On the One Dimensional Poisson Random Geometric Graph

Abstract

Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process, and edges exist between two points if and only if their distance is less than a fixed given threshold. We compute explicitly the distribution of the number of connected components of this graph. The proof relies on inverting some Laplace transforms.

1. Motivation

As technology goes on [1–3], one can expect a wide expansion of the so-called sensor networks. Such networks represent the next evolutionary step in building, utilities, industry, home, agriculture, defense, and many other contexts [4].

These networks are built upon a multitude of small and cheap sensors which are devices with limited transmission capabilities. Each sensor monitors a region around itself by measuring some environmental quantities (e.g., temperature, humidity), detecting intrusion and so forth, and broadcasts its collected information to other sensors or to a central node. The question of whether information can be shared among the whole network is then of crucial importance. Mathematically speaking, sensors can be abstracted as points in 𝐑², 𝐑³, or a manifold. The region a sensor monitors is represented by a circle centered at the location of the sensor. In what follows, it is assumed that the broadcast radius, that is, the distance at which a sensor can communicate with another sensor, is equal to the monitoring radius. Two questions are then of interest: can any two sensors communicate using others as hopping relays and is the whole region covered by all the sensors? The recent works of Ghrist and his collaborators [5, 6] show how, in any dimension, algebraic topology can be used to answer these questions. Their method consists in building the so-called simplicial complex associated to the configuration of points and the radius of communication. Then, simple algebraic computations yield the Betti numbers: the first Betti number, usually denoted as 𝛽₀, is the number of connected components; the second number 𝛽₁ is the number of coverage holes. Thus, we have a satisfactory deployment whenever 𝛽₀=1 and 𝛽₁=0. Trying to pursue their work for random settings, we quickly realized that the dimension of the ambient space played a key role. We then first began by the analysis of dimension 1, which appeared to be the simplest situation. In this case, there is no need of algebraic topology so we will not go further in the description of this line of thought even if it was our first motivation.

In dimension 1, the only question of interest is that of the connexity but it can take different forms. Imagine we are given [0,1] as a domain in which 𝑛 points {𝑥₁,…,𝑥_𝑛} are drawn. For a radius 𝑟, one can wonder whether [0,1]⊂∪_{𝑖=1,…,𝑛}[𝑥_𝑖−𝑟,𝑥_𝑖+𝑟], or one can investigate whether [𝑥_𝑖−𝑟,𝑥_𝑖+𝑟]∩[𝑥_𝑖+1−𝑟,𝑥_𝑖+1+𝑟]≠∅ for all 𝑖=1,…,𝑛−1. The second situation is less restrictive since we do not impose that the frontier of the interval is to be covered. Depending on the application, we have in mind, both questions are sensible. A slightly different but somehow close problem is that of the circle: consider now that the points are dispatched along a circle of unit perimeter 𝐶₁ and ask again whether 𝐶₁⊂∪_{𝑖=1,…,𝑛}𝐵(𝑥_𝑖,𝑟) where 𝐵(𝑥,𝑟) is the 2-dimensional ball of center 𝑥 and radius 𝑟. Several years ago, this problem has been thoroughly analyzed ([7] and references therein) for a fixed number of i.i.d. arcs over the circle. A closed form formula can be given for the probability of coverage as a function of the number and of the common law of the arcs length. Some variations of this problem have been investigated since then, see, for instance, [8]. More recently, in [9], algorithms are devised to determine whether a domain can be protected from intrusion by a “belt” of sensors (namely, a ring or the border of a rectangle). There is no performance analysis in this work which is focused on algorithmic solutions for this special problem of coverage. Still motivated by applications to sensor networks, [10] considers the situation where sensors are actually placed in a plan, have a fixed radius of observation, and analyse the connectivity of the trace of the covered region over a line. Some recent results of Kahle [11, 12] are actually hardly linked to our results: the motivation is the same, studying the Betti numbers of some random simplicial complexes, but the results are only asymptotic and valid in dimension greater than 2.

Our main result is the distribution of the number of connected components for a Poisson distribution of sensors in a bounded interval. We could not use the method of [7] since the number of gaps does not determine the connectivity of the domain. For instance, one may have only one gap at the “beginning” which means that all the points are pairwise within the threshold distance and, thus, that the network is connected, or one may have only one gap in the “middle” which means that there is a true hole of connectivity.

Actually, our method is very much related to the queueing theory. Indeed, clusters, that is, sequence of neighboring points, are the strict analogous of busy periods—see Section 2. As will appear below, our analysis turns down to be that of an M/D/1/1 queue with a preemption: when a customer arrives during a service, it preempts the server, and, since there is no buffer, the customer who was in service is removed from the queuing system. This analogy led us to use standard tools of queueing theory: Laplace transform and renewal processes—see, for instance, [13, 14]. This works perfectly, and, with a bit of calculus, we can compute all the characteristics we are interested in. It is worthwhile to note that a queueing model (namely the M/G/∞) also appears in [10].

The paper is organized as follows: Section 2 presents the model and defines the relevant quantities to be calculated. The calculations and analytical results are presented in Section 3. For our situation, we find results analogous to that of [7]. In Section 4, two other scenarios are presented, considering the number of incomplete clusters and clusters placed in a circle. In Section 5, numerical examples are presented and analyzed.

2. Problem Formulation

Let 𝐿>0, we assume that we are given a Poisson process, denoted by 𝑁, of intensity 𝜆 on [0,𝐿]. Let (𝑋_𝑖,𝑖≥1) be the atoms of 𝑁. We, thus, know that the random variables, Δ𝑋_𝑖=𝑋_𝑖+1−𝑋_𝑖 are i.i.d. and exponentially distributed. We fix 𝜖>0. Two points, located, respectively, at 𝑥 and 𝑦, are said to be directly connected whenever |𝑥−𝑦|≤𝜖. For 𝑖<𝑗, two points of 𝑁, say 𝑋_𝑖 and 𝑋_𝑗, are indirectly connected if 𝑋_𝑙 and 𝑋_𝑙+1 are directly connected for any 𝑙=𝑖,…,𝑗−1. A set of points directly or indirectly connected is called a cluster, a complete cluster is a cluster which begins and ends within [0,𝐿]. The connectivity of the whole network is measured by the number of clusters.

