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

## Authors

Frank Ball

Peter Neal

### 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.

Journal Article Type Article May 26, 2017 Annals of Applied Probability 1050-5164 1050-5164 Institute of Mathematical Statistics (IMS) Peer Reviewed 27 2 Ball, F., & Neal, P. (2017). The asymptotic variance of the giant component of configuration model random graphs. Annals of Applied Probability, 27(2), doi:10.1214/16-AAP1225 https://doi.org/10.1214/16-AAP1225 Random graphs, configuration model, branching processes, variance http://projecteuclid.org/euclid.aoap/1495764374 Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf

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