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Giant Components for Two Expanding Graph Processes

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Mathematics and Computer Science II

Part of the book series: Trends in Mathematics ((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.

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References

  • Adamic L. and Huberman B., “Growth Dynamics of the World Wide Web,” Nature, vol. 401, p. 131, 1999.

    Article  Google Scholar 

  • 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 

  • 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 

  • K. B. Athreya, “On a characteristic property of PĂ©lya’s urn,” Stud. Sci. Math. Hung. vol. 4, pp. 31–35, 1969.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Cooper C. and Frieze A. “A General Model for Web Graphs,” Proceedings of ESA 2001 2001.

    Google Scholar 

  • D. Defays, “Etude du comportement asymptotique de schĂ©mas d’urnes,” Bull. Soc. Roy. Sci. Liège vol. 43, pp. 26–34, 1974.

    MathSciNet  MATH  Google Scholar 

  • L. Devroye, “Branching processes in the analysis of the heights of trees,” Acta Informatica vol. 24, pp. 277–298, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  • L. Devroye, “Applications of the theory of records in the study of random trees,” Acta Informatica vol. 26, pp. 123–130, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • M. Dondajewski and J. Szymañski, “On the distribution of vertex-degrees in a strata of a random recursive tree,” Bulletin de l’AcadĂ©mie Polonaise des Sciences SĂ©rie des Sciences MathĂ©matiques vol. 30, pp. 205–209, 1982.

    MATH  Google Scholar 

  • M. Dwass, “The total progeny in a branching process,” Journal of Applied Probability vol. 6, pp. 682–686, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  • J. L. Gastwirth, “A probability model of a pyramid scheme,” The American Statistician vol. 31, pp. 79–82, 1977.

    MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • N. L. Johnson and S. Kotz Urn Models and Their Application John Wiley, New York, N.Y., 1977.

    Google Scholar 

  • Kim J. Kohayakawa Y. McDiarmid C. Reed B. Spencer J. Vu V. “An expanding graph process,” in preparation 2002.

    Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • H. M. Mahmoud Evolution of Random Search Trees John Wiley, New York, 1992.

    MATH  Google Scholar 

  • A. Meir and J. W. Moon, “Path edge-covering constants for certain families of trees,” Utilitas Mathematica vol. 14, pp. 313–333, 1978.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • J. W. Moon, “On the maximum degree in a random tree,” Michigan Mathematical Journal vol. 15, pp. 429–432, 1968.

    Article  MathSciNet  Google Scholar 

  • 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 

  • H. S. Na and A. Rapoport, “Distribution of nodes of a tree by degree,” Mathematical Biosciences vol. 6, pp. 313–329, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • G. PĂłlya, “Sur quelques points de la thèorie de probabilitĂ©,” Ann. Inst. Henri Poincarè vol. 1, pp. 117–161, 1931.

    Google Scholar 

  • J. Szymanski, “On a nonuniform random recursive tree,” Annals of Discrete Mathematics vol. 33 pp. 297–306, 1987.

    MathSciNet  Google Scholar 

  • 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 

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© 2002 Springer Basel AG

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Devroye, L., McDiarmid, C., Reed, B. (2002). Giant Components for Two Expanding Graph Processes. In: Chauvin, B., Flajolet, P., Gardy, D., Mokkadem, A. (eds) Mathematics and Computer Science II. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8211-8_10

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  • DOI: https://doi.org/10.1007/978-3-0348-8211-8_10

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9475-3

  • Online ISBN: 978-3-0348-8211-8

  • eBook Packages: Springer Book Archive

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