The number of points in the interval [0,𝑥] is denoted by 𝑁_𝑥=∑^∞𝑛=0𝟏_{{𝑋𝑛≤𝑥}}. The random variable 𝐴_𝑖 given by 𝐴_𝑖=𝑋¹inf𝑋_𝑗if𝑖=1,∶𝑋_𝑗>𝐴_𝑖−1,𝑋_𝑗−𝑋_𝑗−1>𝜖if𝑖>1,(2.1) represents the beginning of the 𝑖th cluster, denoted by 𝐶_𝑖. In the same way, the end of this same cluster, 𝐸_𝑖, is defined by 𝐸_𝑖=inf𝑋_𝑗+𝜖∶𝑋_𝑗>𝐴_𝑖,𝑋_𝑗+1−𝑋_𝑗>𝜖.(2.2) So, the 𝑖th cluster, 𝐶_𝑖, has a number of points given by 𝑁_𝐸𝑖−𝑁_𝐴𝑖. We define the length 𝐵_𝑖 of 𝐶_𝑖 as 𝐸_𝑖−𝐴_𝑖. The intercluster size, 𝐷_𝑖, is the distance between the end of 𝐶_𝑖 and the beginning of 𝐶_𝑖+1, which means that 𝐷_𝑖=𝐴_𝑖+1−𝐸_𝑖, and Δ𝐴_𝑖 is the distance between the first points of two consecutive clusters 𝐶_𝑖, given by Δ𝐴_𝑖=𝐴_𝑖+1−𝐴_𝑖=𝐵_𝑖+𝐷_𝑖.

Remark 2.1. With this set of assumptions and definitions, we can see our problem as an 𝑀/𝐷/1/1 preemptive queue, see Figure 1. In this nonconservative system, the service time is deterministic and given by 𝜖. When a customer arrives during a service, the served customer is removed from the system and replaced by the arriving customer. Within this framework, a cluster corresponds to what is called a busy period, the intercluster size is an idle time, and 𝐴_𝑖+𝐷_𝑖 is the length of the 𝑖th cycle.

Figure 1 

Queueing representation of the model. A down arrow denotes that user 𝑖 starts his service. An up arrow indicates that user 𝑖 leaves the system without having finished the service. A double up arrow illustrates that the service of user 𝑖 finishes. Beginning and end of the 𝑖th busy period, respectively, 𝐴𝑖 and 𝐸𝑖, are also shown.

The number of complete clusters in [0,𝐿] corresponds to the number of connected components 𝛽₀(𝐿) (since, in dimension 1, it coincides with the Euler characteristics of the union of intervals, see [5]) of the network. The distance between the beginning of the first cluster and the beginning of the (𝑖+1)th one is defined as 𝑈_𝑖=∑^𝑖𝑘=1Δ𝐴_𝑘. We also define Δ𝑋₀=𝐷₀=𝑋₁. Figure 2 illustrates these definitions.

Figure 2 

Definitions of the relevant quantities of the network. Distance between points, distance between clusters, clusters size, interclusters size, beginning and ends of clusters.

For the sake of completeness, we recall the essentials of Markov’s process theory needed to go along, for further details we refer, for instance, to [13, 14]. In what follows, for a process 𝑋, (ℱ^𝑋𝑡,𝑡≥0) is the filtration generated by the sample-paths of 𝑋: ℱ^𝑋𝑡=𝜎{𝑋(𝑠),𝑠≥𝑡}.(2.3)

Definition 2.2. A process (𝑋(𝑡),𝑡≥0) with values in a denumerable space 𝐸 is said to be Markov whenever 𝔼𝐹(𝑋(𝑡+𝑠))∣ℱ^𝑋𝑡=𝔼[𝐹(𝑋(𝑡+𝑠))∣𝑋(𝑡)],(2.4) for any bounded function 𝐹 from 𝐸 to 𝐑, any 𝑡≥0 and 𝑠≥0.
Equivalently, a process 𝑋 is Markov if and only if, given the present (i.e., given 𝑋(𝑡)), the past (i.e., the sample-path of 𝑋 before time 𝑡) and the future (i.e., the sample-path of 𝑋 after time 𝑡) of the process are independent.

Definition 2.3. A random variable 𝜏 with values in 𝐑⁺∪{+∞} is an ℱ^𝑋-stopping time whenever, for any 𝑡≥0, the event {𝜏≤𝑡} belongs to ℱ^𝑋𝑡.

The point is that (2.4) still holds when 𝑡 is replaced by a stopping time 𝜏: given 𝑋(𝜏), the past and the future of 𝑋 are independent. 𝑋 is then said to be strong Markov. This property always holds for the Markov processes with values in a denumerable space but is not necessarily true for the Markov processes with values in an arbitrary space.

From now on, the Markov process under consideration is 𝑁, the Poisson process of intensity 𝜆 over [0,𝐿].

Lemma 2.4. For any 𝑖≥1, 𝐴_𝑖 and 𝐸_𝑖 are stopping times.

Proof. Let us consider the filtration ℱ^𝑁𝑡=𝜎{𝑁_𝑎,𝑎≤𝑡}. For 𝑖=1, we have 𝐴₁≤𝑡⟺𝑋₁≤𝑡⟺𝑁_𝑡≥1∈ℱ^𝑁𝑡.(2.5)
Thus, 𝐴₁ is a stopping time. For 𝐴₂, we have 𝐴₂>𝑡⟺_𝑛≥1𝑁^𝑡=𝑛,^𝑛_𝑗=1Δ𝑋^𝑗≥𝜖,^𝑛_𝑘_=𝑗+1Δ𝑋_𝑘≤𝜖∈ℱ^𝑁𝑡,(2.6) so 𝐴₂ is also a stopping time. We proceed along the same line for others 𝐴_𝑖 and as well for 𝐸_𝑖 to prove that they are stopping times.

Since 𝑁 is a (strong) Markov’s process, the next corollary is immediate.

Corollary 2.5. The set {𝐵_𝑖,𝐷_𝑖,𝑖≥1} is a set of independent random variables. Moreover, 𝐷_𝑖 is distributed as an exponential random variable with mean 1/𝜆, and the random variables {𝐵_𝑖,i≥1} are i.i.d.

