Fault-Resilient Distributed Detection and Estimation Over

1758 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 17, NO. 3, SEPTEMBER 2020

Fault-Resilient Distributed Detection and EstimationOver a SW-WSN Using LCMV Beamforming

Om Jee Pandey , Member, IEEE, Ved Gautam, Ha H. Nguyen , Senior Member, IEEE,

Mahendra K. Shukla , Member, IEEE, and Rajesh M. Hegde, Senior Member, IEEE

Abstract—Recent technology advancement has resulted in opti-mistic view toward the practicability of wireless sensor networks(WSNs) in the context of Internet of Things (IoT) and CyberPhysical Systems (CPS). However, to realize their full benefits ina broad range of commercial applications, there are still manytechnical hitches that need to be overcome. In this paper, weaddress three vital technical issues in a WSN: (1) distributed eventdetection, (2) distributed parameter estimation, and (3) network’srobustness. We make use of a recent development in socialnetworks called small world characteristics and propose novelfault-resilient distributed detection and estimation methods overa small world WSN (SW-WSN). In particular, a small worldWSN has been developed by mounting antenna arrays on sen-sor nodes for the purpose of beamforming. A low-complexityoptimization problem for beamforming is formulated by intro-ducing a new parameter Flow between node pairs. Additionally,a new beamforming algorithm is also proposed which optimizesthis flow, leading to optimal beam parameters. The proposedmethod yields a lower average path length and a higher averageclustering coefficient of the network. Experiments are conductedusing simulations and real node deployments over a WSN testbed.Analysis and experimental results obtained demonstrate that theproposed SW-WSN model achieves faster convergence rates forboth distributed detection and distributed estimation while beingresilient to node failures when compared to results obtained usingstate-of-the-art methods.

Index Terms—Small world wireless sensor networks (SW-WSNs), small world characteristics (SWC), beamforming, dis-tributed event detection, parameter estimation, robustness.

I. INTRODUCTION

W IRELESS Sensor Networks (WSNs) are responsible forvarious sensing and control tasks in Internet of Things

(IoT), Cyber Physical Systems (CPS), and context-aware

Manuscript received August 26, 2019; revised November 20, 2019and February 21, 2020; accepted April 15, 2020. Date of publicationApril 21, 2020; date of current version September 9, 2020. This work wassupported in part by a research grant from Natural Sciences and EngineeringResearch Council of Canada (NSERC). The associate editor coordinatingthe review of this article and approving it for publication was J.-H. Cho.(Corresponding author: Om Jee Pandey.)

Om Jee Pandey is with the Department of Electronics and CommunicationEngineering, School of Engineering and Applied Sciences, SRM UniversityAP, Amaravati 522502, India (e-mail: [email protected]).

Ved Gautam is with the Department of Products, Urvija AI Private Ltd.,Bengaluru 560061, India (e-mail: [email protected]).

Ha H. Nguyen and Mahendra K. Shukla are with the Department ofElectrical and Computer Engineering, University of Saskatchewan, Saskatoon,SK S7N 5A9, Canada (e-mail: [email protected]; [email protected]).

Rajesh M. Hegde is with the Department of Electrical Engineering,Indian Institute of Technology Kanpur, Kanpur 208016, India (e-mail:[email protected]).

Digital Object Identifier 10.1109/TNSM.2020.2988994

pervasive systems [1], [2]. WSNs involve measurement andcomputational units distributed geographically to monitor theenvironment in their proximity. WSNs serve as a bridgebetween physical and cyber worlds and find applications inindustrial automation [2], smart homes [3], smart agricul-ture [4], video surveillance [4], traffic monitoring [5], smarthealthcare [6], and smart energy grids [7].

Many applications of WSNs involve distributed signal pro-cessing [8], [9], including soil moisture estimation in afarm, forest fire detection, industrial monitoring, wirelessmultimedia transmission, and noise level estimation in adense populated area. Therefore, distributed signal process-ing using WSNs has received much attention in recent years[10], [11]. The two main divisions of distributed processingare distributed detection and distributed estimation [12]–[14].Distributed detection [15], [16] involves consensus in thenetwork for an event detection in a distributed manner. It uti-lizes the data of neighboring nodes to reach consensus in thenetwork. Similarly, distributed estimation [17], [18] involvesestimation of a vector of parameters at every node using mea-surements of the neighboring nodes. Distributed estimationschemes are classified into incremental and diffusion meth-ods. The requirement of cyclic paths in the network renders theincremental method in many practical scenarios useless. Thediffusion method, on the other hand, allows communication ofa node with its neighbors, thus practically more feasible. Thispaper is concerned only with the diffusion estimation method.

Along with distributed processing, reliability is anotherimportant aspect of WSNs [13], [19], [20]. Critical applica-tions of WSNs demand the networks to be reliable. Reliabilityof a network greatly depends on the robustness of the networkto node failure. Different metrics such as k-connectivity andpartial k-connectivity have been proposed in the literature toasses robustness of a network [21], [22].

Numerous methods have been proposed to address the afore-mentioned problems in WSNs. In [23], iterative distributeddetection is proposed to improve consensus for event detec-tion. In [24]–[29], it has been found that the network topologyhas a major influence on the performance of distributed pro-cessing. The more “well connected” the network is, the betterits consensus performance can be achieved. In [24], [30], [31],a systematic study is performed to analyze performance ofthe network topology in distributed estimation. These methodsinvolve introduction of small world phenomena to improve thenetwork performance. Small world phenomena were discov-ered in social networks by Stanley Milgram [32]. It revealed

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PANDEY et al.: FAULT-RESILIENT DISTRIBUTED DETECTION AND ESTIMATION OVER SW-WSN USING LCMV BEAMFORMING 1759

that the human society is a small world network havingshort path lengths. Watts and Strogatz in [33] demonstratedhow rewiring a few links in a regular graph can trans-form the graph into a small world graph. This procedureis widely used in wireless networks [34], [35] to improvethe network performance on robustness [19], distributedprocessing [24], [25], latency [36], [37], localization [38],energy-efficiency [39], [40], time synchronization [41], datagathering [42], and required bandwidth [35].

A. Motivation

Centralized event detection and parameter estimationpresent several challenges over a WSN. For instance, theexchange of a large amount of information in a multi-hopfashion leads to reduced network lifetime and poor band-width utilization. It also reduces the network performance withrespect to decision latency and robustness to the node failure.Distributed detection and estimation over a WSN improvesthe network performance in various contexts by only utilizingthe data of neighboring nodes [11], [16]. The performanceof distributed detection and estimation highly depends on thenetwork topology [24]–[31].

Small world characteristics (SWC) can be introduced in aWSN, leading to a small world WSN (SW-WSN). A SW-WSNis a well connected network with reduced hop counts betweennode pairs. The well-connected property of a SW-WSN leadsto spectral phase transition phenomena and results in improveddistributed detection and estimation performance [24]. In theliterature, most of the works utilize random links to cre-ate a SW-WSN without considering its practicability in realtime applications. Introduction of random links also results inpoor network performance. To address this issue, the authorsin [43], [44] have recently developed SW-WSNs via differ-ent beamforming techniques, which are more realistic thanadding random links. Motivated by the works in [43], [44],in this paper we propose a novel ranking scheme basedon the structural importance of the sensor nodes to developSW-WSNs using Linearly-Constrained Minimum Variance(LCMV) beamforming [45], [46]. LCMV beamforming is atechnique in which the radiation pattern of antenna elementsmounted on a sensor node can be made directional. In general,the use of beamforming methods leads to reduced neigh-borhood connectivity within the network. It also increasesthe development complexity of a SW-WSN and reduces thenetwork performance in terms of robustness to node failure.However, the use of LCMV beamforming results in desiredside lobes, leading to higher network connectivity. Hence,the primary objective of this work is to develop and analyzea low-complexity SW-WSN using LCMV beamforming forfault-resilient distributed detection and estimation.

