Open Access
2023 Largest component of subcritical random graphs with given degree sequence
Matthew Coulson, Guillem Perarnau
Author Affiliations +
Electron. J. Probab. 28: 1-28 (2023). DOI: 10.1214/23-EJP921


We study the size of the largest component of two models of random graphs with prescribed degree sequence, the configuration model (CM) and the uniform model (UM), in the (barely) subcritical regime. For the CM, we give upper bounds that are asymptotically tight for certain degree sequences. These bounds hold under mild conditions on the sequence and improve previous results of Hatami and Molloy on the barely subcritical regime. For the UM, we give weaker upper bounds that are tight up to logarithmic terms but require no assumptions on the degree sequence. In particular, the latter result applies to degree sequences with infinite variance in the subcritical regime.

Funding Statement

M.C. was supported by the FPI grant PRE2018-083621 associated to the project MTM2017-82166-P from the Spanish Ministerio de Economía y Competitividad. G.P was supported by the Spanish Agencia Estatal de Investigación under projects MTM2017-82166-P and PID2020-113082GB-I00, and by the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M).


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Matthew Coulson. Guillem Perarnau. "Largest component of subcritical random graphs with given degree sequence." Electron. J. Probab. 28 1 - 28, 2023.


Received: 2 December 2021; Accepted: 13 February 2023; Published: 2023
First available in Project Euclid: 24 February 2023

arXiv: 2111.11780
MathSciNet: MR4551874
zbMATH: 1517.05158
Digital Object Identifier: 10.1214/23-EJP921

Primary: 05C80 , 05C82 , 60C05 , 60F05

Keywords: component structure , configuration model , largest component , Local limit theorems , random graph with given degree sequence

Vol.28 • 2023
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