3. Calculations

Theorem 3.1. The Laplace transform of the distribution of 𝐵_𝑖 is given by 𝔼𝑒^{−𝑠𝐵𝑖}=𝜆+𝑠𝜆+𝑠𝑒^{(𝜆+𝑠)𝜖}⋅(3.1)

Proof. Since Δ𝑋_𝑗 is an exponentially distributed random variable, 𝔼𝑒^{−𝑠Δ𝑋𝑗}𝟏_{{Δ𝑋𝑗≤𝜖}}=^𝜖0𝑒^−𝑠𝑡𝜆𝑒^−𝜆𝑡𝑑𝑡=𝜆𝑠+𝜆1−𝑒^{−(𝑠+𝜆)𝜖}.(3.2)
Hence, the Laplace transform of the distribution of 𝐵₁ is given by 𝔼𝑒^−𝑠𝐵1=^∞_𝑛=1=𝔼𝑒^−𝑠𝐵1,𝑁_𝐸1=𝑛_∞_𝑛=1𝔼𝑒^{∑−𝑠(𝑛−1𝑗=1Δ𝑋𝑗+𝜖)}𝟏_{{Δ𝑋𝑛>𝜖}𝑛−1}_𝑗=1𝟏_{{Δ𝑋𝑗≤𝜖}}=_∞_𝑛=1𝔼𝑒^{−𝑠Δ𝑋1}𝟏_{{Δ𝑋1≤𝜖}}^𝑛−1𝔼𝑒^{−𝑠Δ𝑋𝑛}𝟏_{{Δ𝑋𝑛>𝜖}}𝑒^−𝑠𝜖=^∞_𝑛=0𝑠+𝜆𝜆1−𝑒^{−(𝑠+𝜆)𝜖}^𝑛𝑒^−𝑠𝜆𝑒^−𝑠𝜖=𝜆+𝑠𝑠𝑒^𝜆𝜖𝑒^𝑠𝜖+𝜆.(3.3)
Using Corollary 2.5, we have 𝔼[𝑒^−𝑠𝐵1]=𝔼[𝑒^{−𝑠𝐵𝑖}], which concludes the proof.

From this result, we can immediately calculate the Laplace transform of the distribution of Δ𝐴_𝑖. Since Δ𝐴_𝑖=𝐵_𝑖+𝐷_𝑖, we have 𝔼[𝑒^{−𝑠Δ𝐴𝑖}]=𝔼[𝑒^{−𝑠(𝐵𝑖+𝐷𝑖)}], and using Corollary 2.5: 𝔼𝑒^{−𝑠Δ𝐴𝑖}=𝔼𝑒^{−𝑠𝐵𝑖}𝔼𝑒^{−𝑠𝐷𝑖}=𝜆𝜆+𝑠𝑒^{(𝜆+𝑠)𝜖}⋅(3.4)

Corollary 3.2. The Laplace transform of the distribution of 𝑈_𝑛, for 𝑛≥0, is given by 𝔼𝑒^{−𝑠𝑈𝑛}=𝜆^𝑛𝜆+𝑠𝑒^{(𝜆+𝑠)𝜖}^𝑛⋅(3.5)

Proof. We use Corollary 2.5 and Theorem 3.1 to calculate the Laplace transform of the distribution of 𝑈_𝑛 since 𝑈_𝑛=∑^𝑛𝑖=1(𝐵_𝑖+𝐷_𝑖): 𝔼𝑒^{−𝑠𝑈𝑛}=^𝑛_𝑖=1=𝔼𝜆+𝑒𝑠^{−𝑠𝐵𝑖}𝔼𝑒^{−𝑠𝐷𝑖}𝜆+𝑠𝑒^{(𝜆+𝑠)𝜖}^𝑛𝜆+𝜆𝑠^𝑛,(3.6) hence, the result.

Let us define the function 𝑝_𝑛 as 𝑝_𝑛∶𝑥∈𝐑⁺⟼𝑝_𝑛(𝑥)=Pr𝛽₀(𝑥)=𝑛,(3.7) that is, 𝑝_𝑛(𝑥) is the probability of having 𝑛 clusters in the interval [0,𝑥]. Since, for all 𝑥∈𝐑₊, 0≤𝑝_𝑛(𝑥)≤1, the Laplace transform of 𝑝_𝑛 with respect to 𝑥, ℒ𝑝_𝑛(𝑠)=^∞0𝑒^−𝑠𝑥𝑝_𝑛(𝑥)𝑑𝑥,(3.8) is well defined.

Theorem 3.3. For any 𝑛≥0, the Laplace transform of 𝑝_𝑛 is given by ℒ𝑝_𝑛(𝑠)=𝜆^𝑛𝑒^{(𝜆+𝑠)𝜖}𝑠𝑒^{(𝜆+𝑠)𝜖}+𝜆^𝑛⋅(3.9)

Proof. We note that; see Figure 3, 𝛽₀(𝑥)≥𝑛⟺Δ𝑋⁰Δ𝑋₀+𝑈_𝑛−1+𝐵_𝑛≤𝐿if𝑛≥1,<∞if𝑛=0.(3.10)
Hence, Pr𝛽₀Pr𝛽₀(𝑥)=0=1−PrΔ𝑋₀+𝐵₁≤𝑥,(3.11)(𝑥)=𝑛=PrΔ𝑋₀+𝑈_𝑛−1+𝐵_𝑛≤𝑥−PrΔ𝑋₀+𝑈_𝑛+𝐵_𝑛+1≤𝑥.(3.12) Let 𝑌_𝑛=Δ𝑋⁰0,+𝑈_𝑛−1+𝐵_𝑛,iiff𝑛𝑛≥=10,,(3.13) then we have ℒPr𝑌_𝑛≤⋅(𝑠)==^∞0Pr𝑌_𝑛≤𝑥𝑒^−𝑠𝑥𝑑𝑥_∞0^𝑥0=1𝑑𝑃_𝑌𝑛(𝑦)𝑒^−𝑠𝑥𝑑𝑥=1𝑠𝔼𝑒^{−𝑠𝑌𝑛}=1𝑠𝔼𝑒^{−𝑠Δ𝑋0}𝔼𝑒^{−𝑠𝑈𝑛−1}𝔼𝑒^{−𝑠𝐵𝑛}𝑠𝑒^𝜆𝜖𝑠𝑒𝜆^𝑛𝑠𝜖+𝜆^𝑛,(3.14) for 𝑛≥1, where we used Corollary 2.5 in the third line. For 𝑛=0, the Laplace transform is trivial and given by ℒ{Pr(𝑌₀≤⋅)}(𝑠)=1/𝑠. Substituting (3.14) in the Laplace transform of both sides of (3.12) yields ℒ𝑝_𝑛(𝑠)==ℒP𝑒r𝑌_𝑛≤⋅(𝑠)−ℒPr𝑌_𝑛+1≤⋅(𝑠)_𝜖𝜆𝑒^𝜖𝑠𝜆^𝑛𝑒^𝜖𝜆𝑠𝑒^𝜖𝑠+𝜆^𝑛+1,𝑛≥0.(3.15) The proof is, thus, complete.

