Research Repository

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

## Authors

FRANK BALL frank.ball@nottingham.ac.uk
Professor of Applied Probability

PETER NEAL Peter.Neal@nottingham.ac.uk
Professor of Statistics

### 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.

### Citation

Ball, F., & Neal, P. (2017). The asymptotic variance of the giant component of configuration model random graphs. Annals of Applied Probability, 27(2), https://doi.org/10.1214/16-AAP1225

Journal Article Type Article Jun 15, 2016 May 26, 2017 Jun 22, 2016 May 26, 2017 Annals of Applied Probability 1050-5164 1050-5164 Institute of Mathematical Statistics (IMS) Peer Reviewed 27 2 https://doi.org/10.1214/16-AAP1225 Random graphs, configuration model, branching processes, variance http://eprints.nottingham.ac.uk/id/eprint/34282 http://projecteuclid.org/euclid.aoap/1495764374 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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