Novel scaling limits for critical inhomogeneous random graphs


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 τ(3,4)\tau\in(3,4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by n(τ2)/(τ1)n^{-(\tau-2)/(\tau-1)}, converge to hitting times of a "thinned" L\'{e}vy 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 n2/3n^{-2/3} converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812-854] for the Erd\H{o}s-R\'{e}nyi 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].Comment: Published in at the Annals of Probability ( by the Institute of Mathematical Statistics (

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