### On the Largest Component of a Hyperbolic Model of Complex Networks

#### Abstract

We consider a model for complex networks that was introduced by Krioukov et al. In this model, $N$ points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an edge if they are within a certain hyperbolic distance. The $N$ points are distributed according to a

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. The present paper focuses on the evolution of the component structure of the random graph. We show that

*quasi-uniform*distribution, which is a distorted version of the uniform distribution. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering.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. The present paper focuses on the evolution of the component structure of the random graph. We show that

**(a)**for $\alpha > 1$ and $\nu$ arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order;**(b)**for $\alpha < 1$ and $\nu$ arbitrary with high probability there is a "giant" component of linear order, and**(c)**when $\alpha=1$ then there is a non-trivial phase transition for the existence of a linear-sized component in terms of $\nu$.#### Keywords

Random graphs; Hyperbolic plane;Giant component