A Tractable Framework for Exact Probability of Node Isolation and Minimum Node Degree Distribution in Finite Multihop Networks
Abstract
This paper presents a tractable analytical framework for the exact calculation of the probability of node isolation and the minimum node degree distribution when sensor nodes are independently and uniformly distributed inside a finite square region. The proposed framework can accurately account for the boundary effects by partitioning the square into subregions, based on the transmission range and the node location. We show that for each subregion, the probability that a random node falls inside a disk centered at an arbitrary node located in that subregion can be expressed analytically in closedform. Using the results for the different subregions, we obtain the exact probability of node isolation and minimum node degree distribution that serves as an upper bound for the probability of connectivity. Our theoretical framework is validated by comparison with the simulation results and shows that the minimum node degree distribution serves as a tight upper bound for probability of connectivity. The proposed framework provides a very useful tool to accurately account for the boundary effects in the design of finite wireless networks.
I Introduction
Wireless multihop networks, also refereed to as wireless sensor networks and wireless ad hoc networks, consist of a group of sensor nodes deployed over a finite region [1, 2, 3, 4, 5]. The nodes operate in a decentralized manner without the need of any fixed infrastructure, i.e., the nodes communicate with each other via a singlehop wireless path (if they are in range) or via a multihop wireless path. In most of the applications, such wireless networks are formed by distributing a finite (small) number of nodes in a finite area, which is typically assumed to be a square region [6, 7, 8, 9].
Connectivity is a basic requirement for the planning and effective operation of wireless multihop networks [10, 11]. The connectivity is a most general notion of connectivity and an important characteristic of wireless multihop networks [12, 9, 13]. The network being connected ensures that there exists at least independent multihop paths between any two nodes. In other words, connected network would still be connected if nodes forming the network fail. The probability of node isolation, defined as the probability that a randomly selected node has no connections to any other nodes, plays a key role in determining the overall network connectivity (1connectivity) [12, 14]. The minimum node degree distribution, which is the probability that each node in the network has at least neighbours, is crucial in determining the connectivity of the network [12].
For largescale wireless sensor networks, assuming Poisson distributed nodes in an infinite area, the connectivity properties such as probability of isolation, average node degree, connectivity have been well studied [15, 16, 12, 17, 18, 19, 20, 21]. When the node locations follow an infinite homogeneous Poisson point process and assuming all nodes have the same transmission range, it has been shown that the network becomes connected with high probability (close to ) at the same time the minimum node degree of the network approaches [15]. This fact was used to approximate the probability of connectivity by the minimum node degree distribution in [12]. It was also used to determine the asymptotic value of the minimum transmission range for connectivity for a uniform distribution of nodes in a unit square and disk [9].
Ia Related Work
Since many practical multihop networks are formed by distributing a finite number of nodes in a finite area, there has been an increasing interest to model and determine the connectivity properties in finite multihop networks [12, 14, 22, 23, 9, 24, 25, 26, 27]. This is also due to the fact, established earlier in [12, 14] and recently in [25], that the asymptotic connectivity results for largescale networks provide an extremely poor approximation for finite wireless networks. This poor approximation is due to the boundary effects experienced by the nodes near the borders of the finite region over which the nodes are deployed. Since the nodes located close to the physical boundaries of the network have a limited coverage area, they have a greater probability of isolation. Therefore, the boundary effects play an important role in determining the overall network connectivity.
Different approaches have been used in the literature, to try to model the boundary effects including (i) using geometrical probability [28] and dividing the square region into smaller subregions to facilitate asymptotic analysis of the transmission range for connectivity [9, 22] and to find mean node degree in different subregions [29], (ii) using a cluster expansion approach and decomposing the boundary effects into corners and edges to yield high density approximations [27] and (iii) using a deterministic grid deployment of nodes in a finite area [30] to approximate the boundary effects with random deployment of nodes [25]. The above approaches provide bounds, rather than exact results, for the probability of node isolation and/or probability of connectivity. For a wireless network deployed over a finite area, the existing results for connectivity and minimum node degree are asymptotic (infinite ) [31, 9]. An attempt was made in [17] to study the minimum node degree and connectivity by circumventing modeling of the boundary effects but the results were shown to be valid for large density (number of nodes) only. Therefore, it is still largely an open research problem to characterize the boundary effects and to find general frameworks for deriving the exact results for the probability of node isolation and the minimum node degree distribution, when a finite number of nodes are independently and uniformly distributed inside a finite region.
