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Hopdistance relationship analysis with quasiUDG model for node localization in wireless sensor networks
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 99 (2011)
Abstract
In wireless sensor networks (WSNs), location information plays an important role in many fundamental services which includes geographic routing, target tracking, locationbased coverage, topology control, and others. One promising approach in sensor network localization is the determination of location based on hop counts. A critical priori of this approach that directly influences the accuracy of location estimation is the hopdistance relationship. However, most of the related works on the hopdistance relationship assume the unitdisk graph (UDG) model that is unrealistic in a practical scenario. In this paper, we formulate the hopdistance relationship for quasiUDG model in WSNs where sensor nodes are randomly and independently deployed in a circular region based on a Poisson point process. Different from the UDG model, quasiUDG model has the nonuniformity property for connectivity. We derive an approximated recursive expression for the probability of the hop count with a given geographic distance. The border effect and dependence problem are also taken into consideration. Furthermore, we give the expressions describing the distribution of distance with known hop counts for inner nodes and those suffered from the border effect where we discover the insignificance of the border effect. The analytical results are validated by simulations showing the accuracy of the employed approximation. Besides, we demonstrate the localization application of the formulated relationship and show the accuracy improvement in the WSN localization.
1 Introduction
In recent years, wireless sensor networks (WSNs) which generally consist of a large number of small, inexpensive and energy efficient sensor nodes have become one of the most important and basic technologies for information access [1]. WSNs have been widely used in military, environment monitoring, medicine care, and transportation control. Spatial information is crucial for sensor data to be interpreted meaningfully in many domains such as environmental monitoring, smart building failure detection, and military target tracking. The location information of sensors also helps facilitate WSN operation such as routing to a geographic field of interests, measuring quality of coverage, and achieving traffic load balance. In many monitoring applications, the sensor nodes must be aware its location to explain 'what happens and where'.
While specialized localization devices exist such as GPS, given the large number of sensor nodes involved in building a single WSN, it is cost ineffective to equip every sensor node with such a sophisticated device. Therefore, seeking for an alternative localization technology in WSNs has become one major research in WSNs [2]. Over the past few years, many localization algorithms have been proposed to provide sensor localization [3]. These localization protocols can be divided into two categories: rangebased and rangefree. The former is defined by methods that use absolute pointtopoint distance estimates (range) or angle estimates for computing locations. The latter makes no assumption about the availability or validity of such information. Recently, rangefree localization methods have attracted much attention because no extra sophisticated device for distance measurement is needed for each sensor node. Despite the challenge in obtaining virtual coordinates purely based on radio connectivity information [4, 5], attempts have been made in developing a practical solution to achieve localization. A few representative protocols of this rangefree scheme include DVHop [6], APIT [7], DRLS [8], MDSMAP [9], and LSSOM [10]. Most of the rangefree localization schemes, such as DVHop, need to compute the average distance per hop to estimate a node's location. In other words, the performance of these localization schemes relies on the accuracy of the employed hopdistance relationship. Since the determination of an accurate hopdistance relationship depends on various complex factors such as node deployment, node density, and wireless communication technology that cannot be easily quantified, the deduction process is tedious and unlikely to produce an exact close form relationship using, say the geometric methods [11].
Due to lack of any predetermined infrastructure and selforganized nature, in most cases, the sensor nodes are randomly and independently deployed in a bounded area. For simplicity, the vast majority of studies based on the idealized unitdisk graph (UDG) network model, where any two sensors can directly communicate with each other if and only if their geographic distance is smaller than a predetermined radio range. Examples of these research include georouting protocols [12, 13], localization algorithms [8, 14], and topology control techniques [15, 16]. Similarly, most of the works related to the hopdistance relationship have been investigated assuming the UDG model [11, 17–23]. The probability that two randomly selected stations with a known distance can communicate in K or less hops with omnidirectional antennas has been analyzed by Chandler [17]. Bettestetter and Eberspacher, derived the probability of the distance of two randomly chosen nodes deployed in a rectangular region within one or two hops [18]. However, when