Novel scaling limits for critical inhomogeneous random graphs

Details Novel scaling limits for critical inhomogeneous random graphs Novel scaling limits for critical inhomogeneous random graphs

2009-09-08

arxiv.org/abs/0909.1472v2

Mathematics

Probability

Main results the same but paper revised

Scientific paper

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' L\'evy process, a special case of the general multiplicative coalescents studied by Aldous and Limic in \cite{Aldo97} and \cite{AldLim98}. Our results should be contrasted to the case where the degree exponent $\tau$ satisfies $\tau>4$, so that the third moment is finite. There, instead, we see that the sizes of the components rescaled by $n^{-2/3}$ converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in \cite{Aldo97} for the Erd\H{o}s-R\'enyi random graph and extended to the present setting in \cite{BhaHofLee09a,Turo09}.

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