Connectivity in Sub-Poisson Networks

TL;DR

This paper investigates how sub-Poisson point processes, which are more regular than Poisson processes, influence connectivity in continuum percolation models like Gilbert's graph and SINR graphs, extending classical phase transition results.

Contribution

It extends the classical phase transition results for Gilbert's and SINR graphs from Poisson to sub-Poisson point processes, broadening understanding of network connectivity in more regular point distributions.

Findings

Sub-Poisson processes exhibit similar phase transition behavior as Poisson processes in percolation models.

Perturbed lattices are examples of sub-Poisson processes, ranging from regular grids to clustered patterns.

The results apply to models relevant for wireless network connectivity analysis.

Abstract

We consider a class of point processes (pp), which we call {\em sub-Poisson}; these are pp that can be directionally-convexly ($dcx$) dominated by some Poisson pp. The $dcx$ order has already been shown useful in comparing various point process characteristics, including Ripley's and correlation functions as well as shot-noise fields generated by pp, indicating in particular that smaller in the $dcx$ order processes exhibit more regularity (less clustering, less voids) in the repartition of their points. Using these results, in this paper we study the impact of the $dcx$ ordering of pp on the properties of two continuum percolation models, which have been proposed in the literature to address macroscopic connectivity properties of large wireless networks. As the first main result of this paper, we extend the classical result on the existence of phase transition in the percolation of the Gilbert's graph (called also the Boolean model), generated by a homogeneous Poisson pp, to the class of homogeneous sub-Poisson pp. We also extend a recent result of the same nature for the SINR graph, to sub-Poisson pp. Finally, as examples we show that the so-called perturbed lattices are sub-Poisson. More generally, perturbed lattices provide some spectrum of models that ranges from periodic grids, usually considered in cellular network context, to Poisson ad-hoc networks, and to various more clustered pp including some doubly stochastic Poisson ones.