Bi-Objective Community Detection (BOCD) in Networks using Genetic Algorithm

Bi -Objective C ommunity De tection (BO CD) i n Networ k s using Ge netic Alg orithm Rohan Ag raw al Jay pee I nstitute o f I nfo rmation Tec hno logy , Com puter Scienc e Depar tmen t, No ida - 20130 7 , Uttar Pr adesh, I ndia rohan.ag rawa l.89@ jiitu.org Abstract . A l ot of researc h ef fo rt has been put into co mmunity detectio n fro m all co rners o f ac ademic int erest suc h as phy sics, ma them atics and co mputer scienc e. I n this pape r I have propo sed a Bi -Objectiv e Gene tic Algo rithm fo r com munity det ection whic h maxim izes m o dularity and co mm unity sco re. Then the results obta ined fo r both benchmark and rea l life data sets are co mpared with other al go rithm s us ing t he mo dularity and MNI perform ance metrics. The results show that th e B OCD alg o rit hm is capa bl e o f s uc cessfully det ecting com munity structure in both rea l l ife and sy nthe tic dataset s, as wel l as improv ing upon the perf orm ance o f prev io us techni ques. Keywo rds : Com munity Structure, Com muni ty de tection, Gene tic Algo rithm, Multi- objec tive Gene tic A lgo r ithm, Multi -o bj ect ive optimiz atio n, modulari t y , No rmalized Mutu al I nf orm ation , Bi -o bj ectiv e Gene t ic A lgo r ithm 1 Introduction In the co ntext of n etwo rks, commu nity str uctu re r ef ers to t he occurr e nce of gr oups of nodes in a n e tw ork that are more de n se l y c onnec ted than with the r est of the n ode s in the netwo r k. The inhomo geneo us co n n ectio ns suggest that th e n etw ork has ce r tai n natural d ivisio n w ithi n it. The occurrence of commu n ity s tr uctu re is quite co mmon in re al n etw o r ks. A n example o f th e occurrence of communit y str ucture i n r eal netwo r ks i s th e a ppea rance of groups in so cial n etw o r ks. Let’s take th e examp le of a s ocial netw o rkin g site. Le t a node represe nt an indiv idual and let the edge r eprese nt fri endship relatio n b etween tw o in di v iduals . If man y student s in a particular class or schoo l a re fri ends amo n g themselv es, th e n th e n etw ork g ra p h w ill ha v e man y conn ect ions betwe en them . Thus one c o mmunity could be identi fie d as a schoo l community . Ot her co mmunities could be r elated to w o r k, family , co lleges or commo n i nt e r ests. Other examples are citati o n n etw o rks which fo r m c ommu ni tie s by research to pics. Spo rt teams f orm communities on th e b asis of th e div ision in which they pl a y , as the y w ill play more of ten w ith teams that are i n the same d iv ision/co mm unity as them. Now let us c o n sider the pote nt ial appl icatio ns of th e detection of communities in netwo r ks. Communit ies in a soc ial n etw o rk might h elp us find real soc ial gr oupi ngs, perhaps by i nt e r est or b ackgrou nd. Com m unities can hav e c oncrete appl icatio ns. Cluste rin g Web clients w ho h ave simil ar inte rests and are geo gr aphically n ea r to each other may impr ov e th e perfo rm ance of services pr ov ided on the Wo rld Wide We b, i n that eac h cluste r of clients could b e served by a dedicated mi rror se rver [1] . Identify in g cluste rs of custo mers wit h simi lar interests in the n etw ork o f pur chase relatio nships be twee n custome r s an d produc t s of online r etaile r s enables to set up eff i cient reco mmen d ation sy stems [2], that be tter guide c ustome rs thr oug h the lis t of items of the retaile r and e nhance th e b usiness oppo rtunities. Cluste rs o f large graphs can be used to cr eate dat a st