Figure 3 

Illustration of the condition equivalent to 𝛽0≥𝑛.

Lemma 3.4. Let 𝑚 be an positive integer. For any 𝑥>0, when 𝜖→0, 𝔼[𝛽^𝑚0]→𝔼[𝑁^𝑚𝐿].

Proof. Since there is almost surely a finite number of points in [0,𝑥], for almost all sample-paths, there exists 𝜂>0 such that Δ𝑋_𝑗≥𝜂 for any 𝑗=1,…,𝑁_𝑥. Hence, for 𝜖<𝜂, 𝛽₀(𝑥)=𝑁_𝑥. This implies that 𝛽₀(𝑥) tends almost surely to 𝑁_𝑥 as 𝜖 goes to 0. Moreover, it is immediate by the very definition of 𝛽₀(𝑥) that 𝛽₀(𝑥)≤𝑁_𝑥. Since, for any 𝑚, 𝔼[𝑁^𝑚𝑥] is finite, the proof follows by dominated convergence.

Let Li_𝑡(𝑧), 𝑧,𝑡∈𝐑, 𝑧<1, be the polylogarithm function with parameter 𝑡, defined by Li_𝑡(𝑧)=^∞_𝑘=1𝑧^𝑘𝑘^𝑡⋅(3.16) For 𝑚 a positive integer, consider the function of 𝑥𝑀^𝑚𝛽₀∶𝑥⟼𝔼𝛽^𝑚0(𝑥)=^∞_𝑖=0𝑖^𝑚𝑝_𝑖(𝑥).(3.17) Its Laplace transform is given by ℒ𝑀^𝑚𝛽₀(𝑠)=^∞0𝔼𝛽₀(𝑥)^𝑚𝑒^−𝑠𝐿𝑑𝑥.(3.18)

Corollary 3.5. Let 𝛼 be defined as follows: 𝛼=𝑒^𝜖𝜆𝜆𝑠𝑒^𝜖𝑠.(3.19) The Laplace transform of the 𝑚th moment of 𝛽₀(𝐿) is ℒ𝑀^𝑚𝛽₀(𝑠)=𝛼𝑠(𝛼+1)Li^−𝑚𝛼+11,(3.20) which converges, provided that 𝛼>0.

Proof. Applying the Laplace transform of both sides of (3.17), we get ℒ𝑀^𝑚𝛽₀(𝑠)=^∞_𝑖=1=𝑖^𝑚𝑒ℒ_𝜖𝜆𝑝_𝑖(𝑠)/𝜆𝑒^𝜖𝑠𝑒^𝜖𝜆/𝜆𝑠𝑒^𝜖𝑠+1^∞_𝑖=1𝑖^𝑚𝑒^𝜖𝜆/𝜆𝑠𝑒^𝜖𝑠+1^𝑖=𝛼𝑠(𝛼+1)Li^−𝑚𝛼+11,(3.21) concluding the proof.

We define {^𝑚𝑘} as the Stirling number of second kind [15]; that is, {^𝑚𝑘} is the number of ways to partition a set of 𝑚 objects into 𝑘 groups. They are intimately related to polylogarithm by the following identity (see [16]) valid for any positive integer 𝑚, Li_−𝑚(𝑧)=^𝑚_𝑘=0(−1)^𝑚+𝑘𝑘!^𝑚+1𝑘+1(1−𝑧)^𝑘+1⋅(3.22)

Corollary 3.6. The 𝑚th moment of the number of clusters on the interval [0,𝐿] is given by 𝑀^𝑚𝛽₀(𝐿)=^𝑚_𝑘=1⎧⎪⎨⎪⎩𝑚𝑘⎫⎪⎬⎪⎭𝐿𝜖−𝑘^𝑘𝜆𝜖𝑒^−𝜖𝜆^𝑘𝟏_{{𝐿/𝜖>𝑘}}.(3.23)

Proof. Using (3.22) in the result of Corollary 3.5, we get ℒ𝑀^𝑚𝛽₀(𝑠)=𝛼𝑠^𝑚_𝑘=0(−1)^𝑚+𝑘𝑘!^𝑚+1𝑘+1(1+𝛼)^𝑘=1𝛼^𝑘+1(𝛼+1)𝑠^𝑚_𝑘=0𝑐_𝑘,𝑚1𝛼^𝑘,(3.24) where the coefficients 𝑐_𝑘,𝑚 are integers given by 𝑐_𝑘,𝑚=^𝑚_𝑗=𝑘(−1)^𝑗𝑗!⎧⎪⎨⎪⎩𝑚𝑗++11⎫⎪⎬⎪⎭⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠.(3.25) Using the following identity of the Stirling numbers [17], ^𝑚_𝑗=0(−1)^𝑗𝑗!⎧⎪⎨⎪⎩𝑚𝑗++11⎫⎪⎬⎪⎭=0,(3.26) we find that 𝑐_0,𝑚=0 for 𝑚 a positive integer. So, we can write the Laplace transform of the moments as ℒ𝑀^𝑚𝛽₀(𝑠)=^𝑚_𝑘=1𝑐_𝑘,𝑚𝜆𝑒^−𝜖𝜆^𝑘𝑠^𝑘+1𝑒^𝑘𝑠𝜖,(3.27) and apply the inverse of the Laplace transform in both sides of (3.12) to obtain 𝑀^𝑚𝛽₀(𝐿)=ℒ⁻¹^𝑚_𝑘=1𝑐_𝑘,𝑚𝜆𝑒^−𝜖𝜆^𝑘=𝑠^𝑘+1𝑒^𝑘𝑠𝜖(𝐿)_𝑚_𝑘=1𝑐_𝑘,𝑚𝜆𝑒^−𝜖𝜆^𝑘ℒ⁻¹1=𝑠^𝑘+1𝑒^𝑘𝑠𝜖(𝐿)_𝑚_𝑘=1𝑐_𝑘,𝑚𝑘!(𝐿−𝑘𝜖)^𝑘𝜆𝑒^−𝜖𝜆^𝑘𝟏_{{𝐿>𝑘𝜖}}.(3.28) According to Lemma 3.4, when 𝜖→0, we obtain 𝑀^𝑚𝛽₀(𝐿)=𝔼𝑁^𝑚𝐿=^𝑚_𝑘=1𝑐_𝑘,𝑚𝑘!(𝐿𝜆)^𝑘𝟏_{𝐿>0}.(3.29) Hence, for any 𝜆>0, ^𝑚_𝑘=1𝑐_𝑘,𝑚𝑘!(𝐿𝜆)^𝑘𝟏_{𝐿>0}=^𝑚_𝑘=1⎧⎪⎨⎪⎩𝑚𝑘⎫⎪⎬⎪⎭(𝐿𝜆)_𝑘𝟏_{𝐿>0},(3.30) which shows that 𝑐_𝑘,𝑚=⎧⎪⎨⎪⎩𝑚𝑘⎫⎪⎬⎪⎭𝑘!.(3.31) Thus, we have proved (3.23) for any positive integer 𝑚.