B. Contributions

The main contributions of this paper are as follows. Anovel method for the development of a SW-WSN usingLCMV beamforming is first proposed. The method controlsthe antenna gains in various directions leading to the optimal

TABLE ILIST OF USED PARAMETERS/SYMBOLS

SWC. A method to compute optimal beamforming parame-ters for individual nodes is also developed utilizing trafficflow optimization. The method of traffic flow optimizationuses the parameter Flow, which defines the data flow betweentwo selected nodes within the network. The computation ofFlow between node pairs is performed using nodes between-ness and centrality measures. The parameter Flow is furtherused to reduce the complexity of SW-WSN development.Subsequently, novel distributed detection and estimation algo-rithms are proposed over the developed SW-WSN. Extensiveinvestigation into the distributed detection, parameter estima-tion, and network robustness over a real WSN testbed isalso carried out. Detailed comparison with the state-of-the-art methods is also presented to highlight the advantages ofthe proposed methods.

C. Notations

The parameters and symbols used in this paper are listedand explained in Table I.



D. Organization of the Paper

The rest of the paper is organized as follows. Section IIdiscusses related research. Section III presents concepts andterminologies involved in describing a SW-WSN. Section IVdescribes the network model and problem formulations. Theproposed methods of distributed detection and estimationalong with corresponding algorithm development are describedin Section V. Section VI presents performance evaluation.Concluding remarks and future work are given in Section VII.

II. RELATED RESEARCH

In recent years, with the emergence of wireless sensornetworks and ad hoc networks, research on distributed signalprocessing algorithms has drawn significant attention (see [14],[16] and references therein). In an earlier work [47], theproblem of finding the maximum likelihood estimator of acommonly observed model based on data collected by a sen-sor network is addressed. First, the authors propose an iterativealgorithm to relax the need of complete data sharing. Then, amethod is proposed where the local data of a sensor is used toobtain a sub-optimal estimate. In [48], bandwidth-constraineddistributed estimation is considered where quantization is anessential part of the estimation process. Efficient distributedparameter estimation methods are developed in [49] based onthe distributed sequential Bayesian estimation method. In [50],distributed parameter estimation in WSNs is performed witha total bit rate constraint. In [51], distributed estimation inlarge WSNs via a locally optimum approach is proposed.Subsequently, in [52] the authors examine new versions ofthe diffusion least-mean squares (LMS) algorithm, namelyAdapt-then-Combine (ATC) and Combine-then-Adapt (CTA).

In [28], the authors investigate the effect of networktopology on performance of the ATC diffusion LMS estima-tion algorithm. Results obtained illustrate that the estimationperformance greatly depends on the network topological prop-erties. Further, in [53] the authors illustrate that inferring thestructure of the graph characterizing the statistical dependen-cies among the observed data can provide vital information forsensor network topology development. Moreover, the authorsfind a network topology that minimizes the energy require-ment to achieve the consensus. They illustrate that the networktopology has a very crucial role in the design of an efficientsensor network. In a recent work [54], the authors investigatethe problem of non-fragile estimation for a class of complexnetworks with switching topologies and quantization effects.

In the context of distributed detection, various works havealso been carried out. In an earlier work [55], the detectionproblem is addressed jointly with system-resource constraints.In [24], an algorithm for ultrafast consensus is proposedwhich studies the convergence of the differential equationx (t) = Lx (t), where x(t) is a decision variable and L isthe Laplacian of a communication graph of agents. In [26],an iterative detection procedure is proposed. The goal of theiterative algorithm is for every sensor state to converge to theglobal average local log likelihood ratios (LLRs). In [56],a consensus about the event is obtained via collaborativeexchange of data among the sensors over a communication

network. In [57], opposite to the conventional method whichutilizes a bank of fixed parallel access channels for sendingthe observed data to the fusion center, the authors explore thepossibility of employing a common multiple access channel,which drastically reduces the required bandwidth and detec-tion delay. Further, the detection algorithms proposed in [58]use the diffusion LMS and recursive least squares (RLS) algo-rithms to obtain a parameter estimate which is further used toexecute a hypothesis test.

Regarding detection problems in WSNs, the network topol-ogy also plays an important role. For instance, in [25], theauthors address the issue of graph connectivity. They spec-ify the connectivity pattern among sensor nodes leading toefficient detection. For this, the authors utilize small worldnetwork engines which allows fast convergence for the dis-tributed detection algorithm. In [26], the authors use the theoryfrom small world networks in designing the network topology.The results illustrate that the consensus is obtained with mini-mal communications and processing cost. In [27], the authorspresent the optimal topology design problem for distributedinference in sensor networks. A topology specifies the graphthat decides the connection established among the sensors.In [29], the authors show that a more fast learning rate can beobtained in a well-connected network.

Robustness toward node failure is another challengingproblem in a WSN. Strong robustness enhances the networklifetime and quality of service. In order to address the problemof robustness, various methods have also been investigated inthe literature. In [21], a method to place redundant sensornodes to establish multi-connectivity is proposed. This workexplores how to add as few nodes as possible to a sensornetwork such that the resulting network is k-connected or par-tially k-connected. In [22], the authors propose a resiliencefactor to measure the network level of robustness and pro-tection against targeted attacks. In addition, they proposedstrategies to improve resilience by simple alterations in thenetwork topology. Recently, in [20] the authors propose arobust wireless sensor and actuator network to maintain thecontrol stability of multiple plants over the spatial-temporalchanges of wireless networks.

Apart from the methods discussed above which deal withestimation and detection problems separately, various meth-ods are proposed in literature which solve these problemsjointly. For instance, in [8] the authors proposed a hid-den Markov random field (HMRF) framework for distributeddetection and estimation in sensor networks. In this work,the authors proposed novel algorithms for distributed detec-tion and estimation of the hidden random field from thenoisy measurements. The proposed methods are less com-plex and applicable to several sensing environments, where thenodes communicate only with their neighbors. However, theproposed methods are too simplistic for many applications andtherefore development of more general process models needto be investigated. In [53], [59], the authors illustrated thatthe topology of a network plays a significant role in determin-ing performance of distributed algorithms in terms of energyexpenditure and latency. They proposed a method to optimizethe network topology in order to minimize energy consumption



or to match the graph describing the statistical dependenciesamong the variables observed by the nodes. Recently, in [16] ajoint problem for detection, estimation, and tracking has beeninvestigated in the context of CPS. In this work, a distributedconsensus algorithm for sparse events detection in CPS isproposed. The devices in a CPS compute the LLR from localobservation via a consensus approach to iteratively optimizethe consensus LLRs for the whole CPS system. However, thecomplexity of the method is high due to the iterative processused to obtain the global decision.

III. SMALL WORLD—WIRELESS SENSOR NETWORKS:CHARACTERISTICS AND DEFINITIONS

Small world phenomena were first observed in [32] in socialconnectivity of people. This work provides a theory of “sixdegrees of separation”. In [33], small world phenomena havebeen investigated in graphs. In [34], [37]–[39], [42]–[44],small world characteristics (SWC) are observed in wirelessnetworks. In general, small world WSNs (SW-WSNs) are char-acterized by low Average Path Length (APL) and high AverageClustering Coefficient (ACC). For an undirected WSN with Nnodes, representing a regular network, the APL is given byLWSN = 1

N (N−1)

∑Ni �=j di ,j , where the sum is over all pairs

of distinct nodes and di ,j is the length of the shortest pathbetween nodes i and j. On the other hand, in a WSN havingSWC the APL, LSW-WSN, between two randomly selectednodes grows proportionally to the logarithm of the number ofnodes in the network [33], i.e., LSW-WSN ∝ ln(N ). The APLfor a network with random link addition [60] is given by,

Lrand(N ,Ł) ∼N 1/d

Kf (NKŁ), (1)

where f is a universal scaling function and is given by

f (a) =

{const, if a � 1ln(a)/a, if a � 1

and each node has at least K neighbors. The probability oflink addition [33], [60], denoted by Ł, is given by

Ł =# links added

# possible links−# existing links. (2)

From the above equations it is noted that if Ł is small (Ł → 0),the network will have the property of a regular network. Onthe other hand, if Ł is large (Ł → 1), then the networkwill have random network characteristics. A general SW-WSNcorresponds to Ł in between 0 and 1.

