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The Phase Transition in the Configuration Model

Published online by Cambridge University Press:  02 February 2012

OLIVER RIORDAN
OLIVER RIORDAN*
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK and Department of Mathematical Sciences, University of Memphis, TN 38152, USA (e-mail: riordan@maths.ox.ac.uk)
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Abstract

Let G = G(d) be a random graph with a given degree sequence d, such as a random r-regular graph where r ≥ 3 is fixed and n = |G| → ∞. We study the percolation phase transition on such graphs G, i.e., the emergence as p increases of a unique giant component in the random subgraph G[p] obtained by keeping edges independently with probability p. More generally, we study the emergence of a giant component in G(d) itself as d varies. We show that a single method can be used to prove very precise results below, inside and above the ‘scaling window’ of the phase transition, matching many of the known results for the much simpler model G(n, p). This method is a natural extension of that used by Bollobás and the author to study G(n, p), itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 265 - 299
Copyright
Copyright © Cambridge University Press 2012

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References

[1]Aldous, D. (1997) Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812–854.CrossRefGoogle Scholar
[2]Bender, E. A. and Canfield, R. E. (1978) The asymptotic number of labeled graphs with given degree sequences. J. Combin. Theory Ser. A 24 296–307.CrossRefGoogle Scholar
[3]Bollobás, B. (1980) A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311–316.CrossRefGoogle Scholar
[4]Bollobás, B. (1984) The evolution of random graphs. Trans. Amer. Math. Soc. 286 257–274.CrossRefGoogle Scholar
[5]Bollobás, B. and Riordan, O. (2009) Random graphs and branching processes. In Handbook of Large-Scale Random Networks (Bollobás, B., Kozma, R. and Miklós, D. eds), Vol. 18 of Bolyai Soc. Math. Studies, pp. 15–115.Google Scholar
[6]Bollobás, B. and Riordan, O. (2012) Asymptotic normality of the size of the giant component via a random walk. J. Combin. Theory Ser. B 102 53–61.CrossRefGoogle Scholar
[7]Brown, B. M. (1971) Martingale central limit theorems. Ann. Math. Statist. 42 59–66.CrossRefGoogle Scholar
[8]Cramér, H. (1938) Sur un nouveau théorème-limite de la théorie des probabilités. Actualités Scientifiques et Industrielles 736 5–23.Google Scholar
[9]Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. (2010) Diameters in supercritical random graphs via first-passage percolation. Combin. Probab. Comput. 19 729–751.CrossRefGoogle Scholar
[10]Doob, J. L. (1953) Stochastic Processes, Wiley/Chapman and Hall.Google Scholar
[11]Fountoulakis, N. (2007) Percolation on sparse random graphs with given degree sequence. Internet Mathematics 4 329–356.CrossRefGoogle Scholar
[12]Goerdt, A. (2001) The giant component threshold for random regular graphs with edge faults. Theoret. Comput. Sci. 259 307–321.CrossRefGoogle Scholar
[13]Janson, S. (2009) On percolation in random graphs with given vertex degrees. Electron. J. Probab. 14 87–118.CrossRefGoogle Scholar
[14]Janson, S. and Luczak, M. J. (2009) A new approach to the giant component problem. Random Struct. Alg. 34 197–216.CrossRefGoogle Scholar
[15]Kang, M. and Seierstad, T. G. (2008) The critical phase for random graphs with a given degree sequence. Combin. Probab. Comput. 17 67–86.CrossRefGoogle Scholar
[16]Karp, R. M. (1990) The transitive closure of a random digraph. Random Struct. Alg. 1 73–93.CrossRefGoogle Scholar
[17]Łuczak, T. (1990) Component behavior near the critical point of the random graph process. Random Struct. Alg. 1 287–310.CrossRefGoogle Scholar
[18]Martin-Löf, A. (1986) Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Probab. 23 265–282.CrossRefGoogle Scholar
[19]McDonald, D. R. (1979) On local limit theorem for integer valued random variables. Teor. Veroyatnost. i Primenen. 24 607–614. See also Theory Probab. Appl. 24 (1980), 613–619.Google Scholar
[20]Molloy, M. and Reed, B. (1995) A critical point for random graphs with a given degree sequence, Random Struct. Alg. 6 161–179.CrossRefGoogle Scholar
[21]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
[22]Nachmias, A. and Peres, Y. (2007) Component sizes of the random graph outside the scaling window. ALEA Lat. Amer. J. Probab. Math. Stat. 3 133–142.Google Scholar
[23]Nachmias, A. and Peres, Y. (2010) Critical percolation on random regular graphs. Random Struct. Alg. 36 111–148.CrossRefGoogle Scholar
[24]Petrov, V. V. (1975) Sums of Independent Random Variables, translated from the Russian by Brown, A. A., Vol. 82 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer.Google Scholar
[25]Pittel, B. (2008) Edge percolation on a random regular graph of low degree. Ann. Probab. 36 1359–1389.CrossRefGoogle Scholar
[26]Pittel, B. and Wormald, C. (2005) Counting connected graphs inside-out. J. Combin. Theory Ser. B 93 127–172.CrossRefGoogle Scholar
[27]Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 323–339.CrossRefGoogle Scholar