This regular paper was presented as part of the main technical program at IFIP WMNC'2013

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A New Spectral Classification for Robust Clusteringin Wireless Sensor Networks

Brahim Elbhiri ∗ §, Sanaa El Fkihi § †, Rachid Saadane § ‡, Nourdine Lasaad §, Ali Jorio §, Driss Aboutajdine §∗ EMSI-Rabat, Morocco

elbhiri@emsi.ac.ma§Laboratoire de Recherche en Informatique et Telecommunication -unite associee au CNRST- FSR, Rabat, Morocco

Email: elbhirij@yahoo.fr, assad.noureddine@gmail.com, jorio.ali@gmail.com, aboutaj@fsr.ac.ma† ENSIAS, Av Mohammed Ben Abdallah Regragui, Madinat Al Irfane, BP 713, Rabat, Maroc

Email: elfkihi@ensias.ma‡ CSI, EHTP, Km 7, Oasis Route El jadida, Casablanca, Maroc

Email: rachid.saadane@ehtp.ac.ma

Abstract—Wireless sensor network has recently become anarea of attractive research interest. It consists of low-cost, lowpower, and energy-constrained sensors responsible for monitoringa physical phenomenon and reporting to sink node where the end-user can access the data. Saving energy and therefore extendingthe wireless sensor network lifetime, involves great challenges.For these purposes, clustering techniques are largely used. Usingmany empirical successes of spectral clustering methods, wepropose a new algorithm that we called Spectral Classification forRobust Clustering in Wireless Sensor Networks (SCRC-WSN).This protocol is a spectral partitioning method using graphtheory technics with the aim to separate the network in a fixedoptimal number of clusters. The cluster’s nodes communicatewith an elected node called cluster head, and then the clusterheads communicate the information to the base station. Definingthe optimal number of clusters and changing dynamically thecluster head election probability are the SCRC-WSN strongestcharacteristics. In addition our proposed protocol is a centralizedone witch take into account the node’s residual energy to definethe cluster heads.We studied the impact of node density on the robustness of theSCRC-WSN algorithm as well as its energy and its lifetime gains.Simulation results show that the proposed algorithm increasesthe lifetime of a whole network and presents more energyefficiency distribution compared to the Low-Energy AdaptiveClustering Hierarchy (LEACH) approach and the CentralizedLEACH (LEACH-C)one.

Index Terms—Graph theory, Spectral classification, EnergyConsumption, Clustering, Wireless Sensor Networks.

I. INTRODUCTION

Wireless Sensor Networks (WSNs) are usually self-organized networks composed of a large number of wirelesssensor nodes with a small size, a low battery capacity, alow processing power, a limited buffer capacity, and a low-power radio. The WSN has been widely studied and employedin many applications such as monitoring environments andembedded systems( [28], [1], [2]). It consists of spatiallyrandom distributed autonomous devices over a wide area usingsensors to cooperatively monitor physical or environmentalconditions, such as temperature, sound, vibration, pressure,motion or pollutants, at different locations [1]. This networkcontains a large number of nodes which sense data from an

inaccessible area and send their reports towards a processingcenter called ”Base Station (BS)”.In wireless sensor networks, sensor nodes are usually battery-powered, but it is not practical to recharge or replace thebatteries of all the sensors. Indeed, the number of sensor nodesin a given WSN, is too large on the other hand, these nodesare positioned in remote, battlefield, desert or hostile areas.Since sensor nodes are power-constrained devices, frequentand long-distance transmissions should be kept to minimumin order to extend the network lifetime [2], [3]. Moreover,in a WSN, a large part of energy is consumed when thewireless communications are established [9]. Therefore, directcommunications between nodes and the base station are notencouraged. Due to these assessments and in order to enhancethe network lifetime, particular innovative techniques thatimprove energy efficiency are highly required. One effectiveapproach consists in dividing the network into several clusters;each one of them elects one node as its cluster head [4].The cluster head collects data from its own cluster sensorsand aggregates these data which will be transmitted to thebase station. Thus, only some nodes of a WSN have totransmit data over a long distance while the rest of nodeswill need to ensure short-distance transmission. Consequently,more energy is saved and the overall network lifetime can beimproved. Basically, any clustering algorithm involves clustermanagement which consists of defining the suitable numberof clusters, selecting the cluster head for each cluster andcontrolling the data transmission within clusters and fromcluster heads to the base station [1]. Many energy-efficientrouting protocols are designed based on the clustering structurewhere the cluster heads are rotated and periodically elected [5],[6], otherwise the cluster heads die quickly. These techniquescan be extremely effective in broadcast and data query [7],[8].Spectral methods [25] for clustering have attracted attentionover the past few years in many applications, such as imagesegmentation [26] and social network analysis [27]. Theyusually involve taking the top eigne vectors of some matrixbased on the distance between points (or other properties) and

