Irregular Cellular Learning Automata and Its Application to Clustering in Sensor Networks

Transcription

1 Irregular Cellular Learnng Automata and Its Applcaton to Clusterng n Sensor Networks M. Esnaashar 1, M. R. Meybod 1,2 1 Soft Computng Laboratory, Computer Engneerng and Informaton Technology Department Amrkabr Unversty of Technology, Tehran, Iran (Esnaashar,mmeybod)@aut.ac.r 2 Insttute for Studes n Theoretcal Physcs and Mathematcs (IPM) School of Computer Scence, Tehran, Iran Abstract In the frst part of ths paper, we propose a generalzaton of cellular learnng automata (CLA) called rregular cellular learnng automata (ICLA) whch removes the restrcton of rectangular grd structure n tradtonal CLA. Ths generalzaton s expected because there are a number of applcatons whch cannot be adequately modeled wth rectangular grds. One category of such applcatons s n the area of wreless sensor networks. In these networks, nodes are usually scattered randomly throughout the envronment, so no regular structure can be assumed for modelng ther behavor. In the second part of the paper, based on the proposed model we desgn a clusterng algorthm for sensor networks. Smulaton results show that the proposed clusterng algorthm s very effcent and outperforms smlar exstng methods. Index Terms Irregular cellular learnng automata, Cellular learnng automata, Sensor networks, Clusterng algorthm 1. Introducton Cellular automata are mathematcal models for systems consstng of large number of smple dentcal components wth local nteractons. CA s a non-lnear dynamcal system n whch space and tme are dscrete. It s called cellular because t s made up of cells lke ponts n a lattce or lke squares of checker boards, and t s called automata because t follows a smple rule [2]. The smple components act together to produce complcated patterns of behavor. Cellular automata perform complex computatons wth a hgh degree of effcency and robustness. They are especally sutable for modelng natural systems that can be descrbed as massve collectons of smple objects nteractng locally wth each other [3, 4]. Informally, a d-dmensonal CA conssts of an nfnte d-dmensonal lattce of dentcal cells. Each cell can assume a state from a fnte set of states. The cells update ther states synchronously on dscrete steps accordng to a local rule. The new state of each cell depends on the prevous states of a set of cells, ncludng the cell tself, and consttutes ts neghborhood [5]. The state of all cells n the lattce s descrbed by a confguraton. A confguraton can be descrbed as the state of the whole lattce. The rule and the 14

