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February 2018 Law of large numbers for the largest component in a hyperbolic model of complex networks
Nikolaos Fountoulakis, Tobias Müller
Ann. Appl. Probab. 28(1): 607-650 (February 2018). DOI: 10.1214/17-AAP1314


We consider the component structure of a recent model of random graphs on the hyperbolic plane that was introduced by Krioukov et al. The model exhibits a power law degree sequence, small distances and clustering, features that are associated with so-called complex networks. The model is controlled by two parameters $\alpha$ and $\nu$ where, roughly speaking, $\alpha$ controls the exponent of the power law and $\nu$ controls the average degree. Refining earlier results, we are able to show a law of large numbers for the largest component. That is, we show that the fraction of points in the largest component tends in probability to a constant $c$ that depends only on $\alpha,\nu$, while all other components are sublinear. We also study how $c$ depends on $\alpha,\nu$. To deduce our results, we introduce a local approximation of the random graph by a continuum percolation model on $\mathbb{R}^{2}$ that may be of independent interest.


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Nikolaos Fountoulakis. Tobias Müller. "Law of large numbers for the largest component in a hyperbolic model of complex networks." Ann. Appl. Probab. 28 (1) 607 - 650, February 2018.


Received: 1 September 2016; Revised: 1 March 2017; Published: February 2018
First available in Project Euclid: 3 March 2018

zbMATH: 06873692
MathSciNet: MR3770885
Digital Object Identifier: 10.1214/17-AAP1314

Primary: 05C80, 05C82
Secondary: 60D05, 82B43

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.28 • No. 1 • February 2018
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