Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-23T21:00:43.402Z Has data issue: false hasContentIssue false
  • English
  • Français

The Critical Phase for Random Graphs with a Given Degree Sequence

Published online by Cambridge University Press:  01 January 2008

M. KANG  and
T. G. SEIERSTAD
M. KANG
Affiliation:
Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany (e-mail: kang@informatik.hu-berlin.de, seiersta@informatik.hu-berlin.de)
T. G. SEIERSTAD
Affiliation:
Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany (e-mail: kang@informatik.hu-berlin.de, seiersta@informatik.hu-berlin.de)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 1 , January 2008 , pp. 67 - 86
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Athreya, K. B. and Ney, P. E. (1972) Branching Processes, Vol. 196 of Die Grundlehren der Mathematischen Wissenschaften, Springer, New York.CrossRefGoogle Scholar
[2]Bender, E. A. and Canfield, E. R. (1978) The asymptotic number of labeled graphs with given degree sequences. J. Combin. Theory Ser. A 24 296–307.CrossRefGoogle Scholar
[3]Bollobás, B. (1984) The evolution of random graphs. Trans. Amer. Math. Soc. 286 257–274.CrossRefGoogle Scholar
[4]Bollobás, B. (2001) Random Graphs, 2nd edn, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[5]Chung, F. and Lu, L. (2002) Connected components in random graphs with given expected degree sequences. Ann. Comb. 6 125–145.CrossRefGoogle Scholar
[6]Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.Google Scholar
[7]Flajolet, P. and Sedgewick, R. (2006) Analytic Combinatorics, book in preparation.Google Scholar
[8]Łuczak, T. (1990) Component behavior near the critical point of the random graph process. Random Struct. Alg. 1 287–310.CrossRefGoogle Scholar
[9]Łuczak, T. (1992) Sparse random graphs with a given degree sequence. In Random Graphs, Vol. 2 (Poznań, 1989), Wiley–Interscience, pp. 165–182.Google Scholar
[10]McKay, B. D. (1985) Asymptotics for symmetric 0–1 matrices with prescribed row sums. Ars Combin. 19 15–25.Google Scholar
[11]Molloy, M. and Reed, B. (1995) A critical point for random graphs with a given degree sequence. In Proc. Sixth International Seminar on Random Graphs and Probabilistic Methods in Combinatorics and Computer Science, ‘Random Graphs '93’ (Poznań, 1993), Vol. 6, pp. 161–179.CrossRefGoogle Scholar
[12]Molloy, M. and Reed, B. (1998) The size of the giant component of a random graph with a given degree sequence. Combin. Probab. Comput. 7 295–305.CrossRefGoogle Scholar
[13]Newman, M. E. J., Strogatz, S. H. and Watts, D. J. (2001) Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64 026118.CrossRefGoogle ScholarPubMed
[14]Wormald, N. C. (1978) Some problems in the enumeration of labelled graphs. PhD thesis, Newcastle University.Google Scholar