Node localization over small world WSNs using constrained average path length reduction

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Node Localization over Small World WSNs using Constrained Average Path Length Reduction Om Jee Pandey, Rajesh M. Hegde PII: DOI: Reference:

S1570-8705(17)30182-8 10.1016/j.adhoc.2017.10.010 ADHOC 1597

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Ad Hoc Networks

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22 March 2017 9 October 2017 17 October 2017

Please cite this article as: Om Jee Pandey, Rajesh M. Hegde, Node Localization over Small World WSNs using Constrained Average Path Length Reduction, Ad Hoc Networks (2017), doi: 10.1016/j.adhoc.2017.10.010

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Node Localization over Small World WSNs using Constrained Average Path Length Reduction

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Om Jee Pandey, and Rajesh M. Hegde

Department of Electrical Engineering Indian Institute of Technology, Kanpur, India Email: {[email protected]; [email protected]}

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Abstract

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Small world characteristics have been observed and studied in social networks in the context of information dissemination in important applications like disease spread among communities. However, the significance of small world characteristics in wireless sensor network (WSN) applications has hitherto not been investigated. This work investigates the significance of introducing small world characteristics in a conventional WSN for improving the node localization accuracy. In this context, a novel constrained iterative average path length reduction algorithm is proposed to introduce small world characteristics into a conventional WSN. This method utilizes a novel frequency selective approach for the introduction of small world phenomena. The frequency selective approach utilizes two frequency bands for link addition and pruning to create an optimized small world WSN. Subsequently, a constrained least squares node localization method over the small world WSN is developed using both model-based and model-free signal propagation approaches. The proposed constrained iterative average path length reduction method provides reduced average path length and a high average clustering coefficient when compared to conventional WSN. Constrained least squares node localization over the small world WSN yields improved localization performance when compared to node localization over conventional WSN. Performance analysis in terms of localization error, power consumption, bandwidth and anchors required for node localization is also discussed. Cramer-Rao lower bound analysis for node location parameters is also used to provide insights into the error bounds obtained using the proposed method. The experimental results obtained are motivating enough for the introduction of small world phenomena in conventional WSN for applications like node localization and related applications.

Preprint submitted to Ad Hoc Networks

October 17, 2017

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Keywords: Wireless sensor network (WSN), small world network, constrained iterative average path length reduction, sensor node localization

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1. Introduction

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Localization in Wireless Sensor Network (WSN) is a challenging problem [1, 2, 3, 4, 5, 6]. The location of nodes in a 2D/3D space can be computed, with the help of few anchor nodes whose locations are known and a set of physical measurements carried out. Node localization is important since the scheduling and routing operations of a WSN are optimal when the locations of all nodes are known. Accurate positional information of nodes also helps in acquiring the value of location tagged parameters such as pressure, temperature, humidity and geographic coordinates from a given site. It can also detect occurrence of events and tracking of the objects in a given network [1]. Localization methods utilizing physical measurements like distance or angle measurements between anchors and nodes fall into the category of range based approaches. Some examples of these methods are Received Signal Strength (RSS), Time of Arrival (ToA), Time Difference of Arrival (TDoA) and Angle of Arrival (AoA) based localization methods [5, 7, 8, 9, 10]. RSS based localization is popular since it can reuse the existing wireless infrastructure and presents a cost-effective solution. RSS based techniques use a pre-assumed signal propagation model, and the sample data are used for calibration of parameters in the model. If the assumed signal propagation model is inaccurate, calibration of model parameters cannot correct it. Therefore, instead of using samples to calibrate the parameters of a fixed selected model, polynomial regression is used to determine the mathematical relationship between the RSS and distance [11]. Recently, machine learning methods have been utilized to estimate the signal propagation function and its parameters which lead to improved localization accuracy across the network [12]. Localization accuracy, energy efficiency, bandwidth utilization, average path length (APL) and the number of anchors required for localization are the important parameters in WSNs [13, 14, 15, 16, 17, 18]. Network lifetime depends on the energy consumption of each node, and also on the energy consumption across the network [14]. Nodes installed with GPS (anchors) will increase the cost of the network [17, 18]. These nodes will also deplete their energy very quickly, which will reduce the network lifetime drastically. In general, localization methods are designed such that number of anchor nodes are reduced. Utilizing limited bandwidth assigned to WSNs, such as the industrial, scientific and medical (ISM) 2

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band, is also critical. Issues of cognitive bandwidth utilization by sensor nodes have also been addressed in literature [19], which improves the bandwidth utilization by the network. In WSN, sensor nodes transfer the data cooperatively, using multiple hops over a network. The large number of hops required for information transmission is one reason for large APL, high energy depletion in a network leading to large localization error [20, 21, 22]. In order to improve the performance of above parameters, several methods are proposed [13, 14, 15, 16, 18, 19, 23]. However, introduction of small world phenomena in WSN has hitherto not been used in this context. Introduction of small world phenomena into a regular network will bring down APL, while maintaining high average clustering coefficient (ACC), which are the characteristics of small world networks. To reduce APL of the network, links that are longer in length (> 25 % of the network diameter) [23], can be created in a regular network to create a network that exhibits small world characteristics [23, 24, 25, 26, 27, 28, 29]. Conventional WSN with small world characteristics is called as small world WSN in this work. Small world WSN reduces average energy expenditure per sensor node, and the non uniformity in the energy expenditure across the sensor nodes. Small world WSN also reduces number of anchors and bandwidth required for localization with improved localization accuracy [20, 21, 22]. In literature, both wired and wireless methods for introduction of small world phenomena have been discussed [30, 31, 25, 32, 33, 34, 35, 36, 37]. Introduction of wired links for introducing small world characteristics are not preferred since they are not feasible in real world networks such as in mobile ad hoc networks [32]. In practice, small world characteristics are introduced into the WSN utilizing nodes with two radios, a short range and a long range [36]. The long range radio is dedicatedly used for establishing a link between distant nodes. Also, sometimes directional antennas between nodes having the highest degree is used to introduce long links [33, 34]. Increasing the transmission radius of a node by introducing variable modulation scheme is also proposed in [37]. In this work, a frequency selective method that utilizes two frequency bands for link addition and pruning is proposed. This method can be used to create an optimized small world WSN in practical scenario, utilizing multi-radios [31] on each node.

1.1. Motivation Localization error performance can be improved using the concept of small world phenomena [20, 21, 22]. This improvement comes as a trade off with increased power consumption and bandwidth requirement in the network which has hitherto not been investigated. In [21], a cognitive small world approach has been 3

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described in the context of conventional WSN. A fixed link length with a particular probability of link addition [21] is utilized herein prior minimizing a cost function. The cost function selected in this work is the normalized sum of desired localization error, power consumption and bandwidth requirement in the localization process. Cognitive small world WSN approaches are based on introducing links with random probability of link addition. This method [21], creates a large number of links with no significant APL reduction. On other hand this method increases the power consumption and bandwidth requirement in the network, with no significant reduction in localization error. In [21], frequency selective introduction of small world phenomena among randomly selected node pairs has not been considered. Apart from the aforementioned approaches there are very few approaches in literature that utilize the small world characteristics in the context of node localization in WSN.