the hop counts are larger than two, only simulation results are available. The distribution parameters are computed by the iterative formula which extends from [19] with a linear formation. Ekici et al. [20] studied the probability of the khop distance in two dimensional network based on the approximated Gaussian distribution. Dulman et al. [11] derived the relationship between the number of hops separating two nodes and the physical distance between them in one and twodimensional topologies considering the UDG model. In the study, the approximated approach based on a Markov Chain in twodimensional case is rather complicated to compute. Zhao and Liang [21] collected the hopdistance joint distribution from Monte Carlo simulations in a circular region and proposed an attenuated Gaussian approximation for the conditional probability distribution function (pdf) of the Euclidean distance given a known hop count. Ta et al. [22] provided a recursive equation for the two randomly located sensor nodes that are khop neighbors given a known distance in homogeneous wireless sensor networks. Ma et al. [23] proposed a method to compute the conditional probability that a destination node has hopcount h with respect to a source node given that the distance between the source and the destination is d.
Despite the current efforts, no fixed communication range exists in actual network environment for the reasons such as multipath fading and antenna issues. Therefore, a certain level of deviation occurs between the intended operation and actual operation in wireless sensor networks when the UDG model is assumed in a protocol design. To deal with this problem, a practical model called the quasi Unitdisk Graph (quasiUDG) model is proposed recently [24]. The quasiUDG model can be characterized by two parameters, the radio range R and the quasiUDG factor α. For any two nodes in the quasiUDG model, if their distance is longer than R, no direct communication link exists between the two. Otherwise, if their distance is between αR and R, a communication link exists with a probability of p_{ l } , and p_{ l } = 1 when their distance is shorter than αR. Given this newly proposed practical property of connectivity, it warrants an investigation of the hopdistance relationship with the quasiUDG model for the rangefree localization schemes to capture practical connectivity characteristics.
In this paper, we focus on exploiting the connectivity property of the quasiUDG model and analyze the relationship between the hop counts separating two nodes and their geographic distance with a specific node density in a WSN. We seek approximation technique to provide a scalable solution for the twodimensional case. We further demonstrate the application of the developed hopdistance relationship to a rangefree localization scheme.
In our WSN setup, we consider that sensor nodes are deployed into a circular region S_{ b } with the radius R_{ b } , where the deployment position follows a Poisson point process with a certain density λ. We set ${p}_{l}=\frac{\alpha}{1\alpha}\left(\frac{R}{d}1\right)$ such that a longer distance between two nodes has a lower probability to form a direct communication link. With this setup, we formulate the probability that a pair of nodes with a known distance resulting a particular hop count. Additionally, we also develop the probability that a pair of nodes with a known distance gives a particular hop count. Finally, in our analysis, we present a quantitative evaluation for the border effect of geographic distance distribution with a given hop count.
The rest of this paper is organized as follows. In Section 2, we present our analytical model deriving an approximate recursive formula for the hopdistance relationship considering the quasiUDG model. Section 3 extends our analytical model by taking the border effect and dependence problem into consideration. Section 4 formulates the probability distribution of distance with known hop counts. In Section 5, we demonstrate the use of our developed hopdistance relationship by applying the relationship to a least squares (LS) based localization algorithm. Finally, we report results in Section 6 and draw important conclusions in Section 7.
2 The probability of the hop count given a known distance
In general, the hopdistance relationship is influenced by the density of sensor nodes and their deployment strategy, as well as the radio communication characteristics. Considering the more practical quasiUDG model, it is recognized that the formulation for the hopdistance relationship with the consideration of quasiUDG model is tedious and unlikely to produce an exact close form. We seek approximation using a recursive approach to derive an approximated hopdistance relationship. In this section, we focus on analyzing the probability that a particular pair of sensor nodes forms a certain hop count with a known distance.
Suppose that N sensor nodes are deployed randomly in circular region S_{ b } with a radius R_{ b } . The number of nodes in any region is a Poisson random variable with an average node density of $\lambda =\frac{N}{{S}_{b}}=\frac{N}{\left(\pi {R}_{b}^{2}\right)}$. Assume that the communication range of a node is R, the communication model between any pair of nodes follows the quasiUDG model with a factor of α where 0 < α < 1.
With the quasiUDG model, the communication area between two nodes with the distance d can be further divided into three cases shown as follows.

