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April 2017 The asymptotic variance of the giant component of configuration model random graphs
Frank Ball, Peter Neal
Ann. Appl. Probab. 27(2): 1057-1092 (April 2017). DOI: 10.1214/16-AAP1225

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 $R_{n}/n\stackrel{p}{\longrightarrow}\rho$ as $n\rightarrow \infty$. We show that for a sequence of well behaved configuration model random graphs with a deterministic degree sequence satisfying $0<\rho <1$; there exists $\sigma^{2}>0$, such that $\operatorname{var}(\sqrt{n}(R_{n}/n-\rho))\rightarrow \sigma^{2}$ as $n\rightarrow \infty$. 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

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Frank Ball. Peter Neal. "The asymptotic variance of the giant component of configuration model random graphs." Ann. Appl. Probab. 27 (2) 1057 - 1092, April 2017. https://doi.org/10.1214/16-AAP1225

Information

Received: 1 March 2015; Revised: 1 March 2016; Published: April 2017
First available in Project Euclid: 26 May 2017

zbMATH: 1367.05192
MathSciNet: MR3655861
Digital Object Identifier: 10.1214/16-AAP1225

Subjects:
Primary: 05C80

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 2 • April 2017
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