A Novel Clustering Algorithm Based Upon Games on Evolving Network

A Nov el Clustering Algorithm Based Upon Games on Ev olving Network Qiang Li 1 ∗ , Zhuo Chen 2 † , Y an He 1 ‡ , Jing-ping Jiang 1 § 1 College of Electrical Engineering , Z hejiang Uni versity , Hang Zhou, Zhejiang, 31002 7, China 2 Department of Automation, Shanghai Jiao T ong Univ ersity , Shanghai, 200240, China October 1, 2018 Abstract This paper introduces a model based upo n game s on an e volving n etwork, and de velops three clustering algorithms according to it. In the clustering algorithms, data points for clustering are regarded as players who can make decisions in games. On the network describing relationships among data points, an edge-remov ing- and-re wiring (ER R) function is employed to explore in a neighborho od of a data point, which remo ves edges connecting to neighbors with small pay o ff s, and cre- ates new edges to neighbors with larger payo ff s. As such, the conn ections among data points vary over time. During the evolu tion of network, some strategies are spread in the network. As a consequen ce, clusters are formed automatically , in which data points wi th the same e volutionarily stable strategy are collected as a cluster , so the number of e volutionarily st able strategies indicates the number of clusters. Moreov er , the experimental results hav e demonstrated that data points in datasets are clustered reasonably and e ffi ciently , and the comparison with other algorithms also provides an indication of the e ff ectiveness of the propo sed algo- rithms. K eywords : Unsupervised learning, data clustering, ev olutionary game theory , e vo- lutionarily stable strategy 1 Introd uction Cluster analysis is an imp ortant branch of Pattern Recognition , whic h is widely used in many fields such as pattern an alysis, data mining, inf ormation retriev al and image segmentation. For the past th irty y ears, many excellent clustering algorithms have been presented , s ay , K -means [1], C4.5 [2 ], support vector clustering (SVC) [3], spec- tral clustering [ 4], etc., in which the data points f or clusterin g are fixed, and various ∗ anjuh@c qu.edu.cn † czcq@sj tu.edu.cn ‡ heya n@zju.edu.cn § eejia ng@zju.edu.cn 1 function s are designed to fin d separating h yperplan es. In recen t years, howev er, a sig- nificant change has b een made. Some researchers th ought a bout that why not those data points could move by themselves, just like agents or something , and co llect together automatically . Theref ore, following their ideas, they created a few exciting algorithms [5, 6, 7, 8, 9], in which d ata points move in space accord ing to certain simple local rules preset in advance. Game the ory came into bein g with the book nam ed ”Theory of Gam es and Eco- nomic Behavior” by Jo hn von Neum ann and Osk ar M orgenstern [10] in 1940. In this period, Cooperative Game was widely studied. T ill 195 0’ s, John Nash pub lished two well-known pap ers to presen t the th eory of no n-coop erati ve g ame, in which he pro- posed the concept o f Nash eq uilibrium, and proved the existence o f equilib rium in a finite non-coo perative game [11, 12]. Altho ugh non-cooper ati ve g ame w as established on the rigorous math ematics, it requ ired that player s in a game must be perf ect r atio- nal o r even hyper-rational. If th is assumption could n ot ho ld, the Nash equilib rium might not be reached some times. On th e o ther hand, ev olutio nary game th eory [1 3] stems from the research es in b iology which ar e to analyze the con flict and co opera- tion b etween animals o r plants. It di ff ers from classical game theo ry by focusing on the dyn amics of strategy cha nge mo re th an the prope rties of strategy equilib ria, and does not require per fect ration al p layers. Besides, an impo rtant co ncept, e volutionarily stable strategy [13, 14], in evolutionary gam e theo ry was defined and intro duced by John Maynard Smith and George R. Price in 1973, which was of ten used to explain the ev olution of social behavior in animals. T o the best of o ur kn owledge, the p roblem of data clustering has n ot been in vesti- gated based on ev olutionar y game theory . So , if data points in a dataset are considere d as player s in g ames, cou ld clusters be for med autom atically by playing gam es amon g them? This is the que stion that we attempt to answer . In ou r clustering algor ithm, each player h opes to maximize h is own payo ff , so he c onstantly adju sts his strategies by observing n eighbors’ payo ff s. In the cou rse of strategies evolving, some strategies are spread in the network o f p layers. Finally , some p arts will be formed au tomatically in each of which the same strategy is used . According to di ff er ent strategies played, data points in the dataset can be n aturally collec ted as se veral di ff er ent clu sters. Th e remain - der of this paper is organized as follows: Section 2 introd uces some basic concep ts an d methods abou t the e volutionary game theory and evolutionary game on graph. In Sec- tion 3, the mod el based upon games o n ev olvin g network is p roposed and describe d specifically . Section 4 g i ves thr ee a lgorithms based on this m odel, and th e a lgorithms are elaborated and analyzed in detail. Sectio n 5 introduce s tho se datasets used in the experiments briefly , and then demonstrates experimental results o f the algo rithms. Fur- ther , the re lationship between the numb er of clusters and the nu mber of nearest neigh- bors is discussed, and th ree edg e-removing- and-rewiring (ERR) functions em ployed in the clustering algorithms are compared . T he conclusion is giv en in Section 6. 