Asymptotic normality of the $k$-core in random graphs

Details Asymptotic normality of the $k$-core in random graphs Asymptotic normality of the $k$-core in random graphs

2006-12-28

arxiv.org/abs/math/0612827v2

Annals of Applied Probability 2008, Vol. 18, No. 3, 1085-1137

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AAP478 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/07-AAP478

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 ${{n\to \infty}}$. 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 $G(n,m)$.

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