WSNs are spatial graphs, where links are created usingradio connectivity. There is a limit on the radio range of asensor node, hence in such networks the long-range connec-tions are generally absent. Thus, these networks are clusteredbut they do not experience small world phenomena. In [60],it is illustrated that the average distance between nodes,l(N, Ł), decreases drastically as long as Ł is non-zero. ForŁ = 0, a linear chain of sites is obtained so that l(N , 0) =N (N+2K−2)4K (N−1)

∼ N4K grows like N. For Ł = 1, l(N, 1) grows

like ln(N)/ln(2K − 1).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 clus-ter is quantified by the clustering coefficient (CC). To be morespecific, CC for a node i in the network is defined as [60]Ci =

2eiki (ki−1)

, where ki is the total number of neighbors ofnode i and ei is the total links between neighbors of node i.ACC of WSNs is given by C = 1

N

∑Ni=1 Ci . Therefore, a

general SW-WSN exhibits low APL and high ACC.

IV. NETWORK MODEL AND PROBLEM FORMULATION

In this section, the network model used for the develop-ment of small world wireless sensor networks is described first.Subsequently, problem formulations for distributed detectionand distributed estimation are presented.

A. Network Model

In this work, a WSN is considered as a graph representedby G = (N , E ,P,B,L). Here, N represents the total numberof sensor nodes {s1, s2, . . . , sN } in the network. E indi-cates a set of M conventional links {lm = (im , jm ), m =1, 2, . . . ,M }, where, (i , j ) ∈ E whenever sensor node si cancommunicate with sensor node sj . P is the set of received sig-nal strengths (RSSs) {γ12, γ13, . . . , γN (N−1)} between nodepairs. The node pairs can communicate with each other whenthe corresponding RSS, γij , lies above or equal to a cer-tain threshold γ, i.e., γij ≥ γ. B denotes the set of Bnodes {s1, s2, . . . , sB} used for beamforming for the devel-opment of a SW-WSN. L represents the set of T new links{lt = (it , jt ), t = 1, 2, . . . ,T}, introduced in the network,where L /∈ E . In the case of a conventional WSN, B andL are null sets {∅}. In this work, we have mainly focusedon a static network, however, the network graph G can beconsidered as both static and dynamic networks. In case ofa dynamic network, graph G can be recognized as snap-shotseries of static graphs G1,G2,G3, . . . ,GT , at correspondingtime instants [61]. Moreover, it is assumed that the changein network topology is slow enough so that the time-varyingsensor network graph G can be recognized as snap-shotseries of static graphs at corresponding time instants for thedevelopment of a SW-WSN. All sensor nodes are uniformlydistributed over a 2-dimensional geographical area of lengthL and width W. Sensor nodes provide information on senseddata, observed phenomena, list of its neighbors, and distancesto adjacent nodes at every time instant. An equal cost for allcommunication links, i.e., c(li ) = c(lj ), ∀ li , lj ∈ {E ,L} isconsidered. All nodes are mounted with antenna arrays and arecapable of forming a desired beam pattern. Initially, all nodestransmit using omnidirectional beams with the radio range r0.Subsequently, a set of B nodes is selected for beamforming,which forms long directional links.

B. Problem Formulations

In this section, problem formulation of distributed detec-tion is discussed first. Subsequently, problem formulation fordistributed parameter estimation is described.

1) Detection Problem: The problem of distributed detec-tion considered in this paper is a binary hypothesis testingproblem [23], [25]. The state of the environment takes one of



Fig. 1. The model of a WSN consisting of multiple-antennas mountedsensor nodes. Sensor nodes sense the environment within their proximity.Conventional links are red. New links created using directional beamformingare illustrated in green.

two possible hypotheses, null hypothesis H0 and alternativehypothesis H1. Conventionally, H0 indicates that the target isabsent, while H1 indicates that the target is present. The truestate H of the environment is observed by N nodes, which col-lect N real observations, o = (o1, o2, . . . , oN ), as illustratedin Fig. 1. Node observations are considered to be indepen-dent and identically distributed with known conditional densityf (o|Hi ), i = 0, 1. The Gaussian shift-in-mean models for theobservations at sensor node sn under H0 and H1 are given by

sn =

{on = μ1 + zn : H0

on = μ2 + zn : H1.(3)

In the above equation, μ1 and μ2 are the signal means underH0 and H1, respectively, zn is Gaussian noise with zero meanand variance σ2, i.e., zn ∼ N (0, σ2). The goal of distributeddetection is for all nodes to reach a global common decision Habout the true state H. It is considered that the statistics storedat each sensor node are updated in an iterative manner to reacha consensus about the global statistics based on informationexchange over the network. The detection performance ismeasured by the average probability of error, expressed as,

Pe = Pr(H �= H

)= π0 Pe,0+π1 Pe,1, where, (4)

Pe,0 = Pr(H = H1|H0

), (5)

Pe,1 = Pr(H = H0|H1

), (6)

and π0 and π1 are the prior probabilities of H0 and H1, respec-tively. In a centralized network all observations are sent tothe fusion center. The optimum fusion rule can be written asa function of the sum of local log likelihood ratios (LLRs),because the observations are independent [23], [25]. Therefore,

R =N∑

n=1

Λn

Hn=1≷

Hn=0

logπ0π1

= δ, (7)

is the global LLR. In Equation (7), δ is the fusion threshold andΛn is the local LLR at the nth node, given by log

Pr(on |H1)Pr(on |H0)

.

In distributed detection, the statistics stored at each nodeare updated in an iterative manner to reach a consensus aboutthe global statistics. This updating is performed by sharing thecurrent statistics only with neighboring nodes. Initially, nodesperform observations o1, o2, . . . , oN , and then every nodesn ,n = 1, 2, . . . ,N , computes the LLR, Λn = log

Pr(on |H1)Pr(on |H0)

of its measurement on . These LLR values serve as the initialstates for the decision making process. If at a given discretetime instant k, the value stored in the node sn is given byvn [k ], then,

vn [k ] = cn,nvn [k − 1] +N∑

i=1,i �=n

cn,ivi [k − 1], ∀n (8)

where, ci ,j is the weight corresponding to the link (i, j). Notethat, ci ,j = 0, if (i , j ) /∈ {E ,L}. A compact expression in amatrix form can be written as,

v[k ] = Cv[k − 1]. (9)

In Equation (9), v[k] is a N × 1 vector of all current statesand C is a weight matrix corresponding to the existing linksin the network. v[k − 1] is a N × 1 vector of all previousstates. The iterative process continues until vn [k ] of all nodesconverges to the global average LLR, R/N. The final decisionis then obtained locally at each node according to its currentstate given by,

vn [k ]Hn=1≷

Hn=0

δ/N , (10)

where Hn denotes the decision of node sn . The updating rulein terms of initial states can be written as,

v[k ] = C(k)v[0]. (11)

It is clear that the weight matrix C needs to be designedin such a way that the iterative process converges fast.Equation (11) shows that this can be achieved when C(k)

converges to all-ones matrix, i.e., limk→∞

C(k) = 1N 1N×N .

Hence, the distributed detection problem can be seen as aoptimal topology design through the optimal weight matrix.Mathematically, the distributed detection problem can beformulated as,

C∗(α∗,L∗) = arg minC(α,L)

∥∥∥∥C

(k)(α,L)− 1

N1N×N

∥∥∥∥

2

F

, (12)

where the weight matrix C contains elements

ci ,j =

⎧⎨

⎩

α, if (i , j ) ∈ {E ,L}1− αdi , if i = j0, otherwise,

(13)

where di is the degree of node i. This can be written as C =IN − αL, where L is the Laplacian matrix corresponding tograph G. The elements of matrix L are given by,

li ,j =

⎧⎨

⎩

−1, if(i , j ) ∈ {E ,L}di , if i = j0, otherwise

(14)



2) Estimation Problem: The distributed estimation problemtries to solve an unknown vector parameter by using only mea-surements from neighboring nodes [28], [30], [62]. At everydiscrete time instant k, the nth node takes a scalar measure-ment Sn [k ] and a 1 × M dimension regression vector rn,k .The objective is to estimate an M × 1 unknown vector p(0)

using the data Sn [k ] and rn,k . Each node i has access to timerealizations {Si [k ], ri ,k}, i = 1, 2, . . . ,N , as shown in Fig. 1.The vector parameter follows standard model given by,

Sn [k ] = rn,kp(0) + zn [k ], (15)

where zn [k ] is zero-mean Gaussian noise with variance σ2v ,nand independent over space and time.

