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

Ball, Frank; Neal, Peter

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
Acceptance Date Jun 15, 2016
Publication Date May 26, 2017
Deposit Date Jun 22, 2016
Publicly Available Date May 26, 2017
Journal Annals of Applied Probability
Print ISSN 1050-5164
Electronic ISSN 1050-5164
Publisher Institute of Mathematical Statistics (IMS)
Peer Reviewed Peer Reviewed
Volume 27
Issue 2
DOI https://doi.org/10.1214/16-AAP1225
Keywords Random graphs, configuration model, branching processes, variance
Public URL http://eprints.nottingham.ac.uk/id/eprint/34282
Publisher URL http://projecteuclid.org/euclid.aoap/1495764374
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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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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