Theorem 3.7. For any 𝑛, 𝐿, 𝜆, and 𝜖, we have 𝑃𝑟𝛽₀(𝐿)=𝑛=1𝑛!^{⌊𝐿/𝜖⌋−𝑛}_𝑖=0(−1)^𝑖𝑖!(𝐿−(𝑛+𝑖)𝜖)𝜆𝑒^−𝜆𝜖^𝑛+𝑖.(3.32)

Proof. Since 𝛽₀(𝐿)≤𝑁_𝐿 and since 𝔼[𝑒^𝑠𝑁𝐿] is finite for any 𝑠∈𝐑, we have, for any 𝑠≥0, 𝔼𝑒^{−𝑠𝛽0(𝐿)}=^∞_𝑘=0(−1)^𝑘𝑠^𝑘𝑘!𝔼𝛽^𝑘0(𝐿).(3.33) Rearranging the terms of the right-hand side, and substituting 𝑀^𝑚𝛽₀(𝐿), by the result of (3.23), we obtain 𝔼𝑒^{−𝑠𝛽0(𝐿)}=^∞_𝑘=0⎛⎜⎜⎝(𝐿−𝑘𝜖)_𝑘𝜆𝑒^−𝜆𝜖^𝑘𝟏_{∞{𝐿>𝑘𝜖}}_𝑗=𝑘(−𝑠)^𝑗𝑗!⎧⎪⎨⎪⎩𝑘𝑗⎫⎪⎬⎪⎭⎞⎟⎟⎠⋅(3.34) Furthermore, it is known (see [17]) that ^∞_𝑗=𝑘𝑥^𝑗𝑗!⎧⎪⎨⎪⎩𝑘𝑗⎫⎪⎬⎪⎭=𝑘1!(𝑒^𝑥−1)^𝑘.(3.35) Hence, 𝔼𝑒^{−𝑠𝛽0(𝐿)}=^∞_𝑘=0(𝐿−𝑘𝜖)^𝑘𝜆𝑒^−𝜆𝜖^𝑘𝟏_{{𝐿>𝑘𝜖}}(𝑒^−𝑠−1)^𝑘𝑘!⋅(3.36) By inverting the Laplace transforms, we get ^∞_∞𝑘=0_𝑖=𝑘(−1)^𝑖𝑖!⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠𝛿^{(𝑘−𝑛)}(𝑘𝜖−𝐿)^𝑘𝜆𝑒^−𝜆𝜖^𝑘𝟏_{{𝐿>𝑘𝜖}},(3.37) where 𝛿_𝑎 is the Dirac measure at point 𝑎. After some simple algebra, we find the expression of the probability that an interval contains 𝑛 complete clusters: Pr𝛽₀(𝐿)=𝑛=𝑝_𝑛(𝐿)=1𝑛!^{⌊𝐿/𝜖⌋−𝑛}_𝑖=0(−1)^𝑖𝑖![𝐿−(𝑛+𝑖)𝜖]𝜆𝑒^−𝜆𝜖^𝑛+𝑖,(3.38) concluding the proof.

Lemma 3.8. For 𝑥≥0, 𝑝_𝑛(𝑥) has the three following properties.(i)𝑝_𝑛(𝑥) is differentiable.(ii)lim_𝑥→∞𝑝_𝑛(𝑥)=0.(iii)lim_𝑥→∞𝑑𝑝_𝑛(𝑥)/𝑑𝑥=0.

Proof. Let 𝑗 be a nonnegative integer. The function is obviously differentiable when 𝑥/𝜖≠𝑗. Besides, we have lim_{𝑥→𝜖𝑗+}𝑝_𝑛(𝑥)−lim_{𝑥→𝜖𝑗−}𝑝_𝑛(𝑥)=lim_{𝑥→𝜖𝑗+}(−1)^𝑗𝑗!(𝑥−(𝑛+𝑗)𝜖)𝑎1^𝑛+𝑗⋅(3.39) Since the right-hand term function of 𝑥 is zero as well as its derivative for all 𝑗, the function is also derivable when 𝑥/𝜖=𝑗, which proves (i). Items (ii) and (iii) are direct consequences of the Final Value theorem in the Laplace transform of 𝑝_𝑛 and its derivative.

The expression of 𝑝_𝑛 gives us a Laplace pair between the 𝑥 and 𝑠 domains: 𝟏_{𝑥≥0}𝑛!^{⌊𝑥/𝜖⌋−𝑛}_𝑖=0(−1)^𝑖𝑖!(𝑥−(𝑛+𝑖)𝜖)𝑎1^ℒ𝑛+𝑖⟺𝑎𝑒^𝜖𝑠(𝑎𝑠𝑒^𝜖𝑠+1)^𝑛+1⋅(3.40) We can use this relation to find the distributions of 𝐵_𝑖 and 𝑈_𝑛.

Theorem 3.9. The probability density functions of 𝐵_𝑖 and 𝑈_𝑛, denoted, respectively, by 𝑓_𝐵𝑖(𝑥) and 𝑓_𝑈𝑛(𝑥), are given: 𝑓_𝐵𝑖(𝑥)=𝜆𝑒^−𝜖𝜆𝑝₀(𝑥−𝜖)+𝑒^−𝜖𝜆𝑑𝑑𝑥𝑝⁰𝑓(𝑥−𝜖)𝟏_{𝑥>𝜖},(3.41)_𝑈𝑛(𝑥)=𝜆𝑒^−𝜖𝜆𝑝_𝑛−1(𝑥−𝜖)𝟏_{𝑥>𝜖},(3.42) where the expressions of 𝑝₀(𝑥−𝜖) and (𝑑/𝑑𝑥)𝑝₀(𝑥−𝜖) are straightforwardly obtained from (3.32).