2 Related work Cooperation is commonly observed in g enomes, cells, multi-cellular organisms, social insects, and human soc iety , but Darw in’ s T heory of Evolution implies fierce com pe- tition for existence am ong selfish an d un related individuals. In past decades, many e ff orts have bee n de voted to understand ing the mechanism s behind the emergence and maintenan ce of coop eration in t he context of e volutionary game theory . Evolutionary game theo ry , which c ombines the tradition al gam e theory with the 2 idea o f ev olutio n, is b ased on the assumptio n of b ounded ratio nality . On the contrary , in classical game th eory player s are supposed to be perfe ctly rational or hyper-rational, and al ways cho ose optim al stra tegies in complex environments. Finite informatio n and cognitive lim itations, h owe ver , o ften ma ke ratio nal decision s inaccessible. Besides, perfect rationality may cause the so-c alled backward induc tion par adox [15 ] in finitely repeated games. On the oth er hand , as the relaxatio n of perfect rationality in classi- cal game theory , bo unded rationality means people in games n eed only part rationality [16], which explains wh y in many cases people respo nd or play instin cti vely accor ding to heuristic ru les and social n orms rath er than adopting the strategies ind icated by ra- tional g ame theo ry [ 17]. So, various dynam ic rules can be defined to character ize the bound edly rationa l beha vio r of playe rs in e volutionary game the ory . Evolutionary stability is a cen tral concept in ev olutio nary game theory . In biolo gi- cal situations the e volutionary stability provides a robust c riterion for strategies again st natural selection . Furthermore, it also means that any small group of individuals who tries some altern ati ve strategies gets lower payo ff s than those who stick to the o riginal strategy [18]. Sup pose that ind i vidu als in an infinite an d homo genous population who play symm etric g ames with equal pr obability are rando mly match ed and all emp loy the same strategy A . Nev erthe less, if a small gro up o f m utants with population share ǫ ∈ ( 0 , 1) who plays some other strategy app ear in the whole grou p of individuals, they will receive lower pa yo ff s. Therefore, the strategy A is said to b e ev olutionar y stable for any mu tant strategy B , if an d only if the in equality , E ( A , (1 − ǫ ) A + ǫ B ) > E ( B , (1 − ǫ ) A + ǫ B ), holds, where the function E ( · , · ) deno tes the pay o ff for play ing strategy A again st strategy B [1 9]. In addition, the cooperation mech anism and spatial-temporal dy namics related to it have long been in vestigated within the framework of ev olutiona ry ga me theor y based on the pr isoner’ s dilemma (PD) game o r snowdrift ga me which mo dels inter actions between a pair of playe rs. In ear ly days, the iterated PD game was widely studied , in which a player interacted with all othe r player s. By ro und ro bin interaction s among players, strategies in the po pulation b egan to e volve accord ing to their payo ff s. As a re- sult, the strategy of u ncondition al defectio n w as always ev olution ary stable [20] while pure coop erators could not sur viv e. Nevertheless, the T it-for-T at strategy is ev olution - ary stable as well, which promotes coopera tion based on recipr ocity [21 ]. Recently , ev olution ary dynamics in structured p opulations has attracted much at- tention, where the s truc tured population denotes an infinite and well-mixed p opulation which simplifies the analytical description o f the e volution process. In r eal pop ulations, individuals are m ore likely a ff ected by their neigh bors than t ho se who are far a way , but the spa tial structure of pop ulation is omitted in the iterated PD game. T o study the spatial e ff ects upon strategy fr equencies in th e p opulation, Nowak and May [22] have introdu ced the spatial PD gam e, in wh ich play ers are lo cated on the vertices o f a two- dimensiona l lattice, whose edges represent con nections amo ng th e correspon ding play- ers. Instead of play ing with all other c ontestants, each p layer only inter acts with his neighbo rs. Without any strate gic co mplexity the stable co existence o f coopera tors and defectors can b e achieved. Howe ver , the mod el presented in [22] assum es a noise f ree en viro nment. T o c haracterize the e ff ect of noise, Szab ´ o and T oke [2 3 ] have presented a stochastic update rule that permits irrational choice. Besides, Perc and Szolno ki [24] account for socia l diversity by stochastic variables that determine the mapping o f game payo ff s to individual fitness. Furthermore, many o ther works ce ntered o n th e lattice structure h a ve a lso been don e. For example, V ukov and Szab ´ o [25] have p resented a hier archical lattice and shown that for di ff e rent hierarc hical lev els the highe st f re- quency of coo perators may occ ur at the top or middle layer . For more details about 3 ev olutionar y gam es on gr aphs, see [17, 26, 27] and references therein. Y et, as imitation s of real social networks, the ev olutionar y game on lattices assumes that ther e is a fixed neigh borhoo d for each play er . Ne vertheless, th is assump tion does not always hold for most of real social networks. Unlike models mention ed above, the relationships am ong player s (d ata p oints) in our model ar e r epresented b y a weigh ted and directed network, which means that players are not located on a regular lattice any more. And the ne twork will evolve ov er time because each player is allowed to app ly an ed ges-removing -and-rewiring (ERR) function to chan ge his co nnections between him an d his neighbo rs. Furthermore , the payo ff matrix of any two players in the pro- posed mo del is also time- varying instead of a con stant payo ff matrix, fo r instance, the payo ff ma trix in PD game. As a consequen ce, when the evolutionarily stable strate- gies emerge in the network , it will be observed that only a few players (d ata po ints) receive considerable connec tions, while most of them have only one co nnection. Nat- urally , p layers (d ata p oints) are divided in to several p arts (c lusters) accor ding to their ev olutionar ily stable strategies. 