then using them to cluster the various points [24]. In case ofWSN, our contribution consists in using the spectral clusteringso as to extend the network lifetime. Hence, in this paperwe propose the Spectral Classification for Robust Clusteringin Wireless Sensor Networks (SCRC-WSN) protocol. Thismethod is a spectral partitioning one using the graph theorytechnics in order to separate the network to a fixed and anoptimal number of clusters. The cluster’s nodes communicatewith an elected node called cluster head. The latter collects,processes and transmits the information to the base station.The strongest SCRC-WSNs characteristics are summed up inthe choice of an optimal number of clusters and the fact thatthe cluster head election probability is changing dynamically.The remainder of this paper is organized as follows: Section IIpresents some previous work related to clustering approachesin WSNs. Section III exhibits the problem outline. The detailsand the properties of the proposed SCRC-WSN approach, aregiven in section IV. Section V evaluates the performance ofthe proposed approach compared to others algorithms. Finally,the conclusion and perspectives are drawn in section VI.

II. RELATED WORK

In a WSN, the large part of energy is consumed when thewireless communications are established [9]. With the aim toface this problem, several communication protocols have beenproposed. In particular, different techniques were proposed toguarantee transmissions providing efficient energies in ad hocnetworks. Thus, many communication models and protocolsthat are designed for specific applications, queries, and topolo-gies, exist.The use of clustering is one of the effective ways to dealwith the above problem. A clustering technique consists individing the network into several clusters (or classes). Then,in each cluster, one node is selected as the cluster head.This latter collects and aggregates data from its own areaand transmits it to the base station. Hence, only some nodesare required to transmit data over a long distance and therest of nodes will need to ensure short-distance transmission.Consequently, more energy will be saved and the overallnetwork lifetime could be prolonged. Nonetheless, there aretwo WSN categories: Homogeneous and heterogeneous. Inthe first category of networks, all nodes have the same initialenergy whereas they have different energies in the secondWSN set. Because of this, two kinds of clustering schemes canbe defined: the homogeneous clustering and the heterogeneousone.

In the last decade, there has been significant recent growthin the research activity of WSN clustering. A consider-able number of techniques has been proposed, such asLEACH [10], LEACH-E [11], LEACH-C [11], HEED (Hy-brid, Energy-Efficient Distributed clustering) [13], EDEEC(Equitable Distributed Energy-Efficient Clustering) [19], andthe SBDEEC(Stochastic and Balanced Distributed Energy-Efficient Clustering) [20]. In this paper, we give a briefdescription of recent works in homogeneous WSN. As to

the other heterogeneous clustering algorithms one can referto exhaustive surveys found in [28], [30], and [29].

• The LEACH algorithm [10] is considered as a fundamen-tal method in homogeneous clustering technics. LEACHchooses cluster heads periodically and distributes con-sumed energy uniformly by role rotation. Under the het-erogeneous network, this protocol will become poor andnot efficient. Moreover, The main problem of LEACHprotocol is the random selection of cluster heads. Becauseof this problem the probability that the determined clusterheads are unbalanced exists. Indeed, all the elected clusterheads may be located in a small part of the networkmaking the rest of this last unreachable.

• The HEED (Hybrid, Energy-Efficient Distributed clus-tering) protocol proposed in [13], is another distributedcluster based protocol in which the election of clusterhead is stochastically made and depends upon the residualenergy of nodes. In heterogeneous WSNs, there is aprobability that the lower energy nodes could own largerelection probability than the higher energy of nodes.

