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

TL;DR

This paper proves that in a continuum space scale-free percolation model, nodes have power-law degree distributions and the clustering coefficient is self-averaging, reflecting properties of real-world spatial networks.

Contribution

It establishes quenched results for degree distribution and clustering coefficient in a spatial inhomogeneous random graph model, extending understanding of spatial network properties.

Findings

Degree distributions follow a power law almost surely.

The clustering coefficient is self-averaging and positive.

Results hold for almost all realizations of the point process.

Abstract

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 $\mathbb R^d$. 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 the annealed clustering coefficient of one point, which is a strictly positive quantity.