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1.2. Contributions The main contributions of this paper are as follows. A novel method to introduce the small world characteristics into a regular network using constrained iterative average path length reduction is proposed. Introduction of additional links in this method can lead to other network issues, like reduced ACC. In order to address this, a novel frequency selective algorithm for introduction of small world characteristics between each pair of nodes in the network is developed. Performance of proposed iterative method is also compared with other non-iterative methods which introduce small world characteristics in a conventional WSN. The proposed iterative method provides better performance in terms of APL reduction and maintenance of high ACC, when compared to non-iterative methods. Performance improvements in terms of localization error, power consumption and bandwidth utilization are also noted. These performance improvements are noted due to the optimal utilization of additional links to create a small world network. Additionally, Cramer-Rao lower bound (CRLB) analysis for node location parameters obtained from the proposed method is also performed. Results of CRLB analysis illustrate that the estimates of location obtained from the small world WSN are reasonably better when compared to that obtained from a conventional WSN. The remainder of the paper is as follows. Section 2 presents known definitions and properties of small world WSN. This is followed by a discussion on the development of small world WSN using constrained iterative average path length reduction. The frequency selective introduction of links between each pair

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of nodes in the network is also enumerated in this section. In Section 3, localization in small world WSN using constrained least squares (CLS) method is discussed. Performance evaluation of the proposed localization method over small world WSN (SW-WSN) and CRLB analysis for node location parameters are also presented in Section 4 prior to concluding the work. 2. Small world WSN development using Iterative Path Length Reduction

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In this section the characteristics of small world networks are first enumerated. Subsequently, the development of the constrained iterative average path length reduction method is described. A novel frequency selective method to introduce small world characteristics is also discussed herein.

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2.1. Characteristics of Small World-Wireless Sensor Networks Small world phenomena was first observed in [38], in social connectivity of people. This work provides a theory of “six degrees of separation”. In [39], small world phenomena has been investigated in graphs which concludes that the typical distance between any two nodes in a random graph scales as the logarithm of the number of nodes. Hence, random graphs are small worlds as well. Small world phenomena in a network leads to short path distance between any two nodes in the network despite the large size of the network. The distance between two nodes is defined as the number of edged along the shortest path connecting them. In [23, 24], small world characteristics are observed in wireless networks. Small world characteristics exhibit low APL and high ACC. Introduction of new links between nodes reduces APL of WSNs while maintaining high ACC. APL is the average of the shortest path lengths among pairs of nodes in the network. Path length between any two nodes is the number of hops for transmission of information between them. ACC is the average of the clustering coefficient of the nodes in the network. Clustering coefficient (CC) is a measure of degree to which nodes in a network tend to cluster together. Figure 1, illustrates a WSN and small world WSN (SW-WSN). In Figure 1a, data transmission from node A to node K requires cooperation of many nodes, while in case of SW-WSN in Figure 1b, it will be accomplished via a single node only. In case of SW-WSN data will move from node A to node P via one link and then from node P to node K using another link. Figure 1 (see page 34) goes here.

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In general SW-WSNs are characterized by their APL and ACC. For an undirected WSN representing a regular network the APL, LWS N is given by N

X 1 di j N(N − 1) i, j

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LWS N =

where the sum is over all pairs of distinct nodes and di j is the length of the shortest path between nodes i and j. On the other hand in a WSN having small world characteristics, APL, LS W−WS N between two randomly chosen nodes grows proportionally to the logarithm of the number of nodes in the network [40]

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LS W−WS N ∝ ln (N).

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APL for a network with random link addition [41, 40] is given as N 1/d f (NK p) K where f is a universal scaling function and is given as ( ) u, if u  1 f (u) = ln(u), if u  1

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Lrand (N, p) ∼

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and each vertex has at least K neighbors. Probability of link addition is denoted by p, for converting WSN into SW-WSN. Probability of link addition is defined as # links added . (4) p= # possible links - # links existing

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Thus a large scale WSNs with small world characteristics will have a small APL. It can be seen that if the value of p is small (p → 0), network will have the property of regular network, while if the value is large (p → 1), then the network will achieve random network characteristics. SW-WSNs contain the value of p, in between 0 and 1. WSNs are spatial graphs, where links are determined by radio connectivity. There is a limitation on radio range, therefore these networks do not have longlength connections. Hence even though, these networks are highly clustered, they do not experience small world phenomena. In [41, 40], it is shown that the mean distance between nodes l(N, p) decreases very rapidly as soon as p is non-zero. For p = 0, a linear chain of sites is obtained, so that l(N, 0) =

N(N + 2K − 2) N ∼ 4K(N − 1) 4K 6

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growing like N. For p = 1, l(N, 1) grows like ln (N)/ln (2K − 1).

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A common property of small world networks is that cliques form, representing circles of acquaintances in which every node knows every other node. This inherent tendency to cluster is quantified by the clustering coefficient. CC for a node i in the network is defined as [41, 40] 2ei (7) Ci = ki (ki − 1) where, ki is the total number of neighbors of node i and ei is the total links between neighbors of node i. This can also be understood as, CC of node i is the ratio between the number ei of edges that actually exist between these ki nodes and the total number ki (k2i −1) . It can also be expressed as Ci = 3N∆ (i)/N3 (i)

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P where, N∆ (i) = k> j ai j a jk aik is triangle numbers which include node i and N3 (i) = P k> j ai j a jk is triple numbers which include node i. In these expressions, ai j denotes the link between ith and jth nodes. ACC of WSNs is given by N 1X ¯ C= Ci N i=1

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2.1.1. Significance of Link Addition on Average Path Length and Average Clustering Coefficient in Wireless Sensor Network The analysis of variation of APL and ACC in a regular network with addition of new links is very important in the study of small world WSN (SW-WSN). This analysis can help understand the nature of these variations and therefore optimize the introduction of new links into a regular network. An optimized approach to link addition can then be used to convert a regular network to a SW-WSN. In order to illustrate variation of APL and ACC, two experimental scenarios are considered. In the first scenario the Intel Berkeley data set with 54 sensor nodes in a dimension of 41 m × 31 m is considered [42]. Eight links are introduced to convert this network into a SW-WSN. Similarly, a real field deployment scenario as shown in Figure 7 and 8 is also considered. For both these experimental data sets variation of APL and ACC with number of links is analyzed herein. A tabular variation in the values of APL and ACC for conventional WSN and SW-WSN for the aforementioned experimental data sets is shown in Table 1.

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Table 1 (see page 41) goes here.

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It is seen in Table 1 that, APL decreases significantly when compared to the change in ACC in the case of SW-WSN. The APL in case of real field node deployment decreases by 1.2617 when nine links are used. This decrease in the APL is noted to be a significant decrease in APL. On the other hand the ACC decreases by 0.0206 for SW-WSN indicating a minor change.

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2.2. Constrained Iterative Average Path Length Reduction for Developing Small World-Wireless Sensor Network In this Section, a problem formulation is given first. Subsequently, solution to aforementioned constrained optimization problem is described next. The methodology for converting conventional WSN into SW-WSN using proposed constrained iterative average path length reduction is then discussed. An algorithm for introducing frequency selective small world phenomena is also described.