In a centralized estimation method the problem can beexpressed as a L2-norm minimization problem. Let Rc =col{r1, r2, . . . , rN }, be an (N × M) global matrix and S =col{S1,S2, . . . ,SN }, be an (N × 1) data vector. Then we seekan M × 1 vector p(0) that solves

minp(0)

E∥∥∥S − Rcp

(0)∥∥∥2

2. (16)

In the case of distributed estimation, the task is to devisea robust distributed method that can approach the solutionp(0) and provides a good estimate of that vector to eachnode in the network. By utilizing the diffusion LMS Adapt-Then-Combine (ATC) algorithm [52], [62], the problem ofdistributed estimation can be reformulated as

ψn [k ] = φn [k − 1] + μnr�n,k

(Sn [k ]− rn,kφn [k − 1]

)

(17)

φn [k − 1] =∑

l∈Nn,[k−1]

cn,l ψl [k − 1]. (18)

In the above equations, at any given time [k − 1], it isassumed that node n has access to a set of unbiased estimatesψl [k − 1] from its neighborhood Nn . The local estimates arefused at the nth node using local combiners cn,l resultingin φn [k − 1]. cn,l is the set of local combiners which con-tains information about the network topology and satisfies thefollowing conditions,

∑

l∈Nn

cn,l = 1, and cn,l = 0, ∀l /∈ Nn . (19)

After obtaining an aggregate estimate φn [k − 1] forp(0), the fusion of φn [k − 1] into the local adaptiveprocess is performed to update it to ψn [k ]. Here, μnis the local step size. In terms of global quantities wedefine the following, zk = col{z1[k ], z2[k ], . . . , zN [k ]},ψ [k ] = col{ψ1[k ],ψ2[k ], . . . ,ψN [k ]}, φ[k − 1] =col{φ1[k − 1],φ2[k − 1], . . . ,φN [k − 1]}, Rk =col{r1,k , r2,k , . . . , rN ,k}, Sk = col{S1[k ], S2[k ], . . . ,SN [k ]},

and p(0) = col{p(0), p(0), . . . , p(0)} (N times). Therefore,the model in Equation (15) can be rewritten as

Sk = Rkp(0) + zk . (20)

Also, with these relations and expressions, Equations (17)and (18) can also be written in a matrix form as,

{φ[k − 1] = G ψ [k − 1]ψ [k ] = φ[k − 1] +DR�

k (Sk −Rkφ[k − 1]),

where G = C⊗IM is a NM × NM transition matrix and C is aN × N diffusion combination matrix with entries [cn,l ]. Here,D = diag{μ1IM , μ2IM , . . . , μN IM } is a diagonal matrix col-lecting the local step sizes. Now the global error vector isgiven by,

ψ [k ] = p(0) − ψ [k ]. (21)

Utilizing the previous equations, the global error is written as,

ψ [k ] = G(INM −DR�kRk )ψ [k − 1]−GDR�

k zk . (22)

In Equation (22), taking expectation on both sides results in

E[ψ [k ]

]= G(INM −DRr )E

[ψ [k − 1]

], (23)

where Rr = diag{Rr ,1,Rr ,2, . . . ,Rr ,N } is block diagonal.To ensure the convergence of error vector E [ψ [k ]], it requiresthat λ(GB) < 1 with B = INM −DRr . Thus, in an adaptivenetwork the convergence of distributed estimation is depen-dent on the space-time data statistics (represented by B) andnetwork topology (represented by G). Therefore, ultimatelythe problem is related to efficient network topology develop-ment leading to fast convergence of parameter estimation. Thefinal problem formulation is expressed as,

C∗ = argminC

∥∥∥[GB][k ] −ONM×NM

∥∥∥2

F(24)

subject to: CqN = qN , (25)

where, qN = col{1, 1, . . . , 1} and O is a matrix of zeros.

V. FAULT-RESILIENT DISTRIBUTED DETECTION AND

ESTIMATION OVER SMALL WORLD WSNS

In this section, the introduction of small world characteris-tics (SWC) using directional beamforming is discussed first.Subsequently, LCMV beamforming leading to higher neigh-borhood connectivity for sensor nodes is described. LCMVbeamforming exercises a radio area constraint on sensor nodesleading to power efficient small world WSN (SW-WSN) devel-opment. In addition, a traffic flow optimization algorithm forSW-WSN development is also discussed in this section. Thetraffic flow optimization algorithm leads to a low-complexitySW-WSN development with optimal SWC. Finally, the algo-rithm development for fault-resilient distributed detection andestimation over SW-WSNs is presented.

S∗B×1(ri ,Δθi , αi ) = arg maxSB×1(ri ,Δθi ,αi )

∣∣∣∣LSW(SB×1(ri ,Δθi , αi ))

LW(0)− CSW(SB×1(ri ,Δθi , αi ))

CW(0)

∣∣∣∣ (26)



A. Introduction of SWC Using Directional Beamforming

Various strategies involving introduction of SWC in WSNscan be found in literature. Some of them involves usage ofadditional wires [35], [39], mobile sensor nodes [63], and highcapacity heterogeneous sensor nodes [36], [40]. Introductionof SWC using these methods has several disadvantages [44].In terms of the risk of having wired shortcuts, a WSN is notpractical in applications such as battlefield surveillance andaccident detection. Breakage of wired shortcuts will lead toloss of SWC in the network. Additionally, to compute thefixed length of long-range links a prior in mobile ad hocnetworks is also not feasible. Utilization of mobile sensornodes results in increased data latency and network deploy-ment cost. In addition, it yields several unwanted shortcuts inthe network, resulting in the formation of a random network.In order to address these issues, in this work, a novel direc-tional beamforming technique is used for the introduction ofSWC.

Directional beamforming involves mounting an antennaarray on top of sensor nodes to act as a directional antenna.Beamforming is achieved by combining the elements ofantenna array in such a way that signals transmitted experienceconstructive interference at particular angles. Various meth-ods [45], [46] have been developed for transmitting a signal ofinterest toward specific directions while suppressing its energyin other directions. Beamforming leads to the formation of aradiation pattern in the form of lobes at various angles. Thedirection in which there is larger field strength than in othersis called the main lobe and others are called side lobes. Theselobes can also be characterized on the basis of their beamlength (r), beam width (Δθ), and direction in which the beamis directed (α). In this work, these parameters for an individualnode are computed to yield a low-complexity SW-WSN devel-opment with optimal SWC. The selection of a set of nodesused for beamforming and the corresponding optimal beam-forming patterns are obtained utilizing the parameter Flowbetween node pairs. The beamforming pattern minimizes theAPL of the network while maintaining high ACC.

B. LCMV Beamforming for Neighborhood Connectivity

It has been shown in [44] that with beamforming, reduc-tion in network APL is accompanied by loss in connectivity,which is exhibited by a decrease in ACC of the network. Thisdecrease in ACC is due to the fact that beamforming node“loses” links to closely located neighbors that are not withinthe main lobe or side lobes of the radiation pattern. Hence,for maintaining a high clustering coefficient, neighboring con-nectivity of beamforming nodes should be maintained. Twopopular directional antenna models, namely sector model [44],and the keyhole model [43] have been proposed to simplifythe radiation pattern. However, both models fail to capturemost important features of a directional antenna, i.e., sidelobes and nulling capability. In recent literature, a more real-istic model called Iris antenna model [45], has been proposedwhich can approximate the realistic antenna model into a sim-plified model while maintaining its properties. In this paper,we use Iris antenna model [45] to realize beam pattern of a

beamforming node and utilizes a popular spatial filtering tech-nique called LCMV beamforming to ensure that connectivityto the neighborhood nodes is maintained.

LCMV beamforming is specifically used for controllingantenna gains in various directions, whose mathematical modelis given by,

minw

wHxxHw

subject to: VHw = c.