If d ≤ αR, then the two nodes can communicate directly.

If αR < d ≤ R, then the two nodes can communicate with a probability p_{ l }, which is set to (R/d  1)α/(1  α). It means that a longer distance between two nodes has a lower probability to form a direct communication link.

If d > R, then the two nodes cannot communicate directly.
The quasiUDG model is illustrated with an example shown in Figure 1. In the figure, we assume that there are two nodes u and v, their distance is d_{ uv } , and their communication probability is P. Let Φ _{ h } (d) be the probability that a particular pair of nodes with d distance apart is h hops away from each other. In the following, we shall first derive Φ _{ h } (d) for the case of h = 1 and then h ≥ 2.
2.1 The case of h= 1
For the case of h = 1, owing to the quasiUDG model, Φ_{1} (d) is obviously
2.2 The case of h≥ 2
We first note that two nodes, named O_{1} and O_{2}, have no direct link but may communicate through h  1 relay nodes. This gives rise to two possibilities, where

O_{2} is not the mhop neighbor of O_{1} if m < h.

Within the communication range of O_{2}, there is a least one (h  1)hop neighbor of O_{1} that has a direct link with O_{2}.
For m < h, the probability, P_{ N } , that O_{2} is not the mhop neighbor of O_{1} can be obtained as
We shall now consider the second possibility in the following. Considering two circles which one centered at O_{1} having a radius of r and the other centered at O_{2} having a radius of R. We denote the distance between the two centers as d and refer the common region of the two circles as S. The quantity P_{ r } (S) is defined as the probability that in the area S, there is no (h  1)hop neighbor of O_{1} that can communicate with O_{2} directly. A differential increment of dr on r can obtain a differential incremental region of dS. Assume that the probability Φ_{ h }(d) of any pair of nodes is independent and statistically identical, we have P_{ r } (S + dS) = P_{ r } (S)P_{ r } (dS). In the following subsections, we calculate P_{ r } (dS) based on three conditions, which are d > R, $\frac{1+\alpha}{2}R<d<R$, and $\alpha R<d\frac{1+\alpha}{2}R$.
2.2.1 O_{1}falls outside the communication range of O_{2}where d> R
In Figure 2, we see that dS can be further divided into many differential regions rdrdθ. Since dr and dθ are infinitesimal, the probability that there exists more than one sensor node in the region rdrdθ can be ignored, and the probability that a single sensor node located within rdrdθ can be approximated as λrdrdθ.
We term the circular region centered in O_{2} with the radius αR as $\mathcal{C}\left({O}_{2}\right)$, and the annulus region centered in O_{2} with the larger radius R and the smaller one αR as $\mathcal{A}\left({O}_{2}\right)$. There are two cases needed to be taken into consideration, which are

When dS falls into $\mathcal{A}\left({O}_{2}\right)$ as shown in Figure 2(a), r satisfies d  R ≤ r ≤ d  αR or d + αR ≤ r ≤ d  R. With the definition of the quasiUDG model, every differential region rdrdθ of dS has a corresponding probability p_{ l }to communicate with O_{2}. Therefore, P_{ r }(dS) is given by (3) where
$${P}_{r}\left(dS\right)=12{\Phi}_{h1}\left(r\right)\lambda rdr\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta .$$(3)
As illustrated in Figure 2(a), we can get the following relationship

When dS covers both $\mathcal{C}\left({O}_{2}\right)$ and $\mathcal{A}\left({O}_{2}\right)$, r will be bounded by d  αR ≤ r < d + αR. The part rdrdθ that falls within $\mathcal{C}\left({O}_{2}\right)$ is surely a onehop neighbor of O_{2}. When that part falls within $\mathcal{A}\left({O}_{2}\right)$, it has a corresponding probability p_{ l }that it has a direct link with O_{2}. Then P_{ r }(dS) can be determined by
$${P}_{r}\left(dS\right)=12{\Phi}_{h1}\left(r\right)\lambda rdr\left[{\phi}_{1}+\underset{\phi 1}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right]$$(6)
and
2.2.2 O_{1}falls within the communication range of O_{1}and d satisfies $\frac{1+\alpha}{2}R<d<R$
We use the foregoing strategy for this derivation. We notice that there are three cases needed to be treated individually which are given as follows.