• Another homogenous clustering technique is the cen-tralized LEACH (LEACH-C) described in [11]. In thisalgorithm, the different cluster heads are elected by theBS. The latter starts by receiving all information abouteach node regarding its location and energy level, andso it runs the algorithm to form firstly the cluster headsand then the clusters. Here, the number of cluster headsis limited and the selection of the cluster heads is alsorandom but the base station makes sure that any node withless energy does not become a cluster head. However,LEACH-C is not feasible for larger networks for tworeasons: (1) Nodes that are far away from the base station,will have problem sending their states to the BS. (2) Asthe role of cluster heads rotates all the time, the far nodeswill not reach the base station in quick time increasingthe latency and delay.

• In the PEGASIS (Power-Efficient Gathering in SensorInformation Systems) protocol [12], all network becomelike a one chain which is calculated by nodes or by thebase station. Only one node of the chain aggregates alldata and sends it to the Sink. The main problem withthis protocol is based on the requirement of the globalknowledge of the network topology. It assumes that allnodes maintain a complete database about the locationof all other nodes in the network. Moreover, the strategyof which the node locations are obtained is not outlined.Note also that this protocol is applied only for a fixedtopology. Indeed, a movement of some nodes can affectextremely the protocol functionality.

The spectral methods for clustering have recently started toget a great attention in many research areas. These methodsmake use of the spectrum of the adjacency matrix of the datato cluster a considered set of elements. They are consideredas powerful techniques in data analysis. For this reason, wepropose to include these concepts to manage a given WSN

and to improve its lifetime.

III. PROBLEM OUTLINE

Periodically, the wireless sensor network nodes sense theenvironment and transmit the data to the base station. Thislast analyzes the data and gives some conclusions about theactivities in the supervised area. The most advantages ofWSNs are the easiness of deployment, the low installationcosts, the possibility to distribute the tiny sensors over awide region, the tolerance of error and the aptitude to self-configuring. In this section, the problem speech is introduced.To this end, we make a few statements about the networkscheme and we introduce the radio model used in this study.In our work, we use the energy mode and analysis that arepresented in [10] and [11]. The radio energy dissipation modelis illustrated in figure 1.

Fig. 1. Radio Energy Dissipation Model

In this figure, ETx(L, d) presents the energy spent totransmit L-bits over a distance d, and ERx is the energy spentto process an L-bit message. The parameter Eelec denotes theenergy per bit dissipated to run both the transmitter and thereceiver circuits. This parameter depends on many factors suchas the digital coding, the modulation, the filtering, and thespreading of the signal [11].The spent energy by the radio transmitter is given by

ETx(L, d) =

{L · Eelec + L · Efs · d2 if d < do

L · Eelec + L · Emp · d4 if d ≥ do(1)

where Efs and Emp present the amplifier energy respectivelyin a free space model (with d2 power loss) and in a multipathfading (with d4 power loss) channel models. They depend ond, where d is the distance between the transmitter and thereceiver. If the distance d is less than a threshold do, thenthe free space model is used; otherwise, the multipath modelis used. The value of the threshold do has been given byHeinzelman et al. in [11]. It is defined as follows:

do =

√EfsEmp

(2)

Given a WSN, its nodes are equipped with sensing, dataprocessing and radio transmission units. Nevertheless, theWSN power is highly limited. Once a sensor node runs outits energy, it is considered as died. Due to the sensor’s limitedpower, innovative techniques improving energy efficiency toextend the network lifetime are highly required. The hierar-chical routing approaches are mostly used. The main idea of

these algorithms consists in a random/rotate selection of thecluster heads, and a balancing energy consumption through thenetwork. Furthermore, considering a WSN, the total dissipatedenergy during a round (ERound) is determined by:

ERound =

K∑k=1

ECHk+

N−K∑j=1

ENCHj(3)

ECHkis the consumed energy when the cluster head of the

cluster labelled k, receives, aggregates, and transmits data tothe base station. Whereas ENCHj

is the consumed energy bya non cluster head labelled j.

On one hand, the ECHkis defined by the next equation:

ECHk= ECHtoBSk

+ EReceptk + EAggregk (4)

Where,

ECHtoBSk= L.Eelec + L.Emp.d

4toBSk

(5)EReceptk = |Sk|.L.Eelec (6)EAggregk = |Sk|.L.EDA (7)

with• ECHtoBSk

is the energy consumed when the cluster headof the cluster labelled k transmits data to the base station.