Here, x is the received signal, which is a combination of trans-mitted signal (s) and noise (n). If a transmitter is sending asignal s, which is directed in some direction specified by thevector v, then x = vs + n, where, n is Gaussian noise, wis the weight vector for a linear combination of the signalsfrom different antennas. (·)H means the Hermitian operationon (·). In the above mathematical model, the variance of thereceived signal is minimized so that the noise component ofthe received signal is reduced. V is a Q×C constraint matrix,where Q represents the number of antennas mounted on theantenna array, and C represents the number of constraints.Vector c keeps the information about the gain of a beam-forming node. In this work, the weight vector c is evaluatedusing the desired radiation pattern for antennas based on thereceived signal strength (RSS) measurement model.

Energy efficiency has always been a critical issue in WSNsdue to the limited powers at sensor nodes. The primary moti-vation behind the use of LCMV beamforming is that thenew links between node pairs can be introduced while usingthe same amount of power as with omnidirectional sensornodes. In this work, LCMV beamforming is used with a con-straint on radio transmission area covered by a sensor node.The radio area constraint during signal transmission is as fol-lows. Consider a node i with a directional antenna having lomnidirectional neighbors (omnidirectional radio range r0). Itoperates in the mode of directional transmission and omnidi-rectional reception, i.e., the receiving gain is unity (Gr = 1).For maintaining neighborhood connectivity, the signal betweena node and its omnidirectional neighbors should be receivedsuccessfully, i.e., the power attenuation β is no greater thanthe threshold β0 [45]. Therefore, the effective aperture (Ae )of an antenna in the direction of a neighborhood node can beexpressed in terms of omnidirectional range and is given by,

Ae =λ2r

ρ0

4πβ0, (27)

where ρ is the path loss exponent and λ is the wavelength ofthe signal. In Equation (27), the shadowing effect is consideredto be zero. Hence, the half power beam width of the side lobes(θs ) is given by,

θs = tan−1

⎛

⎝ λ

2π

√

rρ−20

β0

⎞

⎠. (28)

Therefore, if a node has K neighborhood nodes then the mini-mum radio area required by the node to maintain neighborhoodconnectivity is given by,

A′ = K × θs2

× r20 . (29)



Fig. 2. An illustration of LCMV beamforming performed by sensor nodes.Sensor nodes are denoted using black dots. The omnidirectional radio areascovered by sensor nodes are represented using blue circles. Directionalbeamforming consisting of the main lobe and side lobes is shown in red.

The remaining area A−A′, can be used to model the main lobeof the radiation pattern. Here, A is the total radio area coveredby the sensor node. Fig. 2 depicts the radio area A and A′ cov-ered by a sensor node: directional radio transmission utilizingLCMV beamforming is illustrated in red, whereas radio areacovered using omnidirectional transmission is shown in blue.

After addressing the challenges of neighboring connectivityusing LCMV beamforming, optimal beamforming parametersare obtained by maximizing the difference between normalizedAPL and ACC. Let the total sensor nodes used for beamform-ing be B = N × ν, where, N is the total number of sensornodes in the network and ν is the percentage of nodes usedfor beamforming. Let, LSW(SB×1(ri ,Δθi , αi )), ∀i ∈ B, andLW(0) denote the APLs of a SW-WSN and a regular WSNrespectively. Similarly, CSW(SB×1(ri ,Δθi , αi )), ∀i ∈ B andCW(0) denote the ACCs of a SW-WSN and a regular WSN,respectively. Let SB×1(ri ,Δθi , αi ) be a B × 1 vector whoseelement Si (ri ,Δθi , αi ) is a 2-dimensional representation ofradiation pattern corresponding to the i th beamforming node(1 ≤ i ≤ B). Then, the computation of optimal beamformingpattern (S∗B×1(ri ,Δθi , αi )) is carried out using Equation (26),as shown at the bottom of the p. 6. Finding the solution of theoptimization problem given in Equation (26) is quite challeng-ing as the relationship between quantities is quite complex.Hence, we develop a low-complexity technique to find anoptimal radiation pattern for every node. To this end, we resorton SW-WSNs with traffic flow optimization. This is discussedin the next section.

C. A Low-Complexity Small World WSN Development UsingTraffic Flow Optimization

This section describes the development of a SW-WSNusing a low-complexity traffic flow optimization method. Themethod results in optimal beamforming parameters for individ-ual nodes. Optimal parameters are computed utilizing trafficflow between node pairs. The parameter Flow is used to

Fig. 3. An illustration of (a) Direct Flow and Indirect Flow between sen-sor nodes (b) radio area covered by the main lobe leading to optimal beamparameters is shown using yellow rectangular box.

measure the traffic between a selected node pair. Specifically,Flow between any two nodes u and v of the network is givenby [64],

Flow(u, v) =∑

s,t∈N

σ(s , t)u,vσ(s , t)

, (30)

where σ(s , t)u,v is the number of shortest paths betweennodes s and t, which pass through nodes u and v, and σ(s , t)is the total number of shortest paths between nodes s and t.Similar to betweenness centrality [64], the Flow between twonodes is high if there is a larger number of times both nodeslie between the shortest path of randomly selected node pairsin the network. Flow through any node can be divided intotwo different categories, “Direct Flow”, (FlowD) and “IndirectFlow”, (FlowID), as illustrated in Fig. 3. Direct Flow existsbetween nodes which are directly connected to each other. Onthe other hand, Indirect Flow exists between nodes which areseparated by more than one hop. Direct Flow from a node uis given by,

FlowD(u) =∑

v∈Nu

Flow(u, v), (31)

where Nu denotes the set of neighborhood nodes of node u.The parameter Flow between a node pair illustrates the

importance of the link existing between them for the pur-pose of data transmission across the network. It representsthe number of shortest paths between other pair of nodes thatpasses through the given node pair. This implies that the para-menter Flow plays an important role in governing the APLof the network. Therefore, a decrease in distance (number ofhops) between a node pair having higher Flow also results ina decrease in the APL of the entire network. Hence, in thiswork the beamforming is performed in such a manner thatthe nodes having higher Flow values have shorter distancesbetween them. Direct Flow for node u is given by,

FlowD(u) = mru + lΔθu + cαu , (32)

where, ru , Δθu , and αu are the beam length, beam width, andbeam direction, respectively, corresponding to the uth node.We focus our attention toward finding the optimal values forthese parameters for every node such that the node pair with ahigher Flow gets connected leading to maximization of DirectFlow for every node. This is illustrated in Fig. 3.



Fig. 3(a) illustrates the beamforming by the uth node. Thesensor nodes in green are falling in the main lode and sidelobes of the uth node. The nodes inside the main lobe areshown within yellow beam and leads to Direct Flow. Thenodes which are neither in the main lobe nor in a side lobe areindicated in red and cyan. These nodes lead to Indirect Flow.The sensor nodes which are within the beam width of the mainlobe while out of range of the main lobe are in cyan. Fig. 3(b)illustrates the radio area covered by the main lobe of the uth

sensor node with a yellow rectangular box. The parameterscorresponding to this box result in optimal beam parameters.

Let τ(u, v) be the length of the shortest path betweennodes u and v. Distances from node u to all other nodes arerepresented in a vector Ru = col [du (1), du (2), . . . , du (N )].Angles between node u and other nodes are collected ina vector Θu = col [θu (1), θu (2), . . . , θu (N )], where θu(i)denotes the azimuthal angle between node u and node i.Flows from node u to all other nodes in the network areFlowu = col [Flow(u, 1),Flow(u, 2), . . . ,Flow(u,N )]. Notethat the sizes of all vectors Ru , Θu , Flowu are (N−1)×1. Asthe parameters of the main lobe vary, neighborhood nodes ofnode u also change and hence FlowD(u) changes. Therefore,the optimal parameters are obtained by maximizing FlowD(u),which is given by,

⎧⎨

⎩

S∗u (r

∗u ,Δθ

∗u , α

∗u ) = arg max

S(ru ,Δθu ,αu )FlowD(u)

Subject to:A− A′ = r2u2 Δθu = constant

where, recall that, A and A′ are the total radio area andradio area covered by side lobes of node u, respectively.The algorithmic development of the proposed traffic flowoptimization method yielding optimal beam parameters isgiven in Algorithm 1. In Algorithm 1, first the number of short-est paths passing through the selected node pair, (u, v), is cal-culated using the breadth-first search [65]. Subsequently, flowfrom node u to other nodes is computed by varying node v.Finally, optimal parameters r∗u , Δθ∗u , α∗u , corresponding tonode u are obtained by maximizing its direct flow.

