Advertisement

Abstract

The web graph has been the focus of much recent attention, with several stochastic models proposed to account for its various properties. A survey of these models is presented, focussing on the models which have been defined and analyzed rigorously.

Keywords

Random Graph Preferential Attachment Degree Sequence Random Graph Model Small World Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, A., Mitzenmacher, M.: Towards compressing web graphs, In: Proceedings of the Data Compression Conference (2001)Google Scholar
  2. 2.
    Aiello, W., Chung, F., Lu, L.: A random graph model for massive graphs. Experimental Mathematics 10, 53–66 (2001)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Aiello, W., Chung, F., Lu, L.: Random evolution in massive graphs. In: Abello, J., et al. (eds.) Handbook on Massive Data Sets, pp. 97–122. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar
  4. 4.
    Albert, R., Barabási, A., Jeong, H.: Diameter of the World-Wide Web. Nature 401, 130 (1999)CrossRefGoogle Scholar
  5. 5.
    Albert, R., Barabási, A.: Emergence of scaling in random networks. Science 286, 509–512 (1999)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Baldi, P., Frasconi, P., Smyth, P.: Modeling the Internet and the Web, Probabilistic Methods and Algorithms. John Wiley & Sons, Ltd., Chichester (2003)Google Scholar
  7. 7.
    Barabási, A.: Linked: How Everything Is Connected to Everything Else and What It Means. Perseus Publishing, Cambridge (2002)Google Scholar
  8. 8.
    Bollobás, B.: Modern Graph Theory. Springer, New York (1998)zbMATHGoogle Scholar
  9. 9.
    Bollobás, B.: Random graphs, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 73. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  10. 10.
    Bollobás, B., Riordan, O.: Mathematical results on scale-free graphs. In: Bornholdt, S., Schuster, H. (eds.) Handbook of graphs and networks. Wiley-VCH, Berlin (2002)Google Scholar
  11. 11.
    Bollobás, B., Riordan, O.: Robustness and vulnerability of scale-free random graphs. Internet Mathematics 1, 1–35 (2004)CrossRefGoogle Scholar
  12. 12.
    Bollobás, B., Riordan, O.: The diameter of a scale-free random graph. Combinatorica 24, 5–34 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bollobás, B., Borgs, C., Chayes, T., Riordan, O.: Directed scale-free graphs (submitted)Google Scholar
  14. 14.
    Bollobás, B., Riordan, O., Spencer, J., Tusnády, G.: The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279–290 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bonato, A., Janssen, J.: Infinite limits of copying models of the web graph. Internet Mathematics 1, 193–213 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Bonato, A., Janssen, J.: Limits and power laws of models for the web graph and other networked information spaces. In: López-Ortiz, A., Hamel, A.M. (eds.) CAAN 2004. LNCS, vol. 3405, pp. 42–48. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., Wiener, J.: Graph structure in the web. Computer Networks 33, 309–320 (2000)CrossRefGoogle Scholar
  18. 18.
    Buckley, P.G., Osthus, D.: Popularity based random graph models leading to a scale-free degree sequence (submitted)Google Scholar
  19. 19.
    Caldarelli, G., De Los Rios, P., Laura, L., Leonardi, S., Millozzi, S.: A study of stochastic models for the Web Graph, Technical Report 04-03, dipartimento di Informatica e Sistemistica, Universita’ di Roma “La Sapienza” (2003)Google Scholar
  20. 20.
    Cameron, P.J.: The random graph. In: Graham, R.L., Nešetřil, J. (eds.) Algorithms and Combinatorics, pp. 333–351. Springer, Heidelberg (1997)Google Scholar
  21. 21.
    Cameron, P.J.: The random graph revisited. In: European Congress of Mathematics, Barcelona, vol. I, pp. 267–274 (2000); Progr. Math., vol. 201. Birkhäuser, Basel (2001)Google Scholar
  22. 22.
    Chakrabarti, S.: Mining the Web, Discovering Knowledge from Hypertext Data. Morgan Kauffman Publishers, San Francisco (2003)Google Scholar
