Novel scaling limits for critical inhomogeneous random graphs

• Autores: Shankar Bhamidi, Remco van der Hofstad, Johan S. H. Van Leeuwaarden
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 40, Nº. 6, 2012, págs. 2299-2361
• Idioma: inglés
• DOI: 10.1214/11-AOP680
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• Resumen

  • 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 τ. We investigate the case where τ∈(3,4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by n−(τ−2)/(τ−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 τ>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].