If 0 < r < R  d, dS will be the annulus region and the entire section of dS will fall within $\mathcal{A}\left({O}_{2}\right)$, which gives
$${P}_{r}\left(dS\right)=12{\Phi}_{h1}\left(r\right)\lambda rdr\underset{0}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta $$(8) 
If Rd ≤ r < dαR or d+αR ≤ r < R+d, dS will not be the annulus region but the entire section of dS will still fall within $\mathcal{A}\left({O}_{2}\right)$. Then we can obtain P_{ r }(dS) by (3).

If dαR ≤ r < d+αR, dS will cover both $\mathcal{C}\left({O}_{2}\right)$ and $\mathcal{A}\left({O}_{2}\right)$. In this case, we can determine P_{ r }(dS) by (6).
2.2.3 O_{1}falls within the communication range of O_{2}and d satisfies $\alpha R<d\frac{1+\alpha}{2}R$
There are four cases needed to be considered when O_{1} falls within the communication range of O_{2} and d satisfying the condition $\alpha R<d\frac{1+\alpha}{2}R$, which are

If 0 < r < dαR, dS will be the annulus region and the entire section of dS will fall within $\mathcal{C}\left({O}_{2}\right)$. Then we can determine P_{ r }(dS) by (8).

If dαR ≤ r < Rd, dS will still be the annulus region but it covers both $\mathcal{C}\left({O}_{2}\right)$ and $\mathcal{A}\left({O}_{2}\right)$. Therefore, we have
$${P}_{r}\left(dS\right)=12{\Phi}_{h1}\left(r\right)\lambda rdr\left[{\phi}_{1}+\underset{{\phi}_{1}}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right]$$(9) 
If Rd ≤ r < d+αR, dS will not be will the annulus region and it covers both $\mathcal{C}\left({O}_{2}\right)$ and $\mathcal{A}\left({O}_{2}\right)$. The probability P_{ r }(dS) can be obtained by (6).

If d+αR ≤ r < R+d, dS will fall within the region $\mathcal{A}\left({O}_{2}\right)$, and hence we can compute P_{ r }(dS) by (3).
2.3 Determination of Φ _{ h }(d) for h≥ 2
Consider that P_{ r } (dS) only depends on r with a specific d, we set P_{ r } (dS) = 1  g(r). From P_{ r } (S + dS) = P_{ r } (S)P_{ r } (dS), the expression of P_{ r } (S) can be obtained by the following linear differential equation where
Therefore, with (2) and (10), the probability Φ_{ h }(d) with h ≥ 2 can be obtained as
where knowing d, Ω(d) can be determined by one of the following expressions, which are