  23. 23.
    Chung, F., Dewey, G., Galas, D.J., Lu, L.: Duplication models for biological networks. Journal of Computational Biology 10, 677–688 (2003)CrossRefGoogle Scholar
  24. 24.
    Chung, F., Lu, L.: Connected components in random graphs with given degree sequences. Annals of Combinatorics 6, 125–145 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Chung, F., Lu, L.: The average distances in random graphs with given expected degrees. Internet Mathematics 1, 91–114 (2004)CrossRefGoogle Scholar
  26. 26.
    Chung, F., Lu, L.: The small world phenomenon in hybrid power law graphs. In: Ben-Naim, E., et al. (eds.) Complex Networks, pp. 91–106. Springer, Heidelberg (2004)Google Scholar
  27. 27.
    Chung, F., Lu, L.: Coupling on-line and on-line analyses for random power law graphs (submitted)Google Scholar
  28. 28.
    Cooper, C., Frieze, A.: On a general model of web graphs. Random Structures Algorithms 22, 311–335 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Cooper, C., Frieze, A., Vera, J.: Random deletions in a scale free random graph process (submitted)Google Scholar
  30. 30.
    Diestel, R.: Graph theory. Springer, New York (2000)CrossRefGoogle Scholar
  31. 31.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks: From biological nets to the Internet and WWW. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  32. 32.
    Dorogovtsev, S.N., Mendes, J.F.F., Samukhin, A.N.: Structure of growing networks with preferential linking. Physical Review Letters 85, 4633–4636 (2000)CrossRefGoogle Scholar
  33. 33.
    Drinea, E., Enachescu, M., Mitzenmacher, M.: Variations on random graph models for the web, technical report, Department of Computer Science, Harvard University (2001)Google Scholar
  34. 34.
    Erdős, P., Rényi, A.: On random graphs I. Publ. Math. Debrecen 6, 290–297 (1959)MathSciNetGoogle Scholar
  35. 35.
    Erdös, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17–61 (1960)zbMATHGoogle Scholar
  36. 36.
    Fabrikant, A., Koutsoupias, E., Papadimitriou, C.: Heuristically optimized Trade-offs: a new paradigm for power laws in the internet. In: Proceedings of the 34th Symposium on Theory of Computing (2002)Google Scholar
  37. 37.
    Flaxman, A., Frieze, A., Vera, J.: Adversarial deletions in a scale free random graph process (submitted)Google Scholar
  38. 38.
    Gilbert, E.N.: Random graphs. Annals of Mathematical Statistics 30, 1141–1144 (1959)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Henzinger, M.R.: Algorithmic challenges in web search engines. Internet Mathematics 1, 115–126 (2004)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Janson, S., Luczak, T., Ruciński, A.: Random Graphs. John Wiley and Sons, New York (2000)zbMATHGoogle Scholar
  41. 41.
    Kleinberg, J., Kumar, S.R., Raghavan, P., Rajagopalan, S., Tomkins, A.: 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, p. 1. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  42. 42.
    Kumar, R., Raghavan, P., Rajagopalan, S., Tomkins, A.: Trawling the web for emerging cyber-communities. In: Proceedings of the 8th WWW Conference (1999)Google Scholar
  43. 43.
    Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: Stochastic models for the web graph. In: Proceedings of the 41th IEEE Symp. on Foundations of Computer Science, pp. 57–65 (2000)Google Scholar
  44. 44.
    Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: The web as a graph. In: Proc. 19th ACM SIGACT-SIGMOD-AIGART Symp. Principles of Database Systems, Publ., Dordrecht (2002)Google Scholar
  45. 45.
    Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. Internet Mathematics 1, 226–251 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Strogatz, S.H., Watts, D.J.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Anthony Bonato
    • 1
  1. 1.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada

Personalised recommendations