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2.2.1. Problem formulation Consider a network with dimension L × W, having set and n of unknown nodes o anchors represented as X = {xi | i = 1, 2, ..., r} and A = a j | j = 1, 2, ..., t respectively. xi and a j ∈ R2 denote the 2D coordinates of ith node and jth anchor as [xni , yin ]T and [xaj , yaj ]T . Let n = r + t denote the total number of nodes in the netS work, where Z = X A. The coordinates of anchors are assumed to be known. The problem is to find minimum number of large links (> 25 % of the network diameter) [23] to be introduced in the network such that APL of the network is optimal. Link addition is carried out in an iterative manner and stopped after a particular numbern of iterations when the APL attains its optimal value. o Let H(0) = hi j (0) , denote the minimum hop count matrix for the given i, j conventional WSN. The matrix is updated after each iteration of link addition. The APL at kth iteration with minimum hop count hi j (k), between nodes i and j is given by X 1 hi j (k). (10) H(k) = n(n − 1) i, j Let N(k) = {(im , jm ), | m = 1, 2, ..., p} denote the ordered set of p node pairs having the smallest entries in D(k) = −logH(k) + exp(−C(k)). Here, C(k) and H(k) are the ACC matrix and minimum hop count matrix for WSN at kth iteration, 8

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Subject to

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respectively. N(k) can be obtained using the matrix D(k) by arranging its entries in the ascending order and choosing the first p entries. Let the link introduced th between ith and by πi, j . The set of such additional links is o n j node be denoted given by L = πik , jk |k = 1, 2, ..., L , where L denotes the maximum number of links that are allowed to be added. Therefore, the constrained optimization problem is formulated as Find : arg min H(L), |L| ≤ L, n n o o H(L)|(i∗ , j∗ ) −H(L) ≤ optimal × Hmax (0) . m m

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Where, |L| denote the cardinality of set of long links, L. optimal , represents the optimal threshold on APL reduction that depends on the ACC and APL reduction at each iteration. The values of optimal can be obtained using Equations (12) and (13). Equations (12) and (13) indicate that link addition is allowed till APL reduction is more than ACC reduction at each iteration. This condition leads to an optimal value, optimal = 0. Similarly, if ACC reduction is equal or n more than APLo reduction then link addition is stooped. In this case optimal = H(k)|(i∗m , j∗m ) −H(k) /|Hmax (0)|. These conditions lead to development of an efficient SW-WSN. Therefore, optimal = 0, if ¯ (i∗m , j∗m ) ¯ H(k)|(i∗m , j∗m ) C(k)| C(k) H(k) − − (12) < ¯ Hmax (0) Cmax (0) C¯ max (0) Hmax (0) n o else, optimal = H(k)|(i∗m , j∗m ) −H(k) /|Hmax (0)|, if ¯ (i∗m , j∗m ) ¯ C(k)| H(k)|(i∗m , j∗m ) C(k) H(k) − − (13) ≥ . ¯ Hmax (0) Cmax (0) C¯ max (0) Hmax (0)

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In Equations (12) and (13), it is assumed that C¯ min (∞) ≈ 0 and Hmin (∞) ≈ 0. Therefore normalization of parameters are possible. Here, C¯ min (∞) and Hmin (∞) are the lowest ACC and APL after the introduction of a large number of links into the network. C¯ max (0) and Hmax (0) are representing the maximum ACC and APL for the conventional WSN. H(k) and H(k)|(i∗m , j∗m ) are the actual APLs at kth iteration, before and after the optimal link π(i∗m , j∗m ) is added into the network. Optimal link π(i∗m , j∗m ) is obtained at each iteration of link addition using Equation (14) π(i∗m , j∗m ) = arg max H(k) − H(k)|(im , jm ) (im , jm )∈N

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¯ and C(k)| ¯ (i∗m , j∗m ) are the actual ACCs at kth iteration, before and Similarly, C(k) after the optimal link π(i∗m , j∗m ) is added into the network. The optimization problem given in (11) can be solved using Algorithm 1. It must be noted that at each iteration only one long link is introduced in the network. A heuristic matrix D(k) is obtained using minimum hop count matrix H(k) and averaged clustering coefficient matrix C(k). H(k) is obtained using fixed RSS measurement model [12], ! d Pr (dB) = 10 log10 (ρ) − 10ηe log10 + N(0, σN dB ) (15) Pt d0 and utilizing the Dijkstra Algorithm [43]. In Equation (15), Pt and Pr are the transmitted and received power respectively. d0 is the reference distance, while d is the actual distance between the sender and the receiver. ρ is constant path   λ 2 with λ as a wavelength of the signal. ηe is the path loss factor which is 4πd 0 loss exponent, determined from the measurement model. The typical values of ηe is between 2 and 6, varying with the antenna orientation and environment factors (indoor vs outdoor, obstructed vs open space). N(0, σN dB ) is a Gaussian random variable with zero mean and σ2 N dB variance that models the random variation of the received signal strength. C(k) is the ACC matrix obtained by averaging the clustering coefficient of individual node pair. The reduction in APL at kth iteration is computed as, (APLR ) = H(k) − H(k)|(im , jm ) . The solution to the aforementioned constrained optimization problem is initiated by first computing the best link producing maximum APL reduction at each iteration by Equation (16). Thereafter, compute number of links to be created for significant APL reduction by Equation (17). This set of optimal links of long length are introduced in the conventional WSN and SW-WSN is created.

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Figure 2 (see page 34) goes here.

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The block diagram for developing a SW-WSN using proposed constrained iterative average path length reduction method is shown in Figure 2. Design and deployment of conventional WSN is carried out in the first band of frequency ( f1 − f2 ) with set of unknown nodes and anchor nodes first. Subsequently, cluster heads are assigned to each cluster, which are having maximum residual energy in each cluster. The cluster heads generate incidence matrix at gateway. In the incidence matrix G Gi j ∈ {0, 1}, ∀i, j and Gi j = 1, if RS S (i, j) ≥ γ s 10

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Algorithm 1 Development of Small World-Wireless Sensor Networks using a Constrained Iterative Average Path Length Reduction

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Input : Sets A and X, with known IDs, ∀ a j , xi ∈ R2 . a j = [xaj , yaj ]T , ∀ j = 1, 2, ..., t, are known ¯ ¯ Output : SW-WSN with H(L) << H(0), C(L) ≈ C(0) Data: Pi j , ∀ i = 1, 2, ..., r and ∀ j = 1, 2, ..., t, using Equation (15) , L = φ, k=0 Each n o cluster head consist of incidence matrix I Ii j an element in I is known using Equation in [12], Ii j ∈ {0, 1} , ∀i, j, i, j indicates the connectivity of node pair, depending over, Pi j ≤ γ s or ≥ γ s Gateway consists of incidence matrix G of the entire network using all available I matrices in the network Start Iteration: Using G find C(k) and H(k) by utilizing Equations (7), (8), (9), n (10), o (15) and Dijkstra Algorithm [43] over the network (k = 0 for WSN) hi j (k) an element in H(k), is the minimum hops between node i and j, n o i, j ci j (k) an element in C(k), is averaged clustering coefficient between node

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i and j, in the current network at kth iteration ¯ and H(k) using Equations (9) and (10) Compute C(k) Best Node Pair Selection: Find ordered set of p node pairs, N(k) = {(im , jm )| m = 1, 2, ..., p} having the smallest entries in D(k) obtained using heuristic model of D(k), utilizing H(k) and C(k) Virtual shortcuts will be created, ∀ (im , jm ) ∈ N, s.t. m = 1, 2, ..., p, iteratively, Compute (i∗ , j∗ ) = arg max H(k) − H(k)| (16) m

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End of Node Pair Selection 13: Select, optimal using Equations (12) and (13) 14: If for pair (i∗m , j∗m ) n o n o H(k)|(i∗ , j∗ ) −H(k) > optimal × Hmax (0) , m m

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introduce π(i∗m , j∗m ) in the set L for the node pair (i∗m , j∗m ) in the band ( f2 − f3 ) and S generate L =L π(i∗m , j∗m ) , k = k + 1 15: Find new incidence matrix G0 and update the matrices H(k), C(k) and D(k) go to step 4, else 16: Stop link introduction in the network and obtain L End Iteration 17: Small World-Wireless Sensor Network is created 11

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Gi j = 0,

if RS S (i, j) < γ s or i = j

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where γ s , is the threshold of received power for connectivity between node pairs. Incidence matrix G changes due to small world phenomena introduction at each iteration. Introduction of small world phenomena is carried out in the second band, ( f2 − f3 ). Introduction of small world phenomena is continue iteratively until the APL reduction attains its optimal value. Node pairs selection for introducing links in the second band is carried out using heuristic model of minimum hop count matrix and averaged clustering coefficient matrix. From these pairs, only one pair is selected for small world phenomena introduction at that iteration. The selected node pair is producing maximum APL reduction. Dijkstra algorithm is used to find the minimum hop count matrix. Change in incidence matrix is an iterative process until an optimal APL reduction is obtained. APL computation at kth iteration is given by Equation (10). This entire process leads to introduction of small world characteristics in the conventional WSN and creates a small worldwireless sensor networks.

