The Annals of Applied Probability

Central limit theorems in the configuration model

A. D. Barbour and Adrian Röllin

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We prove a general normal approximation theorem for local graph statistics in the configuration model, together with an explicit bound on the error in the approximation with respect to the Wasserstein metric. Such statistics take the form $T:=\sum_{v\in V}H_{v}$, where $V$ is the vertex set, and $H_{v}$ depends on a neighbourhood in the graph around $v$ of size at most $\ell$. The error bound is expressed in terms of $\ell$, $|V|$, an almost sure bound on $H_{v}$, the maximum vertex degree $d_{\max}$ and the variance of $T$. Under suitable assumptions on the convergence of the empirical degree distributions to a limiting distribution, we deduce that the size of the giant component in the configuration model has asymptotically Gaussian fluctuations.

Article information

Ann. Appl. Probab., Volume 29, Number 2 (2019), 1046-1069.

Received: October 2017
Revised: July 2018
First available in Project Euclid: 24 January 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60F05: Central limit and other weak theorems
Secondary: 05C80: Random graphs [See also 60B20]

Configuration random graph model giant component central limit theorem Stein’s method


Barbour, A. D.; Röllin, Adrian. Central limit theorems in the configuration model. Ann. Appl. Probab. 29 (2019), no. 2, 1046--1069. doi:10.1214/18-AAP1425.

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