For d > hR or d < αR :
$$\Omega \left(d\right)=0;$$(12) 
For R < d ≤ hR :
$$\begin{array}{lll}\hfill \Omega \left(d\right)& ={\int}_{dR}^{d\alpha R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ +{\int}_{d\alpha R}^{d+\alpha R}{\Phi}_{h1}\left(r\right)r\left({\phi}_{1}+\underset{{\phi}_{1}}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right)dr\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ +{\int}_{d+\alpha R}^{d+R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$$(13) 
For $\frac{1+\alpha}{2}R<d\le R$:
$$\begin{array}{lll}\hfill \Omega \left(d\right)& ={\int}_{0}^{Rd}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)\phantom{\rule{2.77695pt}{0ex}}d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ +{\int}_{Rd}^{d\alpha R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ +{\int}_{d\alpha R}^{d+\alpha R}{\Phi}_{h1}\left(r\right)r\left({\phi}_{1}+\underset{{\phi}_{1}}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right)dr\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ +{\int}_{d+\alpha R}^{d+R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$$(14) 
For $\alpha R<d\le \frac{1+\alpha}{2}R$:
$$\begin{array}{lll}\hfill \Omega \left(d\right)& =\phantom{\rule{2.77695pt}{0ex}}{\int}_{0}^{d\alpha R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ +\phantom{\rule{2.77695pt}{0ex}}{\int}_{d\alpha R}^{Rd}{\Phi}_{h1}\left(r\right)r\left({\phi}_{1}+\underset{{\phi}_{1}}{\overset{\pi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right)dr\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ +\phantom{\rule{2.77695pt}{0ex}}{\int}_{d\alpha R}^{d+\alpha R}{\Phi}_{h1}\left(r\right)r\left({\phi}_{1}+\underset{{\phi}_{1}}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta \right)dr\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ +\phantom{\rule{2.77695pt}{0ex}}{\int}_{d+\alpha R}^{d+R}{\Phi}_{h1}\left(r\right)r\underset{0}{\overset{\phi}{\int}}\frac{\alpha}{1\alpha}\left(\frac{R}{l}1\right)d\theta dr.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$$(15)
3 The border effect and dependence problem
In the above analysis, we do not consider borders of a WSN. However, in a realistic scenario, the deployment area of WSNs is finite and hence borders exist. It is known that the probability Φ_{ h }(d) derived assuming that both involved nodes are not near the border of a WSN may give a slightly different result when one or both of them fall near the border. This is known as the border effect. One common handling of the border effect is to consider the toroidal distance metric in the simulation experiment where a node closed to the border can communicate directly with some nodes at the opposite border [25]. While this special setup eliminates the border effect, it creates discrepancy between the study and practical setups which may lead to a certain level of errors.
Clearly, nodes which are closer to the border cover smaller regions than those at least d away from the border, and therefore intuitively the quantity for Ω(d) should be smaller with the consideration of the border effect. Apparently, the border effect gives a different level of impacts in the measure of Φ_{ h }(d) with a different distance between an involved node and the border. However, it is tedious to derive all cases considering the border effect. For simplicity, we take two key cases of the border effect into consideration. Assuming the center of deployment area is O, we consider two annulus near the border in the following.

The first annulus, called ${\mathcal{A}}_{1}\left(o\right)$, is between the circles with radius of R_{ b }R and R_{ b }αR.