• EReceptk is the total energy consumed when the clusterhead of the cluster k receives data from its own clusternodes.

• EAggregk is the total energy needed, by the cluster headof the cluster k, to process data.

• |Sk| is the cardinal of the set enclosing nodes of thecluster labelled k.

Assuming that the nodes are uniformly distributed in a squarearea M ×M , the average distance between the cluster-headsand the base station (dtoBS) is given by [11]:

dtoBS = 0.765M

2(8)

Hence,K∑k=1

ECHk= L((N +K).Eelec +N.EDA +K.Emp.d

4toBS)

(9)On other hand, in the network there are N −K non cluster

heads. Some of the lasts operate on the free space mode whilethe others operate on the amplification mode. Let l be thenumber of last category of nodes. We have:

N−K∑j=1

ENCHj=

l∑i=1

L(Eelec + Empd4i ) +

N−K−l∑j=1

L(Eelec + Efsd2j ) (10)

N−K∑j=1

ENCHj = (N−K)L.Eelec +

(

l∑i=1

d4i ).L.Emp + (

N−K−l∑j=1

d2j ).L.Efs (11)

From eq.(9) and eq.(11), we conclude that the total dissipatedenergy during a round in a given WSN is defined by:

Eround = L(2N.Eelec +N.EDA +K.Emp.d4toBS +

(

l∑i=1

d4i ).Emp + (

N−K−l∑j=1

d2j ).Efs) (12)

where dj < do and di ≥ do.In a given WSN, the challenge is to reduce the total

consumed energy of each round (given by eq.(12)) with theaim to enlarge the whole network lifetime. Besides, twomain problems must be considered when defining a clusteringmethod. They are summed up in: (1) what is the optimalnumber of clusters (K)? and (2) how the cluster heads canbe selected? To deal with these questions, we propose a newprotocol using a spectral clustering approach. The later isbased on the graph theory classification which is an attractiveand easy method to implement. The next section explains indetails the proposed solution.

IV. SCRC-WSN APPROACH

In this section, a new homogenous clustering protocol forWSN is presented. We call it Spectral Classification for a Ro-bust Clustering in Wireless Sensor Networks (SCRC-WSN).In this study, we consider a network with N nodes, uniformlydistributed within a M × M square region. Moreover, weassume that the network topology remains unchanged overtime and the base station is placed in the network center.In addition, as to avoid that each node needs to know theglobal knowledge of the network (which is quite unrealisticfor WSN), in the proposed SCRC-WSN algorithm, the basestation collects the different node positions and applies theclustering process. We note that each node knows its ownlocation, which can be obtained at a low cost from GlobalPositioning System(GPS) or by using some localization system[31], [32]. Then, the WSN nodes transmit their location in ashort message to the Base Station.

The three main steps of the proposed SCRC-WSN protocolare summed up in figure 2.

Fig. 2. Overview of the SCRC-WSN Algorithm

The following will provide further explanations and detailsof the different steps of our proposed algorithm aiming atproviding a power-efficient WSN.

Let us first start by fixing some notations:• N is the total number of nodes in the WSN ;• K denotes the number of clusters in the network ;• Engi is the energy of the node i (i ∈ {1, 2, ..., N}) ;• (xi, yi) is the geographical position of the node i ;• |Sk| the total node number of the cluster k (k ∈{1, 2, ...,K}).

A. Clustering step

The objectives of the current step are to define the optimalnumber of clusters and to form them.

Each WSN can be represented by its corresponding undi-rect graph G(V,E). Where V is the set of vertices (nodes)representing different sensor nodes, and E is the edge setenclosing all dependencies between the nodes. Each vertexof V is identified by an index i ∈ {1, ..., N}. eij indicates theedge between the node i and the node j. We assume that Gis a graph without loops or multiple edges. Let A ∈ <N×Nbe the adjacency matrix of the graph G. Each value of A isassociated to each pair of the graph nodes (i, j). This value isof gaussian type and the matrix A is given by eq.(13).

A = [aij ] =

{exp( 1

2σ2 d2(i, j)) i 6= j

0 otherwise (13)

The total weight of edges incident to node i is given by dii =∑Nj=1 aij . The degree matrix D ∈ <N×N of G is a diagonal

matrix defined by D = [dij ]. The N ×N Laplacian matrix ofthe graph is defined by eq.(14).

