Janson, Svante and Luczak, Malwina J. (2008) Symptotic normality of the k-core in random graphs. The annals of applied probability, 18 (3). pp. 1085-1137. ISSN 1050-5164
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).
|Additional Information:||© 2008 Institute of Mathematical Statistics|
|Uncontrolled Keywords:||cores, random graphs, balls and bins, central limit theorem|
|Library of Congress subject classification:||Q Science > QA Mathematics|
|Sets:||Departments > Mathematics|
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