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The asymptotic variance of the giant component of configuration model random graphs

Research output: Contribution to journalJournal articlepeer-review

Published
<mark>Journal publication date</mark>1/04/2017
<mark>Journal</mark>Annals of Applied Probability
Issue number2
Volume27
Number of pages37
Pages (from-to)1057-1092
Publication StatusPublished
<mark>Original language</mark>English

Abstract

For a supercritical configuration model random graph it is well known that, subject to mild conditions, there exists a unique giant component, whose size $R_n$ is $O (n)$, where $n$ is the total number of vertices in the random graph. Moreover, there exists $0 < \rho \leq 1$ such that $R_n/n \convp \rho$ as $\nr$. We show that for a sequence of {\it well-behaved} configuration model random graphs with a deterministic degree sequence satisfying $0 < \rho < 1$, there exists $\sigma^2 > 0$, such that $var (\sqrt{n} (R_n/n -\rho)) \rightarrow \sigma^2$ as $\nr$. Moreover, an explicit, easy to compute, formula is given for $\sigma^2$. This provides a key stepping stone for computing the asymptotic variance of the size of the giant component for more general random graphs.