The second annulus, called ${\mathcal{A}}_{2}\left(o\right)$, is between the circles with radius of R_{ b }R and R_{ b }αR.
We set an average metric ζ(h) which varies from 0 to 1 for each hop to determine the decrement of Ω(d). For the circle area with the radius R_{ b }  R, which can be called $\mathcal{C}\left(o\right)$, we can set ζ(h) = 1 accordingly.
Another factor we have to consider is the dependence. The hopdistance relationship derived as aforesaid relies on an implicit independence assumption, that is the probability Φ_{ h }(d) of any pair of nodes is independent and statistically identical. However as pointed in [22], the events that those nodes with the direct link to O_{2} are h  1 hops away from O_{1} are not mutually independent for cases when h > 2, and the calculation of Φ_{h1}(r) should include appropriate dependence conditions. For example, as shown in Figure 3, nodes O_{1} and O_{2} are d distance apart and h hops away from each other where h = 3. The probability that node M_{1} is a 2hop neighbor of node O_{1} is the probability that there is at least one node located in the area S_{1} offering packet relay between nodes O_{1} and M_{1}. Here, the area S_{1} is the intersect area between the circles with the centers O_{1} and M_{1}. Similarly, the probability that node M_{2} is a 2hop neighbor of node O_{1} is the probability that there is at least one node located in the area S_{2} which can directly communicate with nodes O_{1} and M_{2}. Here, the area S_{2} is the intersect area between the circles with the centers O_{1} and M_{2}. It is obvious in the figure that the areas S_{1} and S_{2} share a common area S_{12} indicating that the calculated probabilities are not independent.
To include the impact of the dependence, we add a new factor, namely ξ(h), into the expression of Ω(d). Both factors ζ(h) and ξ(h) are added to allow Ω(d) to reflect a practical setup, and they can be estimated by statistical results via experiments. With the inclusion of ζ(h) and ξ(h) into the expression of ω(h), (11) becomes
4 Distance distribution with known hop counts
In this section, assume that sensor nodes are randomly deployed in a circular region, we derive equations to determine the probability density function of distance d with a known hop count ${f}_{\mathcal{H}}\left(d\right)$.
Theorem 4.1 The probability density function for the distance d between two nodes randomly deployed in a circular region with the radius R_{ b }is${f}_{\mathcal{D}}\left(d\right)$, where
We provide the proof of Theorem 4.1 in Appendix A. According to Theorem 4.1, we can obtain the probability density function of distance between any two nodes in the areas $\mathcal{C}\left(o\right)$, ${\mathcal{A}}_{1}\left(o\right)$, and ${\mathcal{A}}_{2}\left(o\right)$. Their probability density functions of distance are ${f}_{{\mathcal{D}}_{c}}\left(d\right)$, ${f}_{{\mathcal{D}}_{{\mathcal{A}}_{1}}}\left(d\right)$, and ${f}_{{\mathcal{D}}_{{\mathcal{A}}_{2}}}\left(d\right)$, respectively. We also term them as ${f}_{\mathcal{D}*}\left(d\right)$, in general, where the symbol * is appropriately substituted by either ${\mathcal{A}}_{1}$, ${\mathcal{A}}_{2}$ or $\mathcal{C}$. Their expressions are given in (18), (19) and (20) in the following.
where Λ(R, r, d) is given by
By the Bayes' formula, given ${f}_{\mathcal{D}*}\left(d\right)$ and Φ_{ h }(d), we can obtain the expression ${f}_{\mathcal{H}*}\left(d\right)$ which is the probability density function of the geographical distance d when the hop count h is known to be H*. This expression is determined by
where r_{0} = 0 when h = 1, and r_{0} = αR when h > 1.
5 Localization Applications
With the development of the hopdistance relationship for the quasiUDG model, in this section, we show the application of this new relationship to a particular localization algorithm using LS based localization algorithms [26], and we call this newly designed localization algorithm enhance weighted least squares (EWLS).
In a particular localization scenario in WSNs, we assume that there is a number of nodes whose locations are known, and they shall be called anchor nodes. Other nodes that have no knowledge of their locations are called unknown nodes. Consider that an unknown node j can obtain the location x _{ i }, hop h_{ ji } and average hopdistance c_{ i } of an anchor node i. The distance between nodes j and i can be calculated as d_{ ji } = c_{ i }h_{ ji } . In our test scenario, we place an anchor node o in the center and add several other anchor nodes in the map.