IB Contributions
In the above context, we address the following open questions in this paper for a wireless network of nodes, which are uniformly distributed over a square region:

How can we accurately account for the boundary effects to determine the exact probability of node isolation?

How can we incorporate the boundary effects to find the minimum node degree distribution?
In this paper, addressing the above two open questions, we present a tractable analytical framework for the exact calculation of the probability of node isolation and the minimum node degree distribution in finite wireless multihop networks, when nodes are independently and uniformly distributed in a square region. Our proposed framework partitions the square into unequal subregions, based on the transmission range and the location of an arbitrary node. Using geometrical probability, we show that for each subregion, the probability that a random node falls inside a disk centered at an arbitrary node located in that subregion can be expressed analytically in closedform. This framework accurately models the boundary effects and leads to an exact expression for the probability of node isolation and the minimum node degree distribution, which can be easily evaluated numerically. We show that the minimum node degree distribution can be used as an upper bound for the probability of connectivity.
Since the connectivity depends on the number of nodes deployed over the finite region and the transmission range of each node [31, 9], the transmission range must be large enough to ensure that the network is connected but small enough to minimize the power consumption at each node and interference between nodes [32, 12], which in turn maximizes the network capacity. This fundamental tradeoff between the network connectivity and the network capacity leads to the following network design question:

Given a network of nodes distributed over a square region, what is the minimum transmission range such that a network is connected with a high probability or alternatively, what is the minimum number of nodes for a given transmission range such that the network is connected?
Addressing this network design problem, we show through an example how the proposed framework can be used to determine the minimum transmission range required for the network to be connected with high probability.
The rest of the paper is organized as follows. The system model, problem formulation and connectivity properties of a wireless network are presented in Section II. The proposed framework to evaluate the probability of node isolation and the minimum node degree distribution is provided in Section III. The boundary effects in the different regions formed with the change in transmission range are presented in Section IV. The validation of the proposed framework via simulation results and the design example are presented in Section V. Finally, Section VI concludes the paper.
Ii System Model and Problem Formulation
Iia Distribution of Nodes and Node Transmission Model
Consider nodes which are uniformly and independently distributed inside a square region , where denotes the two dimensional Euclidean domain. Let and , for , denote the side and vertex of the square, respectively, which are numbered in an anticlockwise direction. Without loss of generality, we assume that the first vertex of the square is located at the origin and we consider a unit square region defined as
(1) 
Let denote the position of an arbitrary node inside the square . The node distribution probability density function (PDF) can be expressed as
(2) 
We define as a measure of the physical area of the square region, where and the integration is performed over the two dimensional square region . Note that since we assume a unit square.
We assume that each sensor node has a fixed transmission range and the coverage region of a node located at is then a disk of radius centered at the node. Note that the coverage area . The number of nodes inside the coverage area of a certain node are termed as its neighbors.
IiB Connectivity Properties
In this subsection, we define the key connectivity properties of a multihop network, which are considered in this paper.
Definition 1 (Conditional Probability of Connectivity)
Let the cumulative distribution function (CDF) denote the conditional probability of connectivity that a randomly placed node according to uniform probability density function (PDF) is connected to a node located at . Mathematically,
(3) 
Definition 2 (Probability of Node Isolation)
Let denote the probability of node isolation that any node in the network is isolated. Assuming that the probability of node isolation is independent for each node, the probability that a given node at is isolated is given by , which can be averaged over all possible locations to evaluate as
(4) 
Definition 3 (Minimum Node Degree)
For a uniform distribution of nodes in a square region, define the minimum node degree as the minimum of number of neighbors of any node in the region. Let the discrete random variable denote the minimum node degree. The associated PDF, termed as the minimum node degree distribution is given by
(5) 
The details of the formulation of are provided in Appendix A.
Definition 4 (connected network)
A network of nodes is said to be connected (or connected) if there exists at least one path between any pair of randomly chosen nodes.
Definition 5 (connected network)
A network of nodes is said to be connected () if there exist at least mutually independent paths between any pair of randomly chosen nodes. In other words, a network is connected if the network stays connected with the removal of any nodes. Let denote the probability that the network of nodes (each with transmission range ) is connected.