2 ntal confguraton of the CA specfy the evoluton of CA that tells how each confguraton s changed n one step. Formally, a CA can be defned as follows: Defnton 1. A d-dmensonal cellular automata s a structure A = ( Z d, Φ, N, F ), where Z d s a lattce of d-tuples of nteger numbers. Each cell n the d-dmensonal lattce Z d s represented by a d-tuple (z 1, z 2,..., z d ). Φ = {1,..., m} s a fnte set of states. N = { x x,..., } s a fnte subset of Z d called the neghborhood vector, where d 1, 2 xm x Z. The Neghborhood vector determnes the relatve poston of the neghborng lattce cells from any gven cell u n the lattce Z d. The neghbors of a partcular cell u are the set of cells. We assume that there exsts a neghborhood functon { u + x = 1,2,... m} N (u) that maps a cell u to the set of ts neghbors, that s N ( u) = ( u + x1, u + x2,..., u + xm). For sake of smplcty, we assume that the frst element of the neghborhood vector (.e. x ) s 1 equal to d-tuple (0, 0,..., 0) or equvalently u + x1 = u. The neghborhood functon N (u) must satsfy the followng two condtons: - u N(u ) for all u Z d. - u1 N ( u2) u2 N ( u1) for all u u Z d 1, 2 F : Φ m Φ s the local rule of the cellular automata. It computes the new state for each cell from the current states of ts neghbors. On the other hand, learnng automaton s a smple entty operates n an unknown random envronment. In a smple form, the automaton has a fnte set of actons to choose from and at each stage, ts choce (acton) depends upon ts acton probablty vector. For each acton chosen by the automaton, the envronment gves a renforcement sgnal wth fxed unknown probablty dstrbuton. The automaton then updates ts acton probablty vector dependng upon the renforcement sgnal at that stage, and evolves to the some fnal desred behavor. A class of learnng automata s called varable structure learnng automata and are represented by quadruple { α, β, p, T} n whch α = { α 1, α 2, L, α r } represents the acton set of the automata, β = { β1, β 2, L, β r } represents the nput set, p = { p1, p2, L, pr} represents the acton probablty set, and fnally p( n + 1) = T[ α( n), β ( n), p( n)] represents the learnng algorthm. Let α be the acton chosen at tme n, then the recurrence equaton for updatng p s defned as for favorable responses, and p ( n + 1) = p ( n) + a.(1 p ( n)) p j ( n + 1) = p j ( n) a. p j ( n) p ( n + 1) = (1 b). p ( n) b p j ( n + 1) = + (1 b) p j ( n) r 1 j j j j for unfavorable ones. In these equatons, a and b are reward and penalty parameters respectvely. If a = b, learnng algorthm s called L 1 R P, f a << b, t s called L 2 R ε P, and f b = 0, t s called L 3. R I (1) (2) 1 Lnear Reward-Penalty 2 Lnear Reward epslon Penalty 3 Lnear Reward Inacton 15

3 Cellular learnng automata [1], whch s a combnaton of cellular automata (CA) and learnng automata (LA), s a powerful mathematcal model for many decentralzed problems and phenomena. The basc dea of CLA, whch s a subclass of stochastc CA, s to use learnng automata to adjust the state transton probablty of stochastc CA. A CLA s a CA n whch a learnng automaton s assgned to every cell. The learnng automaton resdng n a partcular cell determnes ts acton (state) on the bass of ts acton probablty vector. Lke CA, there s a rule that the CLA operates under. The rule of the CLA and the actons selected by the neghborng LAs of any partcular LA determne the renforcement sgnal to the LA resdng n a cell. The neghborng LAs of any partcular LA consttute the local envronment of that cell. The local envronment of a cell s nonstatonary because the acton probablty vectors of the neghborng LAs vary durng evoluton of the CLA. The basc dea of CLA, whch s a subclass of stochastc CA, s to use learnng automata to adjust the state transton probablty of stochastc CA. A d- dmensonal CLA s formally defned below. Defnton 2. A d-dmensonal cellular learnng automata s a structure A = ( Z d, Φ, A, N, F ) where Z d s a lattce of d-tuples of nteger numbers. Φ s a fnte set of states. A s the set of LA each of whch s assgned to one cell of the CLA. N = { x x,..., } s a fnte subset of Z d called the neghborhood vector, where d 1, 2 xm x Z. F : Φ m β s the local rule of the cellular learnng automata, where β s the set of values that the renforcement sgnal can take. It computes the renforcement sgnal for each LA based on the actons selected by the neghborng LA. It computes the new state for each cell from the current states of ts neghbors. CLA has found many applcatons such as mage processng [6, 7, 8, 9], rumor dffuson [10], modelng of commerce networks [8], channel assgnment n cellular networks [11] and VLSI placement [12], to menton a few. In the frst part of ths paper, we propose a generalzaton of cellular learnng automata (CLA) called rregular cellular learnng automata (ICLA) whch removes the restrcton of rectangular grd structure n tradtonal CLA. Ths generalzaton s expected because there are applcatons whch cannot be adequately modeled wth rectangular grds. One category of such applcatons s n the area of wreless sensor networks. In these networks, nodes are usually scattered randomly throughout the envronment, so no regular structure can be assumed for modelng ther behavor. In the second part of the paper, based on the proposed model, we desgn a clusterng algorthm for sensor networks. The proposed clusterng algorthm tres to maxmze the probablty of selectng nodes wth hgher energy and hgher number of neghbors as cluster heads. To evaluate the performance of the proposed algorthm several experments have been conducted usng dfferent clusterng algorthms and the results are compared wth the proposed algorthm Smulaton results show that the proposed clusterng algorthm s very effcent n terms clusterng formaton n the network) and outperform smlar exstng methods. The rest of ths paper s organzed as follows. In secton 2 rregular cellular learnng automata s presented. Secton 3 gves an overvew on clusterng algorthms for wreless sensor networks. The proposed ICLA based clusterng algorthm s descrbed n secton 4. Smulaton results are gven n secton 5. Secton 6 s the concluson. 16