L = D− A. (14)

Let us denote λi (with i = 1, ..., N ) the eigenvalue of thematrix L. By definition, the row sums of L are always zero.Therefore, the Laplacian matrix has always a zero eigenvaluewith associated right eigenvector 1 = [1, 1, ..., 1]T of length N .According to Gersgorin circle theorem [14], the eigenvaluesof L have nonnegative real parts and are all in a circle ofradius N that passes through zero. Moreover, the Laplacianmatrix can be written as follows: L.X = λ.X , where X isthe eigenvectors of the matrix L and λ is the vector of theeigenvalues of L.To define the optimal bi-partitions of a given graph, the secondeigenvectors of graph’s Laplacian, also called Fiedler vector, isused. Here, the eigenvector is seen as a solving discrete graphpartitioning problem and it can be shown that cuts based onthe second eigenvector give a guaranteed approximation tothe optimal division [21]. Thus, two clusters are identified bysplitting the sorted second eigenvector in two based on thesplit point equal to zero. Here, the split point is the same asthe one used in the clustering algorithm called bi-partitioning.Then, the G set of vertices will be defined by V = V +

⋃V −

such that V − and V + are the vertex sets of the new subgraph(cluster). Remark that V +

⋂V − = φ.

Nonetheless, the most important question raised by the pro-posed strategy concerns the condition that must be consideredfor stopping decomposition. With the aim to respond to thisquestion, we consider the total consumed energy in each round(given in eq.12). We note that by considering K cluster, theconsumed energy depends on the distances between the clusterhead and the non cluster heads of each cluster. i.e. Eround isminimal if

(

l∑i=1

d4i ).Emp + (

N−K−l∑j=1

d2j ).Efs (15)

is minimal. However, it is known that the amplifier energy in amultipath fading channel models is greater than the amplifierone in a free space model (i.e. Efs ≤ Emp). In addition, wehave dj < do ≤ di. Thus, to minimize the formula given ineq.15, all non cluster head nodes must operate in a free spacemodel.Let θdo be defined the next equation:

θdo = exp(1

2σ2d2o) (16)

The objective function that allows to decide whether to decom-pose or not a given graph, is defined by the adjacency matrixA. The allowed threshold to this function is θdo . Hence, if atleast one element of A is greater than θdo , the considered graphwill be split into two subgraphs (optimal bi-partitions) by usingthe second eigenvector of graph’s Laplacian (as describedabove).

In view of the fact that the majority of analysis in spectralgraph partitioning appears to deal with partitioning the graphinto exactly two parts, these methods can be adapted and willbe typically applied recursively to find the optimal number ofclusters [23]. The basic idea consists in partitioning recursivelythe graph into two parts and reapply the same procedure tothe subgraphs. Consequently, the optimal number of groups(clusters) is directly controlled by the threshold θdo allowed tothe objective function. Figure 3 shows the recursive algorithmin a hierarchical divisive manner.

Fig. 3. Recursive hierarchical clustering.

For Each cluster we define an identification k, the set ofcluster nodes Sk, and the total number of nodes |Sk| in thiscluster. Moreover, we specify each node on the network by anew id number in its appropriate cluster. Here, for each cluster,the node with the smallest number id in the whole network willtake the smallest id in the suitable cluster. Figure 10 shows indetails this technic of nodes id specification.

Fig. 4. The id nodes specification on the appropriate cluster in the Clusteringstep.

We notice that in the SCRC-WSN algorithm we deter-mine the clusters before specifying the cluster heads. Inthis algorithm, the optimal number of cluster partitions is aswell defined automatically. So, our algorithm is completelydifferent from the others (such as LEACH, SEP, ...). In orderto define the number of cluster heads and cluster’s partitioning,the latter protocols run the same technique in each iterationand by the way consume more energy.