We design a simple mechanism to compute the range of distance d_{ ji } . Each anchor node i collects some information to other anchor node k, computes and ranks the average hopdistance c_{i(k)}= d_{ ik }/h_{ ik }, such as c_{i(1)}≥ c_{i(2)}≥ ⋯ ≥ c_{i(n)}. We set the range of average hopdistance as
Following that, the range of distance d_{ ji } can be computed as ${d}_{ji}^{\left(M\right)}={\stackrel{\u0304}{c}}_{i}\times {h}_{ji}$ and ${d}_{ji}^{\left(m\right)}={\underset{}{c}}_{i}\times {h}_{ji}$. With the range of distance d_{ ji } , the variance v_{ h } of the pdf ${f}_{\mathcal{H}}\left(d\right)$, we compute the weights, w_{ i } , of measured distance d_{ ji } as
Finally, we set W = diag(w_{1}, ⋯, w_{ n } ) and compute the location $\widehat{x}$ of an unknown node using the following results, where
and
6 Result discussions
In this section, we compare the analytical and statistical results through simulation experiments to illustrate the performance of our proposed hopdistance model. To illustrate the benefit of applying our model to LSbased localization algorithms, we compared our enhanced algorithm of EWLS to two classical LSbased localization algorithms namely LS [26] and PDM [27].
6.1 Impacts of boarder effects and dependence
We first illustrate the impacts of the boarder effect and dependence problem. In the experiments, we gather statistics of the hop counts with corresponding distance information using Monte Carlo simulations. All the simulation data are collected from several scenarios where N sensor nodes are randomly deployed in a circular region of radius R_{ b } , and the transmission range is set to R with the consideration of the quasiUDG model. The parameters are set to N = 400, R_{ b } = 200, R = 50, α = 0.75, and the result comparisons are listed in Table 1. Let o be the deployment center. The region where nodes are deployed away from the border is denoted as $\mathcal{C}\left(o\right)$, and we term ${\mathcal{A}}_{1}\left(o\right)$ and ${\mathcal{A}}_{2}\left(o\right)$ as the annulus regions in which the distances to o are within (R_{ b }R, R_{ b }αR] and (R_{ b }αR, R_{ b } ], respectively.
In Table 1 we use cumulative absolute difference (CAD) to measure the sum of absolute differences between the analytical results and statistical data. We set $CAD={\sum}_{d}{\Phi}_{h}\left(d\right)Si{m}_{h}$, where Φ_{ h }(d) and Sim_{ h } are the probabilities of two nodes giving a hop count of h with a known distance of d obtained from the analysis and simulation, respectively. Moreover, we denote CAD* as the CAD measurement between analytical results without the border effect consideration and statistical data. For ${\mathcal{A}}_{1}\left(o\right)$ and ${\mathcal{A}}_{2}\left(o\right)$, we can see that the CAD* of each hop is larger than that of CAD because of the impact of the border effect.
6.2 The validation of distribution of distance by a known hop count
We conduct simulation experiments with N = 400, R_{ b } = 200, R = 50, α = 0.75 and present ${f}_{\mathcal{H}*}\left(d\right)$ in Figures 4, 5 and 6 with the statistical data and our analytical results. In all three cases, we note that the numerical results of ${f}_{\mathcal{H}*}\left(d\right)$ given in (21) show excellent agreement with the simulation results. This excellent agreement confirms the accuracy of our model for the estimation of the distance given a known hop count between two sensor nodes.
6.3 Localization accuracy comparisons
In the following, we conduct several simulation experiments to illustrate the performance of our proposed EWLS algorithm. In the simulation, N = 100 sensor nodes are randomly deployed in the circle ${\mathcal{S}}_{b}$ with the radius R_{ b } = 200. The number of anchor nodes is 16 and the communication range of each sensor node is R = 80. The factor α of the quasiUDG model is set to 0.76. In Figure 7(a), even within the communication range R of node 1, the nodes 30, 38, 53, and 63 cannot communicate directly with node 1 due to the considered quasiUDG model. With the network topology illustrated in Figure 7(a), we show the localization errors of EWLS, LS, and PDM in Figure 7. Apparently, the accuracy of EWLS is higher than that of the two classical algorithms where the average localization errors of EWLS, LS, and PDM are 0.26702R, 0.29728R, and 0.28462R, respectively. This confirms that when WSNs exhibit the quasiUDG connectivity behavior, our new hopdistance relationship that captures the behavior offers an improved accuracy in localization.
In the following, we further compare the localization accuracy among EWLS, LS and PDM under various scenarios. In these simulation experiments, we set N = 400, and sensor nodes are deployed uniformly in the circle area with the radius R_{ b } = 200. The connectivity of nodes follows the quasiUDG model. The localization error is calculated as $\xi ={\sum}_{j}\parallel {x}_{j}{\widehat{x}}_{j}\parallel /\left(Nn\right)$.
