A Geometric Preferential Attachment Model of Networks II

  • Abraham D. Flaxman
  • Alan M. Frieze
  • Juan Vera
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4863)

Abstract

A detailed understanding of expansion in complex networks can greatly aid in the design and analysis of algorithms for a variety of important network tasks, including routing messages, ranking nodes, and compressing graphs. This has motivated several recent investigations of expansion properties in real-world graphs and also in random models of real-world graphs, like the preferential attachment graph. The results point to a gap between real-world observations and theoretical models. Some real-world graphs are expanders and others are not, but a graph generated by the preferential attachment model is an expander whp .

We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model.

The vertices of Gn are n sequentially generated points x1,x2,...,xn chosen uniformly at random from the unit sphere in Open image in new window. After generating xt, we randomly connect it to m points from those points in x1,x2,...,xt − 1 ....

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aiello, W., Chung, F.R.K., Lu, L.: A random graph model for massive graphs. In: Proc. of the 32nd Annual ACM Symposium on the Theory of Computing, pp. 171–180 (2000)Google Scholar
  2. 2.
    Aiello, W., Chung, F.R.K., Lu, L.: Random Evolution in Massive Graphs. In: Proc. of IEEE Symposium on Foundations of Computer Science, pp. 510–519 (2001)Google Scholar
  3. 3.
    Albert, R., Barabási, A., Jeong, H.: Diameter of the world wide web. Nature 401, 103–131 (1999)Google Scholar
  4. 4.
    Barabasi, A., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Berger, N., Bollobas, B., Borgs, C., Chayes, J., Riordan, O.: Degree distribution of the FKP network model. In: Proc. of the 30th International Colloquium of Automata, Languages and Programming, pp. 725–738 (2003)Google Scholar
  6. 6.
    Berger, N., Borgs, C., Chayes, J., D’Souza, R., Kleinberg, R.D.: Competition-induced preferential attachment. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 208–221. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Blandford, D., Blelloch, G.E., Kash, I.: Compact Representations of Separable Graphs. In: Proc. of ACM/SIAM Symposium on Discrete Algorithms, pp. 679–688 (2003)Google Scholar
  8. 8.
    Bollobás, B., Riordan, O.: Mathematical Results on Scale-free Random Graphs. In: Handbook of Graphs and Networks, Wiley-VCH, Berlin (2002)Google Scholar
  9. 9.
    Bollobás, B., Riordan, O.: The diameter of a scale-free random graph. Combinatorica 4, 5–34 (2004)CrossRefGoogle Scholar
  10. 10.
    Bollobás, B., Riordan, O.: Coupling scale free and classical random graphs. Internet Mathematics 1(2), 215–225 (2004)MATHMathSciNetGoogle Scholar
  11. 11.
    Bollobás, B., Riordan, O., Spencer, J., Tusanády, G.: The degree sequence of a scale-free random graph process. Random Structures and Algorithms 18, 279–290 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., Wiener, J.: Graph structure in the web. In: Proc. of the 9th Intl. World Wide Web Conference, pp. 309–320 (2002)Google Scholar
  13. 13.
    Buckley, G., Osthus, D.: Popularity based random graph models leading to a scale-free degree distribution. Discrete Mathematics 282, 53–68 (2004)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Chung, F.R.K., Lu, L., Vu, V.: Eigenvalues of random power law graphs. Annals of Combinatorics 7, 21–33 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Chung, F.R.K., Lu, L., Vu, V.: The spectra of random graphs with expected degrees. Proceedings of national Academy of Sciences 100, 6313–6318 (2003)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Cooper, C., Frieze, A.M.: A General Model of Undirected Web Graphs. Random Structures and Algorithms 22, 311–335 (2003)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Cooper, C., Frieze, A.M., Vera, J.: Random deletions in a scale free random graph process. Internet Mathematics 1, 463–483 (2004)MATHMathSciNetGoogle Scholar
  18. 18.