B. Cluster head election step

Once the clusters are determined, the next step consists indefining the cluster heads. Note that numbered node id willbe in some random position on the cluster. Thus, the clusterhead in each round of communication, will be at a randomposition on the cluster. It is so important that nodes die atrandom locations of the network. The rational idea behindnodes death at random position is to make the sensor networkrobust to failures. Moreover, by taking in consideration onlythe nodes id (identification) in clusters, the cluster heads willbe determined. Indeed, in the round r of the simulation, weuse the number ck = (rmod |Sk|) to select the suitable clusterhead for the appropriate cluster; where |Sk| represents the total

number of nodes in a defined cluster k. Node with id = ck andresidual energy Engck greater than threshold ΘEng (Engck >ΘEng) will be the cluster head of the cluster k in the roundr. We define ΘEng as the minimum residual energy requiredfor a given node to be a cluster head. It is the summation ofthe energy needed to receive and process data coming fromthe appropriate cluster nodes, and to transmit towards the basestation. This ΘEng is given as follow:

ΘEng = L((|Sk|+ 1).Eelec + |Sk|.EDA + Emp.d4i ) (17)

Where di is the distance between the node i and the basestation. Nevertheless if the residual energy Engck is lessthan this threshold ΘEng (Engck > ΘEng), this node mustsbroadcast a short message informing the node with id = ck+1to be the new cluster head at the iteration r. Consequently, eachcluster head will be able to collect data from the cluster nodesand will transmit the aggregate information to the BS. Thus,the number of the direct transmissions is efficiently reducedand the whole network lifetime is enlarged. In addition, theenergy consumption will be distributed with more equatabilitybetween all nodes.

C. Data transmission

Once clusters and cluster heads are created, each clusterhead knows which nodes it is supervising. Based on thenode’s id in the appropriate cluster, a Time Division Multiple-Access (TDMA) MAC protocol schedule assignment willbe generated automatically. If we suppose that the nodewith the id = i is elected as a cluster head, the node withid = (i + 1 + |Sk|)mod |Sk| will take the first time slotto transmit; where |Sk| is the total number of nodes in thedefined cluster k. Here, we avoid the techniques applied bythe traditional algorithms which consume more energy andask for more synchronization when the cluster heads areelected to assign the TDMA access. Moreover, this techniqueguarantees that there are no collisions among data messagesand also allows the radio components of each non-clusterhead node to be turned off at all times except during theirtransmit time, thus reducing the energy consumed by theindividual sensors [11].Assuming that all nodes can transmit with enough power toreach the base station, if the distance between any node andthe base station is less than the distance between this nodeand its corresponding cluster head, the node will transmitdata directly to the base station. Now, each non cluster headsends its data during their allocated transmission time to itsrespective cluster head. The last must keep its receiver on inorder to receive all the data from the nodes in the cluster.When all data is received, the cluster head node performssignal processing functions to compress the data into a singlesignal. Once this phase is completed, each cluster head cansend the aggregated data directly to the base station. In thissub-phase, each non cluster head can turn off to the sleepmode in order to reduce the consumed energy.

Our proposed Spectral Classification for Robust Clusteringin Wireless Sensor Networks algorithm is summarized up inprocedure (1).

Procedure 1: SCRC-WSN Algorithm• Input : {(Eng1, x1, y1), ...., (EngN , xN , yN )}, θEng , andθdo .

• Initialization: Current number of clusters: K = 1.• Steps :1) /*Clustering*/

• Each node sends its position to the BS.• For each cluster k in {1, 2, ...,K}

– BS constructs the graph and the adjacency matrixrepresenting the network.

– Compute l the number of the adjacency matrixelements greater than θdo .

– If l 6= 0

∗ BS forms the graph Laplacian matrix andcomputes the eigenvalues and the eigenvectorsof this matrix.

∗ BS selects the second eigenvectors (Fiedlervector) of the Laplacian matrix and comparesthe Fiedler vector to 0 to form the new 2× 1clusters.

∗ K = K + 1∗ Define the new node’s ids in the appropriate

cluster.End IfEnd For

2) /*Cluster Head Election*/• For each round r

– For each cluster k in {1, 2, ...,K}∗ Each node of the cluster k computes the valueidch = rmod |Sk|.

∗ While Engidch < θEng

idch = idch + 1

End While.∗ Node with idch will be the cluster head.End For

3) /*Steady State Phase*/– For each cluster k in {1, 2, ...,K}∗ Cluster Time−slot assignments.∗ Collects measurement from appropriate Clus-

ter nodes.∗ Cluster heads send data to the BS.End For

End For• Output : { Global area knowledge, the number of clustersK, and {Sk/k ∈ {1, 2, ...,K}} }

By using the proposed SCRC-WSN Algorithm, the totalconsumed energy (ESCRC−WSN

round ) of each round is given by

eq.(18).