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2.2.2. Algorithm for Development of Small World-Wireless Sensor Network using Iterative Average Path Length Reduction In the proposed algorithm (Algorithm 1), incidence matrix of the network is obtained using the connectivity between gateway and cluster heads. Subsequently, using Dijkstra algorithm, minimum hop count matrix is obtained at each iteration of small world characteristics. Node pairs having maximum hop distance and maximum ACC, help in significant reduction in APL of the network. In the proposed method, single node pair selection among the selected node pairs, having largest hop distance and simultaneously largest ACC is carried out iteratively. Node degrees of selected node pairs play a significant role in APL reduction of the network. Therefore, at each iteration, link addition is carried out corresponding to the pair which produces maximum APL reduction. Best node pair selection for link addition at each iteration is enumerated in Algorithm 1 in steps 7 to 9. As illustrated in Figure 3, the hop distance is large between nodes 12 and 48. But these nodes do not exhibit higher node degrees. Therefore, link addition between node pairs, 12 and 48 is not helpful in significant APL reduction. Hence, during the selection of node pairs, a virtual shortcut is created iteratively between all selected pair of nodes having maximum hop distance. By analyzing the node pair which gives maximum APL reduction, a real link is established between that pair. It may be noted that a frequency selective method of link addition is used to create small world characteristics herein. Finally, addition of new links is stopped after 12

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a fixed number of iterations when APL reduction attains an optimal value. The frequency selective method is discussed in detail in the ensuing section.

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2.2.3. Introduction of Frequency Selective Small world Characteristics Small world characteristics are introduced in this work using a frequency selective link addition method. This frequency selective link addition method utilizes H-nodes. H-nodes enable connectivity between sensors that are separated by large distances as the frequency of transmission increases. H-nodes are sensor nodes with multi-radios [31, 33, 35, 44] mounted on them. These multi radios enable creation of long distance communication links at higher frequency bands since they are high power transmission radios. Radio range extension leads creation of undesired links in the network. These undesired links may lead to increased power consumption and bandwidth requirement of the network. To avoid the undesired links, different frequencies are used for each long link in a certain frequency band. One set of normal links are created at lower frequency band ( f1 − f2 ). Addition of links to introduce small world phenomena is carried out in another band ( f2 − f3 ). The adjacent intervals of frequency lead to maximum utilization of limited bandwidth assigned to WSN. The frequency selective introduction of small world phenomena is enumerated in Algorithm 2.

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Algorithm 2 Algorithm for Introduction of Frequency Selective Small World Phenomena 1: Nodes and anchors are mounted with variable radio range transceivers 2: Operating frequency band of WSN is ( f1 − f3 ) 3: Normal links with shorter radio range (< 25 % of the network diameter) [23], are created in the frequency band ( f1 − f2 ), where f2 < f3 4: Long links for small world phenomena introduction are created in the frequency band, ( f2 − f3 ) 5: First, node pair (i∗m , j∗m ), is computed using Equation (16) 6: Starting from the smallest frequency in the band ( f2 − f3 ), links are created iteratively 7: This process continues for all the long links to be created for small world characteristics introduction

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2.2.4. Small World Performance Evaluation of the Proposed Method The performance of proposed iterative method is compared with conventional non-iterative method in terms of APL and ACC. In the first method small world characteristics are introduced simultaneously into the network. Here, this method is non-iterative. In the second method, links in second band ( f2 − f3 ) are created in an iterative manner. Both methods are shown in Figure 3. Figure 3(a) illustrates the first method, in which links are created simultaneously. Links in second band ( f2 − f3 ) are indicated in blue and are created at same time and are numbered accordingly. Figure 3(b) represents the proposed method in which links are created iteratively. Links are created at different time instants in iterative method they are indicated with different color and their sequence is in ascending order. Figure 4 illustrates reduction in APL and ACC for both the non-iterative and iterative methods. It illustrates a significant reduction in APL using proposed method in comparison to non-iterative method, while indicates a constant ACC for both the methods.

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3. Localization in Small World WSN using Constrained Least Squares Method

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In this Section, significance of small world characteristics in reducing localization error is discussed first. Subsequently, localization over SW-WSN using constrained least squares (CLS) method is also described.

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3.1. Significance of Small World Phenomena in Reducing Localization Error Addition of new long links introduces small world characteristics and confines the search area for an unknown node in the network. Reduction in search area by introduction of small world characteristics is shown by considering a network with dimensions L × W, having t anchors and a single node whose location needs to be determined. The radio ranges of t anchors are r1 , r2 ,...,rt respectively. The region (A), over which the unknown node searched is given by A=

πr12

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πr22

\

πr32

\

...

\

πrζ2

ζ \ = (πri2 ).

(18)

i=1

ζ represents the set of anchors in which the unknown node lies. t represents the set of all anchors in the network. ζ is a subset of anchor set t (ζ ⊂ t). Identification 14

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ζ \ lim (πri2 ) → 0.

ζ→∞

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of anchors which are in range of unknown node is carried out using the RSS measurement model given in Equation (15). If the RSS value received from an anchor is above a certain threshold then that anchor is connected to the unknown node. Region (A) is the common area covered by the anchors in which the node can potentially lie. It is the intersection of areas covered by all anchors connected to a particular unknown node. Hence, the search area involved in the location estimation of unknown nodes is reduced by introduction of small world characteristics in the network and is given by (19)

Figure 5 (see page 36) goes here.

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Equations (18) and (19) illustrate the confinement of search area for an unknown node, which leads to reduce the search area for localization of unknown node. Application of the constraints (considering three anchors at a time) ensures that the search area turns out to be a rectangle. In case of conventional WSN it may be noted that the node whose location needs to be determined is searched over an area which is the sum of region 1 and region 2. However, the search area to locate the unknown node in case of SW-WSN reduces to region 2 only, as can be seen from Figure 5. In Figure 5, a network with dimensions L × W is considered. There are four anchors A1, A2, A3 and A4 in the network with radio range R1 , R2 , R3 and R4 respectively. In case of conventional WSN, an unknown node connected to anchors A1, A2, and A3 is searched over a large area formed by region 1 and region 2 as illustrated in Figure 5. Searching area of unknown node is the intersection region formed by the anchors. In case of SW-WSN, unknown node is additionally connected to anchor A4 and confining the search area to region 2 only. New connection with anchor A4 can be considered as small world phenomena introduction in the network.

