## The Annals of Applied Probability

### Spatial preferential attachment networks: Power laws and clustering coefficients

#### Abstract

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 $\tau>2$. 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 $\tau=3$. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 2 (2015), 632-662.

Dates
First available in Project Euclid: 19 February 2015

https://projecteuclid.org/euclid.aoap/1424355126

Digital Object Identifier
doi:10.1214/14-AAP1006

Mathematical Reviews number (MathSciNet)
MR3313751

Zentralblatt MATH identifier
1310.05185

Subjects
Secondary: 60C05: Combinatorial probability 90B15: Network models, stochastic

#### Citation

Jacob, Emmanuel; Mörters, Peter. Spatial preferential attachment networks: Power laws and clustering coefficients. Ann. Appl. Probab. 25 (2015), no. 2, 632--662. doi:10.1214/14-AAP1006. https://projecteuclid.org/euclid.aoap/1424355126

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