4 2. Irregular Cellular Learnng Automata The dea of rregular cellular automata was suggested n md 80s [13], but due to the computatonally ntensve operatons requred to search rregular neghborhood, t has been receved less attenton snce then. In an nformal way, ICA s a confguraton of ponts n the space wth no pror restrcton. Each pont has a number of other ponts as ts neghbors. The few examples of ICA all use Vorono polygons or the related Delaunay trangulaton to dvde space and determne the neghbors of each pont [14]. Vorono polygons dvde space nto regons surroundng objects such that any pont n an object's polygon s closer to that object than to any other object, whle Delaunay trangulaton s a trangulaton of the ponts n a Vorono dagram where the crcumcrcle of each trangle s an empty trangle. Irregular cellular learnng automata (Fgure 1) s a combnaton of rregular cellular automata (ICA) and learnng automata. We defne ICLA as an undrected graph n whch, each vertex represents a cell whch s equpped wth a learnng automaton. The learnng automaton resdng n a partcular cell determnes ts state (acton) on the bass of ts acton probablty vector. Lke CLA, there s a rule that the ICLA operate under. The rule of the CLA and the actons selected by the neghborng LAs of any partcular LA determne the renforcement sgnal to the LA resdng n a cell. The neghborng LAs of any partcular LA consttute the local envronment of that cell. The local envronment of a cell s non-statonary because the acton probablty vectors of the neghborng LAs vary durng evoluton of the ICLA. An ICLA s formally defned below., where A = ( G < E, V >, Φ, A, F) Defnton 3. An rregular cellular learnng automata s a structure G s an undrected graph, wth V as the set of vertces and E as the set of edges. Φ s a fnte set of states. A s the set of LA each of whch s assgned to one cell of the ICLA. s the local rule of the rregular cellular learnng automata n each vertex j, where F : Φ j β Φ j = { Φ (, j) E} + { Φ j} s the set of states of all neghbors of j and β s the set of values that the renforcement sgnal can take. β computes the renforcement sgnal for LA j based on the actons selected by the neghborng LA. Fgure 1. Irregular cellular learnng automata Note that n the defnton of ICLA, no explct defnton of neghborhood of each cell s gven. Ths s because neghborhood n ICLA s mplctly defned n defnton of the graph G. 17

