Ant Colony Algorithm for Clustering through of Cliques

Transcription

1 Ant Colony Algorthm for Clusterng through of Clques Julo Cesar Ponce Gallegos a, Felpe Padlla Díaz a, Carlos Alberto Ochoa Ortz Zezzatt b, Alejandro Padlla Díaz a, Eunce Esther Ponce de León a y Fatma Sayur Quezada Agulera a a Departamento de Cencas de la Computacon, Unversdad Autónoma de Aguascalentes, Av. Unversdad #940, Aguascalentes, Ags., Méxco {jcponce, fpadlla, meza, apadlla, eponce, fsquezada}@correo.uaa.mx b Insttuto de Ingenería y Tecnología, Unversdad Autónoma de Cudad Juárez, C.J.,Chhuahua. Méxco megamax8@hotmal.com Abstract. Ths work show an Ant Colony Algorthm (ACO) for Clusterng, a technque of group very used n the Data Mnng (DM) s the clusterng, ths algorthm works n the search of maxmal clques whch represent groups (clusters). For ths was used the algorthm base of Ant Colony for the problem of maxmum clque whch already was mplemented and was made a modfcaton to the algorthm so that worked lke clusterng algorthm, n the work the algorthm s descrbed and are the expermental results n ths frst approach. Keywords: Data Mnng, Clusterng, Ant Colony Optmzaton, Maxmal Clque. 1 Introducton The Knowledge Dscovery (KD) and Data Mnng are powerful tools of data analyss, and t s predcted that they are possble to be turned n the more frequently used analytcal tools n the future. The terms Knowledge Dscovery and Data Mnng are used to descrbe the extracton non-trval or mplct, prevously unknown and potentally useful of the data nformaton [10]. The Knowledge Dscovery s a concept that descrbes the process of the search n great volumes of data of patrons who can be consdered knowledge on the data. The most well-known branch of the Knowledge Dscovery s the Data Mnng. The Data Mnng, conssts n the extracton of the hdden nformaton n great data bases, s a new and potental technology. The Data Mnng s a process of Knowledge Dscovery n great and complex data sets, refer the extracton process or mner of you seed amounts of data [6]. On the other hand, the Data Mnng can be used to predct a result for a gven organzaton [5]. The algorthms of clusterng n the Data Mnng are equvalent to the task of dentfyng groups of fles that are smlar between they but dfferent wth the rest [9]. The Data Mnng s a multdscplnary feld wth many technques. Wth whch t s possble to create a model that descrbes the data s beng used.

2 The maxmum clque problem s a problem of combnatory optmzaton that s classfed wthn the NP-Hard problems whch are dffcult to solve. Due to ther complexty the exact conventonal technques (exhaustve) take long tme to provde a soluton, therefore t s necessary to develop heurstc algorthms they solve that t reachng a soluton near the optmal n a reasonable tme. Ths problem has real applcatons eg: Codes Theory, Errors Dagnoss, Computer Vson, Clusterng Analyss, Informaton Retreval, Learnng Automatc, Data Mnng, among others. Therefore t s mportant to use new heurstc and/or meta-heurstcs technques to try to solve ths problem, whch obtan better results n a polnomal tme. They have been used dfferent heurstc to try to solve ths problem, eg: Local Search, Genetc Algorthms, Taboo Search and Ant Colony Optmzaton Algorthms (ACO) [7]. The Ant Colony Optmzaton Algorthms are a bo-nspred meta-heurstc based on the behavor of the natural ants, n as they establsh the most sutable way between the anthll and a food source [2], these have a great varety of applcatons between whch s the Data Mnng. 2 Descrpton of the Clque Problem Gven to a graph nondrected any G= (V, E), n whch V= {1.2,, n} s the set of the vertces of the graph and E s the set of edges. Clque s a set C of vertces where all par of vertces of C s connected wth an edge n G, that s to say C s a complete subgraph. Clque s partal f ths form leaves from another clque, of another form ths s maxmal. The goal of the algorthm s to fnd all the maxmal clques. An example of clque can be observed graphcally n fg 1. Fg. 1. Example of clques contaned n a graph.

3 3 Ant Clque Algorthms (ACA) The ACO Algorthms have been used to solve dfferent problems from combnatory optmzaton [3,8]. The man dea of the ACO s to model the problems to look for the way of mnmum cost n a graph. The ants cross the graph n search of good ways (solutons). Each ant s an agent who has a smple behavor so that not always she fnds qualty ways when ths s alone. The ants fnd better ways as a result of the global cooperaton between the colony. Ths cooperaton s realzed of an ndrect way when depostng the pheromone, a substance that s deposted by an ant n ts route. The general Ant Colony Algorthm for the maxmum clque problem proposed by Fenet and Solnon [4] s showed n fg 2. To ntalze the pheromone sgns To place Ants Randomly Repeat For k =1..nb Ants do: Buld the clque (Soluton) C k Update the pheromone sgns { C 1,..., C nbants } If s the frst teraton to keep the best Soluton If not to compare f the best soluton n the teraton s better than the prevous, f t s thus to replace t Untl Reachng the Number of Cycles or Fndng the optmum soluton Fg. 2. Pseudo code of Ant Clque Algorthm. Intalze the pheromone: The ants communcate through the pheromone whch s deposted n the edges of the graph. The pheromone concentraton n the edge ( v, v ) j s denoted by τ ( v, v ), the ntal pheromone sgn s denoted byc. j Constructon of clques wth the ants: An ntal vertex s selected randomly and teratvely t chooses vertces to add to clque. Of set of canddates (all the vertces that are connected wth all vertces of the partal clque), to see fg 3. Choose the frst vertex randomly C { v } f v f V Canddates { v /( v, f v ) E} Whle Canddate 0 do Choose a vertex v Canddates wth a probablty p( v ), see Ec. (2) C C { v } Canddates Canddates { v /( j v, v ) E j End Whle Return C Fg. 3. Constructon of Clque.