Proof. According to Theorem 3.1, 𝔼𝑒^{−𝑠𝐵𝑖}=1=𝜆𝜆𝑒𝑒^𝜆𝜖_−𝜖𝜆/(𝜆𝜆+𝑠𝑒𝑠)^𝑠𝜖+1𝑒^𝜖𝜆𝜆𝑒^𝜖𝜆/𝜆𝑒𝑠^𝜖𝑠𝑒^𝜖𝑠+1𝑒^−𝜖𝑠+𝑒^−𝜖𝜆𝑠𝑒^𝜖𝜆𝜆𝑒^𝜖𝜆/𝜆𝑒𝑠^𝜖𝑠𝑒^𝜖𝑠+1𝑒^−𝜖𝑠=𝜆𝑒^−𝜖𝜆𝑒^−𝜖𝑠ℒ𝑝₀(⋅)(𝑠)+𝑒^−𝜖𝜆𝑒^−𝜖𝑠𝑠ℒ𝑝₀(⋅)(𝑠).(3.43) Here, using the inverse Laplace transform established in (3.40) and remembering that 𝑝₀(𝑥⁻)=0, we get an analytical expression for 𝑓_𝐵𝑖(𝑥), proving (3.41).
Proceeding in a similar fashion, we can find the distribution of 𝑈_𝑛 by inverting its Laplace transform given by Corollary 3.2 as follows: 𝔼𝑒^{−𝑠𝑈𝑛}=1𝑒^𝜆𝜖/𝜆𝑠e^𝑠𝜖+1^𝑛=𝜆𝑒^−𝜖𝜆𝑒^𝜖𝜆𝜆𝑒^𝜖𝜆/𝜆𝑒𝑠𝑒^{𝜖𝑠𝜖𝑠}+1^𝑛𝑒^−𝜖𝑠=𝜆𝑒^−𝜖𝜆𝑒^−𝜖𝑠ℒ𝑝_𝑛−1(⋅)(𝑠).(3.44) We, thus, have (3.42).

We can also obtain the probability that the segment [0,𝐿] is completely covered by the sensors. To do this, we remember that the first point (if there is one) is capable to cover the interval [𝑋₁−𝜖,𝑋₁+𝜖].

Theorem 3.10. Let 𝑅_𝑚,𝑛(𝑥) be defined as follows: 𝑅_𝑚,𝑛(𝑥)=^{⌊𝑥/𝜖⌋−1}_𝑖=𝑚𝑒^−𝜆𝜖^{𝑖+𝑛𝑖+𝑛}_𝑗=0(𝜆[(1−𝑖)𝜖−𝑥])^𝑗𝑗!⋅(3.45) Then, 𝑃𝑟([0,𝐿]iscovered)=𝑅_0,1(𝐿)−𝑒^−𝜆𝜖𝑅_0,1(𝐿−𝜖)−𝑒^−𝜆𝜖𝑅_1,0(𝐿)+𝑒^−2𝜆𝜖𝑅_1,0(𝐿−𝜖).(3.46)

Proof. The condition of total coverage is the same as ∀𝑥∈[0,𝐿],∃𝑋_𝑖∈[0,𝐿]∣𝑥∈𝑋₁−𝜖,𝑋₁+𝜖,(3.47) which means that {[0,𝐿]iscovered}⟺𝐵₁≥𝐿−𝑋₁∩𝑋₁≤𝜖.(3.48) Hence, Pr([0,𝐿]iscovered)=^𝜖0Pr𝐵₁≥𝐿−𝑋₁∣𝑋₁=𝑥𝑑𝑃_𝑋1(𝑥),(3.49) and since 𝐵₁ and 𝑋₁ are independent Pr([0,𝐿]iscovered)=^𝜖0^{∞𝐿−𝑥}𝑓_𝐵1(𝑢)𝜆𝑒^−𝑥𝜆𝑑𝑢𝑑𝑥.(3.50) The result then follows from Lemma 3.8 and some tedious but straightforward algebra.

4. Other Scenarios

The method can be used to calculate 𝑝_𝑛 for other definitions of the number of clusters. We consider two other definitions: the number of incomplete clusters and the number of clusters in a circle.

4.1. Number of Incomplete Clusters

The major difference with Section 3 is that a cluster is now taken into account as soon as one of the point of the cluster is inside the interval [0,𝐿]. So, for instance, in Figure 3, we count actually 𝑛+1 incomplete clusters. We define 𝛽^0(𝐿) as the number of incomplete clusters on an interval [0,𝐿].

Theorem 4.1. Let 𝐺(𝑘) be defined as 𝐺(𝑘)=(−1)^𝑘𝑒^{𝑘−𝑘𝜆𝜖}_𝑗=0[𝜆(𝑘𝜖−𝐿)]^𝑗𝑗!−𝑒^−𝜆𝐿𝟏^{{𝑇>𝑘𝜖}},(4.1) for 𝑘∈ℕ₊ and 𝐺(−1)=𝑒^−𝜆𝐿. Then, 𝑃𝑟𝛽^0(𝐿)=𝑛=^{⌊𝐿/𝜖⌋+1}_𝑖=𝑛(−1)^𝑖+𝑛⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠(𝐺(𝑖−1)+𝐺(𝑖)),for𝑛≥0.(4.2)