Next we examine the relation between probability and the minimum node degree distribution . Penrose [15] presented in his work on graph theory that a random network for large enough number of nodes, becomes connected at the same instant it acheives the minimum node degree with high probability, that is, serves as an upper bound on , which gets tighter as both and approach one or the number of nodes approaches infinity. Mathematically, we can express this as
(6) 
We note that the minimum node degree distribution is of fundamental importance [12] as (i) it determines the connectivity of the network (), (ii) takes into account the failure of the nodes and (iii) also determines the minimum node degree of the network (). Using (2) and (3), we also note the relationship between and : . Since denotes the probability that each node has at least one neighbor, it has been also referred to as the probability of no isolated node in the literature [12, 14].
IiC Problem Statement
There are two key challenges in evaluating the probability of node isolation in (2) and the minimum node degree distribution in (3). The first challenge is to find the CDF in (3), which requires the evaluation of the overlap area . In [23], it is proposed to find this intersection area using polar coordinates and dividing the square into different radial regions. However, due to the dependance between the polar radius and the polar angle, this approach does not lead to closedform solutions. In [33], an alternative approach is presented for finding the intersection area by first finding the area of circular segments formed outside the sides and vertices and then subtracting from the area of the disk. This approach leads to closedform solutions and is adopted in this work.
The second challenge is to average the CDF given in (3) over the square in order to determine the probability of node isolation in (2) and the minimum node degree distribution in (3). is a function of both the node location and the transmission range . For a unit square, if , then the disk will cover the whole square and hence , irrespective of the node location. For intermediate values of the node range , both and need to be taken into account in determining . This adds further complexity to the task of evaluating (2) and (3). A tractable exact solution to this problem is presented in the next section.
Iii Proposed Framework
Iiia Boundary Effects
We use the approach suggested in [33] in order to quantify the overlap area . The basic building blocks in this approach to characterize the boundary effects are (i) the circular segment areas formed outside each side (border effects) and (ii) the corner overlap areas between two circular segments formed at each vertex (corner effects). We modify the approach in [33] by placing the origin at the vertex , rather than at the center of the square. This leads to a simpler formulation, as discussed below.
Let denote the area of the circular segment formed outside the side , as illustrated in Fig. 1. Using the fact that the area of the circular segment is equal to the area of the circular sector minus the area of the triangular portion, we obtain
(7) 
where denotes the Euclidean distance between and side . Similarly, the areas of the circular segments formed outside the sides , and , respectively, can be expressed as
(8)  
(9)  
(10) 
Let denote the area of the corner overlap region between two circular segments at vertex , as illustrated in Fig. 1. Using the fact that the area of the overlap region is equal to the area of the circular sector minus the area of two triangular portions, we can easily show that
(11) 
where the angle is given by
(12) 
where denotes the absolute value or modulus and . Similarly, the areas of the corner overlap region formed at vertex , and , respectively, can be expressed as
(13)  
(14)  
(15) 
where the angles , and are given by
(16)  
(17)  
(18) 
where , and . We note that the expressions for are valid only when , where denotes the Euclidean distance between and vertex .. For the case when , .
IiiB Tractable Framework
As illustrated in the last subsection, for a given value of the transmission range and the location of the arbitrary node , can be expressed in closedform using (7)(10) and (11)(13)(15). In order to facilitate the averaging of (3) over the whole square region, we divide the square region into different nonoverlapping subregions based on the different border and corner effects that occur in that region. Due to the symmetry of the square, some subregions have the same number of border and corner effects which can be exploited to further simplify the averaging. This will be elaborated in detail shortly.
Let , , denote the type of nonoverlapping subregions and , denote the number of subregion of type . If denotes the conditional probability of connectivity for a node located at , we can write the probability of node isolation in (2) as
(19) 
and the minimum node degree distribution in (3) as
(20) 
We note that the average node degree denoted by can also be determined using our framework as [14]
(21) 
In fact in (IIIB) denotes the cumulative distribution function of the distance between two randomly placed nodes and the closed form analytical results exist in the literature for square, hexagon [6] and convex regular polygons [34].
