Research ArticleAnalysing Topology Control Protocols in Wireless SensorNetwork Using Network Evolution Model
Chiranjib Patra1 Samiran Chattopadhyay2
Matangini Chattopadhyay3 and Parama Bhaumik2
1Department of Information Technology Calcutta Institute of Engineering and Management Tollygunge Kolkata 700040 India2Department of Information Technology Jadavpur University Kolkata 700032 India3School of Education Technology Jadavpur University Kolkata 700032 India
Correspondence should be addressed to Chiranjib Patra chiranjibpatragmailcom
Received 2 October 2014 Accepted 20 April 2015
Academic Editor Kameswara Namuduri
Copyright copy 2015 Chiranjib Patra et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In the study of wireless ad hoc and sensor networks clustering is an important research problem as it aims at maximizing networklifetime and minimizing latency A large number of algorithms have been devised to compute ldquogoodrdquo clusters in a WSN but fewpapers have tried to characterize these algorithms in an analytical manner In this paper we use a local world model to understandand characterize the functioning of three tree based clustering algorithms In particular we have chosen simple tree CDS Rule Kand A3 topology construction protocols Using our theoretical framework based on a complex network model we have also triedto quantify some of the observed features of these algorithms such as number of cluster heads and average degree of the resultantgraph The theoretically obtained measures have reasonably matched with measures obtained by simulation studies
1 Introduction
Wireless sensor networks (WSNs) are made up of a largenumber of sensor nodes These nodes are usually deployedin the environment to monitor several physical phenomenaHowever sensor nodes heavily depend on batteries as theyare the only source of energy in many WSN applications Asa result one major problem in WSNs is known as topologymanagement that leads to energy efficient transmission ofdata In this regard connections are set with nodes that areclose enough for radio signal to arrive with acceptable signalstrength However in order to improve energy efficiencytopology control process helps in reducing the connectionswith other neighbors of the node in the network Topologycontrol is an insistent process in which there is an initializa-tion phase which is common to all WSN deployments In theinitialization phase nodes make use of the revelation processby using maximum transmission power to build the initialtopology The initial network topology includes connections
and nodes that allow direct communication and every nodecommunicates with a subset of the nodes according to thedistance between them
Often the topology of a large wireless network is struc-tured in terms of a hierarchy where the network is viewed asa number of clusters and in each cluster there is a cluster headand other normal members Normal members in a clustercommunicate only with the cluster heads and the clusterheads communicatewith the sink in one ormultihopmannerThere are many challenges in finding out the ldquobestrdquo set ofcluster heads in a given network and in many formulationsthese problems turn out to be intractable Consequentlythere are many algorithms to select the cluster heads andthe clusters in a WSN that minimizes latency and maximizesnetwork lifetime
In many cluster head selection algorithms every nodeis selected as the cluster head in different rounds and theprobability of selecting a node as a cluster-head is the same forall nodes In this method the chances of energy dissipation
Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 693602 8 pageshttpdxdoiorg1011552015693602
2 International Journal of Distributed Sensor Networks
in cluster heads reduce if we consider large homogeneousWSNs There are other approaches where the idea of dom-inating set of graphs is used to devise an algorithm Someof these approaches are tree based Most of these algorithmsare distributed and work with local (at the most 2-hop)information available from any given node In this paper wehave considered three such topology construction protocolsnamely simple tree CDS-Rule K and A3 protocols forfurther exploration
In another development the theory complex networkshave recently received increasing attention for understandingthe topological structure functions and dynamical proper-ties of many real-world networks such as the social networksbiological networks and ad hoc networks One of the mostimportant models that can be used to formally characterizeclustering algorithms is known as B-A model [1] This modelis based on two foundational mechanisms growth andpreferential attachment A new node is added to the networkat each step and connects with an existing nodewith a specificprobability which is related to the degree of the existing nodeThe B-A network has the scale-free property and follows thepower-law distribution
The B-A network model is capable of capturing somebasic mechanism that is responsible for the power-law degreedistribution Still it hadmany limitations Li-Chenmodel [2]has improved upon B-A model This model has been able tobetter capture the dynamics of networks constructed with alocal preferential attachment mechanism
The local preferential attachment model [3] is based onthe common sense that people can collect information easilyfrom their local community than from far away environmentUsing preferential connection as the fundamental basismanyvariations of the scale-free network model have been pro-posed during recent years such as comprehensive multilocal-world model [4] Similar to preferential attachment modelthe physical position neighbourhoodsrsquo model [5] mimics theactual communication network The Poisson growth model[6] uses the number of edges added at each step as a randomvariable that corresponds to Poisson distributionThis modelcan generate many kinds of networks by controlling therandom number
Chen et al [7] have studied an evolving mechanism forformalizing fault-tolerant communication topology amongcluster heads with complex network theory Based on the B-A modelrsquos growth and preferential attachment mechanismsthey not only used a local-world strategy for the networkwhen a new node was added to its local-world but alsoselected a fixed number of cluster heads in the local worldfor the purpose of obtaining a good performance in terms ofrandom error tolerance
Luo et al [8] performed theoretical analysis and con-ducted numerical simulation to explore topology charac-teristics and network performances with different energydistributions among nodes Their results have shown thatthe network is better clustered and average path length fortransmitting data is reducedwhen energy distribution amongnodes is more heterogeneous
In [8] a new dimension is added as the nodes are not onlyallowed to join the network through preferential attachment
but they are also allowed to leave the network or not jointhe network through nonpreferential attachment Furtherthe nodes distinguish themselves as cluster head nodes andnormal nodes which is consistent with the function of manyclustering algorithms in WSNs
In this paper we have tried to provide a formalism forsome algorithms which computes clusters in a WSN usinga modified local network model based on similar modelsproposed in [2 9] and [8] In particular we have used threetree based clustering algorithms namely simple tree CDSRule K and A3 Using our theoretical framework we havealso tried to quantify some of the observed features of thesealgorithms such as number of cluster heads and averagedegree of the resultant graph The theoretically obtainedmeasures have reasonably matched with measures obtainedby simulation studies
The paper is organized as follows Section 2 describesa very brief review of the local-world model In Section 3we have described the application of local world model toprovide a framework to describe the functioning of threeclustering algorithms In Section 4 results of theoreticalresults and simulation studies are jointly presented Section 5concludes the paper
2 A Brief Review of Topology ControlProtocols and Local-World Network Model
In this section we shall discuss very briefly the Li-Chenmodel [2] and two topology control protocols which are usedin this paper
21 A3 Protocol [10] TheA3 protocol uses four types of mes-sages Hello message children recognition message parentrecognition message and sleeping message The sink nodestarts the protocol by transmitting an initial hello messageto its neighboring nodes Nodes accept the message if theyhave not been covered by another node they set their statesas covered select the transmitter as its parent node andanswer back with a parent recognition message If a parentnode does not get any parent recognition messages from itsneighbors it turns offThe parent node sets a timeout periodto accept answers from its neighboring nodes Once timeoutoccurs the parent node sorts the list of neighbor accepting itsmessage in decreasing order of some selection metric Thenparent node broadcasts a children recognition message thatincludes the complete sorted list to all its candidates Once thechildren accept the list they set a timeout period proportionalto their position on the candidate list During that timeoutthey wait for sleeping message from their brothers If a nodeaccepts a sleepingmessage during the time out period it turnsitself off
22 CDS Rule k Protocol [11] The CDS Rule k algorithmutilizes connected dominating set algorithm and pruningrules The idea is to start from a big set of dominatingnodes that produces a minimum criterion and prunes itaccording to a particular rule In the first stage the nodeswill interchange their neighbor databases A nodewill remain
International Journal of Distributed Sensor Networks 3
Local world
M = nodes in local world
m0 = universal set of nodes
m = edges which connect M nodes in local world
Figure 1 Illustration of various parameters in their roles asdescribing the local world and the universe
progressive if there is at least one pair of separated neighborsIn the second stage a node chooses to unmark itself if itdetermines that all its neighbors are covered bymarked nodeswith higher precedence which is given by the degree of thenode in the tree Lower level implies higher precedence Theultimate tree is a more compact version of initial one with allredundant nodes with higher or equal priority removed
23 Local Network Model We have used Li-Chen Model [2]This model is used to form a generalized local-world modelUsing the generalized model we have analyzed clusteringalgorithms of wireless sensor networks In this model eachnode has only local connection information Nodes connectonly in their local world based on their local connectivityThefollowing parameters are required to explain the dynamicswith reference from Figure 1
(1) We start from a small number of nodes 1198980and grow
at each time step 119905(2) When a new node chooses a connection to other
nodes the probability prod119896119894 that a new node is
connected to a node 119894 depends on the degree 119896119894of
node 119894 This probability is defined as follows
prod(119896119894) =
119896119894
sum119895119896119895
(1)
(3) We select 119872 nodes randomly from the existingnetwork which is referred to as local world of the newnode
(4) When a new node arrives we add that node with 119898
edges linking the new node to 119898 nodes in the localworld determined in (3) using preferential attach-ment with a probabilityprodlocal(119896119894) This probability isdefined as follows at every time step 119905 Consider
prod
local119896119894= prod
1015840 119896119894
sum119895119896119895
(2)
whereprod1015840(119894 isin Local minusWorld) = 119872(1198980+ 119905)
After 119905 time steps there will be a network with 119872 = 119905 + 1198980
nodes and119898 lowast 119905 edges
3 Applying Local-World Model in WSNTopology Control Algorithms
To use local world model to capture functioning of WSNclustering algorithm the model has to take into accounttwo types of nodes normal nodes and cluster nodes Thecluster nodes are the cluster heads and the normal nodesare members in a cluster There is only one cluster nodeattached to a normal node in other words the normal nodehas only one edge whichmeans that the normal node cannotrelay data from other nodes A cluster node can integrateand transmit data from other nodes Both of these two typesof nodes can connect to a cluster node and the number ofedges is limited in every cluster node because of its energyconsideration When a new cluster node joins the network itis randomly assigned an initial energy 119864
119894from the interval
[119864min 119864max] The limited number of edges in every clusternode is represented by 119896max 119894 which is based on the initialenergy of the cluster nodes 119864
119894where 119896max [12] is given as
follows
119896max 119894 = 119896max lowast119864119894
119864max (3)
119896max 119894 reflects the ability of having the maximum number ofedges for cluster node 119894
The growth model is described as follows Starting with asmall number of nodes (all of them are cluster nodes) theyrandomly link each otherThis results into an initial network
(1) Growth At every time step a new cluster node or anormal node with one edge enters into the existingnetwork with a probability 119901 or 1 minus 119901 respectivelyIf the new node is a cluster node then it is assigneda random energy value of 119864
119894as discussed A small
number of cluster nodes would cause many sensornodes to link to them which results in faster energyconsumption but a large number of cluster nodeswould be more wasteful in terms of energy efficiencyThus the value of 119901 is assumed to be in the range as0 lt 119901 lt 05
(2) Preferential Attachment A new node arriving at thenetwork links to an old cluster node that is selectedrandomly from the already existing network NodesinWSNs have the constraint of energy and connectiv-ity and only communicate data with the cluster nodesin their local area First119872 cluster nodes are selectedrandomly from the network as the new incomingnodersquos local world then one of the cluster nodes ischosen to link with the new node according to theprobabilityprodlocal(119896119894)
If the new incoming node is a cluster node then theprobability is set as follows
prod
119896119894
= (1minus119896119894
119896max 119894)
119896119894
sum119895isinlocal 119896119894
(4)
In this case when the value of 119896119894is high the probability that
it will be chosen to connect with the new node is higher
4 International Journal of Distributed Sensor Networks
If the new incoming node is a normal node then theprobability is defined as follows
prod
119888119894
= (1minus119896119894
119896max 119894)
119888119894
sum119895isinlocal 119888119894
(5)
where 119888119894is the number of edges of the cluster node 119894 The
greater the value of 119888119894is the higher the probability that it will
be chosen to connect with the new node Only through thisapproach we can adjust the number of cluster nodes that arelinked to one cluster node (cluster head)
Total preferential probability is given as follows
prod
119896119894
= prod
119896119894
+prod
119888119894
(6)