5 In what follows, we consder ICLA wth n cells. The learnng automaton A whch has a fnte acton set α s assocated to cell (for =1,2,,,,n) of the ICLA. Let the cardnalty of α be m. The state of the ICLA represented by p= (p 1, p 2,..., p n ), where s the acton p = ( p1, p2,..., pm ) probablty vector of A. The operaton of the ICLA takes place as the followng teratons. At teraton k, each learnng automaton chooses an acton. Let α be the acton chosen by A. Then all learnng automata α receve a renforcement sgnal. Let be the renforcement sgnal receved by A β β. Ths renforcement sgnal s produced by the applcaton of local rule. Fnally, each LA F( Φ ) β updates ts acton probablty vector on the bass of the suppled renforcement sgnal and the chosen acton by the cell. Ths process contnues untl the desred result s obtaned. 3. Clusterng Algorthms In ths secton, we wll brefly overvew some of the exstng clusterng algorthms for wreless sensor networks. One of the frst clusterng algorthms ntroduced for sensor networks s the low energy adaptve clusterng herarchy (LEACH) algorthm [15]. The operaton of LEACH s separated nto two phases: the setup phase and the steady state phase. Durng the setup phase, a predetermned fracton of nodes p, elect themselves as cluster heads by comparng a chosen random number wth a predefned threshold. After the cluster heads have been elected, they broadcast an advertsement message to the rest of the nodes n the network that they are the new cluster heads. Upon recevng ths advertsement, all the noncluster head nodes decde on the cluster to whch they want to belong, based on the sgnal strength of the advertsement. The noncluster head nodes nform the approprate cluster heads that they wll be members of the cluster. There are algorthms whch elect cluster heads based on a certan crteron such as number of neghbor nodes [16, 17], or remanng energy of the nodes [18]. In the algorthm presented n [19], each node wats for a random duraton. After ths perod, f no message s receved from a certan cluster head, the node states tself as a new cluster head. In some other methods, such as the one n [20], cluster heads are specfed pror to network deployment. These specfed nodes are placed n certan postons and other nodes are scattered around them. After the network deployment, these nodes try to form clusters havng balanced traffc load. Ths means that fewer nodes are assgned to the clusters closer to the snk node and more nodes are assgned to the clusters far from the snk. In the sensor network consdered n [21], two classes of nodes are avalable; sensor nodes and aggregator nodes. Aggregator nodes are placed n certan poston and play the role of cluster heads. In ths paper three dfferent algorthms are presented for scatterng sensor nodes around the aggregators. In [22], snk node s assumed to be a moble node whch queres data from dfferent part of the network based on ts dstance from dfferent nodes. For ths reason, t s requred to have a dynamc clusterng algorthm whch can adapt tself to the changng poston of the moble snk, consderng the overall energy consumpton n the network. A number of dfferent routng algorthms for sensor networks are surveyed n [23, 24]. One major class of these algorthms s the herarchcal routng. In herarchcal routng algorthms, at 18

6 frst a clusterng scheme s appled to form a knd of herarchy and afterward, the process of routng s dvded nto nter and ntra-cluster phases. Some examples of these algorthms are "Fxed sze cluster routng", TEEN 4, and APTEEN 5. AIMRP 6 [25] assumes that the snk node s placed n the center of the network. In the setup phase, snk ntates a Tre message to the nodes n the radus of a around tself. These nodes consttute the frst level. These nodes rebroadcast the Ter message to the nodes n the radus of a around themselves. Ths process contnues untl all nodes n the network fnd ther level n the herarchy to the snk node. In [26] an algorthm called Hybrd Energy-Effcent Approach (HEED) 7 has been gven. Ths algorthm has four prmary goals: () prolongng network lfetme, () termnatng the clusterng process wthn a constant number of teratons/steps, () mnmzng control overhead (to be lnear n the number of nodes), and (v) producng well-dstrbuted cluster heads and compact clusters. Before a node starts executng HEED, t sets ts probablty of becomng a cluster head, CH based on ts resdual energy. Usng ths probablty, each node decdes to become a prob tentatve cluster head or not. Tentatve cluster heads, declare themselves to ther neghbors usng a cluster-head message, and double ther CH probablty. Once prob CH n a node reaches 1, prob that node becomes a fnal cluster head. Nodes whch are not cluster head (tentatve or fnal), upon recevng a cluster-head message, jon to the declared cluster. Nodes whch do not receve ant cluster-head message after a whle, decde to become cluster head. In [27] an enhancement on HEED algorthm n whch, only nodes wth resdual energy hgher than a specfed threshold can become cluster head s presented. Also at the end of the algorthm, when some nodes do not jon to any cluster yet, unlke HEED n whch all these nodes become cluster head, HEED algorthm s executed agan for electng cluster heads among them. 4. The Proposed Algorthm Three man objectves of a good clusterng method for sensor networks are ) producng welldstrbuted cluster heads, ) maxmzng the electon probablty of hgh energy level nodes as cluster heads and ) mnmzng the number of sparse clusters. In ths secton an attempt has been made to desgn a clusterng algorthm to satsfy all three objectves. The frst objectve s mplctly satsfed by the use of ICLA for the clusterng. Ths s because n each neghborhood of the ICLA, at least one cluster head wll be elected and ths wll guarantee the well-dstrbuton of cluster heads among the network. The other two objectves wll be satsfed through a proper selecton of local rule for ICLA. Fgures 2 and 3 gve the pseudo code for the proposed clusterng algorthm. Based on the senor network topology an ICLA s created, that s an ICLA s mapped nto the network n such a way that each node n the sensor network be mapped nto a node n the ICLA. Any two nodes n the sensor networks that are close enough to hear each other s sgnal have an edge n the correspondng graph G of the ICLA. 4 Threshold-Senstve Energy-Effcent Protocol 5 Adaptve Perodc TEEN 6 Address-lght Integrated MAC, and Reportng Protocol 7 Hybrd Energy-Effcent Approach 19