ructures in orde r to effic ientl y st ore th e g raph data and to handle nav igatio n a l que r ies, like pa th s ea rches [3][4]. Ad hoc networks [5] , i.e. self - co n figuring n etw o r ks fo rm ed by commu ni ca tio n n odes acti ng in the sa me regio n an d rapidly c hanging (bec ause th e dev i ce s move , fo r i n stance ), usually have no centrally maintai n ed r outi ng table s that spe cify how node s have to comm u nicate to o t her n odes . Groupi n g t he nodes in to cluste r s enab l es o ne to generate compact rout ing tables while the choice of th e co m munic ation pa ths is stil l eff icient [6] . The a im of community detec tion in grap hs is to ide n tify th e modules by usin g the info r mation e ncode d in the netw ork to pology . Weiss a nd Jac ob son [7] w ere amo ng the fir st to analy ze communit y str ucture. They searched fo r wo rk gr oups wit hin a gove r nment a ge n cy . Alread y in 1927, Stua r t Ric e l ooked for clusters of people in small pol itical bo di es b ased o n the simila rity of their vo ti ng patte rn s [8]. In a paper appe aring in 2002, Girva n and New ma n p r opo se d a n ew algo ri thm , aiming at the identific atio n of e dges l y ing betw ee n c ommunities and thei r suc ce ssive remov al. Af t er a few iterations, this pr oc es s l ed to the isolatio n o f communities [9] . The pape r t r igge red inerte st in this fie ld, a n d many n ew m eth ods have been pr opo se d in previo us y ears. In pa r tic ular, p hy sicists entered the ga me, b r inging in thei r tools and technique s: spin models, optimiz atio n , perco lati o n , random walks, sy n chr o nization, et c., bec a me ingredie n ts o f new origi nal algorithms. T he field h as also taken advantage o f co ncepts and metho ds from co mputer scie nce, no nl ine ar dy namics, so cio l ogy , discrete mathematic s. Genetic a lgorit hms [10] ha ve also bee n used t o optimize modularity . In a standa r d genetic algo r ithm one has a set of can didate solut ions to a prob lem, whic h ar e nume ri cal ly encode d as ch r o moso mes, and an ob jec tive function to b e optimiz ed on the space of so lut ions. T h e ob jec tive f unction p l ay s the r ol e of b iological fitness for the chromo so mes. One usually starts fro m a random se t of can didate so l utio ns, w hic h are p r o gressive l y changed t hroug h ma nipulatio n s i nspi red by biolo gical proc esses regarding r eal c hromos ome s, like p oint m utatio n (ran dom variatio n s of some par ts of th e chromos o me) and c r os sing ov er (generating n ew c hr o mosome s by merging parts of existing chromo somes). T hen, the f i tness of th e n ew pool of can didates i s co m puted a nd the chromos ome s w ith the highest fitness h av e th e greatest chances to surviv e in the n ext ge neratio n . Afte r seve r al ite r ations o nly solutions w it h large fitness surviv e. In a w ork by T asgin et al. [11], pa rtitions are th e chromos ome s an d modula rity is th e f itness f unctio n. Genetic algo r ithms w er e also adopte d by Liu et al . [12]. H ere th e maximum modula rity partition is o btain ed v ia suc ce ssive bipartitions of the g raph, w here eac h bipartitio n is determi n ed b y apply in g a ge n et ic algo rithm to each s ub g raph (starti ng from the o r iginal g raph it se lf), whic h is considered isola ted fr o m th e rest o f th e graph. A bipa rtition is ac ce pted only if it inc reases the tota l modul arity of th e grap h. In 2009, Pizutti [13] propo se d a m ulti-ob j ec tive genetic algorithm fo r th e detectio n of co mmunities in a n etwo r k. T he t w o fitn