Firstly, we focus on the impact of the number of anchor nodes. The factor α of quasiUDG model is set to 0.76 and the communication range R of each sensor node is set to 50. In Figure 8, we can see that the localization error ξ of all three algorithms decreases with the increase of number of anchor nodes. Among them, our proposed EWLS always offers the best performance.
Secondly, we investigate the impact of the parameter α of quasiUDG model. In this scenario, we set the number of anchor nodes to 40 and the parameter α varies from 0.72 to 1. The localization error comparison is given in Figure 9. We observe that when the parameter α increases, the number of neighbor nodes increases and the number of hops between an unknown node and an anchor node decreases. Thus, the localization error decreases, and our proposed EWLS algorithm remains the best among all for all considered α values.
Last we study the impact of the communication range R of each sensor node. We set the parameter α of quasiUDG model to 0.76 and set the number of anchor nodes to 40. Similarly, we compare the localization errors in Figure 10 with a range of R values. We observe that because the number of neighbor nodes of a node increases when its communication range increases, and number of hops between an unknown node and an anchor decreases which leads to a decrease in localization errors. Comparing the results for all algorithms, our proposed EWLS outperforms its peers.
7 Conclusions
The hopdistance relationship information can effectively improve the performance of the protocols for wireless sensor networks in many aspects. However, most studies focus on the UDG model which significantly deviates from the real world. In the paper, we presented an analytical modeling to formulate the hopdistance relationship considering the quasiUDG model. Senor nodes are randomly distributed in a circular region according to a Poisson point process. The probability of a particular hop count given a known distance Ω_{ h }(d) was studied, and the border effect and dependence problem was considered in our analysis. Precisely, we derived the probability density function of a random variable describing the distance between two arbitrary nodes with a given hop count. Simulation results confirmed that our analytical results gave excellent accuracy. From the results, we further illustrated impact of the border effect.
Furthermore, we demonstrated the application of our developed hopdistance relationship considering the quasiUDG model in WSN localizations. We designed a LSbased localization algorithm using our developed relationship and compared its performance with other popular LSbased localization algorithms. We again confirmed that the explicit use of our developed relationship in the computation of localization algorithms improved the localization accuracy.
A Appendix
Suppose that a node x(x, y) is randomly deployed in a circular region with the radius R_{ b } , the joint distribution f_{ x }(x, y) can be obtained from
As the nodes x_{ 1 }(x_{1}, y_{1}) and x_{ 2 }(x_{2}, y_{2}) are selected independently, the joint pdf of x_{ 1 } and x_{ 2 } is
We set x_{ d } = x_{ 1 }  x_{ 2 } and x_{ m } = (x_{ 1 } + x_{ 2 })/2. The joint distribution of x_{ m } and x_{ d } can be obtained as
where the constraints L_{1} and L_{2} are
We set the probability of the geographical distance $\mathcal{D}$ between x_{ 1 } and x_{ 2 } less than d to be $P\left(\mathcal{D}\le d\right)$, and the constraint L_{3} can be expressed by ${L}_{3}:{\mathcal{D}}^{2}={x}_{d}^{2}+{y}_{d}^{2}\le {d}^{2}$, then we have
With L_{1} ∩ L_{2}, then x_{ m } falls into the intersectional region of two circles with centers (x_{ d }/ 2, y_{ d }/ 2) and (x_{ d }/ 2, y_{ d }/ 2). The intersectional area is
Since ${{f}_{x}}_{{d}_{},{x}_{m}}\left({x}_{d},{y}_{d},{x}_{m},{y}_{m}\right)$ is constant, (29) can be rewritten as
Therefore, we have
where 0 < d < 2R_{ b } .
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Acknowledgements
The authors gratefully acknowledge the support of the Program of Introducing Talents of Discipline to Universities ("111 Project") under grant No. B08002, and the support of the National Natural Science Foundation of China (NSFC) under Grants No. 60802016, 60833002 and 60972010, the support by "the Fundamental Research Funds for the Central Universities" under grant No. 2009JBM007.
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Keywords
 Sensor Node
 Wireless Sensor Network
 Localization Error
 Localization Algorithm
 Communication Range