ESCRC−WSNround = L(2N.Eelec+N.EDA+K.Emp.d

4toBS +

(

N−K∑j=1

d2j ).Efs) (18)

where dj < do.In a given WSN, we note that:• If all non cluster heads operate in a non free space mode,

the total dissipated energy during a round is determinedby:

E1round = L(2N.Eelec +N.EDA +

K.Emp.d4toBS + (

N−K∑i=1

d4i ).Emp) (19)

where di ≥ do.• If at least one non cluster head operates in a non free

space mode, the total dissipated energy during a round isdetermined by:

E3round = L(2N.Eelec +N.EDAK.Emp.d

4toBS +

K.Emp.d4toBS + (

l,l 6=0∑i=1

d4i ).Emp + (

N−K−l,l 6=0∑j=1

d2j ).Efs)

(20)where dj < do and di ≥ do.

• If all non cluster heads operate in a free space mode,which is exactly our case, the total dissipated energyduring a round, is as the same as the one given by eq.(18).

Consequently,

ESCRC−WSNround ≤ E2

round ≤ E1round (21)

We conclude that the total dissipated energy during a round isminimal when the SCRC-WSN algorithm is used.

V. SIMULATION RESULTS

All simulations are based on the following protocol. Weconsider many wireless sensor networks with N nodes ran-domly distributed in a 100m× 100m field. The Base Stationis located in the center of the sensing area at location (x=50,y=50). Moreover, we ignore the effect caused by the signalcollision and the interference in the wireless channel. Sincethe nodes have limited energy, they consume their energiesduring the course of simulations. Once a node runs out ofenergy, it is considered as dead and cannot transmit or receivedata. For these simulations, energy is removed whenever anode transmits or receives data and whenever it performs dataaggregation using the radio parameters shown in tableI.

We propose to compar our proposed method to:• The LEACH protocol; it is considered as the basic

protocol of many existing methods.• The LEACH-C protocol; witch is the centralized version

of the LEACH algorithm. In particular, our method isconsidered as a centralized one because clusters aredetermined by the BS.

TABLE IRADIO CHARACTERISTICS USED IN OUR SIMULATIONS

Parameter ValueEelec 50 nJ/bitEfs 10 pJ/bit/m2

Emp 0.0013 pJ/bit/m4E0 0.5 J

EDA 5 nJ/bit/messaged0 88 m

Message size 4000 bitsPopt 0.1

A. Impact of node density on the robustness of SCRC-WSNalgorithm

In this section, we propose to evaluate the robustness of thedifferent compared protocols for different values of the nodedensity N .The figure 5 shows the effects of the node density on thecompared clustering techniques as well as on the network’sstable regions (First Node Dead ”FND”).

Fig. 5. Impact of the node density N on the performances of the comparedalgorithms.

As shown in figure 5, for different values of N rangingfrom 100 to 300, the SCRC-WSN algorithm presents animprovement of the performance compared to the LEACH andthe LEACH-C ones. For N = 200, the mean of the first nodedead occurs at the 1150 round by using SCRC-WSN whereasthis value is about 650 and 630 when the LEACH and theLEACH-C algorithms are used. It follows that even if thenode density increases the SCRC-WSN approach still givesbest results compared to the others protocols.The robustness of the new proposed algorithm is certainly dueto the fact that the clustering process is firstly used before theprocess of the cluster head election. Besides, in the SCRC-WSN approach, the election of the cluster head is based onthe residual energy of each node.We notice that the main problem of the LEACH and theLEACH-C protocols is the random selection of cluster heads.

Because of this problem the probability that the determinedcluster heads are unbalanced exists. Indeed, all the electedcluster heads may be located in a small part of the networkmaking the rest of this last unreachable.

For N = 200, the figure 6 presents the different clusters ofthe network by using the SCRC-WSN algorithm. We note thatthe network is subdivided into six clusters that are correctlydistributed over the sensing area. There is no intersectionbetween the different clusters.

Fig. 6. Network clustering using the SCRC-WSN algorithm with N=200and 100× 100 area

B. Energy and lifetime gains

In these experiments, we run the three compared algorithmsunder many networks with N = 200 nodes. Each node beginswith an amount energy equals to 0.5J of energy and anunlimited amount of data to send to the BS. Figure 7 gives thecurves of the number of nodes alive over time for the LEACH,the LEACH-C, and the SCRC-WSN algorithms.