7 Fgure 2. Pseudo code for the ICLA-Clusterng algorthm. Each node follows ths code separately The proposed algorthm has two phases. In the ntal phase, snk node ntates 'Start- Clusterng' message throughout the network. Each node, upon recevng ths message, rebroadcasts t to ts neghbors and then enters the clusterng phase whch s performed usng ICLA. LA n each node has two dfferent actons a 0 =0 and a 1 =1. a 1 ndcates that the node should declare tself as the cluster head, and a 0 ndcates the opposte decson. At the start of the clusterng phase, each node sets the ntal probablty of these two actons to.5 and enters the clusterng teratons. At each teraton k, the LA n each node chooses one of ts actons a 0 or a 1. Each node then forwards the selected acton of ts LA to ts neghbors. Afterward, node, wats for a specfed duraton to receve ts neghbors messages ndcatng ther selected actons. At the end of ths duraton, each node calculates the response β based on the local rule of the ICLA. Ths local rule s desgned n such a way that to both maxmzng the electon probablty of hgh energy level nodes as cluster heads and mnmzng the number of sparse clusters. The local rule for node can be defned nformally as follows: 20

8 Fgure 3. Pseudo code for the man loop each node follows n the proposed clusterng method - If selected acton of LA n node s a 1 (node declares tself as a cluster head) then o If no neghbor of node declares tself as a cluster head, a favorable response wll be generated. o If some neghbors of node declare themselves as cluster head, but node has more resdual energy and more number of neghbors among them, a favorable response wll be generated. o Otherwse, an unfavorable response wll be generated. - If selected acton of LA n node s a 0 (node doesn't declare tself as a cluster head) then o If no neghbor of node declares tself as a cluster head, au unfavorable response wll be generated. o If some neghbors of node declare themselves as cluster head, but node has more resdual energy and more number of neghbors among them, an unfavorable response wll be generated. o Otherwse, a favorable response wll be generated. The local rule can be defned formally as follows. Let a be the acton selected by node and a j be the acton selected by the neghbor node j (whch can be ether a 0 or a 1 ). Equaton (3) gves the local rule of the ICLA. 0; β = 1; N a ( P a j ) + (1 a )) + j= 1 N (1 a ).( P a j ) (1 a ) 0 j= 1 Otherwse (3) 21