D. Algorithm Development for Fault-Resilient DistributedDetection and Estimation Over Small World WSNs

This section discusses the proposed methods of fault-resilient distributed detection and estimation over SW-WSNs.In these methods, first a network with randomly deployed sen-sor nodes is considered. The data traffic load on an individuallink is assumed to be known. Data traffic load computationis performed using Dijkstra’s algorithm. SWC are introducedin this network at various time instants based on node andnetwork characteristics. Introduction of SWC is carried outusing beamforming from a selected set of nodes. Nodes usedfor the beamforming are obtained using traffic flow measure-ment between node-pairs. Algorithm 2 enumerates the stepsfor the distributed detection and estimation over a SW-WSN.The flow diagram corresponding to Algorithm 2 is presented inFig. 4. In Fig. 4, the procedure for converting a regular WSNinto a SW-WSN is indicated by beamforming toward SW-WSN

Algorithm 1 Computation of Optimal Parameters r∗u , Δθ∗u ,α∗u1. Initialize Flow(u, v) = 0;2. for s, t ∈ N − {u, v} do3. Use breadth-first search [65] to compute τ(s, t) and σ(s, t);4. if τ(s, t)u,v = τ(s, u) + τ(u, v) + τ(v , t) do5. σ(s, t)u,v = σ(s, u)× σ(u, v)× σ(v , t);6. Flow(u, v) = Flow(u, v) + σ(s, t)u,v/σ(s, t);7. end8. end9. Compute Flow(u, v), ∀{u, v} ∈ N using steps 1 to 8.10. Initialize FlowD(u) =

∑

v∈Nu

Flow(u, v) with r∗u = 0,

Δθ∗u = 0, α∗u = 0.11. Sort Ru , Θu , Flowu in the increasing order of θu (i).12. Initialize FlowD(u)j = 0;13. for i ∈ [1,N − 1] do14. Δθmax = 2(A− A′)/(du (i))215. Υ = θu (i)−Δθmax16. � = θu (i) + Δθmax17. for j | θu (j ) ∈ [Υ, �] do18. k | θu (k) = θu (j ) + Δθmax

19. FlowD(u)j∗ =∑

Ω∈[j ,k ] Flow(u,Ω)

20. if FlowD(u)j∗ >= FlowD(u)j

21. FlowD(u)new = FlowD(u)j∗ + FlowD(u)22. α∗ = (θu (k) + θu (j ))/223. r∗u = du (i)24. Δθ∗ = Δθmax25. end26. end27. end27. Return r∗u , Δθ∗u , α∗u

development block. The steps adapted for performance anal-ysis of the proposed method are indicated by distributedprocessing over the developed SW-WSN block.

VI. PERFORMANCE EVALUATION

Performance evaluation of the proposed methods is car-ried out using simulation and real node deployments over aWSN testbed. Detection and estimation performance alongwith network robustness performance are evaluated overboth the data sets. Resilience of the network is analyzedthrough k-connectivity to highlight the applicability of theproposed methods in practice. Performance of the proposedmethods is compared with the existing betweenness central-ity method [44], and bio-inspired clustering method [43].In addition, the complexity of the proposed methods isalso investigated to illustrate their significance in real timeapplications.

A. Experimental Setups

The experimental setups used in the performance evaluationare discussed in this section.

1) Simulated WSN: A network of dimension 40 m × 40 mis considered for simulation. The total number of sensor nodesused for simulation is 100. The sensor nodes are uniformlydistributed over the deployed area. Initially all nodes areperforming omnidirectional transmission and reception. Theconventional radio range of sensor nodes is 8 m. Nodes are



Algorithm 2 Fault-Resilient Distributed Detection andEstimation Over a Small World WSN1. Initialization: Consider a randomly deployed conventional WSNas, G(0) = (N , E ,P,B,L), where the set of new links L, and setof beamforming nodes B, are both null sets {∅}.2. Compute optimal beam parameters for each node: For allexisting nodes in the network, compute the optimal beamformingparameters by solving the traffic flow optimization:

⎧⎨

⎩

S∗u (r

∗u ,Δθ

∗u , α

∗u ) = arg max

S(ru ,Δθu ,αu )FlowD(u)

Subject to: A− A′ = r2u2 Δθu = constant

3. Selection of nodes for beamforming: Beamforming nodes areselected from the network in order to maximize the differencebetween APL and ACC utilizing the Flow parameter:

S∗B×1 = arg maxSB×1

∣∣∣∣LSW(SB×1)

LW(0)− CSW(SB×1)

CW(0)

∣∣∣∣ (33)

4. Small world WSN development using LCMV beamforming:After the selection of nodes for beamforming, use the LCMV filteringmethod to ensure neighborhood connectivity for each node based ondirectional links:

minw

wHxxHw

subject to: VHw = c.

5. Obtain new network topology: Compute the new network topol-ogy G(N , E ,P,B,L) with L,B �= {∅}.6. Analysis of small worldness: The scalar measures [33], [60]

of small worldness [66] are given by, � =LSW(B)LW(0)

< 1, η =

CSW(B)CW(0)

≈ 1, where LW(0) is the APL for a WSN and LSW(B)

is the APL for the corresponding SW-WSN. Similarly, CW(0) is theACC for a WSN and CSW(B) is the ACC for the correspondingSW-WSN.7. Optimal small world characteristics introduction: Repeat steps2 to 6 in an iterative manner for selection of the best B nodes leadingto the optimal SWC.8. Performance analysis:

Distributed detection Convergence rate:T (B)T (0)

Distributed estimation MSD: 1N

∑Ni=1 E‖p(0) − ψn [k ]‖

Fault-resilience k − Connectivity check ∀k9. Fault-resilient distributed detection and estimation over aSW-WSN: Developed network results in fault-resilient distributeddetection and estimation over a SW-WSN.

capable of beamforming using directional antennas. Path lossexponent (α) is 2 and the power attenuation threshold is con-sidered to be 0.5. Wavelength of the transmitted signal is0.125 m.

2) Real Node Deployment Over a WSN Testbed:Performance evaluation is also conducted over a WSN testbedusing real sensor node deployment at a residential site of IITKanpur. The WSN testbed consists of 80 National Instruments(NI) 3230 wireless sensor motes. These motes are uniformlydistributed over an open-space geographical area of dimension33 m × 29 m. For RSS measurement, NI wireless gateway9792 is used. All the data of real WSN testbed is collectedover the NI gateway which is connected through a LAN. The

antenna gain for NI motes and gateway is 1.5 dBi correspond-ing to both the transmitter and receiver. The transmitted poweris 10 dBm. Friis transmission model [38] is used to computethe distance between NI mote-pairs. To compute the RSS atthe receiver node (Pr ), a total of 10 readings are taken betweentransmitter and receiver. The values of Gt and Gr are 1.5 dBi,λ is 0.12491 m, and Pt is 10 dBm. The conventional linkshave a length of 7 m.

