## The Annals of Applied Probability

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

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

Nikolaos Fountoulakis and Tobias Müller

#### Abstract

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

**Source**

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

**Dates**

Received: September 2016

Revised: March 2017

First available in Project Euclid: 3 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1520046096

**Digital Object Identifier**

doi:10.1214/17-AAP1314

**Mathematical Reviews number (MathSciNet)**

MR3770885

**Zentralblatt MATH identifier**

06873692

**Subjects**

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]

**Keywords**

Random graphs hyperbolic plane giant component law of large numbers

#### Citation

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. https://projecteuclid.org/euclid.aoap/1520046096