In [12] authors have considered the expenditure of energyin the process of linking nodes together The disadvantageis that the energy in a cluster node will be exhausted inonly few rounds if self-organization is allowed In fact theenergy consumption will be relatively low if only 119896max 119894 isconsidered to be the limit for a cluster node to connect toothers randomly
Antipreferential Attachment Let us consider a parameter 119911called the deletion rate or antipreferential attachment factorwhich is defined as the rate of links removed divided by therate of links added It is observed that lesser the energy of thenode the more will be the probability of it being deleted Letthis probability be denoted asprodlowast(119896
119894)
For the outgoing cluster nodes we have
lowast
prod (119896119894) asymp
11198980 + 119901 lowast 119905
(7)
For the outgoing normal nodes we have
lowast
prod (119888119894) asymp
11198980 + (1 minus 119901) lowast 119905
(8)
So the total antipreferential probability is given as follows
The antipreferential removal mechanism is more reasonablefor deleting links that are antiparallel with the preferentialconnection It is also consistent with the functioning ofclustering algorithms that runs in rounds in wireless sensornetworksThewireless nodes that do not have enough energythat is the dead nodes are to be removed from the systemThus antipreferential removal phenomenon is reasonable forclustering algorithms
Using mean field theory [9 13] a qualitative analysisof dynamic characterization of a wireless sensor networkcan be given By the mean-field theory the preferential
and nonpreferential attachment may be combined in thefollowing differential equation
International Journal of Distributed Sensor Networks 5
31 Analysis of the Dynamic Equation
311 Case I If 119911 = 0 119872 = 1 that is the new nodeselects node unless it reaches 119896 Moreover the preferentialattachment mechanism does not work The rate of growth of119896119894is as
120575119896119894
120575119905
=
11198980 + 119901 lowast 119905
(18)
The denominator of the above expression is the number ofcluster nodes at time 119905
312 Case II If 119872 = 1198980+ 119901 lowast 119905 this means that the local
total degree of node universetotal degree of nodes local-world
=
1198980 + 119901 lowast 119905 + 119873
1198980 + 119901 lowast 119905
=
1198980 + 119901 lowast 119905 + 1198980 + 119905
1198980 + 119901 lowast 119905
(22)
Therefore
119896 =
2 lowast 1198980 + 119905 + 119901 lowast 119905
1198980 + 119901 lowast 119905
(23)
Similarly we have for 119888119894
119888 =
sum119895119888119895
119888119894
= 119896minus
(1 minus 119901) lowast 119905
1198980 + 119901 lowast 119905
= 2 (24)
As the cluster node will have one such node attached to itselfthe status of that node is either another cluster head or normalhead hence the count value of 119888 is 2
Equations (21) and (23) are used to find the values of 119888 and119896
Finally the following equation is formed after substitutingthe values of sum
119860 + 119861 minus 2 lowast 119911 lowast 119862
)
1119860 (28)
Moreover to find the degree distribution 119875(119896) that is theprobability that a node has 119896 edges) we first calculate thecumulative probability 119875[119896
119894(119905) lt 119896] Suppose that the node
enters into the network at equal time intervals We define theprobability density 119905
119894as follows
119875 (119905) =
11198980 + 119905
(29)
So 119875[119896119894(119905) lt 119896] has the following form
Equation (31) denotes the degree distribution function tounderstand the dynamics of clustering algorithms Puttingback the values of 119860 119861 and 119862 we have the distributionfunction as the degree distribution 119875(119896) The probability thata node has 119896 edges is
119875 (119896) asymp (
119901
119901 minus 1+
1 minus 119901
119901
minus
2 lowast 119911
119901 lowast (1 minus 119901)
)
(1+119901)119901
lowast (119896
lowast
119901
1 + 119901
+
1 minus 119901
119901
minus 2 lowast 119911
119901 lowast (1 minus 119901)
)
(minus2lowast119901minus1)119901
(32)
Next the value of 119911 is computed that maximizes (32)Consider
119889119875 (119896)
119889119911
= 0 (33)
By solving (33) we obtain the value of 119911 as
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(34)
Anti preferential attachment means that the node may notlike to have a particular node as its neighbor but in thiscase of sensor networks no such phenomenon is observedhencewe can state that the anti preferential attachment in thisparticular case tends to zero during the evolution process
By setting the value of 119911 to zero we have the followingrelation
119901 =
12[(
119896 + 3119896 minus 1
)
05minus 1] (35)
Equation (35) states the probability of clustering when theantipreferential attachment factor 119911 is zero
4 Analysis of Topology Control Algorithms
In this section we have carried out simulation of threelocalized topology control protocols namely A3 Simple treeProtocol and CDS Rule K Simulation was carried using theAtarraya [14] simulator Total number of nodes was 100 200300 and 500 respectively and theywere tested for A3 SimpleTree Protocol and CDS Rule K The output was recorded foraverage degree of nodes 119896 for three protocols mentionedearlier
935361
916479
67
58
99
97
4934
44
3024
39
65
82
2194
907054
56
26
28
36 100
59
25
40
84
63
68
5787
32
51
47
27
Figure 2 Clustering as obtained by running simple tree protocol ina network of 100 nodes
100 3 0366 37 40200 3 0366 73 71300 4 0263 79 71400 4 0263 105 96500 4 0263 132 130119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
Figure 2 shows an example of clustering and computationof average 119896 value
Figure 2 shows the cluster heads in red circles with whichnormal nodes are attached Taking the number of all theneighboring nodes of every cluster heads and dividing by thenumber of cluster heads we obtain the average value of 119896 Bysimilar procedure we have obtained the average value of 119896 fordifferent number of nodes
We have also carried out simulation study using simpletree protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 1
Similar simulation study was also carried out using CDS-Rule k Protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 2
Similar simulation study was also carried out using A3Protocol for different number of nodes The number of clus-ters obtained using theoretical calculation (35) is compared
International Journal of Distributed Sensor Networks 7
100 4 0264 27 27200 6 0171 35 38300 10 0101 31 33400 10 0101 41 43500 13 0077 39 43119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
against the number of clusters obtained using simulationstudy which is presented in Table 3
41 Discussion on A3 and Simple Tree Protocol Next we plotthe theoretically calculated number of clusters and experimen-tally calculated number of clusters as shown in Tables 1 and 3for A3 and simple tree protocols
We observe the following
(1) In A3 and Simple Tree protocols the curves fortheoretically calculated number of clusters and exper-imentally calculated number of clusters follow fourthdegree equationThe trend line polynomial options ofMS Excel have been used to study the degree of curvefit of the curves to obtain the following polynomials
119899 = minus 4119864minus 101198734+ 8119864minus 071198733
minus 00001198732+ 0197119873
+ 12 (A3 protocol)
119899 = minus 3119864minus 081198734+ 4119864minus 051198733
minus 00161198732+ 3164119873
minus 148 (Simple tree protocol)
(36)
(2) In Figure 3 there is gap between the theoreticaland experimental curves in case of A3 and simpletree protocol This may be attributed due to thefact that A3 (approximate CDS) only preserves 1-connectivity whereas the simple tree protocol hasmultiple connectivity So A3 protocol requires lessenergy to construct the tree as compared to spanningtree which confirms to the experimental results
42 Discussion on CDS Rule K Consider (34) which ismentioned in the following
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(37)
where 119860 119861 119862 119896 and 119911 have been defined previouslyTable 4 enumerates the value of 119911 for various values of 119896
corresponding to different number of nodes in the networkFrom the results of Table 2 we obtain Figure 4In Table 4 the maximum value of 119911 corresponds to a
probability value 05 which indicates that the clusters havedissociated into one cluster head and one normal node whichideally predicts the case when the energy of the nodes hasdrained out
0
20
40
60
80
100
120
140
0 100 200 300 400 500 600
Simple tree (Th)Simple tree (Exp)
A3 (Th)A3 (Exp)
Number of nodes N
Clus
ter h
eads
(n)
Figure 3Number of nodes versus number of cluster heads in simpletree protocol and A3 protocol
05
101520253035404550
CDS Rule K (Th)CDS Rule k (Exp)
0 100 200 300 400 500 600Number of nodes (N)
Clus
ter h
eads
(n)
Figure 4 Total number of nodes (119873) versus number of cluster headsin CDS-Rule k algorithm
100 4 0263 27 34200 6 0171 35 40300 8 0126 38 49400 10 0101 41 48500 12 0083 42 46119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
In Figure 4 it can be seen that the mechanics of clus-tering in CDS-Rule K is in accordance to the assumptionsconsidered while deducing the distribution function usingmean field theory Unlike A3 and Simple tree protocol CDS-Rule K also exhibits a relation of degree 4 as shown (119899 =
minus2119864minus081198734+3119864minus05119873
3minus0012119873
2+2085119873minus87)The linear
graph as shown in Figure 4 represents the situation when
8 International Journal of Distributed Sensor Networks
Table 4 Value of 119911 for different 119896 values in networks of differentsizes
Nodescollection (119873) 119896 Maxima (119901 119911) Minima (119901 119911)
the value of 119911 isminThis graph is linear due to constant valueof 119901 (05 here) where other graphs were drawn on constant 119911(0 here)
Above experimental results lead us to the followingexplanation of the antipreferential factor 119911
The antipreferential factor can have three distinct values
(a) When 119911 = 0 then the effective degree of nodes is equalto the analytically obtained degree of a node Here thenumber of cluster heads formed will be optimal
(b) When 119911 gt 0 then a node will have fewer connectionsavailable Here the number of cluster heads will behigher
(c) When 119911 lt 0 then a node will have less connectivityHere the number of cluster heads will be highest asthe result will lead to dissociation
5 Conclusions
In this paper we have tried to provide a framework toformally model tree based clustering algorithms in a WSNBased on the formalism we have theoretically calculatedsome parameters such as number of cluster heads and averagenumber of degree for a given algorithm The theoreticalresults tally with results obtained by simulation studies Wehave introduced a factor called 119911 in network evolutionmodelIt seems that the 119911 factor has an impact on functioning ofthese protocols We keep that study as a future exercise Thestudy at the current stage is more suitable for modeling treebased clustering algorithms thatwork on the top of connecteddominating set construction Algorithms that are based onmore parameters have to be modeled appropriately in theframework before application
In recent papers energy distribution has been consideredin network evolution model for wireless sensor networksBut so far the framework of network evolution model hasnot been used to capture the characteristics of clusteringalgorithms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquoAmericanAssociation for theAdvancement of Sciencevol 286 no 5439 pp 509ndash512 1999
[2] X Li and G Chen ldquoA local-world evolving network modelrdquoPhysica A vol 328 no 1-2 pp 274ndash286 2003
[3] L-N Wang J-L Guo H-X Yang and T Zhou ldquoLocal prefer-ential attachment model for hierarchical networksrdquo Physica Avol 388 no 8 pp 1713ndash1720 2009
[4] Z Fan G Chen and Y Zhang ldquoA comprehensive multi-local-world model for complex networksrdquo Physics Letters A vol 373no 18-19 pp 1601ndash1605 2009
[5] Z-H Guan and Z-P Wu ldquoThe physical position neighbour-hood evolving network modelrdquo Physica A vol 387 no 1 pp314ndash322 2008
[6] P Sheridan Y Yagahara and H Shimodaira ldquoA preferentialattachmentmodel with Poisson growth for scale-free networksrdquoAnnals of the Institute of Statistical Mathematics vol 60 no 4pp 747ndash761 2008
[7] L-J Chen D-X Chen L Xie and J-N Cao ldquoEvolution ofwireless sensor networkrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo07) vol556 pp 3005ndash3009 March 2007
[8] X Luo H Yu and X Wang ldquoEnergy-aware topology evolutionmodel with link and node deletion in wireless sensor networksrdquoMathematical Problems in Engineering vol 2012 Article ID281465 14 pages 2012
[9] S Li L Li and Y Yang ldquoA local-world heterogeneous model ofwireless sensor networks with node and link diversityrdquo PhysicaA vol 390 no 6 pp 1182ndash1191 2011
[10] P M Wightman andM A Labrador ldquoA3 a topology construc-tion algorithm for wireless sensor networksrdquo in Proceedings ofthe IEEE Global Telecommunications Conference (GLOBECOMrsquo08) pp 1ndash6 IEEE New Orleans La USA December 2008
[11] J Wu M Cardei F Dai and S Yang ldquoExtended dominatingset and its applications in ad hoc networks using cooperativecommunicationrdquo IEEE Transactions on Parallel and DistributedSystems vol 17 no 8 pp 851ndash864 2006
[12] H L Zhu H Luo H P Peng L X Li and Q Luo ldquoComplexnetworks-based energy-efficient evolution model for wirelesssensor networksrdquo Chaos Solitons and Fractals vol 41 no 4 pp1828ndash1835 2009
[13] A-L Barabasi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 no 1 pp 173ndash187 1999
[14] P M Wightman ldquoAtarraya a topology control simulatorrdquohttpwwwcseusfedusimpedrowatarraya
2 International Journal of Distributed Sensor Networks
in cluster heads reduce if we consider large homogeneousWSNs There are other approaches where the idea of dom-inating set of graphs is used to devise an algorithm Someof these approaches are tree based Most of these algorithmsare distributed and work with local (at the most 2-hop)information available from any given node In this paper wehave considered three such topology construction protocolsnamely simple tree CDS-Rule K and A3 protocols forfurther exploration
In another development the theory complex networkshave recently received increasing attention for understandingthe topological structure functions and dynamical proper-ties of many real-world networks such as the social networksbiological networks and ad hoc networks One of the mostimportant models that can be used to formally characterizeclustering algorithms is known as B-A model [1] This modelis based on two foundational mechanisms growth andpreferential attachment A new node is added to the networkat each step and connects with an existing nodewith a specificprobability which is related to the degree of the existing nodeThe B-A network has the scale-free property and follows thepower-law distribution
The B-A network model is capable of capturing somebasic mechanism that is responsible for the power-law degreedistribution Still it hadmany limitations Li-Chenmodel [2]has improved upon B-A model This model has been able tobetter capture the dynamics of networks constructed with alocal preferential attachment mechanism
