The Annals of Applied Probability

Asymptotic normality of the k-core in random graphs

Svante Janson and Malwina J. Luczak

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We study the k-core of a random (multi)graph on n vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50–62] we used properties of empirical distributions of independent random variables to give a simple proof of the fact that the size of the giant k-core obeys a law of large numbers as n→∞. Here we develop the method further and show that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold. Further, we determine precisely the location of the phase transition window for the emergence of a giant k-core. Hence, we deduce corresponding results for the k-core in G(n, p) and G(n, m).

Article information

Ann. Appl. Probab., Volume 18, Number 3 (2008), 1085-1137.

First available in Project Euclid: 26 May 2008

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]

Cores random graphs balls and bins central limit theorem


Janson, Svante; Luczak, Malwina J. Asymptotic normality of the k -core in random graphs. Ann. Appl. Probab. 18 (2008), no. 3, 1085--1137. doi:10.1214/07-AAP478.

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