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

Published online by Cambridge University Press:  25 February 2021

Joseba Dalmau*
NYU Shanghai
Michele Salvi*
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.


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.

Research Papers
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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