- mr_var_Mar16
Accepted author manuscript, 323 KB, PDF document

- https://projecteuclid.org/euclid.aoap/1495764374
Final published version

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

**The asymptotic variance of the giant component of configuration model random graphs.** / Ball, Frank; Neal, Peter John.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Ball, F & Neal, PJ 2017, 'The asymptotic variance of the giant component of configuration model random graphs', *Annals of Applied Probability*, vol. 27, no. 2, pp. 1057-1092. https://doi.org/10.1214/16-AAP1225

Ball, F., & Neal, P. J. (2017). The asymptotic variance of the giant component of configuration model random graphs. *Annals of Applied Probability*, *27*(2), 1057-1092. https://doi.org/10.1214/16-AAP1225

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

@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",

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journal = "Annals of Applied Probability",

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publisher = "Institute of Mathematical Statistics",

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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

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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 -