The local preferential attachment model [3] is based onthe common sense that people can collect information easilyfrom their local community than from far away environmentUsing preferential connection as the fundamental basismanyvariations of the scale-free network model have been pro-posed during recent years such as comprehensive multilocal-world model [4] Similar to preferential attachment modelthe physical position neighbourhoodsrsquo model [5] mimics theactual communication network The Poisson growth model[6] uses the number of edges added at each step as a randomvariable that corresponds to Poisson distributionThis modelcan generate many kinds of networks by controlling therandom number
Chen et al [7] have studied an evolving mechanism forformalizing fault-tolerant communication topology amongcluster heads with complex network theory Based on the B-A modelrsquos growth and preferential attachment mechanismsthey not only used a local-world strategy for the networkwhen a new node was added to its local-world but alsoselected a fixed number of cluster heads in the local worldfor the purpose of obtaining a good performance in terms ofrandom error tolerance
Luo et al [8] performed theoretical analysis and con-ducted numerical simulation to explore topology charac-teristics and network performances with different energydistributions among nodes Their results have shown thatthe network is better clustered and average path length fortransmitting data is reducedwhen energy distribution amongnodes is more heterogeneous
In [8] a new dimension is added as the nodes are not onlyallowed to join the network through preferential attachment
but they are also allowed to leave the network or not jointhe network through nonpreferential attachment Furtherthe nodes distinguish themselves as cluster head nodes andnormal nodes which is consistent with the function of manyclustering algorithms in WSNs
In this paper we have tried to provide a formalism forsome algorithms which computes clusters in a WSN usinga modified local network model based on similar modelsproposed in [2 9] and [8] In particular we have used threetree based clustering algorithms namely simple tree CDSRule K and A3 Using our theoretical framework we havealso tried to quantify some of the observed features of thesealgorithms such as number of cluster heads and averagedegree of the resultant graph The theoretically obtainedmeasures have reasonably matched with measures obtainedby simulation studies
The paper is organized as follows Section 2 describesa very brief review of the local-world model In Section 3we have described the application of local world model toprovide a framework to describe the functioning of threeclustering algorithms In Section 4 results of theoreticalresults and simulation studies are jointly presented Section 5concludes the paper
2 A Brief Review of Topology ControlProtocols and Local-World Network Model
In this section we shall discuss very briefly the Li-Chenmodel [2] and two topology control protocols which are usedin this paper
21 A3 Protocol [10] TheA3 protocol uses four types of mes-sages Hello message children recognition message parentrecognition message and sleeping message The sink nodestarts the protocol by transmitting an initial hello messageto its neighboring nodes Nodes accept the message if theyhave not been covered by another node they set their statesas covered select the transmitter as its parent node andanswer back with a parent recognition message If a parentnode does not get any parent recognition messages from itsneighbors it turns offThe parent node sets a timeout periodto accept answers from its neighboring nodes Once timeoutoccurs the parent node sorts the list of neighbor accepting itsmessage in decreasing order of some selection metric Thenparent node broadcasts a children recognition message thatincludes the complete sorted list to all its candidates Once thechildren accept the list they set a timeout period proportionalto their position on the candidate list During that timeoutthey wait for sleeping message from their brothers If a nodeaccepts a sleepingmessage during the time out period it turnsitself off
22 CDS Rule k Protocol [11] The CDS Rule k algorithmutilizes connected dominating set algorithm and pruningrules The idea is to start from a big set of dominatingnodes that produces a minimum criterion and prunes itaccording to a particular rule In the first stage the nodeswill interchange their neighbor databases A nodewill remain
International Journal of Distributed Sensor Networks 3
Local world
M = nodes in local world
m0 = universal set of nodes
m = edges which connect M nodes in local world
Figure 1 Illustration of various parameters in their roles asdescribing the local world and the universe
progressive if there is at least one pair of separated neighborsIn the second stage a node chooses to unmark itself if itdetermines that all its neighbors are covered bymarked nodeswith higher precedence which is given by the degree of thenode in the tree Lower level implies higher precedence Theultimate tree is a more compact version of initial one with allredundant nodes with higher or equal priority removed
23 Local Network Model We have used Li-Chen Model [2]This model is used to form a generalized local-world modelUsing the generalized model we have analyzed clusteringalgorithms of wireless sensor networks In this model eachnode has only local connection information Nodes connectonly in their local world based on their local connectivityThefollowing parameters are required to explain the dynamicswith reference from Figure 1
(1) We start from a small number of nodes 1198980and grow
at each time step 119905(2) When a new node chooses a connection to other
nodes the probability prod119896119894 that a new node is
connected to a node 119894 depends on the degree 119896119894of
node 119894 This probability is defined as follows
prod(119896119894) =
119896119894
sum119895119896119895
(1)
(3) We select 119872 nodes randomly from the existingnetwork which is referred to as local world of the newnode
(4) When a new node arrives we add that node with 119898
edges linking the new node to 119898 nodes in the localworld determined in (3) using preferential attach-ment with a probabilityprodlocal(119896119894) This probability isdefined as follows at every time step 119905 Consider
prod
local119896119894= prod
1015840 119896119894
sum119895119896119895
(2)
whereprod1015840(119894 isin Local minusWorld) = 119872(1198980+ 119905)
After 119905 time steps there will be a network with 119872 = 119905 + 1198980
nodes and119898 lowast 119905 edges
3 Applying Local-World Model in WSNTopology Control Algorithms
To use local world model to capture functioning of WSNclustering algorithm the model has to take into accounttwo types of nodes normal nodes and cluster nodes Thecluster nodes are the cluster heads and the normal nodesare members in a cluster There is only one cluster nodeattached to a normal node in other words the normal nodehas only one edge whichmeans that the normal node cannotrelay data from other nodes A cluster node can integrateand transmit data from other nodes Both of these two typesof nodes can connect to a cluster node and the number ofedges is limited in every cluster node because of its energyconsideration When a new cluster node joins the network itis randomly assigned an initial energy 119864
119894from the interval
[119864min 119864max] The limited number of edges in every clusternode is represented by 119896max 119894 which is based on the initialenergy of the cluster nodes 119864
119894where 119896max [12] is given as
follows
119896max 119894 = 119896max lowast119864119894
119864max (3)
119896max 119894 reflects the ability of having the maximum number ofedges for cluster node 119894
The growth model is described as follows Starting with asmall number of nodes (all of them are cluster nodes) theyrandomly link each otherThis results into an initial network
(1) Growth At every time step a new cluster node or anormal node with one edge enters into the existingnetwork with a probability 119901 or 1 minus 119901 respectivelyIf the new node is a cluster node then it is assigneda random energy value of 119864
119894as discussed A small
number of cluster nodes would cause many sensornodes to link to them which results in faster energyconsumption but a large number of cluster nodeswould be more wasteful in terms of energy efficiencyThus the value of 119901 is assumed to be in the range as0 lt 119901 lt 05
(2) Preferential Attachment A new node arriving at thenetwork links to an old cluster node that is selectedrandomly from the already existing network NodesinWSNs have the constraint of energy and connectiv-ity and only communicate data with the cluster nodesin their local area First119872 cluster nodes are selectedrandomly from the network as the new incomingnodersquos local world then one of the cluster nodes ischosen to link with the new node according to theprobabilityprodlocal(119896119894)
If the new incoming node is a cluster node then theprobability is set as follows
prod
119896119894
= (1minus119896119894
119896max 119894)
119896119894
sum119895isinlocal 119896119894
(4)
In this case when the value of 119896119894is high the probability that
it will be chosen to connect with the new node is higher
4 International Journal of Distributed Sensor Networks
If the new incoming node is a normal node then theprobability is defined as follows
prod
119888119894
= (1minus119896119894
119896max 119894)
119888119894
sum119895isinlocal 119888119894
(5)
where 119888119894is the number of edges of the cluster node 119894 The
greater the value of 119888119894is the higher the probability that it will
be chosen to connect with the new node Only through thisapproach we can adjust the number of cluster nodes that arelinked to one cluster node (cluster head)
Total preferential probability is given as follows
prod
119896119894
= prod
119896119894
+prod
119888119894
(6)
In [12] authors have considered the expenditure of energyin the process of linking nodes together The disadvantageis that the energy in a cluster node will be exhausted inonly few rounds if self-organization is allowed In fact theenergy consumption will be relatively low if only 119896max 119894 isconsidered to be the limit for a cluster node to connect toothers randomly
Antipreferential Attachment Let us consider a parameter 119911called the deletion rate or antipreferential attachment factorwhich is defined as the rate of links removed divided by therate of links added It is observed that lesser the energy of thenode the more will be the probability of it being deleted Letthis probability be denoted asprodlowast(119896
119894)
For the outgoing cluster nodes we have
lowast
prod (119896119894) asymp
11198980 + 119901 lowast 119905
(7)
For the outgoing normal nodes we have
lowast
prod (119888119894) asymp
11198980 + (1 minus 119901) lowast 119905
(8)
So the total antipreferential probability is given as follows
The antipreferential removal mechanism is more reasonablefor deleting links that are antiparallel with the preferentialconnection It is also consistent with the functioning ofclustering algorithms that runs in rounds in wireless sensornetworksThewireless nodes that do not have enough energythat is the dead nodes are to be removed from the systemThus antipreferential removal phenomenon is reasonable forclustering algorithms
Using mean field theory [9 13] a qualitative analysisof dynamic characterization of a wireless sensor networkcan be given By the mean-field theory the preferential
and nonpreferential attachment may be combined in thefollowing differential equation
International Journal of Distributed Sensor Networks 5
31 Analysis of the Dynamic Equation
311 Case I If 119911 = 0 119872 = 1 that is the new nodeselects node unless it reaches 119896 Moreover the preferentialattachment mechanism does not work The rate of growth of119896119894is as
120575119896119894
120575119905
=
11198980 + 119901 lowast 119905
(18)
The denominator of the above expression is the number ofcluster nodes at time 119905
312 Case II If 119872 = 1198980+ 119901 lowast 119905 this means that the local
total degree of node universetotal degree of nodes local-world
=
1198980 + 119901 lowast 119905 + 119873
1198980 + 119901 lowast 119905
=
1198980 + 119901 lowast 119905 + 1198980 + 119905
1198980 + 119901 lowast 119905
(22)
Therefore
119896 =
2 lowast 1198980 + 119905 + 119901 lowast 119905
1198980 + 119901 lowast 119905
(23)
Similarly we have for 119888119894
119888 =
sum119895119888119895
119888119894
= 119896minus
(1 minus 119901) lowast 119905
1198980 + 119901 lowast 119905
= 2 (24)
As the cluster node will have one such node attached to itselfthe status of that node is either another cluster head or normalhead hence the count value of 119888 is 2
Equations (21) and (23) are used to find the values of 119888 and119896
Finally the following equation is formed after substitutingthe values of sum
119860 + 119861 minus 2 lowast 119911 lowast 119862
)
1119860 (28)
Moreover to find the degree distribution 119875(119896) that is theprobability that a node has 119896 edges) we first calculate thecumulative probability 119875[119896
119894(119905) lt 119896] Suppose that the node
enters into the network at equal time intervals We define theprobability density 119905
119894as follows
119875 (119905) =
11198980 + 119905
(29)
So 119875[119896119894(119905) lt 119896] has the following form
Equation (31) denotes the degree distribution function tounderstand the dynamics of clustering algorithms Puttingback the values of 119860 119861 and 119862 we have the distributionfunction as the degree distribution 119875(119896) The probability thata node has 119896 edges is
119875 (119896) asymp (
119901
119901 minus 1+
1 minus 119901
119901
minus
2 lowast 119911
119901 lowast (1 minus 119901)
)
(1+119901)119901
lowast (119896
lowast
119901
1 + 119901
+
1 minus 119901
119901
minus 2 lowast 119911
119901 lowast (1 minus 119901)
)
(minus2lowast119901minus1)119901
(32)
Next the value of 119911 is computed that maximizes (32)Consider
119889119875 (119896)
119889119911
= 0 (33)
By solving (33) we obtain the value of 119911 as
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(34)
Anti preferential attachment means that the node may notlike to have a particular node as its neighbor but in thiscase of sensor networks no such phenomenon is observedhencewe can state that the anti preferential attachment in thisparticular case tends to zero during the evolution process
By setting the value of 119911 to zero we have the followingrelation
119901 =
12[(
119896 + 3119896 minus 1
)
05minus 1] (35)
Equation (35) states the probability of clustering when theantipreferential attachment factor 119911 is zero
4 Analysis of Topology Control Algorithms
In this section we have carried out simulation of threelocalized topology control protocols namely A3 Simple treeProtocol and CDS Rule K Simulation was carried using theAtarraya [14] simulator Total number of nodes was 100 200300 and 500 respectively and theywere tested for A3 SimpleTree Protocol and CDS Rule K The output was recorded foraverage degree of nodes 119896 for three protocols mentionedearlier
935361
916479
67