3 Pr oposed model Assume a set X with N play ers, X = { X 1 , X 2 , · · · , X N } , w hich are distributed in a m -dimension al m etric space. In this metric space, there is a distance function d : X × X − → R , wh ich satisfies the co ndition that the clo ser any two player s a re, the smaller the outp ut is. Based o n the distance functio n a distance matr ix is compu ted whose en tries are distances between any two players. Next, a weigh ted a nd dire cted k -ne arest n eighbor ( knn) network , G 0 ( X , E 0 , d ), is fo rmed by addin g k edg es dir ected tow ard its k nearest neig hbors for each player, wh ich represents the initial rela tionships among all players. Definition 1 I f ther e is a set X with N players, X = { X 1 , X 2 , · · · , X N } , the initial weighted and dir ected knn network, G 0 ( X , E 0 , d ) , is cr eated as below .                          X =  X i , i = 1 , 2 , · · · , N  E 0 = S N i = 1 E 0 ( i ) E 0 ( i ) = n e 0  X i , X j  | j ∈ Γ 0 ( i ) o Γ 0 ( i ) =  j     j = a rgmin k X h ∈ X  n d ( X i , X h ) , X h ∈ X o  (1) Her e, players in the set X co rr espond to vertexes in the network G 0 ( X , E 0 , d ) ; dir ected edges in the network r epresent certain r elationships established a mong players and the distances denote the weights over edges; the function, argmin k ( · ) , is to find k nearest neighbo rs of a player , which construct a neighbo r set, Γ 0 ( i ) . It is worth no ting that the distance between a p layer X i and h imself, d ( X i , X i ), is zero acco rding to the defin ed distan ce function, which mea ns th at at the beginn ing he i s one of his k nearest neighb ors. So ther e is an ed ge between th e player X i and himself, namely a self-loop. In practice, the distance is set by d ( X i , X i ) = 1. When th e initial network G 0 is established, we can define a ev olutio nary game, Ω = { X , G 0 , S 0 , U 0 } , on it further . Definition 2 A n evolu tionary game Ω = { X , G 0 , S 0 , U 0 } o n a network G 0 is a 4- tuple: X is a set of p layers; G 0 r epr esents th e initial relationships among players; 4 S 0 = { s 0 ( i ) , i = 1 , 2 , · · · , N } repr esents a set of players’ strate gies; U 0 = { u 0 ( i ) , i = 1 , 2 , · · · , N } r epr esents a set of p layers’ pay o ff s. In ea ch r ound , players choose theirs strate gies simulta neously , and each player can only ob serve its neigh bors’ p ayo ff s, b ut does not know the str ategy pr ofi le o f anyon e of a ll other players in X . F inally , all players update their str ategy pr ofi les sync hr ono usly . In the p roposed m odel, assume each player in X sets up a gro up, and hopes to maximize the p ayo ff of his own gro up in ord er to attract more p layers to join. At the same time , he also joins k gr oups set u p by other p layers, so the initial strategy set s 0 ( i ) ∈ S 0 of a player X i is de fined as his n eighbor set, s 0 ( i ) = Γ 0 ( i ). Howe ver , it is worth no ting that his preferen ce to join each gro up is changea ble, who se initial value is giv en below , P 0 ( i ) = n p 0 ( i , j ) , j ∈ Γ 0 ( i ) o p 0 ( i , j ) = 1 /    Γ 0 ( i )    = 1 / k (2) where P 0 ( i ) is the pr eference set, and the sym bol | · | d enotes the cardin ality of a set. Thus, a player’ s payo ff may be defined as follows. Definition 3 A fter a player X i chooses his strate gies and corr espond ing prefer en ces, he r eceives a payo ff u 0 ( i ) , u 0 ( i ) = P j ∈ Γ 0 ( i ) R ( i , j ) R ( i , j ) = p 0 ( i , j ) × Deg 0 ( j ) / d ( X i , X j ) (3) wher e Deg 0 ( j ) r epresents t he degr e e o f a player X j in the neighb or set, an d the de gr ee is a sum of the inde gree and ou tde gr ee. When all player s ha ve r ecei ved their payo ff s, each one will check h is neig hbors’ payo ff s, and apply an ERR f unction B i ( · ) to change his connec tions and upd ate his neighbo r set. Definition 4 Th e ER R functio n B i ( · ) is a function of payo ff s, who se outpu t is a set with k elements, i.e., an updated neighbor set Γ 1 ( i ) of a player X i . Γ 1 ( i ) = B i  b u 0 ( i )  = ar gma xk j ∈ Γ 0 ( i ) S Υ 0 ( i )   u 0 ( j ) , j ∈ Γ 0 ( i ) S Υ 0 ( i )   b u 0 ( i ) = n u 0 ( j ) , j ∈ Γ 0 ( i ) S Υ 0 ( i ) o , Υ 0 ( i ) = S j ∈ Γ + 0 ( i ) Γ 0 ( j ) Γ + 0 ( i ) = n j | u 0 ( j ) ≥ θ 0 ( i ) , j ∈ Γ 0 ( i ) o , Γ − 0 ( i ) = Γ 0 ( i ) \ Γ + 0 ( i ) (4) wher e θ 0 ( i ) is a p ayo ff th r eshold, Υ 0 ( i ) is ca lled an extended neighbo r set, and the function argma xk ( · ) is to find k neigh bors with the fir st to the k-th lar gest payo ff s in the union Γ 0 ( i ) S Υ 0 ( i ) . The ERR fu nction B i ( · ) expands the view of a playe r X i , an d m akes him able to observe pay o ff s of p layers in the extended ne ighbor set, which pr ovides a ch ance to find players with higher pay o ff s around him. If no players with high er payo ff s are found in the extended n eighbor set, i.e ., min ( { u 0 ( j ) , j ∈ Γ 0 ( i ) } ) ≥ ma x ( { u 0 ( h ) , h ∈ Υ 0 ( i ) } ), then the output of the ERR function is Γ 1 ( i ) = B i ( b u 0 ( i )) = Γ 0 ( i ). Otherwise, a neighbor with the minimal payo ff will be removed together with the correspo nding edge from the neighbo r set and th e edge set, an d replaced by a found player with larger pay o ff . Th is process is repeated till the payo ff s of unco nnected players in the e xtend ed neig hbor set 5 are no larger than those of co nnected neighbors. Since the co nnections among p layers, namely the edge set E 0 in the network G 0 ( X , E 0 , d ), are changed by the ERR functio n, the network G t ( X , E t , d ) will b egin to e volve over time, when t ≥ 1. As such, after the ERR func tion is app lied, th e new prefer ence set P t ( i ) of a player X i needs to be adjusted. Definition 5 Th e n ew pr efer ence set of a player X i ∈ X is formed by means o f