Additionally, new long links reduce the error propagation in the network and improve the localization accuracy. In conventional WSN, node localization is carried out using estimated positions of other nodes. These estimated positions are already erroneous. Therefore, propagation of error happens during node localization [45]. This results in a large localization error for nodes which are at farther 15

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distance from the anchors. It must be noted here that, introduction of small world phenomena helps in reducing localization error propagation and hence the localization error.

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3.2. Basic Framework for Localization in Small World WSN SW-WSN is modeled as an undirected graph G(Z, E, P). Here, Z and E represent the set of sensor nodes and edges between each pair of sensor nodes respectively. Element of adjacency matrix, mi j = 1 indicate the existence of edges between ith and jth nodes. P denotes the signal strength between sensor nodes and h iT anchors. Consider t number of anchors in the set A, where a j ∈ R2 . Also xaj , yaj iT h are known ∀ j = 1, 2, ..., t. The problem of localization is to find xi = xni , yin , ∀i = 1, 2, ..., r, with the help of the set A and the inter-nodal distances. Inter-nodal (Euclidean) distance between sender and receiver is computed using a model-free approach based on machine learning techniques [12]. The use of machine learning techniques do not rely on any fixed signal propagation model and therefore called as model-free localization techniques. Model-free methods overcome the model dependency issue of existing range-based algorithms. Rangebased algorithms use sample data for the calibration of parameters leading to large localization error. Model-free techniques learn from the existing anchors whose positions are known. Therefore, for a network having at least three anchors with known coordinates alongo with the corresponding received signal strength (RSS) n vectors S j | j = 1, 2, ..., t from t anchors, the location coordinates of remaining r unknown nodes can be computed easily. The samples received by an anchor are used to determine the position of the unknown node. It must be noted that the samples received from other anchors are used as training data. The predictor variables are functions of the RSS and the outcome variable is a function of the distance. A regression model is built for each anchor to infer the relation y (log d) ∼ x (log RS S , log (−log RS S )) .

(20)

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In Equation (20), y is the outcome variable and x is the collection of predictor variables. RS S ∈ (0, 1] is normalized by transmitter power. Equation (20) allows each anchor to have a different set of regression coefficients leading to different signal models. After fixing the coefficients, the distance from an unknown node to anchor j is calculated as log d j = α j,0 + α j,1 log RS S + α j,2 log (−log RS S ) | {z } | {z } Polynomial path-loss model

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Exponential model

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where, polynomial path-loss model is given by !−η d e Pr (d) = Pt ρ + N(0, σN dB ) d0

(21)

−d2

Pr (d) = Pt e σ f + N(0, τ)

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where the constant −ηe is the path-loss exponent. Exponential path-loss model is given by [46]

(22)

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where, N(0, τ) denotes an independently generated normal random variable with zero mean and standard deviation τ. Weights are assigned to individual path-loss model, using real signal measurements. Euclidean distance is also computed using log-normal shadowing model [12], which is a model-based technique and usually expressed in dB form is given by ! Pr d (dB) = 10 log10 (ρ) − 10ηe log10 + N(0, σN dB ). (23) Pt d0

Distance d between ith unknown node and jth anchor node will be equal to

xi − a j

. Where k·k is £2 norm, which denotes the Euclidean distance between two vectors. The distances measured herein are used in CLS method to compute the location of unknown nodes.

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3.3. Localization in Small World WSN using CLS Method Consider an unknown node that h iT is connected to three or more anchors. The location of unknown node xni , yin can be computed by solving Hx=z

(24)

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as follows [4].

    z =   

 ζ   xa − xa1 yζa − y1a   ζ  " i# ζ  xa − xa2 ya − y2a  x  , x = in H = 2  .. .. y    . . n  ζ ζ−1 ζ ζ−1  xa − xa ya − ya

di12 − diζ2 − (xa1 )2 − (y1a )2 + (xaζ )2 + (yζa )2 di22 − diζ2 − (xa2 )2 − (y2a )2 + (xaζ )2 + (yζa )2 .. .

ζ 2 ζ 2 2 2 di(ζ−1) − diζ2 − (xaζ−1 )2 − (yζ−1 a ) + (xa ) + (ya )

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(25)

      

(26)

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h iT where coordinates of jth anchor in 2D is xaj , yaj , ∀ j = 1, 2, ..., ζ. Inter-nodal distance between ith node and jth anchor is denoted by di j and measured using Equation (23). This formulation performs a search over the entire network and then estimates the location of the unknown node. This complexity can be reduced using probable region which is a rectangular region utilizing the constraints as follows

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Figure 6 (see page 36) goes here.

xcmin = max (xa1 − R1 , xa2 − R2 , xa3 − R3 , xmin )

(27)

= min (xa1 + R1 , xa2 + R2 , xa3 + R3 , xmax ) = min (y1a + R1 , y2a + R2 , y3a + R3 , ymax ).

(29)

ycmin = max (y1a − R1 , y2a − R2 , y3a − R3 , ymin )

xcmax ycmax

(28) (30)

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(·) In the above equations x(·) and y(·) (·) denote the x and y coordinates. Subscripts, min and max denote the minimum and maximum values of the x and y coordinates. Superscripts, “c”, denotes the probable region in which the unknown node lies. xmin , and ymin are the minimum x and y coordinate values of the network. R1 , R2 and R3 are the radio range of the anchors A1, A2 and A3 respectively. The coordinates of the unknown node can be now computed as

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kHx − zk2

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(31)

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Figure 6 illustrates the formation of probable region using three anchors. The probable region is common area covered by the anchors in which the node can potentially lie. Equation (31), is a convex formulation and can be solved using numerical optimization [47]. 3.4. Algorithm Development for Localization in Small World-Wireless Sensor Network Algorithm 3 enumerates the steps for localization of all unknown nodes in SW-WSN. The Algorithm is iterated until convergence or until it reached the predefined number of iterations. In first iteration coordinates of those sensor which 18

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are directly connected to three or more anchors are calculated. These sensors can be treated as a virtual anchor in the next iteration. This step is repeated until the coordinates of all sensor nodes are obtained.

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Algorithm 3 Localization over Small World WSN 1: Input : H matrix and z vector using Equations (25) and (26) respectively at node xi , where xi ∈ X and xi ∈ R2 h iT 2: Output : Global coordinate of node in 2D/3D space, as xi = xni , yin or h iT xni , yin , zin 3: Data: a j ∈ R2 , ∀ j ∈ 1, 2, ..., t, connected to xi and di j , ∀ j ∈ 1, 2, ..., t, connected to xi using RSS measurement using Equations in [12], as log d j = α j,0 + α j,1 log RS S + α j,2 log (−log RS S ) | {z } | {z } Polynomialpath−lossmodel

Pr (dB) Pt

7:

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At individual blind node xi , ∀i ∈ 1, 2, ..., r, perform RSS measurement from all the elements in the set A if Pr received from a j at xi is ≥ γ s then ith node is connected to jth anchor a j are known at xi via transmission through anchors at unknown node, if they are connected Find di j , at xi from the anchors and nodes which has been localized in the network Using all data from step 1 to 4, compute H and z Use CLS method in Equation (31) Localization completed

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or = 10 log10 (ρ) − 10ηe log10 ( dd0 ) + N(0, σN dB )

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4. Performance Evaluation

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Performance evaluation of the proposed method is carried out in simulated conditions, using real field node deployments, and the Intel lab data set. Localization error analysis is carried out to understand the significance of the method in terms of bandwidth requirement, power consumed, and anchor node requirement. Cramer-Rao lower bound analysis is also presented herein.

