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

TL;DR

This paper proves a law of large numbers for the largest component in a hyperbolic random graph model that captures key features of complex networks, showing the largest component size converges to a constant.

Contribution

It refines previous results by establishing a law of large numbers for the largest component in a hyperbolic graph model, linking it to parameters controlling degree distribution and average degree.

Findings

The fraction of points in the largest component converges to a constant c.

All other components are sublinear in size.

The constant c depends on parameters α and ν.

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 the 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.