58
99
97
4934
44
3024
39
65
82
2194
907054
56
26
28
36 100
59
25
40
84
63
68
5787
32
51
47
27
Figure 2 Clustering as obtained by running simple tree protocol ina network of 100 nodes
100 3 0366 37 40200 3 0366 73 71300 4 0263 79 71400 4 0263 105 96500 4 0263 132 130119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
Figure 2 shows an example of clustering and computationof average 119896 value
Figure 2 shows the cluster heads in red circles with whichnormal nodes are attached Taking the number of all theneighboring nodes of every cluster heads and dividing by thenumber of cluster heads we obtain the average value of 119896 Bysimilar procedure we have obtained the average value of 119896 fordifferent number of nodes
We have also carried out simulation study using simpletree protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 1
Similar simulation study was also carried out using CDS-Rule k Protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 2
Similar simulation study was also carried out using A3Protocol for different number of nodes The number of clus-ters obtained using theoretical calculation (35) is compared
International Journal of Distributed Sensor Networks 7
100 4 0264 27 27200 6 0171 35 38300 10 0101 31 33400 10 0101 41 43500 13 0077 39 43119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
against the number of clusters obtained using simulationstudy which is presented in Table 3
41 Discussion on A3 and Simple Tree Protocol Next we plotthe theoretically calculated number of clusters and experimen-tally calculated number of clusters as shown in Tables 1 and 3for A3 and simple tree protocols
We observe the following
(1) In A3 and Simple Tree protocols the curves fortheoretically calculated number of clusters and exper-imentally calculated number of clusters follow fourthdegree equationThe trend line polynomial options ofMS Excel have been used to study the degree of curvefit of the curves to obtain the following polynomials
119899 = minus 4119864minus 101198734+ 8119864minus 071198733
minus 00001198732+ 0197119873
+ 12 (A3 protocol)
119899 = minus 3119864minus 081198734+ 4119864minus 051198733
minus 00161198732+ 3164119873
minus 148 (Simple tree protocol)
(36)
(2) In Figure 3 there is gap between the theoreticaland experimental curves in case of A3 and simpletree protocol This may be attributed due to thefact that A3 (approximate CDS) only preserves 1-connectivity whereas the simple tree protocol hasmultiple connectivity So A3 protocol requires lessenergy to construct the tree as compared to spanningtree which confirms to the experimental results
42 Discussion on CDS Rule K Consider (34) which ismentioned in the following
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(37)
where 119860 119861 119862 119896 and 119911 have been defined previouslyTable 4 enumerates the value of 119911 for various values of 119896
corresponding to different number of nodes in the networkFrom the results of Table 2 we obtain Figure 4In Table 4 the maximum value of 119911 corresponds to a
probability value 05 which indicates that the clusters havedissociated into one cluster head and one normal node whichideally predicts the case when the energy of the nodes hasdrained out
0
20
40
60
80
100
120
140
0 100 200 300 400 500 600
Simple tree (Th)Simple tree (Exp)
A3 (Th)A3 (Exp)
Number of nodes N
Clus
ter h
eads
(n)
Figure 3Number of nodes versus number of cluster heads in simpletree protocol and A3 protocol
05
101520253035404550
CDS Rule K (Th)CDS Rule k (Exp)
0 100 200 300 400 500 600Number of nodes (N)
Clus
ter h
eads
(n)
Figure 4 Total number of nodes (119873) versus number of cluster headsin CDS-Rule k algorithm
100 4 0263 27 34200 6 0171 35 40300 8 0126 38 49400 10 0101 41 48500 12 0083 42 46119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
In Figure 4 it can be seen that the mechanics of clus-tering in CDS-Rule K is in accordance to the assumptionsconsidered while deducing the distribution function usingmean field theory Unlike A3 and Simple tree protocol CDS-Rule K also exhibits a relation of degree 4 as shown (119899 =
minus2119864minus081198734+3119864minus05119873
3minus0012119873
2+2085119873minus87)The linear
graph as shown in Figure 4 represents the situation when
8 International Journal of Distributed Sensor Networks
Table 4 Value of 119911 for different 119896 values in networks of differentsizes
Nodescollection (119873) 119896 Maxima (119901 119911) Minima (119901 119911)
the value of 119911 isminThis graph is linear due to constant valueof 119901 (05 here) where other graphs were drawn on constant 119911(0 here)
Above experimental results lead us to the followingexplanation of the antipreferential factor 119911
The antipreferential factor can have three distinct values
(a) When 119911 = 0 then the effective degree of nodes is equalto the analytically obtained degree of a node Here thenumber of cluster heads formed will be optimal
(b) When 119911 gt 0 then a node will have fewer connectionsavailable Here the number of cluster heads will behigher
(c) When 119911 lt 0 then a node will have less connectivityHere the number of cluster heads will be highest asthe result will lead to dissociation
5 Conclusions
In this paper we have tried to provide a framework toformally model tree based clustering algorithms in a WSNBased on the formalism we have theoretically calculatedsome parameters such as number of cluster heads and averagenumber of degree for a given algorithm The theoreticalresults tally with results obtained by simulation studies Wehave introduced a factor called 119911 in network evolutionmodelIt seems that the 119911 factor has an impact on functioning ofthese protocols We keep that study as a future exercise Thestudy at the current stage is more suitable for modeling treebased clustering algorithms thatwork on the top of connecteddominating set construction Algorithms that are based onmore parameters have to be modeled appropriately in theframework before application
In recent papers energy distribution has been consideredin network evolution model for wireless sensor networksBut so far the framework of network evolution model hasnot been used to capture the characteristics of clusteringalgorithms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquoAmericanAssociation for theAdvancement of Sciencevol 286 no 5439 pp 509ndash512 1999
[2] X Li and G Chen ldquoA local-world evolving network modelrdquoPhysica A vol 328 no 1-2 pp 274ndash286 2003
[3] L-N Wang J-L Guo H-X Yang and T Zhou ldquoLocal prefer-ential attachment model for hierarchical networksrdquo Physica Avol 388 no 8 pp 1713ndash1720 2009
[4] Z Fan G Chen and Y Zhang ldquoA comprehensive multi-local-world model for complex networksrdquo Physics Letters A vol 373no 18-19 pp 1601ndash1605 2009
[5] Z-H Guan and Z-P Wu ldquoThe physical position neighbour-hood evolving network modelrdquo Physica A vol 387 no 1 pp314ndash322 2008
[6] P Sheridan Y Yagahara and H Shimodaira ldquoA preferentialattachmentmodel with Poisson growth for scale-free networksrdquoAnnals of the Institute of Statistical Mathematics vol 60 no 4pp 747ndash761 2008
[7] L-J Chen D-X Chen L Xie and J-N Cao ldquoEvolution ofwireless sensor networkrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo07) vol556 pp 3005ndash3009 March 2007
[8] X Luo H Yu and X Wang ldquoEnergy-aware topology evolutionmodel with link and node deletion in wireless sensor networksrdquoMathematical Problems in Engineering vol 2012 Article ID281465 14 pages 2012
[9] S Li L Li and Y Yang ldquoA local-world heterogeneous model ofwireless sensor networks with node and link diversityrdquo PhysicaA vol 390 no 6 pp 1182ndash1191 2011
[10] P M Wightman andM A Labrador ldquoA3 a topology construc-tion algorithm for wireless sensor networksrdquo in Proceedings ofthe IEEE Global Telecommunications Conference (GLOBECOMrsquo08) pp 1ndash6 IEEE New Orleans La USA December 2008
[11] J Wu M Cardei F Dai and S Yang ldquoExtended dominatingset and its applications in ad hoc networks using cooperativecommunicationrdquo IEEE Transactions on Parallel and DistributedSystems vol 17 no 8 pp 851ndash864 2006
[12] H L Zhu H Luo H P Peng L X Li and Q Luo ldquoComplexnetworks-based energy-efficient evolution model for wirelesssensor networksrdquo Chaos Solitons and Fractals vol 41 no 4 pp1828ndash1835 2009
[13] A-L Barabasi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 no 1 pp 173ndash187 1999
[14] P M Wightman ldquoAtarraya a topology control simulatorrdquohttpwwwcseusfedusimpedrowatarraya
International Journal of Distributed Sensor Networks 3
Local world
M = nodes in local world
m0 = universal set of nodes
m = edges which connect M nodes in local world
Figure 1 Illustration of various parameters in their roles asdescribing the local world and the universe
progressive if there is at least one pair of separated neighborsIn the second stage a node chooses to unmark itself if itdetermines that all its neighbors are covered bymarked nodeswith higher precedence which is given by the degree of thenode in the tree Lower level implies higher precedence Theultimate tree is a more compact version of initial one with allredundant nodes with higher or equal priority removed
23 Local Network Model We have used Li-Chen Model [2]This model is used to form a generalized local-world modelUsing the generalized model we have analyzed clusteringalgorithms of wireless sensor networks In this model eachnode has only local connection information Nodes connectonly in their local world based on their local connectivityThefollowing parameters are required to explain the dynamicswith reference from Figure 1
(1) We start from a small number of nodes 1198980and grow
at each time step 119905(2) When a new node chooses a connection to other
nodes the probability prod119896119894 that a new node is
connected to a node 119894 depends on the degree 119896119894of
node 119894 This probability is defined as follows
prod(119896119894) =
119896119894
sum119895119896119895
(1)
(3) We select 119872 nodes randomly from the existingnetwork which is referred to as local world of the newnode
(4) When a new node arrives we add that node with 119898
edges linking the new node to 119898 nodes in the localworld determined in (3) using preferential attach-ment with a probabilityprodlocal(119896119894) This probability isdefined as follows at every time step 119905 Consider
prod
local119896119894= prod
1015840 119896119894
sum119895119896119895
(2)
whereprod1015840(119894 isin Local minusWorld) = 119872(1198980+ 119905)
After 119905 time steps there will be a network with 119872 = 119905 + 1198980
nodes and119898 lowast 119905 edges
3 Applying Local-World Model in WSNTopology Control Algorithms
To use local world model to capture functioning of WSNclustering algorithm the model has to take into accounttwo types of nodes normal nodes and cluster nodes Thecluster nodes are the cluster heads and the normal nodesare members in a cluster There is only one cluster nodeattached to a normal node in other words the normal nodehas only one edge whichmeans that the normal node cannotrelay data from other nodes A cluster node can integrateand transmit data from other nodes Both of these two typesof nodes can connect to a cluster node and the number ofedges is limited in every cluster node because of its energyconsideration When a new cluster node joins the network itis randomly assigned an initial energy 119864
119894from the interval
[119864min 119864max] The limited number of edges in every clusternode is represented by 119896max 119894 which is based on the initialenergy of the cluster nodes 119864
119894where 119896max [12] is given as
follows
119896max 119894 = 119896max lowast119864119894
119864max (3)
119896max 119894 reflects the ability of having the maximum number ofedges for cluster node 119894
The growth model is described as follows Starting with asmall number of nodes (all of them are cluster nodes) theyrandomly link each otherThis results into an initial network
(1) Growth At every time step a new cluster node or anormal node with one edge enters into the existingnetwork with a probability 119901 or 1 minus 119901 respectivelyIf the new node is a cluster node then it is assigneda random energy value of 119864
119894as discussed A small
number of cluster nodes would cause many sensornodes to link to them which results in faster energyconsumption but a large number of cluster nodeswould be more wasteful in terms of energy efficiencyThus the value of 119901 is assumed to be in the range as0 lt 119901 lt 05
(2) Preferential Attachment A new node arriving at thenetwork links to an old cluster node that is selectedrandomly from the already existing network NodesinWSNs have the constraint of energy and connectiv-ity and only communicate data with the cluster nodesin their local area First119872 cluster nodes are selectedrandomly from the network as the new incomingnodersquos local world then one of the cluster nodes ischosen to link with the new node according to theprobabilityprodlocal(119896119894)
If the new incoming node is a cluster node then theprobability is set as follows
prod
119896119894
= (1minus119896119894
119896max 119894)
119896119894
sum119895isinlocal 119896119894
(4)
In this case when the value of 119896119894is high the probability that
it will be chosen to connect with the new node is higher
4 International Journal of Distributed Sensor Networks
If the new incoming node is a normal node then theprobability is defined as follows
prod
119888119894
= (1minus119896119894
119896max 119894)
119888119894
sum119895isinlocal 119888119894
(5)
where 119888119894is the number of edges of the cluster node 119894 The
greater the value of 119888119894is the higher the probability that it will
be chosen to connect with the new node Only through thisapproach we can adjust the number of cluster nodes that arelinked to one cluster node (cluster head)
Total preferential probability is given as follows
prod
119896119894
= prod
119896119894
+prod
119888119894
(6)
In [12] authors have considered the expenditure of energyin the process of linking nodes together The disadvantageis that the energy in a cluster node will be exhausted inonly few rounds if self-organization is allowed In fact theenergy consumption will be relatively low if only 119896max 119894 isconsidered to be the limit for a cluster node to connect toothers randomly
Antipreferential Attachment Let us consider a parameter 119911called the deletion rate or antipreferential attachment factorwhich is defined as the rate of links removed divided by therate of links added It is observed that lesser the energy of thenode the more will be the probability of it being deleted Letthis probability be denoted asprodlowast(119896
119894)
For the outgoing cluster nodes we have
lowast
prod (119896119894) asymp
11198980 + 119901 lowast 119905
(7)
For the outgoing normal nodes we have
lowast
prod (119888119894) asymp
11198980 + (1 minus 119901) lowast 119905
(8)
So the total antipreferential probability is given as follows
The antipreferential removal mechanism is more reasonablefor deleting links that are antiparallel with the preferentialconnection It is also consistent with the functioning ofclustering algorithms that runs in rounds in wireless sensornetworksThewireless nodes that do not have enough energythat is the dead nodes are to be removed from the systemThus antipreferential removal phenomenon is reasonable forclustering algorithms