th e below formulation. P t ( i ) = n p t ( i , j ) , j ∈ Γ t ( i ) o p t ( i , j ) =            P h ∈ ( Γ t − 1 ( i ) \ ∆ ) p t − 1 ( i , h )    Γ t ( i ) \ ∆    if j ∈ Γ t ( i ) \ ∆ p t − 1 ( i , j ) otherwise ∆ =  Γ t − 1 ( i ) T Γ t ( i )  (5) Then, the player adjusts h is pr efer ence set P t ( i ) as follows. F irst, he id entifies the neighbo r X m with maximal payo ff in his neig hbor set, m = argma x j ∈ Γ t ( i )   u t − 1 ( j ) , j ∈ Γ t ( i )   (6) Ne xt, each element in th e p r efer ence set is taken its squar e r oot an d the pr eference p t ( i , m ) of joining the gr ou p built by the neigh bor X m becomes negative,        P t ( i ) =  p p t ( i , j ) , j ∈ Γ t ( i )  p p t ( i , m ) = − p p t ( i , m ) , m ∈ Γ t ( i ) (7) Further , let Ave t ( i ) = ( P j ∈ Γ t ( i ) p p t ( i , j )) / | Γ t ( i ) | , thus, the upda ted prefer en ce set is, P t ( i ) =  p t ( i , j ) , j ∈ Γ t ( i )  p t ( i , j ) =  2 × A ve t ( i ) − p p t ( i , j )  2 (8) After the preferen ce set o f each player has been adjusted, an iteration of the mo del is co mpleted. In conclusion , when t ≥ 1, the network representin g relatio nships among players begins t o ev olve ov er time, which also makes a player’ s strategy set and payo ff set become time-varying . There fore, the game on e volving network G t ( X , E t , d ) is rewritten as Ω = { X , G t , S t , U t } . G t ( X , E t , d ) =                        X ( t ) = n X i ( t ) , i = 1 , 2 , · · · , N o Γ t ( i ) = B i ( b u t − 1 ( i )) E t = S N i = 1 E t ( i ) E t ( i ) = n e t  X i , X j  | j ∈ Γ t ( i ) o S t = n s t ( i )    s t ( i ) = Γ t ( i ) , i = 1 , 2 , · · · , N o U t = n u t ( i )    u t ( i ) = P j ∈ Γ t ( i ) p t ( i , j ) × Deg t ( j ) / d ( X i , X j ) , i = 1 , 2 , · · · , N o (9) As the mo del is iterated , s ome strategies are s pre ad in the e volving network, which are played by a grea t nu mber o f p layers. In other words, a certain strategy or se veral strategies in th e strategy set s t ( i ) ∈ S t will b e always p layed by th e p layer X i with the maximal preferen ce. 6 Definition 6 I f a playe r X i ∈ X always or periodically chooses a strate gy b s t ( i , j ) ∈ s t ( i ) with the maximal pr efer enc e ma x ( { p t ( i , j ) , j ∈ Γ t ( i ) } ) ,                b s t ( i , j ) = arg ma x j ∈ Γ t ( i )  p t ( i , j ) , j ∈ Γ t ( i )  b s t ( i , j ) = b s t − 1 ( i , j ) b s t ( i , j ) = b s t − nT ( i , j ) (10) then the strate gy b s t ( i , j ) ∈ s t ( i ) is called th e ev olution arily stable strategy (E SS) of th e player X i ∈ X . Here , the variab le T is a constant period. As a conseq uence, each play er in the network will choose one of ev olution arily stable strategies as his strategy and he is not w illing to change his strategy unilater ally during the iterations. 4 Algorithm and analysis In this section , at first three di ff erent ERR fu nctions ( B 1 i ( · ) , B 2 i ( · ) , B 3 i ( · )) are designed , and then three clustering algo rithms (EG1, E G2, EG3 ) based on them ar e established. Finally , the clusterin g algorithm is elabo rated an d analyzed in detail. 4.1 Clustering algorithms Assume an un labeled dataset X = { X 1 , X 2 , · · · , X N } , in which each instance co nsists of m featur es. In the clustering algor ithms, the relation ships among all data points are represented by a weighted and directed network G t ( X , E t , d ), and each data point in the dataset is considered as a player in the propo sed mo del, who adju sts his strate gy profile in order to maximize h is own pay o ff by o bserving o ther p layers’ pay o ff s in the union Γ t ( i ) ∪ Υ t ( i ). According to the propo sed mode l, after a distan ce function d : X × X − → R is selected, th e initial con nections of data poin ts, G 0 ( X , E 0 , d ), are constru cted by means of Defin ition 1. Th en, th e initial payo ff set U 0 of data p oints is com puted step by step. Finally , an ERR fun ction is app lied to explore in a n eighborh ood o f ea ch d ata point, whic h changes n eighbors in the n eighbor set o f the d ata poin t. T hus, a ne twork G t ( X , E t , d ) will be e volving when t ≥ 1. Howe ver , di ff er ent E RR function s provid e di ff eren t explor ing capacities for data points, i.e., the observable areas of d ata poin ts dep end o n an ERR f unction. As such, there is no doubt that the o btained results vary when di ff erent ERR fun ctions ar e em - ployed. Here, th ree ERR functions ( B 1 i ( · ) , B 2 i ( · ) , B 3 i ( · )) are d esigned, and three clu ster - ing algorithm s based on them are constru cted respecti vely . Algorithm EG1: In Algorithm EG1, an ERR functio n B 1 i ( · ) that is realized mo st easily is used. This function always observes an extended neighb or set formed by ⌈ η × | Γ t − 1 ( i ) |⌉ neighb ors of a data point X i , where the symbol ⌈·⌉ is to take an integer par t of a num ber satisfyin g the integer p art is n o larger than the numb er , and the variable η is called a ra tio of exploration, η ∈ [0 , 1]. Accor ding to Definition 4, a pay o ff thr eshold θ 1 t − 1 ( i ) is set by θ 1 t − 1 ( i ) = f in d α ( { u t − 1 ( j ) , j ∈ Γ t − 1 ( i ) } ) , α = ⌈ (1 − η ) × | Γ t − 1 ( i ) |⌉ firstly , where the function 7 f ind α ( · ) is to find the α -th largest payo ff in the neighb or set Γ t − 1 ( i ). As such , the set Γ t − 1 ( i ) is divided i nto two sets naturally: Γ + t − 1 ( i ) = n j    u t − 1 ( j ) ≥ θ 1 t − 1 ( i ) , j ∈ Γ t − 1 ( i ) o , Γ − t − 1 ( i ) = Γ t − 1 ( i ) \ Γ + t − 1 ( i ) (11) Then, based on the set Γ + t − 1 ( i ), the extended neigh bor set is built, Υ t − 1 ( i ) = S j ∈ Γ + t − 1 ( i ) Γ t − 1 ( j ) . Further, the observable payo ff set of the data point is written as, b u t − 1 ( i ) = n u t − 1 ( j ) , j ∈ Γ t − 1 ( i ) ∪ Υ t − 1 ( i ) o (12) At last, the ERR f unction B 1 i ( · ) is app lied, which mean s that the edge s c onnecting to the ne ighbors with small pa yo ff s are removed