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## Abstract

We consider a model for complex networks that was introduced by Krioukov et al. (Phys Rev E 82 (2010) 036106). In this model, N points are chosen randomly inside a disk on the hyperbolic plane according to a distorted version of the uniform distribution and any two of them are joined by an edge if they are within a certain hyperbolic distance. This model exhibits a power-law degree sequence, small distances and high clustering. The model is controlled by two parameters α and ν where, roughly speaking, α controls the exponent of the power-law and ν controls the average degree.

In this paper we focus on the probability that the graph is connected. We show the following results. For inline image and ν arbitrary, the graph is disconnected with high probability. For inline image and ν arbitrary, the graph is connected with high probability. When inline image and ν is fixed then the probability of being connected tends to a constant inline image that depends only on ν, in a continuous manner. Curiously, inline image for inline image while it is strictly increasing, and in particular bounded away from zero and one, for inline image. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 65–94, 2016

In this paper we focus on the probability that the graph is connected. We show the following results. For inline image and ν arbitrary, the graph is disconnected with high probability. For inline image and ν arbitrary, the graph is connected with high probability. When inline image and ν is fixed then the probability of being connected tends to a constant inline image that depends only on ν, in a continuous manner. Curiously, inline image for inline image while it is strictly increasing, and in particular bounded away from zero and one, for inline image. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 65–94, 2016

Original language | English |
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Pages (from-to) | 65-94 |

Number of pages | 30 |

Journal | Random Structures and Algorithms |

Volume | 49 |

Issue number | 1 |

Early online date | 3 Jun 2016 |

DOIs | |

Publication status | Published - Aug 2016 |

## Keywords

- connectivity
- hyperbolic random graph

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