Fig. 7. Number of nodes alive over time of the compared protocols

This figure shows a significant improvement of our protocolin terms of enlarging the round number of the first node dies.Furthermore, it is obvious that the stable time of SCRC-WSNis extended for the whole network compared the to two otheralgorithms. Here, the first node death occurs over 41% timeslonger than the first node death in LEACH and LEACH-Cprotocols. Moreover we can see that the unstable region of ouralgorithm is larger than the ones of the LEACH and LEACH-Cprotocols.

Figure 8 gives the total network remaining energy in everytransmission round by using the three compared approaches.

Fig. 8. Evolution of the remaining energy in the network when thetransmission rounds succeed.

The network remaining energy decreases rapidly in theLEACH and the LEACH-C protocols than in the SCRC-WSNone. So, it presents, for the two first compared methods, a slopeapproximately -0.088J/Round, compared to -0.075J/Round inSCRC-WSN. Then, the network energy depletion is fast inLEACH and LEACH-C than our algorithm. In addition, wecan see that, in the 1000 first transmission rounds, approxi-mately 88% of the total network energy is consumed in caseof the LEACH and the LEACH-C approaches. Whereas, theSCRC-WSN protocol consumes only 76% of the total energyof the whole network.

Moreover, figure 9 presents the number of received mes-sages by the BS for both LEACH and SCRC-WSN algorithms.It is shown that the delivered messages to the BS by ourprotocol are better than those delivered by the other protocols.In fact, the performances of the new approach are increasedby 60% compared to the LEACH one. This is certainly dueto the fact the new approach assumes that the nearest nodesto the base station transmit the data directly to this last.Remark that by using the SCRC-WSN algorithm, the numberof received messages is less than this number when theLEACH-C protocol is used. This influences the total networkconsumed energy. Hence, in this case, our approach presentsthe optimal solution of the three compared methods.

Fig. 9. Number of message received at base station over time.

Figure 10 shows the performances of the different comparedprotocols by using different initial energies. It gives the thefirst node death round depending on the quantity of the nodeinitial energy. It is shown that for different values of the energy,the SCRC-WSN algorithm presents a significant improvementcompared to the other protocols. We note that the rationalraisons behind this are the ways of choosing the cluster andthe cluster heads of the WSN. In fact, taking into account theresidual energies of nodes and requiring nodes to work in thefree space communication mode, allow to improve the networklifetime.

Fig. 10. Impact of the initial energy quantity on the performances of thethree compared algorithms.

We conclude that the SCRC-WSN algorithm gives a signifi-cant performance improvement in terms of energy and lifetimegains, compared to the LEACH and the LEACH-C protocols.

VI. CONCLUSION

In this paper we have proposed a new approach to deal withthe clustering problem in a given wireless sensor network. Wehave explained in details the proposed SCRC-WSN protocolwhich is a Spectral Classification for Robust Clustering inWireless Sensor Networks. It is an energy-aware adaptiveclustering protocol with an adaptive approach which employsthe graph theory and spectral classification to guarantee robustclustering. Moreover, SCRC-WSN uses this concept whichoffers a better use and optimization of the dissipated energyin the network. In this situation, the network base sration(BS) computes the adjacency and the Laplacian matrixes ofthe network graph in order to run the SCRC-WSN protocol.By using the last, we demonstrated that a robust clusteringdepends only on the node positions, the coverage range, andthe node density. Moreover using a central control algorithmto form the clusters has produce better results throughout thenetwork. With this strategy we avoid the charge of treatmentscomplexity generated by all network nodes. The strategiesintroduced into the SCRC-WSN protocol allow to outperformits performances by saving more energy and enlarging moreefficiently the network lifetime.In addition, we have measured and compared the robustnessbetween the SCRC-WSN algorithm and the LEACH and theLEACH-C ones. It has been shown that the first approachpresents a significant performance improvement in terms ofenergy and lifetime gains, compared to the other ones.Further works remain for studying other spectral classificationtechniques which may be more efficient in this kind ofapplications. Selecting the robust one will be the primordialstep in the coming work.

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