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• mr_var_Mar16

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

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Published

### Standard

In: Annals of Applied Probability, Vol. 27, No. 2, 01.04.2017, p. 1057-1092.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

### Vancouver

Ball F, Neal PJ. The asymptotic variance of the giant component of configuration model random graphs. Annals of Applied Probability. 2017 Apr 1;27(2):1057-1092. doi: 10.1214/16-AAP1225

### Author

Ball, Frank ; Neal, Peter John. / The asymptotic variance of the giant component of configuration model random graphs. In: Annals of Applied Probability. 2017 ; Vol. 27, No. 2. pp. 1057-1092.

### Bibtex

@article{2d7acbddc8084f169e1ac912cc837e67,
title = "The asymptotic variance of the giant component of configuration model random graphs",
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.",
keywords = "Random graphs, configuration model, branching processes, variance",
author = "Frank Ball and Neal, {Peter John}",
year = "2017",
month = apr,
day = "1",
doi = "10.1214/16-AAP1225",
language = "English",
volume = "27",
pages = "1057--1092",
journal = "Annals of Applied Probability",
issn = "1050-5164",
publisher = "Institute of Mathematical Statistics",
number = "2",

}

### RIS

TY - JOUR

T1 - The asymptotic variance of the giant component of configuration model random graphs

AU - Ball, Frank

AU - Neal, Peter John

PY - 2017/4/1

Y1 - 2017/4/1

N2 - 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.

AB - 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.

KW - Random graphs

KW - configuration model

KW - branching processes

KW - variance

U2 - 10.1214/16-AAP1225

DO - 10.1214/16-AAP1225

M3 - Journal article

VL - 27

SP - 1057

EP - 1092

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 2

ER -