B. Analysis of Small Worldness

Small world characteristics of SW-WSN are often deter-mined using APL and ACC. However, analysis of smallworldness [66] of a SW-WSN can be performed by definingsome scalar measures [33], [60]. The scalar measures of smallworldness are given by � =

LSW(B)LW(0)

< 1, η =CSW(B)CW(0)

≈ 1.Here LW(0) and LSW(B) are the APLs of WSN and SW-WSN, respectively. Similarly, CW(0) and CSW(B) quantifythe ACCs of WSN and SW-WSN, respectively. The analysisof SWC is carried out using both the data sets. For observingthe variations in APL and ACC, the fraction of beamform-ing nodes is increased from 0 to 1. Variations in APL andACC using the proposed method are also compared with thatof the existing methods. The results obtained over the simu-lated WSN can be seen in Fig. 5, which show that the APL ofthe proposed method decreases as the fraction of beamform-ing nodes increases. In similar conditions the ACC is almostconstant. It is noted that with 20% of beamforming nodes,the APL of the proposed method is 69.68% when comparedto conventional WSN. This reduction in APL is significantlylower when compared to betweenness centrality and cluster-ing methods. For the betweenness centrality and clusteringmethods this reduction is 94.52% and 76.36%, respectively.The ACCs of the proposed and clustering methods are almostequal to 1. For the betweenness centrality method, there is areduction of 13.44% in ACC when compared to the conven-tional WSN. These variations over a real WSN testbed can beseen in Table II, which clearly illustrates the significance ofour proposed method.

C. Performance Analysis of Distributed Detection

The analysis of distributed detection is performed utilizingthe Bayesian model given in Equation (4), with a prior prob-abilities π0 = π1 = 0.5, which lead to the optimum fusionthreshold of δ = 0. The signal means μ1 and μ2 are modeledas μ1 = −μ2 = 1, while the noise variance σ2 = 0.501. Theperformance of distributed detection is measured in terms ofnumber of iterations required by the weight matrix leading to‖C(k)(α,L) − 1

N 1N×N ‖2F → ε. In case of simulations, theerror (ε) is considered to be 0.01% of that in the conventionalWSN. Variation of the convergence rate for distributed detec-tion can be seen in Fig. 6. In case of a SW-WSN, a phasetransition phenomenon in algebraic connectivity is observed.Algebraic connectivity of a network is the second smallesteigenvalue of its Laplacian matrix and is a measure of speedof solving consensus problems in networks [24]. In a SW-WSN, an increase in algebraic connectivity utilizing addition



Fig. 4. Block diagram illustrating the fault-resilient distributed detection and estimation over a small world WSN.

Fig. 5. Variations of APL and ACC with varying fraction of beamform-ing nodes. The variations are compared among the proposed, betweennesscentrality, and clustering methods.

of new links increases the speed of solving consensus prob-lems. The convergence speed increases by 47.32% using ourproposed method when compared to a conventional WSN.This increase is 28.99% and 11.22%, respectively, using theclustering and betweenness centrality methods. These resultsare noted when 20% of nodes perform beamforming in thenetwork. It can also be seen that as the fraction of beam-forming nodes increases, the betweenness centrality methodresults in better performance compared to the other methods.However, it should be noted that in this case the ACC of thenetwork decreases by 50% compared to a conventional WSN,leading to a random network. The results over a real WSNtestbed can also be seen from Table II. A 79.24% improvementin convergence speed over the real WSN testbed can be seenusing the proposed method when only 20% of nodes performbeamforming. In similar conditions, the convergence speedincreases by 35.69% and 66.27% respectively when using thebetweenness centrality and clustering methods.

D. Performance Analysis of Distributed Estimation

To measure performance of distributed estimation theregression data is generated by regressors with shifted struc-ture of size M, i.e., rn,k = col{rn (k), rn (k − 1), . . . , rn (k −

Fig. 6. Convergence speed of distributed detection with a varying fractionof beamforming nodes. The proposed method is compared with betweennessand clustering methods.

M + 1)} with rn (k) being a time series generated as fol-lows: rn (k) = ζnrn (k − 1) + ξnzn (k) for k > −∞, whereζn ∈ [0, 1) is the correlation index, zn(k) is a spatially inde-pendent white Gaussian process with unit variance, and ξn =√σ2n,k (1− ζ2n ). The scalar observation Sn [k ] is obtained

according to the model given in Equation (15) with unknownM-dimensional vector set as p(0) = col{1, 1, . . . , 1}/√M .The performance analysis of distributed estimation is shownin Fig. 7, which illustrates a reduced mean squared devia-tion (MSD) using the proposed method when compared to aconventional WSN and other existing methods, namely clus-tering, and betweeneess centrality methods. The performanceis observed with a varying fraction of sensor nodes used forbeamforming.

Figs. 7(a), 7(b), 7(c), and 7(d) correspond to 10%, 40%,80%, and 100% nodes used for beamforming. Fig. 7(c) illus-trates a reduced MSD as the number of iterations increases. Itcan be seen that as the number of iterations increases the MSDof the proposed method decreases to a minimum of −160 dB.However for both the betweenness centrality and clusteringmethods the minimum MSD obtained is −153 dB. The dis-tributed estimation performance over a real WSN testbed isshown in Table II. Improved performance using the proposedmethod is also observed for this case of real WSN testbedwhen compared to the existing methods.



Fig. 7. Illustration of estimation performance analysis (MSD in dB) with a varying number of iterations. The results are obtained with a varying fraction ofnodes used for beamfoming (a) 0.1 fraction (b) 0.4 fraction (c) 0.8 fraction and (d) All the nodes are used for beamforming. The results are obtained usingthe proposed method (violet), clustering method (yellow), and betweenness centrality method (blue) over the simulated WSN.

TABLE IIVARIATIONS IN THE NORMALIZED AVERAGE PATH LENGTH (N-APL), NORMALIZED AVERAGE CLUSTERING COEFFICIENT (N-ACC), DISTRIBUTED

DETECTION CONVERGENCE SPEED (T (B)/T (0)), MEAN SQUARED DEVIATION (MSD), AND k-CONNECTIVITY CHECK OVER A REAL WSN TESTBED

E. Network Resilience Analysis

Node-failure resilience of the network is measured usingnetwork responses to a perturbation. However, it is difficult tosee the quantitative assessment of network resilience. In thispaper, we use resilience factor (RF ) [21], [22] to measure thenetwork resilience quantitatively, which is given by

RF =

∑Ni=2 k(i)

N − 2. (34)

In Equation (34), k(i) denotes the percentage of node com-binations that guarantees partial i-connectivity. It is assumed

that all networks considered are 1-connected and n-connected,thus these cases are excluded from observations. Even thoughthe resilience factor gives us good information about networkresilience, the challenging issue is it’s exponential complexity.Therefore, rather than computing k(i) for all i’s, in this paperfor illustrating the network resilience performance we consideri ∈ {2, 3, 4, 5}.

Fig. 8 illustrates the k-connectivity check, which is used asa measure of robustness of the SW-WSN to the node failure.The SW-WSN developed using the proposed method leadsto a high algebraic connectivity [24] and results in a more



Fig. 8. Illustration of robustness analysis (k-connectivity check) with varying fraction of nodes used for beamforming. The results are obtained for differentk-connectivity (a) 2-connectivity check (b) 3-connectivity check (c) 4-connectivity check and (d) 5-connectivity check. The results are obtained using theproposed method (violet), clustering method (yellow), and betweenness centrality method (blue) over the simulated WSN.

robust network in the context of node failures. Connectivityperformance of the proposed method is also compared withthat of state-of-the-art methods. An important aspect to notehere is that, with beamforming, directional nodes becomesignificantly important than omnidirectional nodes. The fail-ure of such nodes will have a considerable impact on thenetwork since the network connectivity depends heavily onthem. Therefore, while doing k-connectivity check it can benoted that as the number of beamforming nodes increases incase of betweenness centrality and clustering methods, thefailure of these nodes results in higher disconnectivity in thenetwork. Fig. 8(a) shows results for 2-connectivity check. Incase of betweenness centrality and clustering methods the dis-connectivity of the network increases to 0.82% and 2.3%respectively when compared to a conventional WSN. Thisresult is obtained when 30% of the nodes perform beam-forming. In case of a conventional WSN the disconnectivityis 0.35%. However, using the proposed method the discon-nectivity in the network remains almost equal to that of theconventional WSN. This is because the proposed methodmakes sure that the node remains connected with its neigh-borhood nodes. Beamforming only introduces new links inthe network. The results of k-connectivity over a real WSNtestbed can be seen in Table II. Improved performance of k-connectivity is also noted using the proposed method whencompared to other existing methods over a real WSN testbed.