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4.1. Experimental Conditions and Data sets The experimental conditions used in the performance evaluation are described in this Section. All three scenarios in which the proposed localization method is evaluated namely, simulated conditions, Intel experimental data set, and the real field node deployments are discussed herein.

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4.1.1. Simulation Conditions The network dimension considered for simulation is 40 m × 30 m. The total number of unknown nodes and anchors are 42 and 12 respectively. Nodes can take any location and follow a uniform distribution within the network. The minimum number of anchors required is 12 in case of WSN. In case of SW-WSN, 9 anchors are considered. Small world phenomena is introduced in the network using proposed method as discussed in Algorithm 1. A total of seven links introduced in the network, significantly reduce the APL of the network. An unconstrained and regular links have a length of 9 m.

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4.1.2. Intel Berkeley Lab Experimental Data Set [42] The Intel Berkeley research lab database [42] is used in the experiments on localization. Fifty four mica2dot nodes are randomly deployed in 41 m × 31 m indoor scenario. TinyOS platform is used for collecting the Intel data using the TinyDB in-network query processing system [42]. Forty eight unknown nodes and six anchors are used from the data set. In this case, minimum number of required anchors are six and three in WSN and SW-WSN respectively. A total of eight long links are added to the existing network, which are significantly reduce the APL of the network. Similarly, unconstrained and normal links of length 9 m are also introduced.

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4.1.3. Real Nodes Deployment Conditions Real node deployment is also conducted at IIT Kanpur. A total of 80 NI 3230 WSN motes are deployed over a 33 m × 29 m dimensional area at residential site of IIT Kanpur. The node deployment and their links are illustrated in Figure 7 and 8 for WSN and SW-WSN respectively. For RSS measurement NI WSN gateway 9792 is used. The gateway is connected through a LAN over which the data is collected. The transmitter and receiver antenna gain is 1.5 dBi for NI motes and the gateway. The transmitted power is 10 dBm. For computing the distance, lognormal shadowing model [12], is used. The values of Gt and Gr are 1.5 dBi, λ is 0.12491 m, and Pt is 10 dBm. For the measurement of Pr a total 5 readings are taken between transmitter and receiver. A total of nine long links added into the 20

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Figure 7 (see page 37) goes here.

Figure 8 (see page 37) goes here.

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4.2. Localization Performance Analysis For each of the experimental condition discussed in the aforementioned sections, localization performance analysis is conducted. Figure 9 illustrates radar plots obtained with five parameters, APL, ACC, localization error, power consumption and bandwidth required in case of conventional WSN and SW-WSN. As shortcuts are introduced in WSN, reduction in area is noted. Figure (9a, 9b), Figure (9c, 9d) and Figure (9e, 9f) illustrate the results under simulated conditions, Intel data set and the real nodes deployment respectively. The range of links introduction is 0 to 9. Results corresponding to SW-WSN in Figure 9 are shown with introduction of 7, 8, and 9 links respectively. In case of SW-WSN the area formed is small in comparison to WSN due to the reduction in parameters. Figure 9 illustrates a drastic reduction in localization error and APL in case of SW-WSN. It also indicates an insignificant increase in power consumption and bandwidth requirement over SW-WSN. ACC is almost same in both WSN and SW-WSN.

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4.2.1. Localization Error Analysis The localization error for simulated condition, Intel data sets and real field nodes deployment scenarios are 3.8935 m, 6.4307 m and 8.3272 m respectively in case of regular WSN utilizing model-based approach. These values for SW-WSN utilizing model-based approach are 1.1116 m, 3.0720 m and 3.8790 m respectively. The number of links created in the second frequency band are 7, 8, and 9 respectively. If model-free method is used in similar conditions then the localization error is 3.1236 m, 5.9988 m and 6.4619 m respectively in case of regular WSN. These values for model-free SW-WSN are 0.8871 m, 2.1166 m and 2.3495 m respectively. Constrained least square (CLS) method is used for localization in 21

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both conditions. In Figure 10, variation of localization error with number of shortcuts introduced are illustrated corresponding to all three experimental scenarios. It is illustrated that not all the shortcuts will reduce the localization error significantly. In a simulated network, shortcuts between pair of nodes 1 and 15, 15 and 52, 1 and 37 & 16 and 24 helps in significant reduction of localization error. Over the Intel data set this reduction is obtained using shortcuts between pair of nodes 27 and 53, 14 and 33. It may be noted that these pairs in case of real field nodes deployment are 31 and 74, 4 and 24, 64 and 74, & 5 and 74.

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Figure 10 (see page 38) goes here.

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Tables 2, 3, 4, and 5 list the mean localization error (L.E) and standard deviation (S.D) obtained after introduction of small world characteristics into the network. Results obtained over simulated network using model-based and modelfree RSS measurement techniques are listed in Table 2, and 3 respectively. The values corresponding to real WSN testbed are listed in Table 4, and 5. Results listed in Tables 2, 3, 4, and 5 illustrate a significant improvement in localization accuracy over small world WSN when compared to conventional WSN. Small world WSN results in improved localization accuracy in terms of localization error and standard deviation due to the introduction of additional long links into the network. Results obtained also illustrate an improved localization performance in case of model-free RSS measurement technique when compared to model-based RSS measurement technique under similar conditions. Model-free measurement techniques result in improved localization performance because they use learning approaches which do not rely on any fixed signal propagation model. For instance, localization error and standard deviation in conventional WSN reduces by 0.7699 m and 0.2416 m respectively under simulation conditions (using CLS method) when model-free technique is used over the model-based technique. The same performance measures corresponding to small world WSN (with 7 links) are 0.2245 m and 0.0944 m as noted from Table 2, and 3. Localization performance improvement over real WSN testbed under various conditions are also listed in Table 4, and 5 for model-based and model-free techniques respectively. Table 2 (see page 41) goes here.

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The computation of mean localization error and standard deviation is carried out utilizing constrained least squares (CLS) [21], total least squares (TLS) [48, 49] and multidimensional scaling (MDS) [50, 51, 22] methods. Results obtained using CLS method illustrate the significance of proposed method when compared to other localization methods. CLS method does not rely on multi-hop distance between node pairs and results in better localization performance when compared to TLS and MDS methods.

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Localization error and standard deviation in case of Intel lab data set over conventional WSN are 6.4307 m and 1.9236 m respectively using CLS method when model-based RSS measurement is used. However, these values are 3.0720 m and 0.6832 m respectively when 8 link is added. Similarly, using model-free RSS measurement a 5.9988 m localization error with 1.8327 m standard deviation is obtained over conventional WSN. This result corresponding to small world WSN with 8 long links are 2.1166 m and 0.5837 m respectively.

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4.2.2. Energy Requirement Analysis Computation of energy required is carried out using the model [52, 53] given in Figure 11. Figure 11 illustrates a energy dissipation model used by the transmitter and receiver. The transmitter dissipates energy to perform radio electronics and amplification. The receiver dissipates energy to execute radio electronics only. Free space (d2 power loss) and multi-path fading (d4 power loss) channel models are used herein. If the distance between transmitter and receiver is less than a threshold “d0 ”, then free space ( f s ) model is used. Otherwise, the multi-path (m p ) model is used. Thus, to transmit an k-bit message for a distance “d”, the energy expended by a radio is given by

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   kEe + k fs d2 , d < d0 ET x (k, d) = ET x−e (k) + ET x−a (k, d) =   kEe + km p d4 , d ≥ d0

.

