Matern Cluster Process with Holes at the Cluster Centers

TL;DR

This paper introduces a new variant of the Matérn cluster process with holes at cluster centers, providing closed-form distributions and functional characterizations in 3D space, inspired by biological nanonetwork applications.

Contribution

It proposes the MCP-H process with holes at cluster centers and derives its key distributions and PGFL in 3D, extending prior MCP models.

Findings

Closed-form distributions for MCP and MCP-H in 3D

Characterization of contact distance distribution

Derivation of probability generating functional

Abstract

Inspired by recent applications of point processes to biological nanonetworks, this paper presents a novel variant of a Mat\'ern cluster process (MCP) in which the points located within a certain distance from the cluster centers are removed. We term this new process the MCP with holes at the cluster center (MCP-H, in short). Focusing on the three-dimensional (3D) space, we first characterize the conditional distribution of the distance between an arbitrary point of a given cluster to the origin, conditioned on the location of that cluster, for both MCP and MCP-H. These distributions are shown to admit remarkably simple closed forms in the 3D space, which is not even possible in the simpler two-dimensional (2D) case. Using these distributions, the contact distance distribution and the probability generating functional (PGFL) are characterized for both MCP and MCP-H.