Using mean field theory [9 13] a qualitative analysisof dynamic characterization of a wireless sensor networkcan be given By the mean-field theory the preferential
and nonpreferential attachment may be combined in thefollowing differential equation
International Journal of Distributed Sensor Networks 5
31 Analysis of the Dynamic Equation
311 Case I If 119911 = 0 119872 = 1 that is the new nodeselects node unless it reaches 119896 Moreover the preferentialattachment mechanism does not work The rate of growth of119896119894is as
120575119896119894
120575119905
=
11198980 + 119901 lowast 119905
(18)
The denominator of the above expression is the number ofcluster nodes at time 119905
312 Case II If 119872 = 1198980+ 119901 lowast 119905 this means that the local
total degree of node universetotal degree of nodes local-world
=
1198980 + 119901 lowast 119905 + 119873
1198980 + 119901 lowast 119905
=
1198980 + 119901 lowast 119905 + 1198980 + 119905
1198980 + 119901 lowast 119905
(22)
Therefore
119896 =
2 lowast 1198980 + 119905 + 119901 lowast 119905
1198980 + 119901 lowast 119905
(23)
Similarly we have for 119888119894
119888 =
sum119895119888119895
119888119894
= 119896minus
(1 minus 119901) lowast 119905
1198980 + 119901 lowast 119905
= 2 (24)
As the cluster node will have one such node attached to itselfthe status of that node is either another cluster head or normalhead hence the count value of 119888 is 2
Equations (21) and (23) are used to find the values of 119888 and119896
Finally the following equation is formed after substitutingthe values of sum
119860 + 119861 minus 2 lowast 119911 lowast 119862
)
1119860 (28)
Moreover to find the degree distribution 119875(119896) that is theprobability that a node has 119896 edges) we first calculate thecumulative probability 119875[119896
119894(119905) lt 119896] Suppose that the node
enters into the network at equal time intervals We define theprobability density 119905
119894as follows
119875 (119905) =
11198980 + 119905
(29)
So 119875[119896119894(119905) lt 119896] has the following form
Equation (31) denotes the degree distribution function tounderstand the dynamics of clustering algorithms Puttingback the values of 119860 119861 and 119862 we have the distributionfunction as the degree distribution 119875(119896) The probability thata node has 119896 edges is
119875 (119896) asymp (
119901
119901 minus 1+
1 minus 119901
119901
minus
2 lowast 119911
119901 lowast (1 minus 119901)
)
(1+119901)119901
lowast (119896
lowast
119901
1 + 119901
+
1 minus 119901
119901
minus 2 lowast 119911
119901 lowast (1 minus 119901)
)
(minus2lowast119901minus1)119901
(32)
Next the value of 119911 is computed that maximizes (32)Consider
119889119875 (119896)
119889119911
= 0 (33)
By solving (33) we obtain the value of 119911 as
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(34)
Anti preferential attachment means that the node may notlike to have a particular node as its neighbor but in thiscase of sensor networks no such phenomenon is observedhencewe can state that the anti preferential attachment in thisparticular case tends to zero during the evolution process
By setting the value of 119911 to zero we have the followingrelation
119901 =
12[(
119896 + 3119896 minus 1
)
05minus 1] (35)
Equation (35) states the probability of clustering when theantipreferential attachment factor 119911 is zero
4 Analysis of Topology Control Algorithms
In this section we have carried out simulation of threelocalized topology control protocols namely A3 Simple treeProtocol and CDS Rule K Simulation was carried using theAtarraya [14] simulator Total number of nodes was 100 200300 and 500 respectively and theywere tested for A3 SimpleTree Protocol and CDS Rule K The output was recorded foraverage degree of nodes 119896 for three protocols mentionedearlier
935361
916479
67
58
99
97
4934
44
3024
39
65
82
2194
907054
56
26
28
36 100
59
25
40
84
63
68
5787
32
51
47
27
Figure 2 Clustering as obtained by running simple tree protocol ina network of 100 nodes
100 3 0366 37 40200 3 0366 73 71300 4 0263 79 71400 4 0263 105 96500 4 0263 132 130119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
Figure 2 shows an example of clustering and computationof average 119896 value
Figure 2 shows the cluster heads in red circles with whichnormal nodes are attached Taking the number of all theneighboring nodes of every cluster heads and dividing by thenumber of cluster heads we obtain the average value of 119896 Bysimilar procedure we have obtained the average value of 119896 fordifferent number of nodes
We have also carried out simulation study using simpletree protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 1
Similar simulation study was also carried out using CDS-Rule k Protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 2
Similar simulation study was also carried out using A3Protocol for different number of nodes The number of clus-ters obtained using theoretical calculation (35) is compared
International Journal of Distributed Sensor Networks 7
100 4 0264 27 27200 6 0171 35 38300 10 0101 31 33400 10 0101 41 43500 13 0077 39 43119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
against the number of clusters obtained using simulationstudy which is presented in Table 3
41 Discussion on A3 and Simple Tree Protocol Next we plotthe theoretically calculated number of clusters and experimen-tally calculated number of clusters as shown in Tables 1 and 3for A3 and simple tree protocols
We observe the following
(1) In A3 and Simple Tree protocols the curves fortheoretically calculated number of clusters and exper-imentally calculated number of clusters follow fourthdegree equationThe trend line polynomial options ofMS Excel have been used to study the degree of curvefit of the curves to obtain the following polynomials
119899 = minus 4119864minus 101198734+ 8119864minus 071198733
minus 00001198732+ 0197119873
+ 12 (A3 protocol)
119899 = minus 3119864minus 081198734+ 4119864minus 051198733
minus 00161198732+ 3164119873
minus 148 (Simple tree protocol)
(36)
(2) In Figure 3 there is gap between the theoreticaland experimental curves in case of A3 and simpletree protocol This may be attributed due to thefact that A3 (approximate CDS) only preserves 1-connectivity whereas the simple tree protocol hasmultiple connectivity So A3 protocol requires lessenergy to construct the tree as compared to spanningtree which confirms to the experimental results
42 Discussion on CDS Rule K Consider (34) which ismentioned in the following
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(37)
where 119860 119861 119862 119896 and 119911 have been defined previouslyTable 4 enumerates the value of 119911 for various values of 119896
corresponding to different number of nodes in the networkFrom the results of Table 2 we obtain Figure 4In Table 4 the maximum value of 119911 corresponds to a
probability value 05 which indicates that the clusters havedissociated into one cluster head and one normal node whichideally predicts the case when the energy of the nodes hasdrained out
0
20
40
60
80
100
120
140
0 100 200 300 400 500 600
Simple tree (Th)Simple tree (Exp)
A3 (Th)A3 (Exp)
Number of nodes N
Clus
ter h
eads
(n)
Figure 3Number of nodes versus number of cluster heads in simpletree protocol and A3 protocol
05
101520253035404550
CDS Rule K (Th)CDS Rule k (Exp)
0 100 200 300 400 500 600Number of nodes (N)
Clus
ter h
eads
(n)
Figure 4 Total number of nodes (119873) versus number of cluster headsin CDS-Rule k algorithm
100 4 0263 27 34200 6 0171 35 40300 8 0126 38 49400 10 0101 41 48500 12 0083 42 46119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
In Figure 4 it can be seen that the mechanics of clus-tering in CDS-Rule K is in accordance to the assumptionsconsidered while deducing the distribution function usingmean field theory Unlike A3 and Simple tree protocol CDS-Rule K also exhibits a relation of degree 4 as shown (119899 =
minus2119864minus081198734+3119864minus05119873
3minus0012119873
2+2085119873minus87)The linear
graph as shown in Figure 4 represents the situation when
8 International Journal of Distributed Sensor Networks
Table 4 Value of 119911 for different 119896 values in networks of differentsizes
Nodescollection (119873) 119896 Maxima (119901 119911) Minima (119901 119911)
the value of 119911 isminThis graph is linear due to constant valueof 119901 (05 here) where other graphs were drawn on constant 119911(0 here)
Above experimental results lead us to the followingexplanation of the antipreferential factor 119911
The antipreferential factor can have three distinct values
(a) When 119911 = 0 then the effective degree of nodes is equalto the analytically obtained degree of a node Here thenumber of cluster heads formed will be optimal
(b) When 119911 gt 0 then a node will have fewer connectionsavailable Here the number of cluster heads will behigher
(c) When 119911 lt 0 then a node will have less connectivityHere the number of cluster heads will be highest asthe result will lead to dissociation
5 Conclusions
In this paper we have tried to provide a framework toformally model tree based clustering algorithms in a WSNBased on the formalism we have theoretically calculatedsome parameters such as number of cluster heads and averagenumber of degree for a given algorithm The theoreticalresults tally with results obtained by simulation studies Wehave introduced a factor called 119911 in network evolutionmodelIt seems that the 119911 factor has an impact on functioning ofthese protocols We keep that study as a future exercise Thestudy at the current stage is more suitable for modeling treebased clustering algorithms thatwork on the top of connecteddominating set construction Algorithms that are based onmore parameters have to be modeled appropriately in theframework before application
In recent papers energy distribution has been consideredin network evolution model for wireless sensor networksBut so far the framework of network evolution model hasnot been used to capture the characteristics of clusteringalgorithms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquoAmericanAssociation for theAdvancement of Sciencevol 286 no 5439 pp 509ndash512 1999
[2] X Li and G Chen ldquoA local-world evolving network modelrdquoPhysica A vol 328 no 1-2 pp 274ndash286 2003
[3] L-N Wang J-L Guo H-X Yang and T Zhou ldquoLocal prefer-ential attachment model for hierarchical networksrdquo Physica Avol 388 no 8 pp 1713ndash1720 2009
[4] Z Fan G Chen and Y Zhang ldquoA comprehensive multi-local-world model for complex networksrdquo Physics Letters A vol 373no 18-19 pp 1601ndash1605 2009
[5] Z-H Guan and Z-P Wu ldquoThe physical position neighbour-hood evolving network modelrdquo Physica A vol 387 no 1 pp314ndash322 2008
[6] P Sheridan Y Yagahara and H Shimodaira ldquoA preferentialattachmentmodel with Poisson growth for scale-free networksrdquoAnnals of the Institute of Statistical Mathematics vol 60 no 4pp 747ndash761 2008
[7] L-J Chen D-X Chen L Xie and J-N Cao ldquoEvolution ofwireless sensor networkrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo07) vol556 pp 3005ndash3009 March 2007
[8] X Luo H Yu and X Wang ldquoEnergy-aware topology evolutionmodel with link and node deletion in wireless sensor networksrdquoMathematical Problems in Engineering vol 2012 Article ID281465 14 pages 2012
[9] S Li L Li and Y Yang ldquoA local-world heterogeneous model ofwireless sensor networks with node and link diversityrdquo PhysicaA vol 390 no 6 pp 1182ndash1191 2011
[10] P M Wightman andM A Labrador ldquoA3 a topology construc-tion algorithm for wireless sensor networksrdquo in Proceedings ofthe IEEE Global Telecommunications Conference (GLOBECOMrsquo08) pp 1ndash6 IEEE New Orleans La USA December 2008
[11] J Wu M Cardei F Dai and S Yang ldquoExtended dominatingset and its applications in ad hoc networks using cooperativecommunicationrdquo IEEE Transactions on Parallel and DistributedSystems vol 17 no 8 pp 851ndash864 2006
[12] H L Zhu H Luo H P Peng L X Li and Q Luo ldquoComplexnetworks-based energy-efficient evolution model for wirelesssensor networksrdquo Chaos Solitons and Fractals vol 41 no 4 pp1828ndash1835 2009
[13] A-L Barabasi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 no 1 pp 173ndash187 1999
[14] P M Wightman ldquoAtarraya a topology control simulatorrdquohttpwwwcseusfedusimpedrowatarraya
4 International Journal of Distributed Sensor Networks
If the new incoming node is a normal node then theprobability is defined as follows
prod
119888119894
= (1minus119896119894
119896max 119894)
119888119894
sum119895isinlocal 119888119894
(5)
where 119888119894is the number of edges of the cluster node 119894 The
greater the value of 119888119894is the higher the probability that it will
be chosen to connect with the new node Only through thisapproach we can adjust the number of cluster nodes that arelinked to one cluster node (cluster head)
Total preferential probability is given as follows
prod
119896119894
= prod
119896119894
+prod
119888119894
(6)
In [12] authors have considered the expenditure of energyin the process of linking nodes together The disadvantageis that the energy in a cluster node will be exhausted inonly few rounds if self-organization is allowed In fact theenergy consumption will be relatively low if only 119896max 119894 isconsidered to be the limit for a cluster node to connect toothers randomly
Antipreferential Attachment Let us consider a parameter 119911called the deletion rate or antipreferential attachment factorwhich is defined as the rate of links removed divided by therate of links added It is observed that lesser the energy of thenode the more will be the probability of it being deleted Letthis probability be denoted asprodlowast(119896
119894)
For the outgoing cluster nodes we have
lowast
prod (119896119894) asymp
11198980 + 119901 lowast 119905
(7)
For the outgoing normal nodes we have
lowast
prod (119888119894) asymp
11198980 + (1 minus 119901) lowast 119905
(8)
So the total antipreferential probability is given as follows
The antipreferential removal mechanism is more reasonablefor deleting links that are antiparallel with the preferentialconnection It is also consistent with the functioning ofclustering algorithms that runs in rounds in wireless sensornetworksThewireless nodes that do not have enough energythat is the dead nodes are to be removed from the systemThus antipreferential removal phenomenon is reasonable forclustering algorithms
Using mean field theory [9 13] a qualitative analysisof dynamic characterization of a wireless sensor networkcan be given By the mean-field theory the preferential
and nonpreferential attachment may be combined in thefollowing differential equation
International Journal of Distributed Sensor Networks 5
31 Analysis of the Dynamic Equation
311 Case I If 119911 = 0 119872 = 1 that is the new nodeselects node unless it reaches 119896 Moreover the preferentialattachment mechanism does not work The rate of growth of119896119894is as
120575119896119894
120575119905
=
11198980 + 119901 lowast 119905
(18)