an d new e dges are created between the data point and found players with larger payo ff s. Hen ce, his new n eighbor s et is Γ t ( i ) = B 1 i ( b u t − 1 ( i )) = ar gma xk j ∈ Γ t − 1 ( i ) ∪ Υ t − 1 ( i )   u t − 1 ( j ) , j ∈ Γ t − 1 ( i ) ∪ Υ t − 1 ( i )   (13) Algorithm EG2: Unlike Algorithm EG1, the ERR function B 2 i ( · ) in Algorithm EG2 adjusts the num- ber of neig hbors dynamically to form an extend ed neighbor set instead of the con stant number of neighbors in Algorithm EG1. Furtherm ore, the payo ff thre shold θ 2 t − 1 ( i ) is set by th e average of neighb ors’ pa yo ff s, θ 2 t − 1 ( i ) = P j ∈ Γ t − 1 ( i ) u t − 1 ( j ) / | Γ t − 1 ( i ) | . Next, the set Γ + t − 1 ( i ) is formed, Γ + t − 1 ( i ) = { j | u t − 1 ( j ) ≥ θ 2 t − 1 ( i ) , j ∈ Γ t − 1 ( i ) } , an d then the new neigh bor set is achiev ed by mean s of the ERR fun ction B 2 i ( · ). In the case, when the payo ff s of all n eighbors ar e eq ual to the p ayo ff thr eshold θ 2 t − 1 ( i ), the output o f the ERR function is Γ t ( i ) = B 2 i ( b u t − 1 ( i )) = Γ t − 1 ( i ). This m ay be viewed as self-pro tecti ve be havior for av oiding a payo ff loss d ue to no enough i nf ormation acquired. Algorithm EG3: The ERR function B 3 i ( · ) used in Algorithm EG3 p rovides more strong ly exploring capacities for the data points with small payo ff s than that fo r those with l arger payo ff s. Generally speaking, for m aximizing their payo ff s, player s with sma ll p ayo ff s of ten seems r adical and show stronger d esire for exploration , because this is the only way to impr ove their payo ff s. On the other hand, those players with large pay o ff s look conservati ve for protecting their payo ff gotten. Formally , the ratio of exploration γ ( i ) of a data point X i ∈ X is g i ven as below: γ ( i ) = ( ma x j ∈ X ( u t − 1 ( j ) ) + min j ∈ X ( u t − 1 ( j ) )) − u t − 1 ( i ) ma x j ∈ X  u t − 1 ( j )  (14) Thus, the data p oint X i ∈ X can ob serve an extended neig hbor set that is fo rmed by ⌈ γ ( i ) ×| Γ t − 1 ( i ) |⌉ n eighbors. Fu rther , th e payo ff thr eshold is set by θ 3 t − 1 ( i ) = f ind β ( i ) ( { u t − 1 ( j ) , j ∈ Γ t − 1 ( i ) } ) , β ( i ) = ⌈ (1 − γ ( i )) × | Γ t − 1 ( i ) |⌉ and then the n e w neig hbor set Γ t ( i ) is built ac- cording to the ERR function B 3 i ( · ). The ERR function br ings abo ut cha nges of the co nnections amo ng data points, so that th e prefe rences need to be ad justed in terms of Definition 5. Wh en the evolution- arily stable strategies app ear in the network, the clustering alg orithm exits. In the end, the data po ints using the same ev olutio narily stable strategy a re g athered together as a cluster, and the num ber of evolutionarily stable strategies indicates the nu mber of clusters. 8 4.2 Analysis of algorithm The process of data clusterin g in t he propo sed algo rithm can be viewed an explanation about the gr oup formation in society . Initially , each data po int (a player) in the dataset establishes a gr oup which cor responds to an initial cluster at the same time th at he join s other grou ps b uilt by k other players. As such, the p reference p ( i , j ) may b e explained as th e level of participation ; Deg ( i ) re presents the total n umber o f p layers in a group; 1 / d ( X i , X j ) denotes the position that the playe r occup ies in a grou p in ord er to identify a player is a president or an average member, and a p layer X i usually occupies the highest po sition in his own gr oup; th e to tal pa yo ff u ( i ) o f a player X i is viewed a s the attraction of a group . Ac cording t o Definition 3, the rew ard of a player X i is associated directly with the pref erence, the total number of player s and his position in a grou p. If a play er X i who occupies an important position joins a group with maximal preferen ce, and the num ber of players in th e gro up is considerable, then the reward R ( i , j ) that the player r ecei ves is also large. On the oth er hand , f or a p layer X h who is an av erag e member in the sam e gro up, i.e., 1 / d ( X i , X j ) > 1 / d ( X h , X j ), alth ough his p reference to join this grou p is as s ame as that of the pla yer X i , p ( i , j ) = p ( h , j ), his reward acqu ired from this gro up is smaller tha n that o f the player X i by m eans of Eq. 3 . Th is seems unfair , but it is consistent with the phenomena observed in society . I n ad dition, the total payo ff u t ( i ) of the player X i is the sum of his all rew ard s, u t ( i ) = P j ∈ Γ t ( i ) R ( i , j ). Certainly , each player is willing to join a gr oup with large attraction, an d q uit a group with little attraction, as is d one b y an E RR fu nction. Next, the p layer find s the group with the largest attraction and incre ases th e lev el of participation , namely th e preferen ce p t ( i , j ) at the same time that oth er preferen ces ar e d ecreased. T o adjust the preferen ce set P t ( i ) of a player X i , the Grover iteration G in th e qu antum search algorithm [ 28], a we ll-known alg orithm in quantum co mputation[2 9 ], is employed, which is a way to adjust the proba bility amplitude of each term in a superposition state. By ad justment, the probability amplitude of the wanted is inc reased, while th e others are reduced. Th is wh ole process may be regar ded as the inver sion about av era ge operation [28]. For our case, eac h elem ent in the preference set needs to be taken its square root first, and then the av erag e Ave t ( i ) of all square roots is obtaine d. Finally , all values are in verted ab out th e average. Th ere are three main reasons tha t we select the modified Grover iteration as the updatin g method of prefere nces: (a) the sum of preferen ces updated retains one, P j ∈ Γ t ( i ) p t ( i , j ) = 1, (b) a certain preference up dated in a play er’ s pre ference set is m uch larger than the oth ers, p t ( i , j ) ≫ p t ( i , h ) , h ∈ Γ t ( i ) \ j , and ( c) it help s player s’ payo ff s to be a p ower law distribution, in wh ich o nly a few players’ payo ff s are far larger than others’ after iterations. Moreover , this is consistent with our o bservations in society , i.e. , a player will change the level of particip ation obviously after he takes p art in activities held by n eighbor s’ gr oups. In other words, for the grou p with large attra ction he shows higher level of par ticipation, whereas he hardly joins those group s with little attraction . After the m odel is iterated several times, a play er X i ∈ X will find th e most attr ac- ti ve group for h im, an d join this group with the largest pref erence, ma x ( { p t ( i , j ) , j ∈ Γ t ( i ) } ). Hence, only a few groups are so attracti ve that alm ost all players join them. On the o ther han d, most of gr oups are closed du e to a lack of players. Those lucky sur viv als not only attract many players but also those p layers in the gro ups show the highest level of participation . Furthe rmore, these surviving group s are th e clusters form ed by player s (data points) automatically , where the players (data points) in a cluster play the same ev olutionar ily stable strategies, and the number of surviving g roups is also the number of clusters. 9 5 Experiments and Discussions T o evaluate these three clustering a lgorithms, five datasets are selected from UCI repos- itory [30], which are Soybea n, Iris, W ine, Ionosphe re an d Breast ca ncer Wisconsin datasets, and experiments are performed on them. 5.1 Experiments In this subsection, firstly these datasets are introd uced briefly , an d then th e experimental results are dem onstrated. The origin al data points in above datasets all ar e scattered in high dim ensional spaces spanned by their features, wh ere th e descriptio n of all datasets is summar ized in T ab le 1. As for Breast d ataset, some lost features are rep laced by random nu mbers, and the W ine dataset is standardized. Finally , this algorith m is co ded in Matlab 6.5. T able 1: Description of datasets. Dataset Instances Features classes Soybean 47 21 4 Iris 1 50 4 3 W ine 178 13 3 Ionosph ere 351 32 2 Breast 699 9 2 Throu ghout all expe riments, data po ints in a dataset are consider ed as play ers in games whose initial po sitions are taken from the d ataset. Next, the network represent- ing initial relatio nships among data po ints ar e created according to Definition 1, after a distance function is selected. This distance function only need s to satisfy the condition that the more similar data poin ts are, the smaller the outpu t of the fu nction is. In the experiments, the distance function applied is as following: d  X i , X j  = e x p  k X i − X j k / 2 σ 2  , i , j = 1 , 2 , · · · , N (15) where th e sym bol k · k rep resents L 2-nor m. The advantage of this functio n is that it not on ly satisfies above requirem ents, b ut also overco mes the drawbacks of E uclidean distance. F or instance, when two poin ts are very close, the ou tput o f Euclid ean distance function app roaches zero, as may make the co mputation of payo ff fail due to the payo ff approa ching infinite. Nevertheless, when Eq.(15 ) is selected as th e distan ce fu nction, it is more convenient to compute the players’ p ayo ff s, si nce its minim um is one an d the reciproca ls of its output are between zero and one , 1 / d ( X i , X j ) ∈ [0 , 1]. In addition , the parameter σ in Eq .(15) takes one and the distance between a data point and itself is set by d ( X i , X i ) = 1. As is analyzed in 4. 2, a data po int X i occupies the highest position in the group established by himself. Three clustering alg orithms ar e ap plied on the five datasets respectively . Becau se the ca pacity of exploration of an E RR fun ction depends in par t on th e num ber k of nearest neighbors, the algorithms are ru n on every da taset at di ff eren t n umbers of near- est neigh bors. Those clusterin g r esults obtained by three algo rithms are co mpared in Fig. 1, in which each point represents a clusterin g accur acy . Th e clustering accuracy is defined as below: 10 Definition 7 c st i is the label which is a ssigned to a data point X i in a dataset by the algorithm, and c i is the actu al label of the da ta p oint X i in th e dataset. S o the clustering accuracy is [31]: accur acy = P n i = 1 λ ( ma p ( c st i ) , c i ) / n λ ( ma p ( c st i ) , c i ) =        1 if ma p ( c st i ) = c i 0 otherwise (16) wher e the mapping function ma p ( · ) ma ps th e label go t by the algorithm to the actual label. As is shown in Fig. 1, the clustering results obtained by Algor ithm EG1 and EG2 are similar at di ff eren t numbers of n earest neighb ors. As a wh ole, the results of Alg o- rithm E G1 are a bit b etter than that of Algor ithm EG2 owing to the stronger capacity of explora tion. Ho wever , for the same dataset th e best result is achieved by Algorithm EG3, which sh ows the stro ngest cap acity of exploratio n. As analyzed ab ove, both the strongest cap acity of explor ation and the stable results cann ot be achieved at the same time, so a trade-o ff is neede d between them . 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 number of nearest neighbors k clustering accuracy Algorithm EG1 Algorithm EG2 Algorithm EG3 (a) Soybe an data set 3 4 5 6 7 8 9 0.7 0.75 0.8 0.85 0.9 0.95 1 number of nearest neighbors k clustering accuracy Algorithm EG1 Algorithm EG2 Algorithm EG3 (b) Iris datase t 3 3.5 4 4.5 5 5.5 6 6.5 7 0.7 0.75 0.8 0.85 0.9 0.95 1 number of nearest neighbors k clustering accuracy Algorithm EG1 Algorithm EG2 Algorithm EG3 (c) W ine dataset 4 6 8 10 12 14 16 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 number of nearest neighbors k clustering accuracy Algorithm EG1 Algorithm EG2 Algorithm EG3 (d) Ionosphere dataset 4 6 8 10 12 14 16 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 number of nearest neighbors k clustering accuracy Algorithm EG1 Algorithm EG2 Algorithm EG3 (e) Breast data set Figure 1: Compar ison of clustering accuracies in Algor ithm EG1 , EG2, and EG3. W e compar e o ur r esults to those results obtained by oth er clustering algo rithms, Kmeans [32 ], PCA-Kmeans [32], LD A-Km [32], on the same dataset. The co mparison is summarized in T able 2. 