    Drinea, E., Enachescu, M., Mitzenmacher, M.: Variations on Random Graph Models for the Web, Harvard Technical Report TR-06-01 (2001)Google Scholar
  19. 19.
    Estrada, E.: Spectral scaling and good expansion properties in complex networks. Europhysics Letters 73(4), 649–655 (2006)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Erdös, P., Rényi, A.: On random graphs I. Publicationes Mathematicae Debrecen 6, 290–297 (1959)MathSciNetGoogle Scholar
  21. 21.
    Fabrikant, A., Koutsoupias, E., Papadimitriou, C.H.: Heuristically Optimized Trade-Offs: A New Paradigm for Power Laws in the Internet. In: Proc. of 29th International Colloquium of Automata, Languages and Programming (2002)Google Scholar
  22. 22.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On Power-law Relationships of the Internet Topology. ACM SIGCOMM Computer Communication Review 29, 251–262 (1999)CrossRefGoogle Scholar
  23. 23.
    Flaxman, A.: Expansion and lack thereof in randomly perturbed graphs. In: Proc. of the Web Algorithms Workshop (to appear, 2006)Google Scholar
  24. 24.
    Flaxman, A., Frieze, A.M., Vera, J.: A Geometric Preferential Attachment Model of Networks, Internet Mathematics (to appear)Google Scholar
  25. 25.
    Gómez-Gardeñes, J., Moreno, Y.: Local versus global knowledge in the Barabási-Albert scale-free network model. Physical Review E 69, 037103 (2004)Google Scholar
  26. 26.
    Hayes, B.: Graph theory in practice: Part II. American Scientist 88, 104–109 (2000)CrossRefGoogle Scholar
  27. 27.
    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, Springer, Heidelberg (1999)CrossRefGoogle Scholar
  28. 28.
    Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: Stochastic Models for the Web Graph. In: Proc. IEEE Symposium on Foundations of Computer Science, p. 57 (2000)Google Scholar
  29. 29.
    Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: The Web as a Graph. In: PODS 2000. Proc. 19th ACM SIGACT-SIGMOD-AIGART Symp. Principles of Database Systems, pp. 1–10 (2000)Google Scholar
  30. 30.
    Li, L., Alderson, D., Doyle, J.C., Willinger, W.: Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications. Internet Mathematics 2(4), 431–523Google Scholar
  31. 31.
    Mihail, M.: private communicationGoogle Scholar
  32. 32.
    Kumar, R., Raghavan, P., Rajagopalan, S., Tomkins, A.: Trawling the Web for emerging cyber-communities. Computer Networks 31, 1481–1493 (1999)CrossRefGoogle Scholar
  33. 33.
    McDiarmid, C.J.H.: Concentration. Probabilistic methods in algorithmic discrete mathematics, 195–248 (1998)Google Scholar
  34. 34.
    Mihail, M., Papadimitriou, C.H.: On the Eigenvalue Power Law. In: Proc. of the 6th International Workshop on Randomization and Approximation Techniques, pp. 254–262 (2002)Google Scholar
  35. 35.
    Mihail, M., Papadimitriou, C.H., Saberi, A.: On Certain Connectivity Properties of the Internet Topology. In: Proc. IEEE Symposium on Foundations of Computer Science, p. 28 (2003)Google Scholar
  36. 36.
    Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. Internet Mathematics 1(2), 226–251 (2004)MATHMathSciNetGoogle Scholar
  37. 37.
    Newman, M., Barabási, A.-L., Watts, D.J.: The Structure and Dynamics of Networks, Princeton University Press (2006)Google Scholar
  38. 38.
    Penrose, M.D.: Random Geometric Graphs. Oxford University Press, Oxford (2003)MATHGoogle Scholar
  39. 39.
    Simon, H.A.: On a class of skew distribution functions. Biometrika 42, 425–440 (1955)MATHMathSciNetGoogle Scholar
  40. 40.
    van der Hofstad, R.: Random Graphs and Complex Networks, unpublished manuscript (2007)Google Scholar
  41. 41.
    Watts, D.J.: Small Worlds: The Dynamics of Networks between Order and Randomness. Princeton University Press, Princeton (1999)Google Scholar
  42. 42.
    Yule, G.: A mathematical theory of evolution based on the conclusions of Dr. J.C. Willis. Philosophical Transactions of the Royal Society of London (Series B) 213, 21–87 (1925)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Abraham D. Flaxman
    • 1
  • Alan M. Frieze
    • 1
  • Juan Vera
    • 1
  1. 1.Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA15213U.S.A.

Personalised recommendations