On Comparison of Clustering Properties of Point Processes

Abstract

In this paper, we propose a new comparison tool for spatial homogeneity of point processes, based on the joint examination of void probabilities and factorial moment measures. We prove that determinantal and permanental processes, as well as, more generally, negatively and positively associated point processes are comparable in this sense to the Poisson point process of the same mean measure. We provide some motivating results on percolation and coverage processes, and preview further ones on other stochastic geometric models, such as minimal spanning forests, Lilypond growth models, and random simplicial complexes, showing that the new tool is relevant for a systemic approach to the study of macroscopic properties of non-Poisson point processes. This new comparison is also implied by the directionally convex ordering of point processes, which has already been shown to be relevant to the comparison of the spatial homogeneity of point processes. For this latter ordering, using a notion of lattice perturbation, we provide a large monotone spectrum of comparable point processes, ranging from periodic grids to Cox processes, and encompassing Poisson point processes as well. They are intended to serve as a platform for further theoretical and numerical studies of clustering, as well as simple models of random point patterns to be used in applications where neither complete regularity nor the total independence property are realistic assumptions.

Spreading and Suppression of Infection Clusters on the Ginibre Continuum Percolation Clusters

“…The maximal connected components are called percolation clusters whose sizes are given by the numbers of points of Ξ included in the clusters. The above defines the standard Gilbert disk model with radius r, which is one of the Boolean percolation models, on the point process Ξ [2,3,4,27,28,29]. It is said that the system percolates if there is at least one infinite cluster, whose size is infinity [30].…”

HetHetNets: Heterogeneous Traffic Distribution in Heterogeneous Wireless Cellular Networks

“…A very inclusive review of Point processes in space domain is conducted in [31]. A fixed distance between points generates perfect homogeneity (deterministic lattice).…”

“…On the contrary, the Poisson distribution generates complete randomness. For generating sub-Poisson patterns, one way is to generate a perfect lattice, then apply a random perturbation on its points [29]- [31].…”

“…In multi dimensions, however, there is no natural ordering of the points, so finding the analogue of the iat is not straightforward. There are many density-based heterogeneity measures in the literature such as Ripley's K-function and pair correlation function [31], but they are all parameterized. For introducing distance-based measures, there is the problem of defining the 'next point' or the 'neighboring points' in multidimensional domains.…”