To receive this message the energy expended by radio is given by ERx (k) = ERx−e (k) = kEe .

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Figure 12 shows the variation of required energy for localization with number of long links introduced. Energy requirement in conventional WSN and SW-WSN using proposed algorithm is almost similar. Energy required in sending data grows with transmission distance proportionally to dn where n is 2 to 4 [52]. All the related parameters of this model are given in [52]. Fewer links are created in the proposed algorithm without affecting energy requirement, while the performance of localization error improves significantly. After addition of seven new long links in the network, power required is -15.5784 dBm and -15.6047 dBm in simulated conditions utilizing model-based and model-free approaches. In the Intel data set and real field deployment, after creating eight and nine links respectively power required is -17.0702 dBm, -17.1277 dBm and -14.4583 dBm, -14.5382 dBm respectively in similar conditions. Power required in conventional WSN in the same context are -15.6980dBm, -15.7167 dBm, -17.2644 dBm, -17.2994 dBm and -14.5750 dBm, -14.6553 dBm respectively. It indicates a very small change in energy requirement from conventional WSN to SW-WSN. In case of model-free approach energy requirement is lower as compared to model-based approach. Figure 12 (see page 39) goes here.

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4.2.3. Bandwidth Requirement Analysis

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q 2 M(d) ˆ , where, v is the velocity Links bandwidth [54], is computed as β = v Kγ s ˆ is positioning accuracy which is defined as inverse of variance of of signal. M(d) distance estimation error. K is number of symbols and γ s is signal to noise ratio (SNR). SNR of a signal is related to inter-nodal distance. Therefore bandwidth is related to distance. The computation of bandwidth requirement is carried out over SW-WSN and conventional WSN. The difference between these two is calculated and compared with conventional WSN. This computation results in percentage increase in bandwidth requirement over SW-WSN as compared to conventional WSN. For different topology this value changes as bandwidth requirement depends over number of existing links and their length. In this work, the computation ˆ = 0.01 m2 . The value of β is carried out assuming K = 10, v = 340 m/s, M(d) of γ s changes for different node pairs as this value depends over distance. The transmitted power assumed is 10 dBm. Additive white Gaussian noise (AWGN) noise is assumed at the receiver. Change in bandwidth requirement is also small in SW-WSN when compared to conventional WSN. As shown in Figure 13, if a total of seven links are introduced into conventional WSN, then bandwidth requirement increases by only 2.056 and 2.485 % respectively utilizing model-based and model-free approaches in the simulated network. These changes for the experimental data set (8 links) and real field nodes deployment (9 links) scenarios are 3.918, 4.174 and 2.453, 2.879 % respectively. Hence, bandwidth requirement also does not increase significantly with link addition using the proposed method. Results obtained also illustrate a reduction in bandwidth requirement when a modelfree technique is used.

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4.3. Cramer-Rao Lower Bound Analysis for Node Location Parameters In this Section, Cramer-Rao lower bound [55, 56] analysis for node location parameters in WSN and SW-WSN is presented. Results are obtained utilizing the model-based signal propagation technique. A comparison of CRLB performance analysis for both networks is also illustrated. The parameter vector whose CRLB h iT is to be computed is ϕ = x y . For Gaussian received signal power observations i.e., p ∼ N(µ(ϕ), C(ϕ)), Fisher information matrix in [55] can be expressed as

dA¯

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dA¯

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I(ϕ) = h ∂µ(ϕ) i 1 h h ∂µ(ϕ) iT (32) ∂C(ϕ) 2 i C−1 (ϕ) + tr C−1 (ϕ) ∂ϕ ∂ϕ 2 ∂ϕ where tr(.) denotes the trace of a matrix, which is sum of its diagonal elements. C is the covariance matrix which does not depend on ϕ. Equation (33) shows the h ∂µ(ϕ) i value of ∂ϕ . The Fisher information matrix for parameters of the node location can be finally reduced to Equation (35).  x−x1  Pt Gt Gr λ2  y−y1  Pt Gt Gr λ2    4   d1  8π2 2  d14  8π2 2   y−y2 Pt Gt Gr λ   x−x2 Pt Gt Gr λ  h ∂µ(ϕ) i  d24 8π2 8π2 d24  =  (33)  .. .. ∂ϕ   . .   x−x      4 A¯ Pt Gt G2r λ2 y−y4 A¯ Pt Gt G2r λ2 

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¯ where where σ2a represents variance corresponding to the ath anchor, ∀ a ∈ A, ¯ A is the set of anchors. CRLB is the sum of diagonal elements of the inverse Fisher information matrix [55, 56]. Hence, CRLB for node location parameter can be written as q (34) CRLB(ϕ) = [I−1 ]1,1 + [I−1 ]2,2 . Figure 15 (see page 40) goes here.

In CRLB analysis, four anchors are placed at the boundary of the network and four other anchors are placed inside the region symmetrically. The nodes are placed inside the convex hull of the network. In order to assess minimum 26

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a=1

a a

A¯ P

a=1

 P G G λ2 2   t t r  8π2   P G G λ2 2  t t r 

(x−xa )(y−ya ) σ2a da8

A¯ P

a=1

(y−ya )2 σ2a da8

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 A¯  P (x−xa )2  Pt Gt Gr λ2 2  8π2 σ2a da8 I(ϕ) =  A¯a=1  P (x−xa )(y−ya )  Pt Gt Gr λ2 2 8π2 σ2 d 8

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variance of localization error a CRLB plot is computed as shown in Figure 15. Figure 15a illustrates the CRLB plot for a conventional WSN. A similar CRLB plot for SW-WSN is illustrated in Figure 15b. It can be noted from Figure 15b that the value of the lower bound obtained for a SW-WSN is better when compare to that obtained in a conventional WSN (Figure 15a). A difference plot between the CRLBs of conventional WSN and SW-WSN is computed and plotted in Figure 15c. The ratio of CRLBs obtained between conventional WSN and SW-WSN is computed and plotted in Figure 15d. It must be noted from Figure 15c that the difference value is of the order of 0.591. On the other hand from Figure 15d, it can be noted that the value of CRLB ratio is equal to 1 at maximum. Therefore, it can be concluded that the performance of localization in SW-WSN is reasonably better in comparison to conventional WSN. Since the x- and y-axes denote the dimensions of the network, the minimum value of variance of the localization error is attained at the center of the CRLB analysis plot. It must be noted here that anchors are placed symmetrically at the boundary and inside the network to compute minimum variance of the localization error. However, placing the anchors randomly inside the network may increase localization error. 5. Conclusion

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In this paper, a novel constrained iterative average path length reduction algorithm is proposed to introduce small world characteristics into the conventional WSN. An algorithm for introduction of frequency selective small world phenomena is also discussed to prune links that can lead to loss of small world characteristics. Frequency selective algorithm uses two different frequency bands. The first band is used for creating conventional links and the second band for link addition in an iterative manner for maintaining the small world characteristics of the network. Performance of constrained iterative average path length reduction algorithm is also compared with non-iterative link addition methods. The proposed method provides significant reduction in average path length and maintains 27

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high average clustering coefficient when compared to non-iterative methods. Constrained least squares node localization over the Small world WSN created using the constrained iterative average path length method yields improved localization performance when compared to node localization over conventional WSN. Performance improvements are noted in terms of localization error, power consumption, bandwidth and anchors required for the node localization. Cramer-Rao lower bound analysis for node location parameters is also used to illustrate the significance of the proposed method. Future work will involve development of localization methods that are designed to utilize the power of small world WSN in the context of data aggregation and node fault detection.

