2019/02/15 by Joseba Dalmau, Dalmau, Joseba, Michele Salvi +1 · 1 citation
Mathematics · Physics and Astronomy · Social Sciences · #05C63 #05C80 (Primary) #05C82 #05C90 (Secondary) #Complex Network Analysis Techniques #FOS: Mathematics #Human Mobility and Location-Based Analysis #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1902.05774
openalex publication_date 2019/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuum space version of scale-free percolation introduced in [DW18]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in \mathbb Rd. 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 the annealed clustering coefficient of one point, which is a strictly positive quantity.