ess functions used we re communit y sco re and co mmunity fitn ess. The algorit hm had th e a dva ntage that it prov ided a set of so lutions based o n the m aximiza tion of both the ev a luatio n f un ct ions. In sec t ion 2, the p rob lem o f Co mm unity De tection w i ll be formulated mathematic ally w ith the introductio n o f two functions. In sectio n 3, a ll the stages of the Genetic Algorithm such as Initializ ation, Fitness Functio ns, Mu tatio n and Crossov er will be elabo rated upon. I n sectio n 4, the expe r imental r esults of BOCD w ill be pr esented an d co m pared with existing Com m unity D etec tion techniques. The Conclusio n w ill be pr ese nted in se ction 5. 2 Pr ob lem Def inition A n etw ork N w can be modele d as a grap h  = ( V,E) whe re V is a set of objects , called nodes o r v ertices, and E is a set of lin ks, called edge s, that connec t t w o eleme n ts of V . A co m munity (or cluster) in a netwo r k is a group of verti ce s ha ving a high density of edge s with in them, and a l o wer density of edges be t w een gr oups. The prob l em of detec t in g k co m munities i n a netw o rk, w here the numb er k is u nk n ow n , can b e formula ted as fin di n g a partitio n ing of th e n ode s in k sub sets tha t ar e hig h ly intra - co n n ecte d a nd sparsely i n te r -co nnected. To deal w ith g raphs , o f ten th e adjace n cy matrix is used. If the netwo r k i s constituted by N n ode s, the graph can be r eprese nted w ith the N × N adjacency matr ix A , w here th e entry at positi o n (i, j) is 1 i f ther e is an edge from node i to node j , 0 o the rwise . Le t us in troduc e the co n cept of Community Score as a def ined in [13] and [14]. Let     be the sub grap h w here n o de i be l ongs to, the deg r ee of i with r es pec t to  can be split as   󰇛󰇜      󰇛  󰇜      󰇛󰇜 . w her e    󰇛  󰇜       is the numb er o f edge s conn ec ting i to the o the r n ode s in  . H er e  is t h e adjace ncy matrix of  .    󰇛  󰇜       is the numb er o f edge s conn ec ting i to the r es t o f the netwo r k. Let   represent t h e fraction o f edges co nnecting i to the ot h er node s in  .            󰇛  󰇜  w her e     is t h e ca rdinality of  . The po wer mean o f  of order  ,  󰇛  󰇜  󰇛  󰇜    󰇛  󰇜     In the co mputation of  󰇛  󰇜 , since        , the expo nent  inc reases th e w eight of nodes h avi ng many conn ectio ns w ith othe r n o des b elonging to the same community , and dimi nis h es the we ight o f t h o se node s havi ng fe w connectio ns inside  . The vo lume   of a co mmunity is de fined a s the number of edges co nn ecting ve r ti ce s inside   ,         The   of  i s d e f i n e d a s   󰇛  󰇜   󰇛  󰇜     The Comm u n ity score of a cluste ring 󰇝       󰇞 of a netwo r k i s def in e d as      󰇛  󰇜    (1) The prob l em of c o mmunity de t ec tion has b ee n fo rmulated in [1 4] a s the prob lem of maximiz ing th e Commu nity Score. Th e other o bje ctive i s to ma xim ize modularity , def i n ed in [15]. Let k be th e numb er of modules found inside a n etw ork. The modula rity is defined as         󰇡    󰇢      (2) w her e   is the total n umb er of edge s joining vertice s inside the module  , a nd    represe nt s the fraction of edges in the n etw o r k tha t co nnect the sam e commu nity .    represe nts the su m of the deg r ees o f th e n ode s o f  . If the numbe r o f w ith in- co m munity edges is no more than rando m, w e will get    . The maximu m value of  is 1, w h ich i n d icates st r o n g commu nity str uctu r e. 3 Algorithm Des cription The var ious stages of the genetic algo ri thm have been described in th e follow in g sub sectio ns. The f ramew o r k used w as NSGA-II in C desc ribe d in [24] . 