The denominator of the above expression is the number ofcluster nodes at time 119905
312 Case II If 119872 = 1198980+ 119901 lowast 119905 this means that the local
total degree of node universetotal degree of nodes local-world
=
1198980 + 119901 lowast 119905 + 119873
1198980 + 119901 lowast 119905
=
1198980 + 119901 lowast 119905 + 1198980 + 119905
1198980 + 119901 lowast 119905
(22)
Therefore
119896 =
2 lowast 1198980 + 119905 + 119901 lowast 119905
1198980 + 119901 lowast 119905
(23)
Similarly we have for 119888119894
119888 =
sum119895119888119895
119888119894
= 119896minus
(1 minus 119901) lowast 119905
1198980 + 119901 lowast 119905
= 2 (24)
As the cluster node will have one such node attached to itselfthe status of that node is either another cluster head or normalhead hence the count value of 119888 is 2
Equations (21) and (23) are used to find the values of 119888 and119896
Finally the following equation is formed after substitutingthe values of sum
119860 + 119861 minus 2 lowast 119911 lowast 119862
)
1119860 (28)
Moreover to find the degree distribution 119875(119896) that is theprobability that a node has 119896 edges) we first calculate thecumulative probability 119875[119896
119894(119905) lt 119896] Suppose that the node
enters into the network at equal time intervals We define theprobability density 119905
119894as follows
119875 (119905) =
11198980 + 119905
(29)
So 119875[119896119894(119905) lt 119896] has the following form
Equation (31) denotes the degree distribution function tounderstand the dynamics of clustering algorithms Puttingback the values of 119860 119861 and 119862 we have the distributionfunction as the degree distribution 119875(119896) The probability thata node has 119896 edges is
119875 (119896) asymp (
119901
119901 minus 1+
1 minus 119901
119901
minus
2 lowast 119911
119901 lowast (1 minus 119901)
)
(1+119901)119901
lowast (119896
lowast
119901
1 + 119901
+
1 minus 119901
119901
minus 2 lowast 119911
119901 lowast (1 minus 119901)
)
(minus2lowast119901minus1)119901
(32)
Next the value of 119911 is computed that maximizes (32)Consider
119889119875 (119896)
119889119911
= 0 (33)
By solving (33) we obtain the value of 119911 as
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(34)
Anti preferential attachment means that the node may notlike to have a particular node as its neighbor but in thiscase of sensor networks no such phenomenon is observedhencewe can state that the anti preferential attachment in thisparticular case tends to zero during the evolution process
By setting the value of 119911 to zero we have the followingrelation
119901 =
12[(
119896 + 3119896 minus 1
)
05minus 1] (35)
Equation (35) states the probability of clustering when theantipreferential attachment factor 119911 is zero
4 Analysis of Topology Control Algorithms
In this section we have carried out simulation of threelocalized topology control protocols namely A3 Simple treeProtocol and CDS Rule K Simulation was carried using theAtarraya [14] simulator Total number of nodes was 100 200300 and 500 respectively and theywere tested for A3 SimpleTree Protocol and CDS Rule K The output was recorded foraverage degree of nodes 119896 for three protocols mentionedearlier
935361
916479
67
58
99
97
4934
44
3024
39
65
82
2194
907054
56
26
28
36 100
59
25
40
84
63
68
5787
32
51
47
27
Figure 2 Clustering as obtained by running simple tree protocol ina network of 100 nodes
100 3 0366 37 40200 3 0366 73 71300 4 0263 79 71400 4 0263 105 96500 4 0263 132 130119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
Figure 2 shows an example of clustering and computationof average 119896 value
Figure 2 shows the cluster heads in red circles with whichnormal nodes are attached Taking the number of all theneighboring nodes of every cluster heads and dividing by thenumber of cluster heads we obtain the average value of 119896 Bysimilar procedure we have obtained the average value of 119896 fordifferent number of nodes
We have also carried out simulation study using simpletree protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 1
Similar simulation study was also carried out using CDS-Rule k Protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 2
Similar simulation study was also carried out using A3Protocol for different number of nodes The number of clus-ters obtained using theoretical calculation (35) is compared
International Journal of Distributed Sensor Networks 7
100 4 0264 27 27200 6 0171 35 38300 10 0101 31 33400 10 0101 41 43500 13 0077 39 43119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
against the number of clusters obtained using simulationstudy which is presented in Table 3
41 Discussion on A3 and Simple Tree Protocol Next we plotthe theoretically calculated number of clusters and experimen-tally calculated number of clusters as shown in Tables 1 and 3for A3 and simple tree protocols
We observe the following
(1) In A3 and Simple Tree protocols the curves fortheoretically calculated number of clusters and exper-imentally calculated number of clusters follow fourthdegree equationThe trend line polynomial options ofMS Excel have been used to study the degree of curvefit of the curves to obtain the following polynomials
119899 = minus 4119864minus 101198734+ 8119864minus 071198733
minus 00001198732+ 0197119873
+ 12 (A3 protocol)
119899 = minus 3119864minus 081198734+ 4119864minus 051198733
minus 00161198732+ 3164119873
minus 148 (Simple tree protocol)
(36)
(2) In Figure 3 there is gap between the theoreticaland experimental curves in case of A3 and simpletree protocol This may be attributed due to thefact that A3 (approximate CDS) only preserves 1-connectivity whereas the simple tree protocol hasmultiple connectivity So A3 protocol requires lessenergy to construct the tree as compared to spanningtree which confirms to the experimental results
42 Discussion on CDS Rule K Consider (34) which ismentioned in the following
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(37)
where 119860 119861 119862 119896 and 119911 have been defined previouslyTable 4 enumerates the value of 119911 for various values of 119896
corresponding to different number of nodes in the networkFrom the results of Table 2 we obtain Figure 4In Table 4 the maximum value of 119911 corresponds to a
probability value 05 which indicates that the clusters havedissociated into one cluster head and one normal node whichideally predicts the case when the energy of the nodes hasdrained out
0
20
40
60
80
100
120
140
0 100 200 300 400 500 600
Simple tree (Th)Simple tree (Exp)
A3 (Th)A3 (Exp)
Number of nodes N
Clus
ter h
eads
(n)
Figure 3Number of nodes versus number of cluster heads in simpletree protocol and A3 protocol
05
101520253035404550
CDS Rule K (Th)CDS Rule k (Exp)
0 100 200 300 400 500 600Number of nodes (N)
Clus
ter h
eads
(n)
Figure 4 Total number of nodes (119873) versus number of cluster headsin CDS-Rule k algorithm
100 4 0263 27 34200 6 0171 35 40300 8 0126 38 49400 10 0101 41 48500 12 0083 42 46119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
In Figure 4 it can be seen that the mechanics of clus-tering in CDS-Rule K is in accordance to the assumptionsconsidered while deducing the distribution function usingmean field theory Unlike A3 and Simple tree protocol CDS-Rule K also exhibits a relation of degree 4 as shown (119899 =
minus2119864minus081198734+3119864minus05119873
3minus0012119873
2+2085119873minus87)The linear
graph as shown in Figure 4 represents the situation when
8 International Journal of Distributed Sensor Networks
Table 4 Value of 119911 for different 119896 values in networks of differentsizes
Nodescollection (119873) 119896 Maxima (119901 119911) Minima (119901 119911)
the value of 119911 isminThis graph is linear due to constant valueof 119901 (05 here) where other graphs were drawn on constant 119911(0 here)
Above experimental results lead us to the followingexplanation of the antipreferential factor 119911
The antipreferential factor can have three distinct values
(a) When 119911 = 0 then the effective degree of nodes is equalto the analytically obtained degree of a node Here thenumber of cluster heads formed will be optimal
(b) When 119911 gt 0 then a node will have fewer connectionsavailable Here the number of cluster heads will behigher
(c) When 119911 lt 0 then a node will have less connectivityHere the number of cluster heads will be highest asthe result will lead to dissociation
5 Conclusions
In this paper we have tried to provide a framework toformally model tree based clustering algorithms in a WSNBased on the formalism we have theoretically calculatedsome parameters such as number of cluster heads and averagenumber of degree for a given algorithm The theoreticalresults tally with results obtained by simulation studies Wehave introduced a factor called 119911 in network evolutionmodelIt seems that the 119911 factor has an impact on functioning ofthese protocols We keep that study as a future exercise Thestudy at the current stage is more suitable for modeling treebased clustering algorithms thatwork on the top of connecteddominating set construction Algorithms that are based onmore parameters have to be modeled appropriately in theframework before application
In recent papers energy distribution has been consideredin network evolution model for wireless sensor networksBut so far the framework of network evolution model hasnot been used to capture the characteristics of clusteringalgorithms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquoAmericanAssociation for theAdvancement of Sciencevol 286 no 5439 pp 509ndash512 1999
[2] X Li and G Chen ldquoA local-world evolving network modelrdquoPhysica A vol 328 no 1-2 pp 274ndash286 2003
[3] L-N Wang J-L Guo H-X Yang and T Zhou ldquoLocal prefer-ential attachment model for hierarchical networksrdquo Physica Avol 388 no 8 pp 1713ndash1720 2009
[4] Z Fan G Chen and Y Zhang ldquoA comprehensive multi-local-world model for complex networksrdquo Physics Letters A vol 373no 18-19 pp 1601ndash1605 2009
[5] Z-H Guan and Z-P Wu ldquoThe physical position neighbour-hood evolving network modelrdquo Physica A vol 387 no 1 pp314ndash322 2008
[6] P Sheridan Y Yagahara and H Shimodaira ldquoA preferentialattachmentmodel with Poisson growth for scale-free networksrdquoAnnals of the Institute of Statistical Mathematics vol 60 no 4pp 747ndash761 2008
[7] L-J Chen D-X Chen L Xie and J-N Cao ldquoEvolution ofwireless sensor networkrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo07) vol556 pp 3005ndash3009 March 2007
[8] X Luo H Yu and X Wang ldquoEnergy-aware topology evolutionmodel with link and node deletion in wireless sensor networksrdquoMathematical Problems in Engineering vol 2012 Article ID281465 14 pages 2012
[9] S Li L Li and Y Yang ldquoA local-world heterogeneous model ofwireless sensor networks with node and link diversityrdquo PhysicaA vol 390 no 6 pp 1182ndash1191 2011
[10] P M Wightman andM A Labrador ldquoA3 a topology construc-tion algorithm for wireless sensor networksrdquo in Proceedings ofthe IEEE Global Telecommunications Conference (GLOBECOMrsquo08) pp 1ndash6 IEEE New Orleans La USA December 2008
[11] J Wu M Cardei F Dai and S Yang ldquoExtended dominatingset and its applications in ad hoc networks using cooperativecommunicationrdquo IEEE Transactions on Parallel and DistributedSystems vol 17 no 8 pp 851ndash864 2006
[12] H L Zhu H Luo H P Peng L X Li and Q Luo ldquoComplexnetworks-based energy-efficient evolution model for wirelesssensor networksrdquo Chaos Solitons and Fractals vol 41 no 4 pp1828ndash1835 2009
[13] A-L Barabasi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 no 1 pp 173ndash187 1999
[14] P M Wightman ldquoAtarraya a topology control simulatorrdquohttpwwwcseusfedusimpedrowatarraya
International Journal of Distributed Sensor Networks 5
31 Analysis of the Dynamic Equation
311 Case I If 119911 = 0 119872 = 1 that is the new nodeselects node unless it reaches 119896 Moreover the preferentialattachment mechanism does not work The rate of growth of119896119894is as
120575119896119894
120575119905
=
11198980 + 119901 lowast 119905
(18)
The denominator of the above expression is the number ofcluster nodes at time 119905
312 Case II If 119872 = 1198980+ 119901 lowast 119905 this means that the local
total degree of node universetotal degree of nodes local-world
=
1198980 + 119901 lowast 119905 + 119873
1198980 + 119901 lowast 119905
=
1198980 + 119901 lowast 119905 + 1198980 + 119905
1198980 + 119901 lowast 119905
(22)
Therefore
119896 =
2 lowast 1198980 + 119905 + 119901 lowast 119905
1198980 + 119901 lowast 119905
(23)
Similarly we have for 119888119894
119888 =
sum119895119888119895
119888119894
= 119896minus
(1 minus 119901) lowast 119905
1198980 + 119901 lowast 119905
= 2 (24)
As the cluster node will have one such node attached to itselfthe status of that node is either another cluster head or normalhead hence the count value of 119888 is 2
Equations (21) and (23) are used to find the values of 119888 and119896
Finally the following equation is formed after substitutingthe values of sum
119860 + 119861 minus 2 lowast 119911 lowast 119862
)
1119860 (28)
Moreover to find the degree distribution 119875(119896) that is theprobability that a node has 119896 edges) we first calculate thecumulative probability 119875[119896
119894(119905) lt 119896] Suppose that the node
enters into the network at equal time intervals We define theprobability density 119905
119894as follows
119875 (119905) =
11198980 + 119905
(29)
So 119875[119896119894(119905) lt 119896] has the following form
Equation (31) denotes the degree distribution function tounderstand the dynamics of clustering algorithms Puttingback the values of 119860 119861 and 119862 we have the distributionfunction as the degree distribution 119875(119896) The probability thata node has 119896 edges is
119875 (119896) asymp (
119901
119901 minus 1+
1 minus 119901
119901
minus
2 lowast 119911
119901 lowast (1 minus 119901)
)
(1+119901)119901
lowast (119896
lowast
119901
1 + 119901
+
1 minus 119901
119901
minus 2 lowast 119911
119901 lowast (1 minus 119901)
)
(minus2lowast119901minus1)119901
(32)
Next the value of 119911 is computed that maximizes (32)Consider
119889119875 (119896)
119889119911
= 0 (33)
By solving (33) we obtain the value of 119911 as
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(34)
Anti preferential attachment means that the node may notlike to have a particular node as its neighbor but in thiscase of sensor networks no such phenomenon is observedhencewe can state that the anti preferential attachment in thisparticular case tends to zero during the evolution process
By setting the value of 119911 to zero we have the followingrelation
119901 =
12[(
119896 + 3119896 minus 1
)
05minus 1] (35)
Equation (35) states the probability of clustering when theantipreferential attachment factor 119911 is zero
4 Analysis of Topology Control Algorithms
In this section we have carried out simulation of threelocalized topology control protocols namely A3 Simple treeProtocol and CDS Rule K Simulation was carried using theAtarraya [14] simulator Total number of nodes was 100 200300 and 500 respectively and theywere tested for A3 SimpleTree Protocol and CDS Rule K The output was recorded foraverage degree of nodes 119896 for three protocols mentionedearlier
