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Giant Components for Two Expanding Graph Processes

  • Luc Devroye
  • Colin McDiarmid
  • Bruce Reed
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

We discuss the emergence of giant components in two random graph models (one directed one undirected). Our study of these models was motivated by an interest in finding random model of the Internet.

Keywords

Random Graph Independent Random Variable Degree Sequence Giant Component Strong Component 
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.

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References

  1. Adamic L. and Huberman B., “Growth Dynamics of the World Wide Web,” Nature, vol. 401, p. 131, 1999.CrossRefGoogle Scholar
  2. Aiello W.,Chung F. and Lu L., “A random graph model for massive graphs,” Proceedings of the 32nd Annual ACM Symposium on Theory of Computing p. 171–180, 2000.Google Scholar
  3. Aiello W.,Chung F. and Lu L., “A random graph model for massive graphs,” Proceedings of the 32nd Annual ACM Symposium on Theory of Computing p. 171–180, 2000.Google Scholar
  4. K. B. Athreya, “On a characteristic property of Pélya’s urn,” Stud. Sci. Math. Hung. vol. 4, pp. 31–35, 1969.MathSciNetzbMATHGoogle Scholar
  5. Barabási A. Albert R. and Jeong H., “Scale-free characteristics of random networks: the topology of the world wide web,” Physics A. vol. 272, pp. 173–187, 1999.CrossRefGoogle Scholar
  6. Broder A. Kumar R. Maghoul F. Raghavan P. Rajagopalan S. Stata R. Tompkins A and Wiener J. “Graph Structure in the Web,” Computer Networks vol. 33, pp. 309–321, 2000.CrossRefGoogle Scholar
  7. Cooper C. and Frieze A. “A General Model for Web Graphs,” Proceedings of ESA 2001 2001.Google Scholar
  8. D. Defays, “Etude du comportement asymptotique de schémas d’urnes,” Bull. Soc. Roy. Sci. Liège vol. 43, pp. 26–34, 1974.MathSciNetzbMATHGoogle Scholar
  9. L. Devroye, “Branching processes in the analysis of the heights of trees,” Acta Informatica vol. 24, pp. 277–298, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  10. L. Devroye, “Applications of the theory of records in the study of random trees,” Acta Informatica vol. 26, pp. 123–130, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  11. L. Devroye and J. Lu, “The strong convergence of maximal degrees in uniform random recursive trees and dags,” Random Structures and Algorithms vol. 6, pp. 1–14, 1995.MathSciNetCrossRefGoogle Scholar
  12. M. Dondajewski and J. Szymañski, “On the distribution of vertex-degrees in a strata of a random recursive tree,” Bulletin de l’Académie Polonaise des Sciences Série des Sciences Mathématiques vol. 30, pp. 205–209, 1982.zbMATHGoogle Scholar
  13. M. Dwass, “The total progeny in a branching process,” Journal of Applied Probability vol. 6, pp. 682–686, 1969.MathSciNetzbMATHCrossRefGoogle Scholar
  14. J. L. Gastwirth, “A probability model of a pyramid scheme,” The American Statistician vol. 31, pp. 79–82, 1977.zbMATHGoogle Scholar
  15. J. L. Gastwirth and P. K. Bhattacharya, “Two probability models of pyramid or chain letter schemes demonstrating that their promotional claims are unreliable,” Operations Research vol. 32, pp. 527–536, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  16. N. L. Johnson and S. Kotz Urn Models and Their Application John Wiley, New York, N.Y., 1977.Google Scholar
  17. Kim J. Kohayakawa Y. McDiarmid C. Reed B. Spencer J. Vu V. “An expanding graph process,” in preparation 2002.Google Scholar
  18. Kleinberg J.,Kumar S.,Raghavan P.,Rajagopalan S.,and Tomkins A. “The web as a graph: Measurements, models and methods,” Proceedings of the International Conference on Combinatorics and Computing 1999.Google Scholar
  19. Kumar S., Raghavan P., Rajagopalan S. and Tomkins A. “Extracting Large-scale Knowledge Bases from the Web,” Proceedings of the 25th VLDB conference 1999.Google Scholar
  20. H. M. Mahmoud and R. T. Smythe, “On the distribution of leaves in rooted sub-trees of recursive trees,” Annals of Applied Probability vol. 1, pp. 406–418, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  21. H. M. Mahmoud Evolution of Random Search Trees John Wiley, New York, 1992.zbMATHGoogle Scholar
  22. A. Meir and J. W. Moon, “Path edge-covering constants for certain families of trees,” Utilitas Mathematica vol. 14, pp. 313–333, 1978.MathSciNetzbMATHGoogle Scholar
  23. Molloy M. and Reed B., “A critical point for random graphs with a given degree sequence,” Random Structures and Algorithms vol. 6, pp. 161–179, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  24. J. W. Moon, “On the maximum degree in a random tree,” Michigan Mathematical Journal vol. 15, pp. 429–432, 1968.MathSciNetCrossRefGoogle Scholar
  25. J. W. Moon, “The distance between nodes in recursive trees,” London Mathematical Society Lecture Notes vol. 13, pp. 125–132, Cambridge University Press, London, 1974.Google Scholar
  26. H. S. Na and A. Rapoport, “Distribution of nodes of a tree by degree,” Mathematical Biosciences vol. 6, pp. 313–329, 1970.MathSciNetzbMATHCrossRefGoogle Scholar
  27. D. Najock and C. C. Heyde, “On the number of terminal vertices in certain random trees with an application to stemma construction in philology,” Journal of Applied Probability vol. 19, pp. 675–680, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  28. B. Pittel, “Note on the heights of random recursive trees and random m-ary search tres,” Random Structures and Algorithms vol. 5, pp. 337–347, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  29. G. Pólya, “Sur quelques points de la thèorie de probabilité,” Ann. Inst. Henri Poincarè vol. 1, pp. 117–161, 1931.Google Scholar
  30. J. Szymanski, “On a nonuniform random recursive tree,” Annals of Discrete Mathematics vol. 33 pp. 297–306, 1987. MathSciNetGoogle Scholar
  31. J. Szymanski, “On the maximum degree and the height of a random recursive tree,” in: Random Graphs 87 (edited by M. Karonski, J. Jaworski and A. Rucinski) pp. 313–324, John Wiley, Chichester, 1990. Google Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Luc Devroye
    • 1
  • Colin McDiarmid
    • 2
  • Bruce Reed
    • 1
  1. 1.McGill UniversityMontrealCanada
  2. 2.Oxford UniversityOxfordUK

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