3.1 Genet ic Rep resentatio n The chromo so me is represented in the form at me n tio ned in [16]. The r ep r es entatio n of an individual consists of N genes, and eac h ge ne ca n t ake a v alue in the r ange {1, …, N}, w her e N is the numbe r of n ode s i n the netwo r k. If a value j i s assig ned to the i th ge n e, this suggests th at i a nd j are in the same c l uster. B ut if i an d j a re already assig n ed, th e n the ge n e is i g n o red. T hus late r ge nes w i ll h av e l ess bearing on cluste r formatio n . The deco ding of t his indiv idual to ob ta in cluste rs can be done in linear time acc o r ding to [17]. Fo r example, co nsider the indiv idual fo r a netwo r k of 34 nodes (N = 34). Th e numbe r in the cu rl y brackets represe nts the i ndex of the elem ent in the indiv idual. {1}2, {2} 3, {3}4, {4}14 , {5} 17, {6}17, {7} 6, {8}14, {9}19, {10}19 , {11}17 , {12}14, {13} 2, {14}9, {15}19 , {16} 9, {17}15 , {18} 8, {19}21, {20}8, {21}27, {22}1, {23}15, {24}26, {25}26 , {26}32, {27}30, {28}26, {29}25, {30} 3, {31}1 9, {32}4, {33}23 , {34}9 We ar e assuming here that if the above in dividual w as stored in an array , th e in dex of the 1 st eleme n t wo ul d be 1 an d not 0. I n the abov e chro moso me, t he eleme nt at index 1 of th e array is 2. T hus node s 1 an d 2 a re in the same c luster. Si milarly th e eleme nt at inde x po sition 2 is 3, th us 2 a nd 3 are put in th e same cluster. Si nce n ode s 1 and 2 ar e already in Cl us ter 1, we have node s 1, 2 and 3 pu t in t he same cluste r. The eleme n t a t index pos itio n 3 i s 4, thus n o des 3 and 4 are a lso in the s a me cluste r. Th e eleme nt at the 5 th index po sitio n is 17. Since n eit h er 5, no r 17 h ave been pr ev iously assig n ed a cluster, they a re put toge th e r in a n ew cluster, Cl uster 2. Thi s proc ess go es on iterativ ely til l the last eleme n t. Fi nally the cluste rs are ma de as fo l low s: Cluste r1: 1, 2, 3, 4 , 14, 8, 1 2, 13, 18, 20, 22 Cluste r2: 5, 17, 6 , 7 , 11 Cluste r3: 9, 19, 10 , 15, 16, 21, 27, 23, 30, 31, 33, 34 Cluste r4: 24, 26, 25 , 32, 28, 29 3.2 Initiali zatio n The populati o n is i nitialized r andomly from v alues betwee n 1 and N, w here N i s th e numbe r of nodes in the ne two r k. 3.3 Fitness Fu nctio ns The a lgorit hm use d h e re is a bi-objec tive optimization, w here b oth f itness functio n s are mi nimized. The first fit ness funct ion is de riv ed from equ at io n (2).        The 2 nd fitness fu nction uses bo th equ atio ns (1) a n d (2).     󰇛    󰇜  󰇛     󰇜  lies in th e range [0 ,1], therefo r e the minimiza tio n of  󰇛    󰇜 h elps in finding the maximum v a lue o f mo dularity . In t he se co n d fit ness f un ctio n, the w eight 10 fo r the Commu nity Str uctu re te rm 󰇛  󰇜 has bee n f ound out e mpi ri ca lly . T he abo ve pair of fitness functions take n to get h e r perfo r ms better than the single objec tive optimizatio n of e ith e r of th e tw o taken sepa r ate ly . 3.4 Cro ssov e r and Mutatio n Simple U nifo r m c r o ssov er is used as th e c rossove r operato r . The cr os sover site is chose n at random. Selec tion s trategy used i s t o urnament sele ctio n, w i th 4 in dividu als co n t es ting in the to urna me nt. Take fo r e. g. Parent 1: 1, 2, 4, 5 , 3, 5, 6 , 1, 9, 4 Parent 2: 3, 6, 3, 2 , 6, 4, 3 , 1, 2, 9 Suppo se th e crossove r site is randomly de cided at 5. T hi s mea n s that the first 5 eleme nt s of Chil d 1 will co me from Par e nt 1, i.e. {1, 2, 4, 5, 3}. The oth er eleme n ts for Child 1 wil l come from Parent 2 , i.e. {4, 3, 1, 2, 9}. Th e beginning eleme n ts fo r Child 2 co me from Parent 2 and th e latte r eleme n ts co me f r o m Par ent 1. Thus the child r e n f ormed a re: Child 1: 1, 2, 4, 5 , 3, 4, 3 , 1, 2, 9 Child 2: 3, 6, 3, 2 , 6, 5, 6 , 1, 9, 4 Mutatio n ope rato r a lso per f orms simple mutatio n , i.e. a ge ne is ch os en a t r ando m and its v alue is si mply changed. 