935361
916479
67
58
99
97
4934
44
3024
39
65
82
2194
907054
56
26
28
36 100
59
25
40
84
63
68
5787
32
51
47
27
Figure 2 Clustering as obtained by running simple tree protocol ina network of 100 nodes
100 3 0366 37 40200 3 0366 73 71300 4 0263 79 71400 4 0263 105 96500 4 0263 132 130119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
Figure 2 shows an example of clustering and computationof average 119896 value
Figure 2 shows the cluster heads in red circles with whichnormal nodes are attached Taking the number of all theneighboring nodes of every cluster heads and dividing by thenumber of cluster heads we obtain the average value of 119896 Bysimilar procedure we have obtained the average value of 119896 fordifferent number of nodes
We have also carried out simulation study using simpletree protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 1
Similar simulation study was also carried out using CDS-Rule k Protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 2
Similar simulation study was also carried out using A3Protocol for different number of nodes The number of clus-ters obtained using theoretical calculation (35) is compared
International Journal of Distributed Sensor Networks 7
100 4 0264 27 27200 6 0171 35 38300 10 0101 31 33400 10 0101 41 43500 13 0077 39 43119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
against the number of clusters obtained using simulationstudy which is presented in Table 3
41 Discussion on A3 and Simple Tree Protocol Next we plotthe theoretically calculated number of clusters and experimen-tally calculated number of clusters as shown in Tables 1 and 3for A3 and simple tree protocols
We observe the following
(1) In A3 and Simple Tree protocols the curves fortheoretically calculated number of clusters and exper-imentally calculated number of clusters follow fourthdegree equationThe trend line polynomial options ofMS Excel have been used to study the degree of curvefit of the curves to obtain the following polynomials
119899 = minus 4119864minus 101198734+ 8119864minus 071198733
minus 00001198732+ 0197119873
+ 12 (A3 protocol)
119899 = minus 3119864minus 081198734+ 4119864minus 051198733
minus 00161198732+ 3164119873
minus 148 (Simple tree protocol)
(36)
(2) In Figure 3 there is gap between the theoreticaland experimental curves in case of A3 and simpletree protocol This may be attributed due to thefact that A3 (approximate CDS) only preserves 1-connectivity whereas the simple tree protocol hasmultiple connectivity So A3 protocol requires lessenergy to construct the tree as compared to spanningtree which confirms to the experimental results
42 Discussion on CDS Rule K Consider (34) which ismentioned in the following
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(37)
where 119860 119861 119862 119896 and 119911 have been defined previouslyTable 4 enumerates the value of 119911 for various values of 119896
corresponding to different number of nodes in the networkFrom the results of Table 2 we obtain Figure 4In Table 4 the maximum value of 119911 corresponds to a
probability value 05 which indicates that the clusters havedissociated into one cluster head and one normal node whichideally predicts the case when the energy of the nodes hasdrained out
0
20
40
60
80
100
120
140
0 100 200 300 400 500 600
Simple tree (Th)Simple tree (Exp)
A3 (Th)A3 (Exp)
Number of nodes N
Clus
ter h
eads
(n)
Figure 3Number of nodes versus number of cluster heads in simpletree protocol and A3 protocol
05
101520253035404550
CDS Rule K (Th)CDS Rule k (Exp)
0 100 200 300 400 500 600Number of nodes (N)
Clus
ter h
eads
(n)
Figure 4 Total number of nodes (119873) versus number of cluster headsin CDS-Rule k algorithm
100 4 0263 27 34200 6 0171 35 40300 8 0126 38 49400 10 0101 41 48500 12 0083 42 46119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
In Figure 4 it can be seen that the mechanics of clus-tering in CDS-Rule K is in accordance to the assumptionsconsidered while deducing the distribution function usingmean field theory Unlike A3 and Simple tree protocol CDS-Rule K also exhibits a relation of degree 4 as shown (119899 =
minus2119864minus081198734+3119864minus05119873
3minus0012119873
2+2085119873minus87)The linear
graph as shown in Figure 4 represents the situation when
8 International Journal of Distributed Sensor Networks
Table 4 Value of 119911 for different 119896 values in networks of differentsizes
Nodescollection (119873) 119896 Maxima (119901 119911) Minima (119901 119911)
the value of 119911 isminThis graph is linear due to constant valueof 119901 (05 here) where other graphs were drawn on constant 119911(0 here)
Above experimental results lead us to the followingexplanation of the antipreferential factor 119911
The antipreferential factor can have three distinct values
(a) When 119911 = 0 then the effective degree of nodes is equalto the analytically obtained degree of a node Here thenumber of cluster heads formed will be optimal
(b) When 119911 gt 0 then a node will have fewer connectionsavailable Here the number of cluster heads will behigher
(c) When 119911 lt 0 then a node will have less connectivityHere the number of cluster heads will be highest asthe result will lead to dissociation
5 Conclusions
In this paper we have tried to provide a framework toformally model tree based clustering algorithms in a WSNBased on the formalism we have theoretically calculatedsome parameters such as number of cluster heads and averagenumber of degree for a given algorithm The theoreticalresults tally with results obtained by simulation studies Wehave introduced a factor called 119911 in network evolutionmodelIt seems that the 119911 factor has an impact on functioning ofthese protocols We keep that study as a future exercise Thestudy at the current stage is more suitable for modeling treebased clustering algorithms thatwork on the top of connecteddominating set construction Algorithms that are based onmore parameters have to be modeled appropriately in theframework before application
In recent papers energy distribution has been consideredin network evolution model for wireless sensor networksBut so far the framework of network evolution model hasnot been used to capture the characteristics of clusteringalgorithms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquoAmericanAssociation for theAdvancement of Sciencevol 286 no 5439 pp 509ndash512 1999
[2] X Li and G Chen ldquoA local-world evolving network modelrdquoPhysica A vol 328 no 1-2 pp 274ndash286 2003
[3] L-N Wang J-L Guo H-X Yang and T Zhou ldquoLocal prefer-ential attachment model for hierarchical networksrdquo Physica Avol 388 no 8 pp 1713ndash1720 2009
[4] Z Fan G Chen and Y Zhang ldquoA comprehensive multi-local-world model for complex networksrdquo Physics Letters A vol 373no 18-19 pp 1601ndash1605 2009
[5] Z-H Guan and Z-P Wu ldquoThe physical position neighbour-hood evolving network modelrdquo Physica A vol 387 no 1 pp314ndash322 2008
[6] P Sheridan Y Yagahara and H Shimodaira ldquoA preferentialattachmentmodel with Poisson growth for scale-free networksrdquoAnnals of the Institute of Statistical Mathematics vol 60 no 4pp 747ndash761 2008
[7] L-J Chen D-X Chen L Xie and J-N Cao ldquoEvolution ofwireless sensor networkrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo07) vol556 pp 3005ndash3009 March 2007
[8] X Luo H Yu and X Wang ldquoEnergy-aware topology evolutionmodel with link and node deletion in wireless sensor networksrdquoMathematical Problems in Engineering vol 2012 Article ID281465 14 pages 2012
[9] S Li L Li and Y Yang ldquoA local-world heterogeneous model ofwireless sensor networks with node and link diversityrdquo PhysicaA vol 390 no 6 pp 1182ndash1191 2011
[10] P M Wightman andM A Labrador ldquoA3 a topology construc-tion algorithm for wireless sensor networksrdquo in Proceedings ofthe IEEE Global Telecommunications Conference (GLOBECOMrsquo08) pp 1ndash6 IEEE New Orleans La USA December 2008
[11] J Wu M Cardei F Dai and S Yang ldquoExtended dominatingset and its applications in ad hoc networks using cooperativecommunicationrdquo IEEE Transactions on Parallel and DistributedSystems vol 17 no 8 pp 851ndash864 2006
[12] H L Zhu H Luo H P Peng L X Li and Q Luo ldquoComplexnetworks-based energy-efficient evolution model for wirelesssensor networksrdquo Chaos Solitons and Fractals vol 41 no 4 pp1828ndash1835 2009
[13] A-L Barabasi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 no 1 pp 173ndash187 1999
[14] P M Wightman ldquoAtarraya a topology control simulatorrdquohttpwwwcseusfedusimpedrowatarraya
Equation (31) denotes the degree distribution function tounderstand the dynamics of clustering algorithms Puttingback the values of 119860 119861 and 119862 we have the distributionfunction as the degree distribution 119875(119896) The probability thata node has 119896 edges is
119875 (119896) asymp (
119901
119901 minus 1+
1 minus 119901
119901
minus
2 lowast 119911
119901 lowast (1 minus 119901)
)
(1+119901)119901
lowast (119896
lowast
119901
1 + 119901
+
1 minus 119901
119901
minus 2 lowast 119911
119901 lowast (1 minus 119901)
)
(minus2lowast119901minus1)119901
(32)
Next the value of 119911 is computed that maximizes (32)Consider
119889119875 (119896)
119889119911
= 0 (33)
By solving (33) we obtain the value of 119911 as
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(34)
Anti preferential attachment means that the node may notlike to have a particular node as its neighbor but in thiscase of sensor networks no such phenomenon is observedhencewe can state that the anti preferential attachment in thisparticular case tends to zero during the evolution process
By setting the value of 119911 to zero we have the followingrelation
119901 =
12[(
119896 + 3119896 minus 1
)
05minus 1] (35)
Equation (35) states the probability of clustering when theantipreferential attachment factor 119911 is zero
4 Analysis of Topology Control Algorithms
In this section we have carried out simulation of threelocalized topology control protocols namely A3 Simple treeProtocol and CDS Rule K Simulation was carried using theAtarraya [14] simulator Total number of nodes was 100 200300 and 500 respectively and theywere tested for A3 SimpleTree Protocol and CDS Rule K The output was recorded foraverage degree of nodes 119896 for three protocols mentionedearlier
935361
916479
67
58
99
97
4934
44
3024
39
65
82
2194
907054
56
26
28
36 100
59
25
40
84
63
68
5787
32
51
47
27
Figure 2 Clustering as obtained by running simple tree protocol ina network of 100 nodes
100 3 0366 37 40200 3 0366 73 71300 4 0263 79 71400 4 0263 105 96500 4 0263 132 130119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
Figure 2 shows an example of clustering and computationof average 119896 value
Figure 2 shows the cluster heads in red circles with whichnormal nodes are attached Taking the number of all theneighboring nodes of every cluster heads and dividing by thenumber of cluster heads we obtain the average value of 119896 Bysimilar procedure we have obtained the average value of 119896 fordifferent number of nodes
We have also carried out simulation study using simpletree protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 1
Similar simulation study was also carried out using CDS-Rule k Protocol for different number of nodes The numberof clusters obtained using theoretical calculation (35) iscompared against the number of clusters obtained usingsimulation study which is presented in Table 2
Similar simulation study was also carried out using A3Protocol for different number of nodes The number of clus-ters obtained using theoretical calculation (35) is compared
International Journal of Distributed Sensor Networks 7
100 4 0264 27 27200 6 0171 35 38300 10 0101 31 33400 10 0101 41 43500 13 0077 39 43119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
against the number of clusters obtained using simulationstudy which is presented in Table 3
41 Discussion on A3 and Simple Tree Protocol Next we plotthe theoretically calculated number of clusters and experimen-tally calculated number of clusters as shown in Tables 1 and 3for A3 and simple tree protocols
We observe the following
(1) In A3 and Simple Tree protocols the curves fortheoretically calculated number of clusters and exper-imentally calculated number of clusters follow fourthdegree equationThe trend line polynomial options ofMS Excel have been used to study the degree of curvefit of the curves to obtain the following polynomials
119899 = minus 4119864minus 101198734+ 8119864minus 071198733
minus 00001198732+ 0197119873
+ 12 (A3 protocol)
119899 = minus 3119864minus 081198734+ 4119864minus 051198733
minus 00161198732+ 3164119873
minus 148 (Simple tree protocol)
(36)
(2) In Figure 3 there is gap between the theoreticaland experimental curves in case of A3 and simpletree protocol This may be attributed due to thefact that A3 (approximate CDS) only preserves 1-connectivity whereas the simple tree protocol hasmultiple connectivity So A3 protocol requires lessenergy to construct the tree as compared to spanningtree which confirms to the experimental results
42 Discussion on CDS Rule K Consider (34) which ismentioned in the following
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(37)
where 119860 119861 119862 119896 and 119911 have been defined previouslyTable 4 enumerates the value of 119911 for various values of 119896
corresponding to different number of nodes in the networkFrom the results of Table 2 we obtain Figure 4In Table 4 the maximum value of 119911 corresponds to a
probability value 05 which indicates that the clusters havedissociated into one cluster head and one normal node whichideally predicts the case when the energy of the nodes hasdrained out
0
20
40
60
80
100
120
140
0 100 200 300 400 500 600
Simple tree (Th)Simple tree (Exp)
A3 (Th)A3 (Exp)
Number of nodes N
Clus
ter h
eads
(n)
Figure 3Number of nodes versus number of cluster heads in simpletree protocol and A3 protocol
05
101520253035404550
CDS Rule K (Th)CDS Rule k (Exp)
0 100 200 300 400 500 600Number of nodes (N)
Clus
ter h
eads
(n)
Figure 4 Total number of nodes (119873) versus number of cluster headsin CDS-Rule k algorithm
100 4 0263 27 34200 6 0171 35 40300 8 0126 38 49400 10 0101 41 48500 12 0083 42 46119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
In Figure 4 it can be seen that the mechanics of clus-tering in CDS-Rule K is in accordance to the assumptionsconsidered while deducing the distribution function usingmean field theory Unlike A3 and Simple tree protocol CDS-Rule K also exhibits a relation of degree 4 as shown (119899 =
minus2119864minus081198734+3119864minus05119873
3minus0012119873
2+2085119873minus87)The linear
graph as shown in Figure 4 represents the situation when
8 International Journal of Distributed Sensor Networks
Table 4 Value of 119911 for different 119896 values in networks of differentsizes
Nodescollection (119873) 119896 Maxima (119901 119911) Minima (119901 119911)
the value of 119911 isminThis graph is linear due to constant valueof 119901 (05 here) where other graphs were drawn on constant 119911(0 here)
Above experimental results lead us to the followingexplanation of the antipreferential factor 119911
The antipreferential factor can have three distinct values