4 The update of the pheromone sgn uses the Ec. (1). τ j ( t + n ) = ρτ j ( t ) + τ j (1) p ( v ) = v [ τ c ( v )] j canddates α α τ c ( v ) j (2) 4 Proposed Algorthm The proposed algorthm s based on the algorthm n [7] the dfference s n the part to choose the soluton wthn the clque constructon process whch s showed n fg 4. To ntalze the pheromone sgns To place Ants Randomly Repeat For k en 1..nb Ants do: Buld the clque (Soluton) C k Update the pheromone sgns { C 1,..., C nbants } If s the frst teraton to keep n lsts all the solutons wthout repeatng no one Else only are added to the lst the solutons that not exst n the lst Untl Reachng the Number of Cycles or Fndng the optmum soluton Fg. 4. Pseudo code of Ant Clusterng Algorthm. In the algorthm all the solutons must be kept snce each of these represents maxmal clque (cluster). 5 Desgn of Experments and Results The ACO Algorthms depend of the α parameters that s the factor of weght (mportance) of the pheromone, and ρ s the percentage of evaporaton of the pheromone. If we decrease the value of α, the ants have less senstvty to the pheromone sgn, and f ρ s ncreased, the evaporaton of the pheromone s slower. When the ablty of exploraton of the ants s ncreased, these can fnd better solutons but ths mples more tme. Takng nto account these parameters the algorthm wth the followng values was executed: Ants number=100, ntal concentraton of the pheromone ϲ =0.01, mportance of the pheromone α =1, factor of evaporaton of the pheromone ρ =0.99, maxmum concentraton that can take the pheromone, number of cycles that executes the Nc=100 algorthm, these values were chosen on the bass of

5 the results obtaned when mplementng a frst algorthm at the begnnng of the 2006 n whch the ants wthn the graph n the vertces wth greater degree were placed [7]. In order to carry out the desgn of experments we took one from benchmark of the DIMACS [1] that s the used ones at the moment at nternatonal level for the problem of maxmum clque. It was decded to solve the problem wth the executon of software wth the parameters before mentoned n the algorthm, n 1 of the 36 benchmarks of the DIMACS that s brock200_2. Whch obtan the greater cluster wthout problem because t s desgned to solve the problem of maxmum clque, and the found amount of clusters depends on the number of teratons whereupon t s ran. 5 Conclusons and Future Work In ths paper s presented an algorthm based on Ant Colony wth a Local Optmzer k-opt, whch was used to obtan clusters n a graph takng nto account the degree of the vertces from ths, whch can ncrease the probablty of fndng groups (clques) greater, thus can be seen that t s possble to mplement algorthms of Ant Colony to realze clusterng n the area of the Data Mnng. The proposed algorthm was executed n 1 benchmark of the DIMACS for the problem of maxmum clque. Ths algorthm s a passage n ths area snce the majorty of the proposed algorthms at the moment works clusterng under a board of two dmensons, whch lmts the relatons and the sze of the applcatons. Future work: It s mportant to make a study of the behavor of the parameters and the algorthm, as well as to make a desgn of ampler experment to determne whch are the best values for the parameters, as well as to already use the algorthm n a real applcaton lke the socal networks. Referencas [1] DIMACS Center for Dscrete Mathematcs and Theoretcal Computer Scence fttp://dmacs.rutgers.edu/pub/challenge/graph/benchmarks/ [2] M. Dorgo, V. Manezzo, and A Colorn (1996) Ant System: Optmzaton by a Colony of Cooperatng Agents. IEEE Transactons on Systems, Man And Cybernetcs Part B: Cybernetcs, 26:1, [3] M. Dorgo, G. D Caro and L.M. Gambardella (1999) Ant algorthms for dscrete optmzaton. Artfcal Lfe, 5(2): [4] S. Fenet and C. Solnon (2003) Searchng for Maxmum Clques wth Ant Colony Optmzaton EvoWorkshops 2003, LNCS 2611, [5] J. Hernández, A. Ochoa, J. Muñoz & G. Burlak (2006). Detectng cheats n onlne student assessments usng Data Mnng, Proceedngs of The 2006 Internatonal Conference on Data Mnng (DMIN 2006), pp , Las Vegas, USA, June 2006, Nevada Cty [6] J. Ponce, A. Hernández, A. Ochoa, F. Padlla, A. Padlla, F. Álvarez y E. Ponce de León (2009), Data Mnng n Web Applcaton, n Book: Data Mnng and Knowledge Dscovery n Real Lfe Applcatons, Edted by: Julo Ponce and Adem Karahoca, ISBN , Hard cover, 436 pages, January 2009, Publsher: IN-TECH

6 [7] J. Ponce, E. Ponce de León, F. Padlla, A. Padlla y A. Ochoa (2006) Algortmo De Colona De Hormgas Para El Problema Del Clque Máxmo Con Un Optmzador Local K-Opt, Hífen, Uruguaana, v. 30, n. 58, pag. 191, ISSN , Novembre, Uruguaana, Brasl. [8] Stutzle T. and Hoos H.H. (2000) MAX MIN Ant System. Journal of Future Generaton Computer Systems, 16: [9] Varan, S. (2006). Crme Pattern Detecton Usng Data Mnng, Oracle Corporaton [10] Wahlstrom K., & Roddck J. (2000). On the Impact of Knowledge Dscovery and Data Mnng, Proceedngs of Australan Insttute of Computer Ethcs Conference (ACE2000), Canberra, Australa, Aprl 2000, Sydney Cty.