Remark 1
The general formulation for in (2) is also indirectly suggested in [25]. However, no guidelines are presented in order to evaluate (2). Hence, the authors in [25] use a deterministic grid deployment of nodes to approximate the boundary effects when nodes are uniformly and independently distributed in a square region. By contrast, we provide a tractable framework for complete and exact characterization of the boundary effects in (19).
Remark 2
While in (3) can be expressed analytically in closedform, the integration in (19) and (IIIB) does not have a closedform due to the factor in the exponent. However, it can be easily evaluated numerically using the explicit closedform expressions for for different transmission ranges and different subregions. It must be noted that numerical evaluation of twofold integrals is widely practiced in the literature [35].
Remark 3
Since the subregions are classified on the basis of the boundary effects, the subregions change with the transmission range . We divide the range over the desired interval , as explained in Section IIC, such that the boundary effects are the same for the different subregions over each subinterval of the transmission range. This is explained in detail in the next section.
Iv Effect of Boundaries for the Different Transmission Range Cases
Iva Transmission Range 
Consider the first case of the transmission range, i.e., , as shown in Fig. 3. This case may be of greatest interest in many practical situations where typically the sensor transmission range is a small fraction of the side length of the square. In this case, we can divide the square into four () types of subregions , , and . As shown in Fig. 3, although there is one subregion of type , there are four subregions of types , and , respectively, which are shaded in the same color for ease of identification, e.g., for an arbitrary node located in any subregion of type , the disk is limited by one side only. Hence, we determine only for the following subregions
It is easy to see that for an arbitrary node located anywhere in subregion , the disk is completely inside the square , i.e., there are no border or corner effects. Hence, . For an arbitrary node located anywhere in subregion , the disk is limited by side , i.e., there is a circular segment formed outside the side . Hence, . For an arbitrary node located anywhere in subregion , the disk is limited by sides and , i.e., there is are two circular segments formed outside the sides and and there is no corner overlap between them. Hence, . For an arbitrary node located anywhere in subregion , the disk is limited by sides and and vertex , i.e., there is are two circular segments formed outside the sides and and there is corner overlap between them. Hence, . The number of subregions of each type and the corresponding closed form are tabulated in Table I. For the sake of brevity, and are denoted as and , respectively in this and subsequent tables.
As increases from to , we can see that the subregions of type and become smaller and the subregions of type and become larger. For the value of range , the subregions of type and approach zero.
Remark 4
The division of the square into subregions for transmission range has been previously shown in [25, Fig. 7], [26, Fig. 2] and [29, Fig. 2] to illustrate the intuitive argument that the nodes situated in boundary subregions experience border effects. However, subregions and are indicated as one subregion in [26, 25]. Using our framework, we show that these are two distinct subregions with unique border and corner effects. In addition, we formulate all the subregions for all possible values of the range. This is different to [29, 36] where only the transmission range was considered for finding the average node degree.
IvB Transmission Range 
For the case of the transmission range in the interval , we have types of subregions, which are shown in Fig. 5 and can be expressed as
The upper limit for this interval of transmission range, i.e., is computed as the range for which the lines , and circle intersect. As approaches , the subregion squeezes to zero. The number of subregions of each type and the corresponding closedform are tabulated in Table II.
IvC Transmission Range 
For the case of the transmission range in the interval , we again have types of subregions, which are shown in Fig. 7 and can be expressed as
The upper limit for this interval of the transmission range, i.e., is determined as the range where the subregion squeezes to zero and is computed as an intersection of the line and two circles and . The number of subregions of each type and the corresponding closedform are tabulated in Table III.
IvD Transmission Range 
For the case of the transmission range in the interval , we have types of subregions, which are shown in Fig. 9 and can be expressed as
The upper limit for this interval of the transmission range, i.e., is determined as the range where the four circles , , and intersect. The number of subregions of each type and the corresponding closedform are tabulated in Table IV.
IvE Transmission Range 
For the case of the transmission range in the interval , we have types of subregions, which are shown in Fig. 11 and can be expressed as
The upper limit for this interval of the transmission range, i.e., corresponds to the length of the side of the square region. For , there is always the effect of the sides of the square on the coverage area of a node irrespective of the location of the node. The number of subregions of each type and the corresponding closedform are tabulated in Table V.
IvF Transmission Range 
For the case of the transmission range in the interval , we have types of subregions, which are shown in Fig. 13 and can be expressed as