The component sizes of a critical random graph with pre-described degree sequence

Abstract : Consider a critical random multigraph $\mathcal{G}_n$ constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $\nu$ (criticality means that the second moment of $\nu$ is finite and equals twice its first moment). We specify the scaling limits of the ordered sequence of component sizes of $\mathcal{G}_n$ in different cases. When $\nu$ has finite third moment, the components sizes rescaled by $n^{-2/3}$ converge to the excursion lengths of a Brownian motion with parabolic drift, whereas when $\nu$ is a power law distribution with exponent $\gamma \in (3,4)$, the components sizes rescaled by $n^{-(\gamma-2)/(\gamma-1)}$ converge to the excursion lengths of a drifted process with independent increments that will be characterized.
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Preprints, Working Papers, ...
2010
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https://hal.archives-ouvertes.fr/hal-00545815
Submitted on : Monday, December 13, 2010 - 5:59:17 AM
Last modification on : Wednesday, October 12, 2016 - 1:03:40 AM

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• HAL Id : hal-00545815, version 1
• ARXIV : 1012.2352

Citation

Adrien Joseph. The component sizes of a critical random graph with pre-described degree sequence. 2010. 〈hal-00545815〉

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