5.2 Discussions In the subsection, firstly , we discuss ho w the number of clusters is a ff ected by t he num- ber k of near est n eighbor s chan ging. Then, fo r the Algor ithm EG1 , the r elationships between the ratio of exploration and the clustering results is in vestigated, w hich pro- vides a way to select the ratio of exploration η . Finally , the rates of co n vergence in three clustering algorithm s are com pared. 11 T able 2: Compar ison of clusterin g accuracies of algorithm. Algorithm Soybean Iris W ine Iono sphere Breast EG1 95.75 % 90% 96.63 % 74.64 % 94. 99% EG2 91.49 % 90.67% 96.07 % 74.64 % 95 .14% EG3 97.87 % 90.67% 97.19 % 74.64 % 94 .99% Kmeans 6 8.1% 89.3% 70 .2% 71% – PCA-Kmeans 72.3% 88.7% 70 .2% 71% – LD A-Km 76.6% 98% 82.6% 71.2% – 5.2.1 Number of nearest neighbors vs. number of clusters The number k of nearest neighbors repr esents th e nu mber of neig hbors to which a data point X i ∈ X co nnects. For a dataset, the num ber k of nearest n eighbors determin es the nu mber of clusters in part. Generally spe aking, the number of clu sters decreases in versely with the n umber k of nearest neighbors. If the number k of nearest neighbors is small, which indicates a da ta po int X i connects to a f e w neighbo rs, in this case the area that the ERR fu nction may explore is also small, i.e., the elements in the u nion of the extend ed neigh bor set Υ t − 1 ( i ) an d the neigh bor set Γ t − 1 ( i ) are only a few . Therefore, when the network e volves over time, strategies are spread only in a small area. Finally many small clusters with ev olu tionarily stable strategies appear in the ne twork. On th e other hand, a big nu mber k of neare st neigh bors provides mo re n eighbors fo r a data point, which implies that the cardinality of the un ion is larger than that wh en a small k is employed . This also means that a larger area can be ob served and explo red by the ERR fu nction, so that big clu sters containing more data p oints are formed b ecause ev olutionar ily stable strategies are spread in larger ar eas. For a dataset, the clustering re sults in di ff erent number k of nearest neighb ors have been illustrated in Fig . 2, in which each d ata po int only con nects to the neighbor with the largest preferen ce, and clusters are r epresented b y di ff erent signs. As is shown in Fig. 2, we can find that only a f e w data p oints receive considerab le co nnections, whereas m ost of data po ints have only one connectio n. T his indicates that when the ev olution o f network is ended , the n etwork formed is chara cterized by the scale-free network [ 33], i. e., win ner takes all. Besides, in Fig. 2 (a), six clu sters are obtain ed by the clu stering algorithm, whe n k = 8. As the n umber k of nearest n eighbors rises, five clusters are obtained when k = 10, three clusters when k = 15. So, if the exact number of clusters is not k nown in advance, di ff erent number of clusters ma y b e ach ie ved by adjusting the number of nearest neighb ors in practice . −5 −4 −3 −2 −1 0 1 2 3 4 5 −4 −3 −2 −1 0 1 2 3 4 (a) k = 8 −5 −4 −3 −2 −1 0 1 2 3 4 5 −4 −3 −2 −1 0 1 2 3 4 (b) k = 10 −5 −4 −3 −2 −1 0 1 2 3 4 5 −4 −3 −2 −1 0 1 2 3 4 (c) k = 15 Figure 2: The number of nearest neighbo rs vs. n umber of clusters. 12 5.2.2 E ff ect of the ratio of explorat ion in Algorithm EG1 For the ERR function B 1 i ( · ) used in Algorithm EG1, its capacity of exploration may be adjusted by setting di ff e rent ratio of exploration η ∈ [ 0 , 1]. I f the r atio of exploratio n is η = 0, then the extended neigh bor set formed w ill b e empty . Hence, the network does not e volve over time , since the ERR fun ction d oes not explor e. On th e other hand, when the ratio of explora tion takes the maximu m η = 1, the ERR function is with the strongest capacity of exploration, because an extended neigh bor set for med by all neigh bors can be observed. T hen we may ask n aturally: what should the ratio of exploratio n be taken? T o answer this question, we com pare those clu stering results at di ff erent η shown in Fig. 3, in which the results are represente d b y th e clustering accuracies. 3 3.5 4 4.5 5 5.5 6 6.5 7 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 number of nearest neighbors k clustering accuracy eta=0.1 eta=0.3 eta=0.5 eta=0.7 eta=0.9 Figure 3: The results of algorith m EG1 at di ff er ent rates of exploration. From Fig. 3, we can see th at when the ratio of exploration is greater than 0.5, the cluster ing results obtained by Algor ithms EG1 fluctu ates strongly , as seems over exploration. On th e contrary , in the case when η ≤ 0 . 5, the clu stering results are stable relativ ely . As a whole, when th e ratio of exploration η = 0 . 5, the best r esults are achieved in Algor ithm E G1. I n conclusio n, a b ig ratio of exploratio n is n ot too good, because it may cause the E RR f unction B 1 i ( · ) over exploration . Ho wever , if the ratio of explo ration is to o small, good r esults ar e not obtaine d because of a lack of exploration. Therefo re, in th e later discussion, the ratio of explor ation in Algorithm EG1 takes η = 0 . 5. 5.2.3 Three E RR funct ions vs. rate of con vergence As fo r Algorithm EG1 , whe n η = 0 . 