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References

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[3] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero III, R. L. Moses, N. S. Correal, Locating the nodes: cooperative localization in wireless sensor networks, Signal Processing Magazine, IEEE 22 (4) (2005) 54–69. [4] G. Mao, B. Fidan, B. D. Anderson, Wireless sensor network localization techniques, Computer networks 51 (10) (2007) 2529–2553.

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[56] H. L. Van Trees, Detection, estimation, and modulation theory, optimum array processing, John Wiley & Sons, 2004.

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Figure 1: Data transmission from node A to node K in (a) WSN via nodes B, C, D, F, H, and J (b) Small world WSN via node P only. Red dots and black lines are representing the sensor nodes and links among them respectively

Does link exist in (f2-f3) band ?

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Figure 2: Block diagram for converting conventional WSN into small world WSN using proposed constrained iterative average path length reduction method

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Real Data Model−based Real Data Model−free Intel Data Model−based Intel Data Model−free Simulated Model−based Simulation Model−free

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Figure 9: Radar plots illustrating the variation of localization error with the parameters, average path length, average clustering coefficient, bandwidth required and power required in case of model-based WSN and SW-WSN. A smaller area in the radar plot indicates an optimal set of parameters. Radar plots for WSN and small world WSN for simulation conditions is shown in (a) and (b). Radar plots obtained on Intel Berkeley data sets are shown in (c) and (d), while plots obtained on real field deployment are shown in (e) and (f).

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WSN (Simulated) SW−WSN (Simulated) WSN (Intel data) SW−WSN (Intel data) WSN (Real data) SW−WSN (Real data)

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Figure 14: The effect of small world phenomena on anchors required for localization utilizing model-based technique. Results obtained for simulated conditions (red), Intel Berkeley lab data set (magenta), and real field deployment (blue) are shown.

(b)

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Figure 15: CRLB plots for (a) Conventional WSN and (b) Small world WSN using RSS. Network dimensions are 30 m × 30 m. The difference CRLB plot between WSN and small world WSN is illustrated in (c). The CRLB ratio plot in WSN and small world WSN is illustrated in (d)

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Table 1: Tabulation of average path length (APL) and average clustering coefficient (ACC) for the networks shown in Figure 1 (NF-1), Intel Berkeley lab data set [42], and real field node deployment

Real Field Deployment

Intel Lab Data Set

NF-1

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Network WSN SW-WSN WSN SW-WSN WSN characteristic APL 3.0277 2.5692 3.4752 2.5353 4.0225 ACC 0.4637 0.4261 0.6256 0.5978 0.6632

2.7608 0.6426

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TLS Method

S.D (m)

MDS Method

L.E (m) S.D (m) L.E (m) 4.0167 3.6612 3.1268 2.8764 2.5678 2.3123 1.8997 1.4561

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1.4758 1.3234 1.2764 1.0134 0.7246 0.5764 0.3562 0.3432

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Links Introduced L.E (m) 0 3.8935 1 3.2736 2 3.0312 3 2.3458 4 1.7900 5 1.6221 6 1.1116 7 1.1116

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Table 2: Variation in localization error (L.E) and standard deviation (S.D) computed using CLS, TLS and MDS localization methods. Results are enumerated over simulated WSN testbed. RSS measurement for distance estimation between pair of nodes utilizes model-based technique.

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1.5102 1.3412 1.1738 1.0069 0.9876 0.7243 0.6633 0.5703

6.1106 5.3213 4.7864 4.2345 4.1011 3.4566 3.1234 2.9867

S.D (m) 2.1302 1.8678 1.5967 1.4435 1.4200 1.2788 1.0011 0.8666

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CLS Method 1.2342 1.1034 0.8812 0.7234 0.6112 0.5890 0.3674 0.2488

L.E (m) S.D (m) L.E (m) 3.4561 3.1256 2.7833 2.4365 1.9764 1.5453 1.3234 1.1106

1.3120 1.2789 1.0146 0.8767 0.7123 0.6132 0.4100 0.3202

5.8780 5.1236 4.6413 4.5432 3.9877 2.5780 2.5611 2.2346

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S.D (m)

MDS Method

S.D (m) 2.0004 1.8156 1.5854 1.4742 1.2412 1.0046 1.0076 0.7886

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Table 3: Variation in localization error (L.E) and standard deviation (S.D) computed using CLS, TLS and MDS localization methods. Results are enumerated over simulated WSN testbed. RSS measurement for distance estimation between pair of nodes utilizes model-free technique.

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Table 4: Variation in localization error (L.E) and standard deviation (S.D) computed using CLS, TLS and MDS localization methods. Results are enumerated over real WSN testbed. RSS measurement for distance estimation between pair of nodes utilizes model-based technique.

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Links Introduced L.E (m) 0 8.3272 1 8.3272 2 7.4675 3 7.4676 4 6.5732 5 6.0732 6 6.0732 7 4.2790 8 4.2790 9 3.8790

S.D (m) 2.6716 2.6134 2.3642 2.2540 2.0210 1.8022 1.7656 1.3540 1.1789 1.0222

TLS Method L.E (m) S.D (m) 9.0089 8.6542 7.8466 7.0234 6.6056 6.1233 5.8490 5.2466 4.9088 4.3467

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MDS Method L.E (m)

S.D (m)

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CLS Method

TLS Method

S.D (m)

6.9604 6.4234 6.0040 6.0130 5.3000 4.9286 4.4060 3.8964 3.4210 2.7498

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L.E (m) S.D (m) L.E (m)

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Table 5: Variation in localization error (L.E) and standard deviation (S.D) computed using CLS, TLS and MDS localization methods. Results are enumerated over real WSN testbed. RSS measurement for distance estimation between pair of nodes utilizes model-free technique.

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2.1067 1.9604 1.8890 1.7428 1.6244 1.5808 1.4860 1.2721 1.0837 0.8721

8.6842 8.5744 7.4368 6.8654 5.3266 5.2100 4.8880 4.1057 4.0085 3.3478

S.D (m) 2.8678 2.7864 2.1348 1.7864 1.4689 1.3460 1.2206 1.0644 0.9668 0.8602

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Om Jee Pandey is currently working toward the Ph.D. degree at the Department of Electrical Engineering, Indian Institute of Technology Kanpur (IITK), India. He is currently working with Sensor Networks lab at Electrical Engineering Department at IIT Kanpur. His research interest lies in the signal processing for wireless networks with specific focus on robust sensor node localization and tracking over wireless ad-hoc networks. He also works on related areas like vehicular tracking, data fusion, data aggregation, autonomous node fault detection in wireless sensor networks. Rajesh M. Hegde is a Professor and P K Kelkar Research Fellow (09-12) with the Department of Electrical Engineering at Indian Institute of Technology Kanpur (IITK). His current areas of research interest include multimedia signal processing, multimicrophone speech processing, pervasive multimedia computing, ICT for socially relevant applications in the Indian context, and applications of signal processing in wireless networks with specific focus on emergency response and transportation applications. He has also worked on NSF funded projects on ICT and mobile applications at the University of California San Diego, USA, where he was a researcher and lecturer in the Department of Electrical and Computer Engineering between 2005-2008. He is also a member of the National working group of ITU-T (NWG-16) on developing multimedia applications. Additional biographic information can be found at the URL: http://home.iitk.ac.in/rhegde

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