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

Ball, Frank; Neal, Peter

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: https://nottingham-repository.worktribe.com/output/861887
• Publisher URL: http://projecteuclid.org/euclid.aoap/1495764374
• Contract Date: Jun 22, 2016