5, the ERR f unction B 1 i ( · ) may observe the extended neighbo r set for med by half of neighbor s. In this case, the payo ff th reshold is θ 1 t − 1 ( i ) = med ian ( { u t − 1 ( j ) , j ∈ Γ t − 1 ( i ) } ). I n Algo rithm EG2, however , the ERR f unction B 2 i ( · ) only can o bserve the extended neigh bor set for med by neigh bors whose payo ff s are g reater than the a verage θ 2 t − 1 ( i ). Generally speaking, for the same k , th e me dian of pay o ff s is smaller than or equal to th e mean, i.e ., θ 1 t − 1 ( i ) ≤ θ 2 t − 1 ( i ). So the explorin g area o f the ERR function B 1 i ( · ) is larger than t hat of the ERR functio n B 2 i ( · ), that is, the number of edges rewired in Algo rithm EG1 is mo re than that in Algorithm EG2 . In additio n, the number of edg es rewired in Algorithm EG3 is largest, since in Algorithm EG3 t he ERR function B 3 i ( · ) provides stronger capacity of exploratio n for players with small payo ff s, which makes mo re edg es are removed and rewired. The comparison of n umber of 13 edges rewired in three algorithm s is illustrated in Fig. 4. As is sho wn in Fig. 4 , at di ff erent number of nearest neigh bors, t he nu mber of edges re wired in Algorithm EG3 is larger than that in other two, and the number of edges rewired in Alg orithm EG1 is larger th an Algo rithm EG2 . F or each o ne of three algor ithms, the number of edges rewired incr eases rapidly with the number of nearest neighbors. 1 2 3 4 5 6 7 8 9 10 0 500 1000 1500 2000 2500 3000 number of iterations number of edges rewired k=3 k=4 k=5 k=6 k=7 (a) EG1 1 2 3 4 5 6 7 8 9 10 0 500 1000 1500 2000 2500 3000 number of iterations number of edges rewired k=3 k=4 k=5 k=6 k=7 (b) EG2 2 4 6 8 10 12 14 0 500 1000 1500 2000 2500 3000 number of iterations number of edges rewired k=3 k=4 k=5 k=6 k=7 (c) EG3 Figure 4: The num ber of ed ges re wired or rates of conv ergence of three algo rithms. Besides, the num ber of iterations ind icates the rate of convergence of an algor ithm. From Fig. 4, we can see that th e r ates of con vergence in Algor ithm EG1 and EG2 are almost the same, and the rate of A lgorithm EG3 is slower sligh tly th an th e o ther two because the ERR function B 3 i ( · ) in Algorithm EG3 explores la rger areas than that in Algorithm E G1 and EG2. For eac h one of thr ee algor ithms, as the nu mber of nearest neighbo rs rises, the num ber of itera tions also rises slightly . When th e algorith m conv erges, the evolutionarily stable strategies appear in the network at the same time. The ch anges of the strategies with the largest prefer ence for each data po int are shown in Fig. 5, wh ere the straigh t lines in the righ t side of figures r epresent the ev olu tionarily stable strategies, and th e numb er of straight line s is the n umber of clusters. As is shown in Fig. 5, the ev olutio narily stable strategies appear in the n etwork after on ly a few iteratio ns, as ind icates the rates of con vergence of algorithm s are fast enoug h. 1 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 120 140 160 180 number of iterations strategy with the largest preference (a) EG1 1 2 3 4 5 6 7 8 9 0 20 40 60 80 100 120 140 160 180 number of iterations strategy with the largest preference (b) EG2 1 2 3 4 5 6 7 8 9 10 11 12 0 20 40 60 80 100 120 140 160 180 number of iterations strategy with the largest preference (c) EG3 Figure 5: Evolutions of strategies with the largest pref erence and ev olution arily stable strategies. 6 Conclusion A m odel b ased up on g ames on an evolving network has been established , which m ay be u sed to explain the form ation of gro ups in so ciety par tly . Following this m odel, three c lustering alg orithms ( EG1, EG2 and EG3) using thr ee d i ff erent ERR function s 14 are constructed, in which each data point in a dataset is regarded as a player in a game. When a distance fu nction is selected, the initial network is created among data points accordin g to Definition 1. T hen, by applying an ERR function, the network will ev olve over time d ue to edges removed and rewired. Hen ce, the prefe rence set of a player needs to be a djusted in ter ms of D efinition 5, and pay o ff s of p layers are rec omputed too. Durin g the network ev olving, certain strategies are spread in the network . Finally , the ev olutio narily stable strategies emerge in the network. Accordin g to evolutionarily stable strategies played by players, those data points with the same e volutionarily stable strategies are collected as a cluster . As such, the c lustering results are ob tained, where the number of ev olution arily stable strategies correspon ds to the numbe r of clusters. The ERR functions ( B 1 i ( · ) , B 2 i ( · ) , B 3 i ( · )) em ployed in three clustering algorithm s pro- vide di ff erent capacities of e xplo ration for these clustering algorithms, i .e., the sizes of areas which they can observe are v arious. So, the clustering results of three algorithm s are di ff e rent. For the ERR function B 1 i ( · ), it is with a c onstant ratio of explor ation η = 0 . 5 b ecause over exploration may occu r when η > 0 . 5, and can observe an ex- tend neig hbor set fo rmed by half o f neig hbors. Th e ERR f unction B 2 i ( · ), however , can observe an extend neighbo r set f ormed by neighbo rs wh ose payo ff s are larger than the av erag e, while the ERR fu nction B 3 i ( · ) provides stronger capa city of exploration for data points with small payo ff s. Besides, th e clustering results of Algor ithm EG1 and EG2 ar e m ore stable than tha t of Algorithm EG3, b ut the best results are achieved by Algorithm EG3 due to the strongest capacity of exploration among three algorithms. In the case when th e exact numb er of clusters is unknown in advance, one can adjust the number k of nearest neighbors to con trol the nu mber of clusters, where the number of clusters decreases in versely with the numb er k of n earest neigh bors. W e ev aluate the clustering alg orithms on five real datasets, experim ental results ha ve demo nstrated that data poin ts in a dataset are clustered reaso nably and e ffi cien tly , and the r ates of conv ergence of three algo rithms are fas t eno ugh. Refer ences [1] J. 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