(a) When 119911 = 0 then the effective degree of nodes is equalto the analytically obtained degree of a node Here thenumber of cluster heads formed will be optimal
(b) When 119911 gt 0 then a node will have fewer connectionsavailable Here the number of cluster heads will behigher
(c) When 119911 lt 0 then a node will have less connectivityHere the number of cluster heads will be highest asthe result will lead to dissociation
5 Conclusions
In this paper we have tried to provide a framework toformally model tree based clustering algorithms in a WSNBased on the formalism we have theoretically calculatedsome parameters such as number of cluster heads and averagenumber of degree for a given algorithm The theoreticalresults tally with results obtained by simulation studies Wehave introduced a factor called 119911 in network evolutionmodelIt seems that the 119911 factor has an impact on functioning ofthese protocols We keep that study as a future exercise Thestudy at the current stage is more suitable for modeling treebased clustering algorithms thatwork on the top of connecteddominating set construction Algorithms that are based onmore parameters have to be modeled appropriately in theframework before application
In recent papers energy distribution has been consideredin network evolution model for wireless sensor networksBut so far the framework of network evolution model hasnot been used to capture the characteristics of clusteringalgorithms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquoAmericanAssociation for theAdvancement of Sciencevol 286 no 5439 pp 509ndash512 1999
[2] X Li and G Chen ldquoA local-world evolving network modelrdquoPhysica A vol 328 no 1-2 pp 274ndash286 2003
[3] L-N Wang J-L Guo H-X Yang and T Zhou ldquoLocal prefer-ential attachment model for hierarchical networksrdquo Physica Avol 388 no 8 pp 1713ndash1720 2009
[4] Z Fan G Chen and Y Zhang ldquoA comprehensive multi-local-world model for complex networksrdquo Physics Letters A vol 373no 18-19 pp 1601ndash1605 2009
[5] Z-H Guan and Z-P Wu ldquoThe physical position neighbour-hood evolving network modelrdquo Physica A vol 387 no 1 pp314ndash322 2008
[6] P Sheridan Y Yagahara and H Shimodaira ldquoA preferentialattachmentmodel with Poisson growth for scale-free networksrdquoAnnals of the Institute of Statistical Mathematics vol 60 no 4pp 747ndash761 2008
[7] L-J Chen D-X Chen L Xie and J-N Cao ldquoEvolution ofwireless sensor networkrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo07) vol556 pp 3005ndash3009 March 2007
[8] X Luo H Yu and X Wang ldquoEnergy-aware topology evolutionmodel with link and node deletion in wireless sensor networksrdquoMathematical Problems in Engineering vol 2012 Article ID281465 14 pages 2012
[9] S Li L Li and Y Yang ldquoA local-world heterogeneous model ofwireless sensor networks with node and link diversityrdquo PhysicaA vol 390 no 6 pp 1182ndash1191 2011
[10] P M Wightman andM A Labrador ldquoA3 a topology construc-tion algorithm for wireless sensor networksrdquo in Proceedings ofthe IEEE Global Telecommunications Conference (GLOBECOMrsquo08) pp 1ndash6 IEEE New Orleans La USA December 2008
[11] J Wu M Cardei F Dai and S Yang ldquoExtended dominatingset and its applications in ad hoc networks using cooperativecommunicationrdquo IEEE Transactions on Parallel and DistributedSystems vol 17 no 8 pp 851ndash864 2006
[12] H L Zhu H Luo H P Peng L X Li and Q Luo ldquoComplexnetworks-based energy-efficient evolution model for wirelesssensor networksrdquo Chaos Solitons and Fractals vol 41 no 4 pp1828ndash1835 2009
[13] A-L Barabasi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 no 1 pp 173ndash187 1999
[14] P M Wightman ldquoAtarraya a topology control simulatorrdquohttpwwwcseusfedusimpedrowatarraya
100 4 0264 27 27200 6 0171 35 38300 10 0101 31 33400 10 0101 41 43500 13 0077 39 43119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
against the number of clusters obtained using simulationstudy which is presented in Table 3
41 Discussion on A3 and Simple Tree Protocol Next we plotthe theoretically calculated number of clusters and experimen-tally calculated number of clusters as shown in Tables 1 and 3for A3 and simple tree protocols
We observe the following
(1) In A3 and Simple Tree protocols the curves fortheoretically calculated number of clusters and exper-imentally calculated number of clusters follow fourthdegree equationThe trend line polynomial options ofMS Excel have been used to study the degree of curvefit of the curves to obtain the following polynomials
119899 = minus 4119864minus 101198734+ 8119864minus 071198733
minus 00001198732+ 0197119873
+ 12 (A3 protocol)
119899 = minus 3119864minus 081198734+ 4119864minus 051198733
minus 00161198732+ 3164119873
minus 148 (Simple tree protocol)
(36)
(2) In Figure 3 there is gap between the theoreticaland experimental curves in case of A3 and simpletree protocol This may be attributed due to thefact that A3 (approximate CDS) only preserves 1-connectivity whereas the simple tree protocol hasmultiple connectivity So A3 protocol requires lessenergy to construct the tree as compared to spanningtree which confirms to the experimental results
42 Discussion on CDS Rule K Consider (34) which ismentioned in the following
119911 =
(1 + 119860 + 119861 minus 119896)
2 lowast 119862
(37)
where 119860 119861 119862 119896 and 119911 have been defined previouslyTable 4 enumerates the value of 119911 for various values of 119896
corresponding to different number of nodes in the networkFrom the results of Table 2 we obtain Figure 4In Table 4 the maximum value of 119911 corresponds to a
probability value 05 which indicates that the clusters havedissociated into one cluster head and one normal node whichideally predicts the case when the energy of the nodes hasdrained out
0
20
40
60
80
100
120
140
0 100 200 300 400 500 600
Simple tree (Th)Simple tree (Exp)
A3 (Th)A3 (Exp)
Number of nodes N
Clus
ter h
eads
(n)
Figure 3Number of nodes versus number of cluster heads in simpletree protocol and A3 protocol
05
101520253035404550
CDS Rule K (Th)CDS Rule k (Exp)
0 100 200 300 400 500 600Number of nodes (N)
Clus
ter h
eads
(n)
Figure 4 Total number of nodes (119873) versus number of cluster headsin CDS-Rule k algorithm
100 4 0263 27 34200 6 0171 35 40300 8 0126 38 49400 10 0101 41 48500 12 0083 42 46119873 = number of nodes 119896 = number of neighbors 119901 = probability of selectinga cluster head (35) 119899(119879) = theoretically calculated number of clusters and119899(119864) = experimentally calculated number of clusters
In Figure 4 it can be seen that the mechanics of clus-tering in CDS-Rule K is in accordance to the assumptionsconsidered while deducing the distribution function usingmean field theory Unlike A3 and Simple tree protocol CDS-Rule K also exhibits a relation of degree 4 as shown (119899 =
minus2119864minus081198734+3119864minus05119873
3minus0012119873
2+2085119873minus87)The linear
graph as shown in Figure 4 represents the situation when
8 International Journal of Distributed Sensor Networks
Table 4 Value of 119911 for different 119896 values in networks of differentsizes
Nodescollection (119873) 119896 Maxima (119901 119911) Minima (119901 119911)
the value of 119911 isminThis graph is linear due to constant valueof 119901 (05 here) where other graphs were drawn on constant 119911(0 here)
Above experimental results lead us to the followingexplanation of the antipreferential factor 119911
The antipreferential factor can have three distinct values
(a) When 119911 = 0 then the effective degree of nodes is equalto the analytically obtained degree of a node Here thenumber of cluster heads formed will be optimal
(b) When 119911 gt 0 then a node will have fewer connectionsavailable Here the number of cluster heads will behigher
(c) When 119911 lt 0 then a node will have less connectivityHere the number of cluster heads will be highest asthe result will lead to dissociation
5 Conclusions
In this paper we have tried to provide a framework toformally model tree based clustering algorithms in a WSNBased on the formalism we have theoretically calculatedsome parameters such as number of cluster heads and averagenumber of degree for a given algorithm The theoreticalresults tally with results obtained by simulation studies Wehave introduced a factor called 119911 in network evolutionmodelIt seems that the 119911 factor has an impact on functioning ofthese protocols We keep that study as a future exercise Thestudy at the current stage is more suitable for modeling treebased clustering algorithms thatwork on the top of connecteddominating set construction Algorithms that are based onmore parameters have to be modeled appropriately in theframework before application
In recent papers energy distribution has been consideredin network evolution model for wireless sensor networksBut so far the framework of network evolution model hasnot been used to capture the characteristics of clusteringalgorithms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquoAmericanAssociation for theAdvancement of Sciencevol 286 no 5439 pp 509ndash512 1999
[2] X Li and G Chen ldquoA local-world evolving network modelrdquoPhysica A vol 328 no 1-2 pp 274ndash286 2003
[3] L-N Wang J-L Guo H-X Yang and T Zhou ldquoLocal prefer-ential attachment model for hierarchical networksrdquo Physica Avol 388 no 8 pp 1713ndash1720 2009
[4] Z Fan G Chen and Y Zhang ldquoA comprehensive multi-local-world model for complex networksrdquo Physics Letters A vol 373no 18-19 pp 1601ndash1605 2009
[5] Z-H Guan and Z-P Wu ldquoThe physical position neighbour-hood evolving network modelrdquo Physica A vol 387 no 1 pp314ndash322 2008
[6] P Sheridan Y Yagahara and H Shimodaira ldquoA preferentialattachmentmodel with Poisson growth for scale-free networksrdquoAnnals of the Institute of Statistical Mathematics vol 60 no 4pp 747ndash761 2008
[7] L-J Chen D-X Chen L Xie and J-N Cao ldquoEvolution ofwireless sensor networkrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo07) vol556 pp 3005ndash3009 March 2007
[8] X Luo H Yu and X Wang ldquoEnergy-aware topology evolutionmodel with link and node deletion in wireless sensor networksrdquoMathematical Problems in Engineering vol 2012 Article ID281465 14 pages 2012
[9] S Li L Li and Y Yang ldquoA local-world heterogeneous model ofwireless sensor networks with node and link diversityrdquo PhysicaA vol 390 no 6 pp 1182ndash1191 2011
[10] P M Wightman andM A Labrador ldquoA3 a topology construc-tion algorithm for wireless sensor networksrdquo in Proceedings ofthe IEEE Global Telecommunications Conference (GLOBECOMrsquo08) pp 1ndash6 IEEE New Orleans La USA December 2008
[11] J Wu M Cardei F Dai and S Yang ldquoExtended dominatingset and its applications in ad hoc networks using cooperativecommunicationrdquo IEEE Transactions on Parallel and DistributedSystems vol 17 no 8 pp 851ndash864 2006
[12] H L Zhu H Luo H P Peng L X Li and Q Luo ldquoComplexnetworks-based energy-efficient evolution model for wirelesssensor networksrdquo Chaos Solitons and Fractals vol 41 no 4 pp1828ndash1835 2009
[13] A-L Barabasi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 no 1 pp 173ndash187 1999
[14] P M Wightman ldquoAtarraya a topology control simulatorrdquohttpwwwcseusfedusimpedrowatarraya
the value of 119911 isminThis graph is linear due to constant valueof 119901 (05 here) where other graphs were drawn on constant 119911(0 here)
Above experimental results lead us to the followingexplanation of the antipreferential factor 119911
The antipreferential factor can have three distinct values
(a) When 119911 = 0 then the effective degree of nodes is equalto the analytically obtained degree of a node Here thenumber of cluster heads formed will be optimal
(b) When 119911 gt 0 then a node will have fewer connectionsavailable Here the number of cluster heads will behigher
(c) When 119911 lt 0 then a node will have less connectivityHere the number of cluster heads will be highest asthe result will lead to dissociation
5 Conclusions
In this paper we have tried to provide a framework toformally model tree based clustering algorithms in a WSNBased on the formalism we have theoretically calculatedsome parameters such as number of cluster heads and averagenumber of degree for a given algorithm The theoreticalresults tally with results obtained by simulation studies Wehave introduced a factor called 119911 in network evolutionmodelIt seems that the 119911 factor has an impact on functioning ofthese protocols We keep that study as a future exercise Thestudy at the current stage is more suitable for modeling treebased clustering algorithms thatwork on the top of connecteddominating set construction Algorithms that are based onmore parameters have to be modeled appropriately in theframework before application
In recent papers energy distribution has been consideredin network evolution model for wireless sensor networksBut so far the framework of network evolution model hasnot been used to capture the characteristics of clusteringalgorithms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquoAmericanAssociation for theAdvancement of Sciencevol 286 no 5439 pp 509ndash512 1999
[2] X Li and G Chen ldquoA local-world evolving network modelrdquoPhysica A vol 328 no 1-2 pp 274ndash286 2003
[3] L-N Wang J-L Guo H-X Yang and T Zhou ldquoLocal prefer-ential attachment model for hierarchical networksrdquo Physica Avol 388 no 8 pp 1713ndash1720 2009
[4] Z Fan G Chen and Y Zhang ldquoA comprehensive multi-local-world model for complex networksrdquo Physics Letters A vol 373no 18-19 pp 1601ndash1605 2009
[5] Z-H Guan and Z-P Wu ldquoThe physical position neighbour-hood evolving network modelrdquo Physica A vol 387 no 1 pp314ndash322 2008
[6] P Sheridan Y Yagahara and H Shimodaira ldquoA preferentialattachmentmodel with Poisson growth for scale-free networksrdquoAnnals of the Institute of Statistical Mathematics vol 60 no 4pp 747ndash761 2008
[7] L-J Chen D-X Chen L Xie and J-N Cao ldquoEvolution ofwireless sensor networkrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo07) vol556 pp 3005ndash3009 March 2007
[8] X Luo H Yu and X Wang ldquoEnergy-aware topology evolutionmodel with link and node deletion in wireless sensor networksrdquoMathematical Problems in Engineering vol 2012 Article ID281465 14 pages 2012
[9] S Li L Li and Y Yang ldquoA local-world heterogeneous model ofwireless sensor networks with node and link diversityrdquo PhysicaA vol 390 no 6 pp 1182ndash1191 2011
[10] P M Wightman andM A Labrador ldquoA3 a topology construc-tion algorithm for wireless sensor networksrdquo in Proceedings ofthe IEEE Global Telecommunications Conference (GLOBECOMrsquo08) pp 1ndash6 IEEE New Orleans La USA December 2008
[11] J Wu M Cardei F Dai and S Yang ldquoExtended dominatingset and its applications in ad hoc networks using cooperativecommunicationrdquo IEEE Transactions on Parallel and DistributedSystems vol 17 no 8 pp 851ndash864 2006
[12] H L Zhu H Luo H P Peng L X Li and Q Luo ldquoComplexnetworks-based energy-efficient evolution model for wirelesssensor networksrdquo Chaos Solitons and Fractals vol 41 no 4 pp1828ndash1835 2009
[13] A-L Barabasi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 no 1 pp 173ndash187 1999
[14] P M Wightman ldquoAtarraya a topology control simulatorrdquohttpwwwcseusfedusimpedrowatarraya