F. Complexity Analysis of Traffic Flow Optimization Method

The time complexity of the breadth-first search algo-rithm [65] toward computation of both the shortest path andthe number of shortest paths is O(M + N). Further, compu-tation of Flow between two nodes results in a complexity of

order O(MN 2 + N 3) or O(N 3) as the network is consid-ered to be sparse. Therefore, the time complexity to computeFlow across the network is O(N 5). The objective functiondeveloped for maximization of Flow is a convex problem andcan be solved in polynomial time. The algorithm involvesmaximization of the Direct Flow for every node leading tooptimal beamforming parameters for each node. In this con-text, sorting of θu (i) has a complexity of O(N × log N).In addition, computation of optimal parameters, FlowD(u),r, Δθ has time complexity of O(a × N). Here a is a con-stant which is obtained by the number of times the inner “forloop” runs. Hence the time complexity for computing optimalbeamforming parameters is O(N × (log N + a)).

VII. CONCLUSION AND FUTURE WORK

In this paper, novel methods of distributed detection and dis-tributed estimation over a SW-WSN have been proposed. Themethods utilize traffic flow between node pairs and result ina robust and low-complexity SW-WSN development. A novelLCMV beamforming technique is used for the introductionof SWC in the network. LCMV beamforming increases thefeasibility of the network utilization in real time applications.Experimental results are obtained over both computer simu-lated WSN and real WSN testbed. Results obtained using theproposed methods are compared with existing methods in lit-erature, namely the betweenness centrality and bio-inspiredclustering methods.

In case of the real WSN testbed, distributed detection speedincreases by 84.21% using the proposed method when com-pared to a conventional WSN. In this condition, only 40%of the nodes are used for beamforming. This increase is by57.14% and 66.46%, respectively, using the clustering and



betweenness centrality methods under similar conditions. TheMSD of parameter estimation reduces to a minimum value of−164.05 dB using the proposed method. This value for theclustering and betweenness centrality methods is −157.33 dBand −155.93 dB, respectively. The MSD in case of theconventional WSN is −126 dB. The 2-disconnectivity and3-disconnectivity using the proposed method are zero in allcases, while a maximum of 9.7 × 10−5 and 8.3 × 10−5

fraction of 4-disconnectivity and 5-disconnectivity are notedusing the proposed method, respectively. The disconnectiv-ity results obtained using the proposed method illustrate asignificant improvement when compared to the clusteringand betweenness centrality methods. To conclude, signifi-cant improvements over distributed detection, distributed esti-mation, and network robustness are obtained utilizing theproposed methods when compared to two existing methods.

Mounting an antenna array on a node generally increasesthe hardware complexity of the node. Therefore, collaborativebeamforming from the group of sensor nodes is interesting andshall be investigated for SW-WSN development in a futurework.

ACKNOWLEDGMENT

The authors would like to thank the Editor and the anony-mous reviewers for constructive comments which greatlyhelped to improve the quality and clarity of this paper.

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Om Jee Pandey (Member, IEEE) received theB.Tech. degree in electronics and communica-tion engineering from Uttar Pradesh TechnicalUniversity, Lucknow, India, in 2008, the M.Tech.degree in digital communications from the ABV-Indian Institute of Information Technology andManagement, Gwalior, India, in 2013, and thePh.D. degree from the Department of ElectricalEngineering, Indian Institute of Technology Kanpur,Kanpur, India, in 2019. He worked as a PostdoctoralFellow with the Communications Theories Research

Group, Department of Electrical and Computer Engineering, University ofSaskatchewan, Saskatoon, SK, Canada. From 2008 to 2011, he worked withEscorts Ltd., and FANUC India Private Ltd. He was a Senior Lecturer with theJaipur Engineering College and Research Center, Jaipur, from 2013 to 2014.He is currently working as an Assistant Professor with the Department ofElectronics and Communication Engineering, SRM University AP, Amaravati.His research interest includes the signal processing for wireless networks witha specific focus on robust sensor node localization and tracking over wire-less ad hoc networks. He also works on related areas, such as low-latencydata transmission, data aggregation, and distributed detection and estimationin wireless sensor networks. He is a regular reviewer for various reputedjournals of IEEE, including the IEEE TRANSACTIONS ON VEHICULAR

TECHNOLOGY, IEEE ACCESS, IEEE INTERNET OF THINGS, and the IEEETRANSACTIONS ON COMMUNICATIONS.

Ved Gautam received the bachelor’s degree inelectrical engineering from the Indian Institute ofTechnology Kanpur, where he had worked as anUndergraduate Researcher with the Sensor NetworksLab, Electrical Engineering Department from 2017to 2019. He is currently working with Urvija AI.His current research interests include distributedcomputation, use of machine learning algorithmsfor enhancement of network performance, and dis-tributed learning in wireless sensor networks.

Ha H. Nguyen (Senior Member, IEEE) receivedthe B.Eng. degree in electrical engineering fromthe Hanoi University of Technology (HUT), Hanoi,Vietnam, in 1995, the M.Eng. degree in elec-trical engineering from the Asian Institute ofTechnology (AIT), Bangkok, Thailand, in 1997,and the Ph.D. degree in electrical engineeringfrom the University of Manitoba, Winnipeg, MB,Canada, in 2001. He joined the Department ofElectrical and Computer Engineering, University ofSaskatchewan, Saskatoon, SK, Canada, in 2001,

where became a Full Professor in 2007. He currently holds the positionof NSERC/Cisco Industrial Research Chair in Low-Power Wireless Accessfor Sensor Networks. He has coauthored, with Ed Shwedyk textbook “AFirst Course in Digital Communications” (Cambridge University Press). Hisresearch interests fall into broad areas of communication theory, wireless com-munications, and statistical signal processing. He was an Associate Editorfor the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and IEEEWIRELESS COMMUNICATIONS LETTERS from 2007 to 2011 and from 2011to 2016, respectively. He currently serves as an Associate Editor for theIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. He is a fellow of theEngineering Institute of Canada and a Registered Member of the Associationof Professional Engineers, and the Geoscientists of Saskatchewan.



Mahendra K. Shukla (Member, IEEE) receivedthe Ph.D. degree in electronics and communicationengineering from the Indian Institute of InformationTechnology Allahabad, Prayagraj, India, in 2018.He is currently working as a Postdoctoral Fellowwith the Communications Theories Research Group,Department of Electrical and Computer Engineering,University of Saskatchewan, Saskatoon, SK, Canada.His research interests include cooperative relay-ing, physical layer security, nonorthogonal multipleaccess, energy harvesting, and signal processing.

He is a regular reviewer of premier journals, including the IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY, the IEEE TRANSACTIONS

ON NETWORK SCIENCE AND ENGINEERING, IEEE SYSTEMS JOURNAL,IEEE COMMUNICATIONS LETTERS, and IET Communications.

Rajesh M. Hegde (Senior Member, IEEE) receivedthe Ph.D. degree in computer science and engineer-ing from the Indian Institute of Technology Madrasin 2005. He joined IIT Kanpur as an AssistantProfessor of EE in May 2008. He became anAssociate Professor in 2012 and a Full Professorof EE in July 2016. He currently holds the UmangGupta Chair position at IIT Kanpur. From 2005 to2008, he worked as a Researcher with the CaliforniaInstitute of Telecommunication and InformationTechnology and concurrently as a Lecturer with the

Department of Electrical Engineering, University of California San Diego,USA, in 2007. Prior to 2005, he was involved with teaching undergraduateengineering courses at various levels in India. He has established two researchLaboratories at IIT Kanpur namely, Multimodal Information Systems Laband Wireless Sensor Networks Lab with funding obtained from BSNL, DST,MietY, LG Soft, Samsung Research, and Indian Space Research Organization.He has published prolifically at several international conferences and journalsin the area of signal processing, communication and networks. He is also amember of the National working groups of ITU-T (NWG-16 and NSG-6) ondeveloping multimedia applications. He actively teaches both undergraduateand post graduate courses related to electronics, digital signal processing, sta-tistical signal processing, array signal processing, wireless sensor networks,and digital speech processing. He was also awarded the P.K. Kelkar ResearchFellowship from 2009 to 2013. He is an affiliate member of IEEE-AASPtechnical committee.


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