The Annals of Applied Probability

Law of large numbers for the largest component in a hyperbolic model of complex networks

Nikolaos Fountoulakis and Tobias Müller

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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.

Article information

Ann. Appl. Probab., Volume 28, Number 1 (2018), 607-650.

Received: September 2016
Revised: March 2017
First available in Project Euclid: 3 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 82B43: Percolation [See also 60K35]

Random graphs hyperbolic plane giant component law of large numbers


Fountoulakis, Nikolaos; Müller, Tobias. Law of large numbers for the largest component in a hyperbolic model of complex networks. Ann. Appl. Probab. 28 (2018), no. 1, 607--650. doi:10.1214/17-AAP1314.

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