9 In (3), N s the number of neghbors of node and P s a parameter whch specfes the sutablty of node for becomng cluster head among ts neghbors. P s defned usng (4) gven below. P = α [ EnergyLevel EnergyLevel Max ] + (4) (1 α) [ N N Max ] K In equaton (4), EnergyLevel s the remanng energy level of the node. α s a coeffcent whch controls the effect of energy level and number of neghbors on the selected cluster heads. Hgher value of α ndcates cluster heads wth more energy level, and lower value of t ndcates cluster heads wth hgher number of neghbors. K s a constant whch controls the sutablty of node for becomng cluster head among ts neghbors. Large values of K ndcate more sutablty whch results n few clusters. Ths may leave some parts of the network wth no cluster. On the other hand, choosng small values of K results n too many clusters whch leads to more number of sparse clusters. Fnally, EnergyLevel Max and N Max are defned usng equatons (5) and (6) respectvely. EnergyLevel Max = Max ( EnergyLevel j ) 1 j N (5) N Max = Max ( N j ) 1 j N In these equatons, EnergyLevel j s the remanng energy level of the neghbor j and N j s the number of neghbors of the node j. Ths nformaton about neghbor nodes are exchanged between nodes perodcally va smple announcement messages. For postve values of P, node would be a better choce than ts neghbors for becomng cluster head, but negatve values of P ndcates that some neghbors are better choces than for ths purpose. Based on the response specfed by the equaton (3), LA resdes n node rewards (penalzes) ts selected acton a usng equatons (7)((8)), respectvely. (6) p ( n + 1) = p ( n) + a.(1 p ( n)) p ( n + 1) = 1 p ( n + 1) j (7) p ( n + 1) = (1 b) p ( n) p ( n + 1) = 1 p (n+ 1) j (8) In the above two equatons, a s the reward rate and b s the punshment rate. In the smulatons conducted, equatons (9) and (10) are used to compute a and b, respectvely. a = a [ Alpha ( N. EnergyLeve l )] + N (1 a ) [ a j ( N EnergyLeve l )] j= 1 (9) 22

10 N b = a [ a j j = 1 ( N EnergyLeve l )] + (1 a ) [ Alpha ( N. EnergyLeve l )] In equatons (9) and (10), Alpha s a parameter whch s defned through equaton (11). Ths parameter s used to control the value of a and b so that they wll be always less than 1. (10) Alpha= mn( EnergyLeve l,1 ( N EnergyLeve l )) (11) Ths process contnues untl the acton probablty of one of the actons of the LA n each node passes a certan threshold near to 1. Nodes for whch, the probablty of selectng acton a 1 of ther LA passes ths threshold, become cluster heads and broadcast messages ndcatng that they are fnal cluster heads. Other nodes wat for the fnal cluster head messages from ther neghbors and jon to the one whch has hgher energy level and hgher number of neghbors. 5. Expermental Results To evaluate the performance of the proposed method several experments have been conducted. In these experments, the proposed method s compared wth the basc HEED clusterng method proposed n [26] and ts extenson gven n [27]. Also, we study the convergence behavor of learnng automata n dfferent nodes of the network. All smulatons have been mplemented usng NS2 smulator. We use IEEE as the MAC layer protocol. Nodes of the network are placed randomly on a 2 dmensonal grd. In all experments, α s set to.3 and K s set to 1.2. Smulatons are performed for 50, 100, 200, 300, 400, and 500 nodes. The results are averaged over 20 runs. Experment 1: In ths experment, we study the cluster formaton behavor of the proposed method n comparson wth two other methods mentoned earler. For ths purpose, we make use of the followng metrcs: total number of clusters formed n the network, number of sparse clusters (whch s defned as the clusters havng less than 3 nodes) and mean number of nodes n each cluster. Fgure 4 depcts the number of clusters formed n the network, number of sparse clusters s depcted n fgure 5 and fgure 6 depcts the mean number of nodes n each cluster. These fgures show that the cluster formaton n the proposed method s better than the other two methods. Ths s because t forms less number of clusters and more nodes n each cluster on average. Also, number of sparse clusters n the proposed method s less than that n two other methods. Ths means that the proposed method can better dstrbute cluster heads among the sensor network. Experment 2: In ths experment, mean energy level of cluster heads at the end of clusterng algorthm and pror to any data transton s studed. Intal energy level of each node s selected randomly between 60% and 100% of the maxmum energy level (full charged). For estmatng the energy consumpton of each node, we assume that sendng a sngle packet wll consume.005% of the maxmum energy level and recevng a sngle packet wll consume.001% of the maxmum energy level. Energy consumpton n dle state s assumed to be 0. Fgure 7 depcts the results for the proposed method n comparson to other two methods. Ths fgure shows that the proposed method outperforms HEED and ts extenson n terms of energy level of cluster heads wth the factor of approxmately

