The Annals of Probability
- Ann. Probab.
- Volume 40, Number 6 (2012), 2299-2361.
Novel scaling limits for critical inhomogeneous random graphs
Shankar Bhamidi, Remco van der Hofstad, and Johan S. H. van Leeuwaarden
Abstract
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].
Article information
Source
Ann. Probab., Volume 40, Number 6 (2012), 2299-2361.
Dates
First available in Project Euclid: 26 October 2012
Permanent link to this document
https://projecteuclid.org/euclid.aop/1351258728
Digital Object Identifier
doi:10.1214/11-AOP680
Mathematical Reviews number (MathSciNet)
MR3050505
Zentralblatt MATH identifier
1257.05157
Subjects
Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20] 90B15: Network models, stochastic
Keywords
Critical random graphs phase transitions inhomogeneous networks thinned Lévy processes multiplicative coalescent
Citation
Bhamidi, Shankar; van der Hofstad, Remco; van Leeuwaarden, Johan S. H. Novel scaling limits for critical inhomogeneous random graphs. Ann. Probab. 40 (2012), no. 6, 2299--2361. doi:10.1214/11-AOP680. https://projecteuclid.org/euclid.aop/1351258728
