)


The geographical threshold graph model is a random graph model with nodes distributed in a Euclidean space and edges assigned through a function of distance and node weights. We study this model and give conditions for the absence and existence of the giant component, as well as for connectivity.


random graph geographical threshold graph giant component connectivity 


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  1. 1.
    Bonato, A.: A survey of models of the web graph. In: López-Ortiz, A., Hamel, A.M. (eds.) CAAN 2004. LNCS, vol. 3405, pp. 159–172. Springer, Heidelberg (2005)Google Scholar
  2. 2.
    Abello, J., Pardalos, P.M., Resende, M.G.C. (eds.): Handbook of massive data sets. Kluwer Academic Publishers, Norwell, MA (2002)zbMATHGoogle Scholar
  3. 3.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. In: SIGCOMM 1999: Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication, pp. 251–262. ACM Press, New York (1999)CrossRefGoogle Scholar
  4. 4.
    Kleinberg, J.M., Kumar, R., Raghavan, P., Rajagopalan, S., Tomkins, A.S.: The web as a graph: Measurements, models, and methods. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 1–17. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: Stochastic models for the web graph. In: FOCS 2000: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, Washington, DC, USA, p. 57. IEEE Computer Society, Los Alamitos (2000)CrossRefGoogle Scholar
  6. 6.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Aiello, W., Chung, F., Lu, L.: A random graph model for massive graphs. In: STOC 2000: Proceedings of the thirty-second annual ACM symposium on Theory of computing, pp. 171–180. ACM Press, New York (2000)CrossRefGoogle Scholar
  8. 8.
    Bollobás, B., Riordan, O., Spencer, J., Tusnády, G.: The degree sequence of a scale-free random graph process. Random Struct. Algorithms 18, 279–290 (2001)zbMATHCrossRefGoogle Scholar
  9. 9.
    Cooper, C., Frieze, A.M.: A general model of undirected Web graphs. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 500–511. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Bradonjić, M., Kong, J.: Wireless Ad Hoc Networks with Tunable Topology. To appear in Proceedings of the 45th Annual Allerton Conference on Communication, Control and Computing (2007)Google Scholar
  11. 11.
    Erdős, P., Rényi, A.: On random graphs. Publ. Math. Inst. Hungar. Acad. Sci. (1959)Google Scholar
  12. 12.
    Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. (1960)Google Scholar
  13. 13.
    Penrose, M.D.: Random Geometric Graphs. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  14. 14.
    Mahadev, N.V.R., Peled, U.N.: Threshold graphs and related topics. Annals of discrete mathematics, vol. 56. Elsevier, New York (1995)zbMATHGoogle Scholar
  15. 15.
    Hagberg, A., Swart, P.J., Schult, D.A.: Designing threshold networks with given structural and dynamical properties. Phys. Rev. E 74, 056116 (2006)Google Scholar
  16. 16.
    Masuda, N., Miwa, H., Konno, N.: Geographical threshold graphs with small-world and scale-free properties. Physical Review E 71, 036108 (2005)Google Scholar
  17. 17.
    Alon, N., Spencer, J.H.: The probabilistic method, 2nd edn. John Wiley & Sons, Inc., New York (2000)zbMATHGoogle Scholar
  18. 18.
    Gupta, P., Kumar, P.R.: Critical power for asymptotic connectivity. In: Proceedings of the 37th IEEE Conference on Decision and Control, vol. 1, pp. 1106–1110 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Milan Bradonjić
    • 1
  • Aric Hagberg
    • 2
  • Allon G. Percus
    • 3
    • 4
  1. 1.Department of Electrical Engineering, UCLA, Los Angeles, CA 90095 
  2. 2.Mathematical Modeling and Analysis, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 
  3. 3.Department of Mathematics, UCLA, Los Angeles, CA 90095 
  4. 4.Information Sciences Group, Los Alamos National Laboratory, Los Alamos, NM 87545 

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