4 Exper i mental Results Bi -Objec tive C ommu ni ty De tection (BO CD) is appli ed on 3 r eal w orld n etw orks, the America n Co llege Foo t b all [19] , B ottlenos e Dolphin [ 26] a nd th e Zac har y Karate Club [18] n etw ork. Th e met h o d is a lso tested on a be n c hm ark generating program propo sed i n [23] w h ich is an ex tensio n of th e b enc hmark propo sed by Girvan and New man in [9]. The expe ri me nts were per fo r med on a Core2duo m achi n e, 2 .0 Giga H z with 3 Mb RA M. The f ra mew ork for M ulti-Obj ective Gen etic Algorit h m used w as NSGA - II w ri tte n in C describ ed in [24] . Th e pa rameters used in co mpiling the code ar e as follow s: Po pulati o n: 200 Generatio n s: 3000 Crossov er Prob abilit y : 0. 7 Mutatio n Prob a b ilit y : 0.03 Fig. 1. The 34 node Zac ha ry Karate C lub Network divided into 2 com munities . Thi s wa s how the club ac t ually br oke into 2 groups . The fi rst group is sho wn by circular no des and t he seco nd by triangular node s. The modul arity of this div isio n is 0.371 . Fig. 2. Div ision of the Zachary Karate Club Netwo r k int o 4 com munit ies by BOC D. Each com munity i s shown w i th a diff erent sy mbo l . The mo dularity of this d ivisio n is 0. 419. The evaluatio n metrics use d were Mo dularit y w h ich was desc ri b ed above, as w ell as No r malize d Mutual Info r matio n whic h w a s described in [ 22]. The r esults o btai n ed by BOCD ar e compa r ed with the fast GN algorithm [2 0] and MOG A-Net [13] o n th e bas is of Modularity an d NM I. 4.1 Zacha ry Karat e Clu b Network This netw o rk was ge ner ated by Zachary [18], who studied the friendship o f 34 memb ers of a ka r ate club ove r a pe ri o d of two y ear s. Du ring thi s period, be cause of disagree ments, the club div ided in two gr oups al most of t he same size. Th e o r igi nal divisio n o f th e club in 2 co mm unities is show n in Figure 1. The BOCD algorithm divide s t he n ode s i nto 4 co mmunities, with this separatio n show in g a hig her v alue o f modula rity th en the original soluti o n itself. As can be seen from T able 1, BOCD perfo r ms be tter than b o th GN and MOGA-Net i n te rms o f modularity . The NM I of the divisio n w as found t o be 0. 695622, which is be tter tha n GN algorithm but not MAGA -Net. As MOGA-Net generates a pa r eto set of results, th ey ha ve a chieved highe r NMI values . 4.2 Am erican Co llege Fo otball Network The Ame r ican Co llege Footb all n etw ork [9] is a n etw ork of 115 t eams, whe re the edge s represe n t th e r egula r seaso n games b etwe en the tw o teams t hey co n n ect. T he teams are div ided into confe ren ce s an d p lay teams w ithin the i r ow n co nference more freque ntly . Th e netw ork has 12 confe r en ce s o r co mmunities. The divisio n obtai n ed by BO CD was better than the result o f MOGA-Ne t an d was ex actly on equal t e rms with the m odula rity value of th e GN algorithm. The NMI of th e division w as fo un d to be 0.878178, w h ich is the hig hest value among t he three algorit hms. 4.3 B o ttlenose Do l phin Ne two rk The n etwo r k of 62 bottlenose dolphi n s l i ving in Do ub tful So un d, New Z ealand, was co m piled in [26] by L usseau from seve n y ears of dolphin behavio r. A t i e be twee n 2 dolphi ns w as es tablished by th eir statist ically f requent asso ciation. The n etwo r k split naturally into 2 l ar ge g roups, t h e numb er o f ties b eing 1 59. The pe r fo rm ance o f BO CD was m uch better than the GN algorithm an d margi nally be tter t han that of MOGA -Net. The NM I of th e divisio n w as fo und t o be 0. 