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

Ball, Frank and Neal, Peter (2016) The asymptotic variance of the giant component of configuration model random graphs. Annals of Applied Probability . ISSN 1050-5164 (In Press)

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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.

Item Type: Article
Keywords: Random graphs, configuration model, branching processes, variance
Schools/Departments: University of Nottingham UK Campus > Faculty of Science > School of Mathematical Sciences
Depositing User: Ball, Prof Frank Granville
Date Deposited: 22 Jun 2016 09:42
Last Modified: 15 Sep 2016 03:02
URI: http://eprints.nottingham.ac.uk/id/eprint/34282

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