### Abstract

Language | English |
---|---|

Pages | 632-662 |

Journal | Annals of Applied Probability |

Volume | 25 |

Issue number | 2 |

Early online date | 19 Feb 2015 |

DOIs | |

Status | Published - 1 Mar 2015 |

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### Cite this

**Spatial preferential attachment networks : Power laws and clustering coefficients.** / Jacob, Emmanuel; Morters, Peter.

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 25, no. 2, pp. 632-662. DOI: 10.1214/14-AAP1006

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TY - JOUR

T1 - Spatial preferential attachment networks

T2 - Annals of Applied Probability

AU - Jacob,Emmanuel

AU - Morters,Peter

PY - 2015/3/1

Y1 - 2015/3/1

N2 - We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent τ>2 . The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value τ=3 . Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.

AB - We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent τ>2 . The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value τ=3 . Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.

UR - http://www.e-publications.org/ims/submission/AAP/user/submissionFile/14054?confirm=dbee9130

UR - http://10.1214/14-AAP1006

U2 - 10.1214/14-AAP1006

DO - 10.1214/14-AAP1006

M3 - Article

VL - 25

SP - 632

EP - 662

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 2

ER -