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Scale-free percolation in continuous space: quenched degree and clustering coefficient

Part of: Graph theory

Published online by Cambridge University Press:  25 February 2021

Joseba Dalmau  and
Michele Salvi Open the ORCID record for Michele Salvi [Opens in a new window]
Joseba Dalmau*
Affiliation:
NYU Shanghai
Michele Salvi*
Affiliation:
LPSM, Université de Paris
*
*Postal address: Institute of Mathematical Sciences, NYU Shanghai, Geography Building, 3663 North Zhongshan Road, Shanghai, China.
**Postal address: Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Université de Paris – Sorbonne Université – CNRS, 1 place Aurélie Nemours, 75013 Paris, France.
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Abstract

Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuous-space version of scale-free percolation introduced in [14]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in $\mathbb{R}^d$. Each vertex is equipped with a random weight, and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph is self-averaging. In particular, it is almost surely equal to the annealed clustering coefficient of one point, which is a strictly positive quantity.

Keywords

Random graphscale-free percolationdegree distributionclustering coefficientsmall worldPoisson point processself-averaging

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C63: Infinite graphs 05C82: Small world graphs, complex networks 05C90: Applications

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 106 - 127
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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