11 Number of Clusters HEED ExtendedHEED ICLA Number of Nodes Fgure 4. Total number of clusters formed n the network for the proposed method and other methods Number of Sparse Clusters HEED ExtendedHEED ICLA Number of Nodes Fgure 5. Number of sparse clusters formed n the network for the proposed method and other methods Experment 3: Ths experment s conducted to better understand the convergence behavor of learnng automata resdng n the nodes of the network. The sensor network used n ths experment s composed of 50 nodes whch are scattered randomly throughout a 2 dmensonal grd of 1000x1000 meter. Fgure 8 depcts ths network and fgure 9 depcts clusters formed n t usng the proposed algorthm. Mean Number of Nodes n Each Cluster HEED ExtendedHEED ICLA Number of Nodes Fgure 6. Mean number of nodes n each cluster for the proposed method and other methods 24

12 Mean Remanng Energy Level n Cluster Heads HEED ExtendedHEED ICLA Number of Nodes Fgure 7. Mean remanng energy level n cluster heads for the proposed method and other methods Fgure 8. Sensor network of 50 nodes n a grd of 1000x1000 meter Fgure 9. Sensor network of fgure 8 whch s clustered usng proposed method We study the acton probabltes of two automata resdng n two nodes; one s a cluster head and the other s a cluster member. Fgure 10 shows convergence of the acton probablty of acton "Be Head" to 1 n the learnng automaton resdng n the cluster head node to the acton "Be Head". Fgure 11 shows convergence of the acton probablty of acton "Be Member" to 1 n the learnng automaton resdng n the other node. 25

13 1.2 1 Be Head Be Member Probablty Tme Fgure 10. Acton probabltes of the learnng automaton n the cluster head node Probablty Be Head Be Member Tme Fgure 11. Acton probabltes of the learnng automaton n the cluster member node 6. Concluson In ths paper we propose rregular cellular learnng automata as a generalzaton to the cellular learnng automata whch removes the restrcton of rectangular grd structure n tradtonal CLA. There are a number of problems and applcatons whch cannot be adequately modeled wth rectangular grds, one of whch s the area of wreless sensor networks. In these networks, nodes are usually scattered randomly throughout the envronment, so no regular structure can be assumed for modelng ther behavor. Based on the proposed ICLA model, we present a novel clusterng algorthm for sensor networks n whch, each node n the network s a cell n the ICLA, and hence equpped wth an LA. Any two nodes that are close enough together to hear each other s sgnal have a common edge n the graph G of the ICLA. LA n each node has two actons one of whch ndcates that the node should declare tself as a cluster head and the other one ndcates the opposte decson. Usng local rule of the ICLA, acton probablty of one of these two actons converges to 1 n each of these LA. Nodes for whch probablty of acton "Be Head" of ther LA converges to 1, becomes cluster head and other nodes become cluster members. It has been shown through smulatons that the proposed clusterng algorthm outperforms other smlar methods n terms of cluster formaton as well as remanng energy level of cluster heads at the end of the clusterng. References [1] H. Beyg, M. R. Meybod, "A Mathematcal Framework for Cellular Learnng Automata", Advances n Complex Systems, Vol. 7, Nos. 3 & 4, pp , September & December

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