Isolation and connectivity in random geometric graphs with self-similar intensity measures

TL;DR

This paper investigates how self-similar and fractal node distributions affect connectivity and isolation in random geometric graphs, revealing that nonuniformity and fractal structures influence the probability of connectivity and the distribution of isolated nodes.

Contribution

It extends the analysis of random geometric graphs to self-similar and fractal distributions, showing how these structures impact connectivity and isolation properties.

Findings

Nonuniform distributions can break the Poisson property of isolated nodes.

Nonuniformity strengthens the link between isolation and connectivity.

Fractal distributions exhibit similar properties to smooth distributions with some differences.

Abstract

Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many nodes and shrinking linking range, the number of isolated nodes is Poisson distributed, and the probability of no isolated nodes is equal to the probability the whole graph is connected. Here we examine these properties for several self-similar node distributions, including smooth and fractal, uniform and nonuniform, and finitely ramified or otherwise. We show that nonuniformity can break the Poisson distribution property, but it strengthens the link between isolation and connectivity. It also stretches out the connectivity transition. Finite ramification is another mechanism for lack of connectivity. The same considerations apply to fractal distributions as smooth, with some technical differences in evaluation of the integrals and analytical arguments.