Proof. The condition of 𝛽^0(𝐿)≥𝑛 is now given by 𝛽^0≥𝑛⟺Δ𝑋⁰Δ𝑋₀+𝑈_𝑛−1≤𝐿if𝑛≥1,<∞if𝑛=0.(4.3) We define 𝑌_𝑛 as 𝑌_𝑛=Δ𝑋⁰0+𝑈_𝑛−1iiff𝑛𝑛≥=10,.(4.4) Repeating the same calculations, we find the Laplace transform of Pr(𝛽^0(⋅)=𝑛): ℒPr𝛽^0(⋅)=𝑛(𝑠)=⎧⎪⎨⎪⎩𝑠+𝜆𝜆𝑒𝜆^𝜖𝜆𝑒^𝜖𝜆/𝜆𝑒𝑠𝑒^{𝜖𝑠𝜖𝑠}+1^𝑛1if𝑛≥1,𝜆+𝑠if𝑛=0.(4.5) With this expression, following the lines of Lemma 3.4, we obtain ℒ𝔼𝛽^0(⋅)^𝑚(𝑠)=^𝑚+1_𝑘=1⎧⎪⎨⎪⎩𝑚𝑘+1⎫⎪⎬⎪⎭(𝑘−1)!1𝑠^𝑘𝜆𝜆+𝑠𝜆𝑒𝑒^{−𝜆𝜖𝑠𝜖}^𝑘−1.(4.6) Then, we write 𝜆𝜆+𝑠𝑠1^𝑘=(−1)^𝑘𝜆^𝑘−11𝜆+𝑠+^𝑘_𝑖=11𝑠^𝑖−𝜆1^𝑘−𝑖,(4.7) to find an expression with a well-known Laplace transform inverse, and, after inverting it, we obtain 𝔼𝛽^0𝑚=^𝑚_𝑘=0⎧⎪⎨⎪⎩𝑚𝑘++11⎫⎪⎬⎪⎭𝑘!𝐺(𝑘).(4.8) Expanding the Laplace transform of the distribution of 𝛽^0(𝐿) in a Taylor series and rearranging terms, we get 𝔼𝑒^{−𝑠𝛽′0(𝐿)}=1+𝐺(0)^∞_𝑗=1(−𝑠)^𝑗𝑗!⎧⎪⎨⎪⎩1𝑗⎫⎪⎬⎪⎭+⎛⎜⎜⎝^∞_𝑘=1𝐺(𝑘)^∞_𝑗=𝑘(−𝑠)^𝑗𝑗!⎧⎪⎨⎪⎩𝑘𝑗++11⎫⎪⎬⎪⎭⎞⎟⎟⎠.(4.9) Now, we use another recurrence that the Stirling numbers obey [17], ⎧⎪⎨⎪⎩𝑘𝑗++11⎫⎪⎬⎪⎭=⎧⎪⎨⎪⎩𝑘𝑗⎫⎪⎬⎪⎭+(𝑘+1)⎧⎪⎨⎪⎩𝑘+𝑗1⎫⎪⎬⎪⎭,(4.10) to get ^∞_𝑗=𝑘𝑥^𝑗𝑗!⎧⎪⎨⎪⎩𝑘𝑗++11⎫⎪⎬⎪⎭=^∞_𝑗=𝑘𝑥^𝑗=1𝑗!⎛⎜⎜⎝⎧⎪⎨⎪⎩𝑘𝑗⎫⎪⎬⎪⎭+(𝑘+1)⎧⎪⎨⎪⎩𝑘+𝑗1⎫⎪⎬⎪⎭⎞⎟⎟⎠𝑘!(𝑒^𝑥−1)^𝑘+1𝑘!(𝑒^𝑥−1)^𝑘+1.(4.11) Hence, 𝔼𝑒^{−𝑠𝛽′0(𝐿)}=1+^∞_𝑘=1(𝐺(𝑘−1)+𝐺(𝑘))(𝑒^−𝑠−1)^𝑘.(4.12) Inverting this expression for any nonnegative integer 𝑛, we have the searched distribution.

4.2. Number of Clusters in a Circle

We investigate now the case where the points of the process are deployed over a circumference, and we want to count the number of complete clusters, which corresponds to calculate the Euler's Characteristic of the total coverage, so we call this quantity 𝜒. Without loss of generality, we can choose an arbitrary point to be the origin.

Theorem 4.2. The distribution of the Euler's Characteristic, 𝜒(𝐿), when the points are deployed over a circumference of length 𝐿 is given by 𝑃𝑟(𝜒(𝐿)=𝑛)=𝑒^−𝜆𝐿𝟏_{𝑛=0}+1−𝑒^−𝜆𝐿𝜆𝑒^−𝜖𝜆×𝑛!_{⌊𝐿/𝜖⌋−𝑛}_𝑖=0(−𝑖1!)^𝑖[𝐿−(𝑛+𝑖)𝜖]𝜆𝑒^−𝜖𝜆^{𝑛+𝑖−1}𝐿+(𝑛+𝑖)𝜆1−𝜖for𝑛≥0.(4.13)

Proof. If there is no points on the circle, 𝜒(𝐿)=0. Otherwise, if there is at least one point, we choose the origin at this point, and we have equivalence between the events: {𝜒(𝐿)≥𝑛}⟺𝑈^𝑛−1Δ𝑋₀+𝐵_𝑛≤𝐿∩𝑁_𝐿>0if𝑛≥1,<∞if𝑛=0.(4.14) In Figure 4, we present an example of this equivalence.
We can define 𝑌_𝑛 as 𝑌_𝑛=𝑈^𝑛−10+𝐵_𝑛iiff𝑛𝑛≥=10,,(4.15) to find the Laplace transform or Pr(𝜒(𝐿)=𝑛): ℒ{Pr(𝜒(⋅)=𝑛)}(𝑠)=1−𝑒^−𝜆𝐿𝜆+𝑠𝜆𝑒𝜆^𝜖𝜆𝑒^𝜖𝜆/𝜆𝑒𝑠𝑒^{𝜖𝑠𝜖𝑠}+1^𝑛⋅(4.16) The number of clusters is almost surely equal to the number of points when 𝜖→0, so 𝔼𝜒(𝐿)^𝑚=1−𝑒^−𝜆𝐿𝜆𝑒^{𝑚−𝜖𝜆}_𝑘=1⎡⎢⎢⎣⎧⎪⎨⎪⎩𝑚𝑘⎫⎪⎬⎪⎭[𝐿−𝑘𝜖]𝜆𝑒_−𝜖𝜆^𝑘−1𝐿+𝑘𝜆1−𝜖𝟏^{{𝐿>𝑘𝜖}}⎤⎥⎥⎦.(4.17) Expanding the Laplace transform in a Taylor series and rearranging terms, as we did previously, yields 𝔼𝑒^{−𝑠𝜒(𝐿)}=1−𝑒^−𝜆𝐿𝜆𝑒^{∞−𝜖𝜆}_𝑘=0⎡⎢⎢⎣[𝐿−𝑘𝜖]𝜆𝑒_−𝜖𝜆^𝑘−1𝐿+𝑘𝜆1−𝜖𝟏^{∞{𝐿>𝑘𝜖}}_𝑗=𝑘(−𝑠)^𝑗𝑗!⎧⎪⎨⎪⎩𝑘𝑗⎫⎪⎬⎪⎭⎤⎥⎥⎦.(4.18) Since ^∞_𝑗=𝑘(−𝑠)^𝑗𝑗!⎧⎪⎨⎪⎩𝑘𝑗⎫⎪⎬⎪⎭=(𝑒^−𝑠−1)^𝑘𝑘!,(4.19) we can directly invert this Laplace transform, add the case where there are no points for 𝜒(𝐿)=0, and the theorem is proved.

