Chemical distance in geometric random graphs with long edges and scale-free degree distribution

TL;DR

This paper investigates the properties of geometric random graphs with scale-free degree distributions, focusing on conditions for ultra-small world behavior and establishing limit theorems for chemical distances in spatially embedded networks.

Contribution

It provides sharp criteria for the absence of ultra-smallness and characterizes the chemical distance limits in a broad class of spatial scale-free random graphs.

Findings

Criteria for absence of ultra-smallness in the graphs

Limit theorem for chemical distances in the ultrasmall regime

Dependence of ultra-smallness boundary on spatial embedding

Abstract

We study geometric random graphs defined on the points of a Poisson process in $d$-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centred in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.