1 Matérn Cluster Process with Holes at the Cluster Centers

2025-04-27 1 0 421.6KB 9 页 10玖币

Matérn Cluster Process with Holes at the Cluster

Centers

Seyed Mohammad Azimi-Abarghouyi and Harpreet S. Dhillon

Abstract

Inspired by recent applications of point processes to biological nanonetworks, this paper presents a novel variant of a Matérn

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.

Index Terms

Stochastic geometry, Matérn cluster process, Poisson hole process, biological nanonetworks, wireless networks.

I. INTRODUCTION

Clustered point patterns appear in many diverse areas of science and engineering, such as geodesy, ecology, biology, and

wireless networks. Owing to their generality and tractability, Poisson cluster processes (PCPs) are often the ﬁrst choice for

modeling such point patterns. As a representative application, PCPs [1, Sec. 3.4] have been used extensively over the last

decade to model a variety of wireless network conﬁgurations over a 2D space [2]–[8]. They are particularly useful in capturing

user hotspots that exhibit point clustering, which cannot be captured using a simpler homogeneous Poisson point process (PPP).

Two of the speciﬁc PCPs that have been of particular interest in the applications are the Thomas cluster process (TCP) [1,

Deﬁnition 3.5] and the Matérn cluster process (MCP) [1, Deﬁnition 3.6]. The formalism for establishing distance distributions

of PCPs is well-known [9] and has been applied extensively to derive key distance distributions for PCPs [2], [10]–[14].

Very recently, 3D PCPs have been used to model and analyze biological nanonetworks by the authors [15]. The main

idea is to model the locations of molecular fusion centers as a PPP that serves as the parent process for the nanomachines

modeled as a PCP around the fusion centers. Since fusion centers have a non-zero size, it is more reasonable to model them

as spheres instead of points, which essentially places exclusion zones (equivalently, holes) around the parent points of the

PCPs where the nanomachines cannot exist. This results in a new variant of PCPs, which is the main topic of this paper.

Our work in [15] also identiﬁed a speciﬁc structure of 3D TCPs that allowed us to express the distance distributions in 3D

TCPs in remarkably simple closed form expressions (which was not possible in the simpler 2D case). This further inspired

us to investigate the distributional properties of the aforementioned variant of PCPs in more detail in this paper. Before we

describe our contributions, please note that the existence of exclusion zones in clustered processes can also be motivated from

the perspective of wireless networks if one needs to ensure a certain minimum distance between the nodes of two different

networks transmitting at the same frequency channel. When the holes are not placed on the cluster centers, [16] considered

such a cluster process model with holes over a 2D space. This essentially generalized the idea of a Poisson hole process (PHP)

[17] from a homogeneous PPP to the PCP with the common theme being that the underlying point process (PPP or PCP) is

thinned by placing holes independently of the underlying point process.

In this work, we ﬁrst derive new distance distribution results for MCPs over 3D spaces, as the counterpart of similar results

obtained for 3D TCPs in [15]. Then inspired by the recent applications of 3D PCPs to biological nanonetworks, we propose

a new point process that we term the Matérn cluster process with holes at the cluster centers (MCP-H), which is a variant of

the MCP obtained by removing MCP points lying within a certain distance from any of the cluster centers (effectively placing

exclusion zones or holes at each cluster center). While we present results for 3D, the deﬁnition and analytical approach can

be easily extended to the n-dimensional space1. We identify speciﬁc structures for both a 3D MCP and a MCP-H that provide

remarkably simple expressions for the distribution of the distance between an arbitrary point of a given cluster to the origin,

conditioned on the location of that cluster. Using these distance distributions, we characterize the contact distance distributions

and probability generating functionals (PGFL) of both MCP and MCP-H that are not only useful on their own but will also

ﬁnd use in the applications of these processes to diverse ﬁelds.

S. M. Azimi-Abarghouyi is with the School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden

(Email: seyaa@kth.se). H. S. Dhillon is with Wireless@VT, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA USA

(Email: hdhillon@vt.edu).

1Note that the 3D case is considered in this work because of its relevance to the underlying application of biological nanonetworks.

arXiv:2210.06065v1 [cs.IT] 12 Oct 2022

Fig. 1: An illustration of a 2D MCP-H, where the red points are the cluster centers and the blue points are the offspring points.

II. SPATIAL MODEL

Consider an MCP ΦMin R3, which is formally deﬁned as a union of offspring points that are located around parent points

(i.e., cluster centers). The parent point process is a PPP Φpwith intensity λp, and the offspring point processes (one per parent)

are conditionally independent. The set of offspring points of x∈Φpis a ﬁnite point process that is denoted by Nx

1, such

that ΦM=∪x∈ΦpNx

1. The points in Nx

1are uniformly distributed in the ball with the center xand radius Rrepresented by

b(x, R), i.e., Nx

1∼Unif {b(x, R)}. Hence, the probability density function (PDF) of an element of this set being at a location

y+x∈R3is

fY1(y) = (3kyk2

R3kyk ≤ R,

0o.w. (1)

The number of points in Nx

1,∀xis Poisson distributed with mean M1.

Along the same lines, the MCP-H is deﬁned as ΦM−H=∪x∈ΦpNx

2, where the set Nx

2is a thinned version of Nx

A point of Nx

1can be thinned either by the exclusion zone at xor by the exclusion zones of any of the other points

of Φp. The resulting ﬁnite point process can be mathematically described by Nx

2∼ N x

1\ ∪z∈Φpb(z, r0), or equivalently

2∼Unif {b(x, R)} \ ∪z∈Φpb(z, r0), where r0is the radius of the exclusion holes and is assumed to be the same for all

the holes. The PDF of an element of Nx

2being at a location y+x∈R3is denoted by fY2(y). A realization of the MCP-H

is illustrated in Fig. 1. The number of points in Nx

2,∀xis Poisson distributed with mean M2.

III. DISTANCE DISTRIBUTIONS

For both MCP and MCP-H, we ﬁrst present the following results on the PDF of the distance of any (arbitrary) element in

the set Nx

1and Nx

2of the cluster centered at x∈Φpto the origin o, respectively. Such PDF derivation is the key intermediate

step required in PCPs for further analyses [2], [4]–[12], [15]. In order to illustrate this concretely, we will use these PDF

results later in derivations of the contact distance distribution and PGFL in Section IV. It is worth reiterating that even though

the formalism for establishing distance distributions of point processes is well-known, our focus here is on demonstrating that

these distributions admit a simple closed form solution in the case of 3D MCP, which is not even the case in the simpler 2D

MCP.

Theorem 1: For the MCP and conditioned on kxk, i.e., the distance of the parent point xfrom the origin, the PDF of the

distances d=ky+xk,∀y∈ N x

1, for kxk ≤ Ris

fd(r|kxk) = 









3r2

R30≤r < R − kxk,

r(R−kxk+r)(R+kxk−r)

R3kxkR− kxk ≤ r < R +kxk,

0r≥R+kxk,

(2)

and for kxk> R

fd(r|kxk) = (3

r(R−kxk+r)(R+kxk−r)

R3kxkkxk − R≤r < R +kxk,

0 0 ≤r < kxk − R, r ≥R+kxk.(3)

Proof: See Appendix A.

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1MatérnClusterProcesswithHolesattheClusterCentersSeyedMohammadAzimi-AbarghouyiandHarpreetS.DhillonAbstractInspiredbyrecentapplicationsofpointprocessestobiologicalnanonetworks,thispaperpresentsanovelvariantofaMatérnclusterprocess(MCP)inwhichthepointslocatedwithinacertaindistancefromtheclustercentersa...

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