Figure 4 

Illustration of the condition equivalent to 𝜒(𝐿)≥𝑛. Since the coverage of the last point on [0,𝐿] overlaps the cluster with a point in zero, they are actually contained in the same cluster.

5. Examples

We consider some examples to illustrate the results of the paper. Here, the behavior of the mean and the variance of 𝛽₀(𝐿) as well as Pr(𝛽₀(𝐿)=𝑛) are presented.

From (3.23), we have that 𝔼[𝛽₀(𝐿)] is given by 𝔼𝛽₀(𝐿)=(𝐿−𝜖)𝜆𝑒^−𝜖𝜆𝟏_{𝐿>𝜖}.(5.1) This expression agrees with the intuition in that there are three typical regions given a fixed 𝜖. When 𝜆 is much smaller than 1/𝜖, the number of clusters is approximatively the number of sensors, since the connections with few sensors will unlikely happen, which can be seen from the fact that 𝔼[𝛽₀(𝐿)]→𝐿𝜆 when 𝜆→0. As we increase 𝜆, the mean number of direct connections overcomes the mean number of sensors, and, at some value of 𝜆, we expect that 𝔼[𝛽₀(𝐿)] decreases, when adding a point is likely to connect disconnected clusters. We remark that the maximum occurs exactly for 𝜖=1/𝜆, that is, when the mean distance between two sensors equals the threshold distance for them to be connected. At this maximum, 𝔼[𝛽₀(𝐿)] takes the value of (𝐿/𝜖−1)𝑒⁻¹. Finally, when 𝜆 is too large, all sensors tend to be connected, and there is only one cluster which even goes beyond 𝐿, so there are no complete clusters into the interval [0,𝐿]. This is trivial when we make 𝜆→∞ in the last equation. Figure 5 shows this behavior when 𝐿=4 and 𝜖=1.

Figure 5 

Variation of the mean number of clusters in function of 𝜆 when 𝐿=4 and 𝜖=1.

The variance can be obtained also by (3.23) as follows: Var𝛽₀(𝐿)=(𝐿−𝜖)𝜆𝑒^−𝜖𝜆𝟏_{𝐿>𝜖}+(𝐿−2𝜖)𝜆²𝑒^−2𝜖𝜆𝟏_{𝐿>2𝜖}−(𝐿−𝜖)²𝜆²𝑒^−2𝜖𝜆𝟏_{𝐿>𝜖},(5.2) and, under the condition that 𝐿>2𝜖Var𝛽₀(𝐿)=(𝐿−𝜖)𝜆𝑒^−𝜖𝜆+𝜖(3𝜖−2𝐿)𝜆²𝑒^−2𝜖𝜆.(5.3)

Figure 6 shows a plot of Var(𝛽₀(𝐿)) in function of 𝜆 for 𝐿=4 and 𝜖=1. We can expect that, when 𝜆 is small compared to 𝜖, the plot should be approximatively linear, since there would not be too much connections in the network, and the variance of the number of clusters should be close to the variance of the number of sensors given by 𝜆𝐿. Since 𝛽₀(𝐿) tends almost surely to 0 when 𝜆 goes to infinity, Var(𝛽₀(𝐿)) should also tend to 0 in this case. Those two properties are observed in the plot. Besides, we find the critical points of this function, and again, 𝜆=1/𝜖 is one of them, and at this value Var(𝛽₀(𝐿))=(𝐿/𝜖)𝑒⁻¹+(3−2𝐿/𝜖)𝑒⁻¹. The other two are the ones satisfying the transcendent equation: 𝜆𝑒^−𝜆𝜖=𝐿−𝜖2𝜖(2𝐿−3𝜖)⋅(5.4) By using the second derivative, we realize that 1/𝜖 is actually a minimum. Besides, if 𝐿≤2𝜖, there is just one critical point, a maximum, at 𝜆=1/𝜖.

Figure 6 

Behavior of the variance of the number of clusters in function of 𝜆 when 𝐿=4 and 𝜖=1.

The last example in the section is performed with the result obtained in Theorem 3.7. We consider again 𝐿=4 and 𝜖=1 to obtain the following distributions: PPrr𝛽𝛽₀₀(𝐿)=0=1−3𝜆𝑒^−𝜆+2𝜆²𝑒^−2𝜆−1/6𝜆³𝑒^−3𝜆,Pr𝛽₀(𝐿)=1=3𝜆𝑒^−𝜆−4𝜆²𝑒^−2𝜆+1/2𝜆³𝑒^−3𝜆,Pr𝛽₀(𝐿)=2=2𝜆²𝑒^−2𝜆−1/2𝜆³𝑒^−3𝜆,Pr𝛽₀(𝐿)=3=1/6𝜆³𝑒^−3𝜆,(𝐿)>3=0.(5.5) Those expressions are simple, and they have at most four terms, since 𝐿=4𝜖. We plot these functions in Figure 7. The critical points on those plots at 𝜆=1/𝜖 are confirmed for the fact that, in function of 𝜆 for every 𝑛, Pr(𝜒(𝐿)=𝑛) can be represented as a sum: ^𝑗_𝑖=0𝑞_𝑖,𝑗𝜆𝑒^−𝜆𝜖^𝑖,(5.6) where the coefficients 𝑞_𝑖,𝑗 are constant in relation to 𝜆. However, (𝜆𝑒^−𝜆𝜖)^𝑖 has a critical point at 𝜆=1/𝜖 for all 𝑖>0, so this should be also a critical point of Pr(𝜒(𝐿)=𝑛). If 𝜆 is small, we should expect that Pr(𝜒(𝐿)=0) is close to one, since it is likely to 𝑁 have no points. For this reason, in this region, Pr(𝜒(𝐿)=𝑛) for 𝑛>0 is small. When 𝜆 is large, we expect to have very large clusters, likely to be larger than 𝐿, so it is unlikely to have a complete cluster in the interval, and, again, Pr(𝜒(𝐿)=0) approaches to the unity, while Pr(𝜒(𝐿)=𝑛) for 𝑛>0 become again small.

Figure 7 

Probabilities of connectiveness, Pr(𝛽0(𝐿)=𝑛), for 𝑛=0,1,2,3, in function of 𝜆 when 𝐿=4 and 𝜖=1.

Acknowledgment

The authors would like to thank the anonymous referee whose constructive remarks helped us to improve the presentation of this paper.