615492, w hich lies in be t w een MOGA-Ne t and GN pe rformance w i se, the be st be i ng MOGA-Net. Table 1. C om pariso n of Modulari ty values for the 3 real da tasets . The f irst column gives t he valu e of m odularit y f o r N MI = 1. T he fo llowin g colu mns g iv e modula rity re sults fo r t he f a st GN , MOGA -Net and BOCD al gorithm s. Datase t Mod. Fo r N MI =1 GN MOGA BOC D Zachary Karate Clu b 0.371 0.380 0.415 0.419 Co llege Foo t ball 0.5 18 0.577 0.515 0.577 Bo ttlenose Do lphin s 0.373 0.495 0.505 0.507 Table 2. Co mpar ison of NMI values fo r the 3 rea l dataset s. The colu mns give NMI re sult s for the fast GN, MOG A-N et and BOCD algorit hms. Datase t GN MOGA BOC D Zachary Karate Clu b 0.692 1.0 0.695 Co llege Foo t ball 0.762 0.795 0.878 Bo ttlenose Do lphin s 0.573 1.0 0.615 4.4 B en chmark T est Netw ork The n etwo r k consists of 128 n odes divided into fo ur c o mmunities of 32 node s each. The av erage de gree of each n ode is 16. The f r action o f edges shared by each node w ith nodes in its o wn c ommunit y is know n as the mixing paramete r. If the value of the mixing parameter µ > 0.5, it suggests that a node w ill have more link to o th e r nodes, outside it s c o mmunity . Th us fin d ing co m munity structure will be difficult for µ = 0.5, as ev i dent from th e fo ll ow in g grap h. Acco r ding to a graph draw n in [13], MOGA -Net c ould achiev e an NMI o f les s than 0 .1 f o r µ = 0.5. Thus o u r al go r it hm perfo r ms b etter in case of a highe r mixing pa rameter. Fig. 3. NMI value s obtai ned by BOCD f or diff erent v alues of mixin g paramete r. H ere r=2.5. Table 3. Modul ari ty and NMI values fo r benc hmark netwo r k w ith increas i ng value of Mixing Param eter . The perfo r manc e fo r µ = 0.2 i s the be st, and expe ctedly de teriora tes as µ is increased . Mixing Param eter (µ) Modula rity NMI 0.2 0.4511 1.0 0.3 0.347 0.792138 0.4 0.5 0.218 0.181 0.559844 0.266481 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0. 5 0.6 NMI Mixing Para meter 5 Conclusions The pa pe r prese nt ed a Bi-Obj ectiv e C ommunity Detection te chnique t hr o ugh the use of Genetic Algorithm. By sim ply co mbining co mmunity s co re an d modularity , th e BO CD algorithm imp r o ve d up o n the perfo r man ce of both t he GN a lgorit hm which used Mo dularit y and M OGA-Ne t w hich use d commu n ity sco re i n t he c o mmunity detec t ion prob lem. R es ults o n real l ife netw o rks as well as s y n thetic b en chma r ks show t h e capability of th is approac h in f in ding out communities w ithin netwo r ks. Futu r e r esea r ch s h o uld aim at decreasing co mputatio nal co mpl exity of Community Dete cting algorit hms and f in ding co mmunit ies in n etw orks w ith a hi gh m ixi ng parameter. Acknow ledgments. I wo ul d like to thank D r S K . Gupta and P ranava Chaud har y , both o f whom have profo undly h elped m e in understa nding Genetic Algo rithm and Multi-Ob jective Optimi zat ion. I wo ul d also li ke to thank Andrea La n cic hinetti for prov iding the NMI ca lculating c ode . The so ftw ar e used for draw ing netw o r k diagrams w as Netdraw [25]. 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