Open Access
November 2012 Novel scaling limits for critical inhomogeneous random graphs
Shankar Bhamidi, Remco van der Hofstad, Johan S. H. van Leeuwaarden
Ann. Probab. 40(6): 2299-2361 (November 2012). DOI: 10.1214/11-AOP680


We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent $\tau$. We investigate the case where $\tau\in(3,4)$, so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by $n^{-(\tau-2)/(\tau-1)}$, converge to hitting times of a “thinned” Lévy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812–854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1–59].

Our results should be contrasted to the case $\tau>4$, so that the third moment is finite. There, instead, the sizes of the components rescaled by $n^{-2/3}$ converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812–854] for the Erdős–Rényi random graph and extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden [Electron. J. Probab. 15 (2010) 1682–1703] and Turova [(2009) Preprint].


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Shankar Bhamidi. Remco van der Hofstad. Johan S. H. van Leeuwaarden. "Novel scaling limits for critical inhomogeneous random graphs." Ann. Probab. 40 (6) 2299 - 2361, November 2012.


Published: November 2012
First available in Project Euclid: 26 October 2012

zbMATH: 1257.05157
MathSciNet: MR3050505
Digital Object Identifier: 10.1214/11-AOP680

Primary: 05C80 , 60C05 , 90B15

Keywords: Critical random graphs , inhomogeneous networks , multiplicative coalescent , Phase transitions , thinned Lévy processes

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 6 • November 2012
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