Replicator Equation: Applications Revisited
aa r X i v : . [ c s . C V ] M a y Replicator Equation: Applications Revisited
Tinsae G.Dulecha
DAIS, Universit`a Ca’ Foscari di Veneziavia Torino 155, 30172 Venezia Mestre, Italy [email protected]
The replicator equation is a simple model of evolution that leads to stable form ofNash Equilibrium, Evolutionary Stable Strategy (ESS). No individuals get an incentiveunilaterally deviating from the equilibrium. It has been studied in connection with Evo-lutionary Game Theory, a theory John Maynard Smith and George R. Price developedto predict biological reproductive success of populations. Replicator equation was orig-inally developed for symmetric games, games whose payoff matrix is skew-symmetricand in general tells us, in a game, how individuals or populations change their strat-egy over time based on the payoff matrix of the game. Consider a large populationof players where each player is assigned a strategy and players cannot choose theirstrategies which means rationality and consciousness don’t enter the picture (playersplay based on a pre-assigned set of strategies). Evolutionary game theory assumes ascenario where a non-rational pairs of players, which play based on a pre-assigned setof strategies, repeatedly drawn from a large population plays a symmetric two-playergame which drives the strategies with lower payoff to extinction . Since players inter-acts with another randomly chosen player in the population, a players expected payoffis determined by the payoff matrix and the proportion of each strategy in the popula-tion. The limiting behavior of the replicator dynamics (i.e., the evolutionary outcome)are the Evolutionary Stable Strategies, a NE with additional stability properties.Let x i ( t ) is the proportion of the population which plays strategy i ∈ N (set ofstrategies) at time t . The state of the population at any given instant is then given by x i ( t ) = [ x , x , ..., x n ] ′ where ′ denotes transposition and n refers the size of availablepure strategies, | N | . With A be the n × n payoff matrix ( biologically measured asDarwinian fitness i.e reproductive success ), the payoff for the i th -strategist, assumingthe opponent is playing the j th strategy, is ( a ij ) , the corresponding i th row and the j th column of A . If the population is in state x, the expected payoff earned by an the i th -strategist is ( A x ) i while the mean payoff over the whole population is x ′ A x . Thegrowth rate of the population is then computed as the product of the current frequencyof strategy with the own fitness relative to the average (the difference in the expectedand the average payoff gives us the per capita rate). The game, which is assumed to beplayed over and over, generation after generation, changes the state of the populationover time until equilibrium is reached, a point x is said to be a stationary (or equilib-rium) point of the dynamical system if ˙ x = 0 where the dot implies derivative. Differentformalization of this selection process have been proposed in evolutionary game theorywhere replicator dynamics is one of the well knowns.Beyond its first emphasis in biological use, evolutionary game theory has been ex-panded well beyond in social studies for behavioral analysis, in machine learning, com-puter vision and others. Its application in the fields of computer science has drawn myattention which is the reason to write this extended abstract.orsello et al. has shown that dominant sets, a well known clustering notion whichgeneralizes the maximal clique problem to edge weighted graphs, can be bijectivelyrelated to Evolutionary Stable Strategies (ESS) [13]. Since the formalization of cluster-ing problem as a game, replicator dynamics has been used to address different problems[13]. Players simultaneously select an object to cluster and receive a payoff proportionalto the similarity of the selected objects. Since clusters are sets of objects that are highlysimilar, the competition induced by the game forces the players to coordinate by select-ing objects from the same cluster. Indeed, by doing so, both players are able to max-imize their expected payoff. The clustering game is a non-cooperative game betweentwo-players, where the objects to cluster form the set of strategies, while the affinitymatrix provides the players payoffs. The dynamics is also able to solve multiple affini-ties by extending the clustering game, which has a single payoff, to a multi-payoff game[10]. Since the introduction of the connection, replicator equation helps dominant setformulation be a robust clustering approach with many interesting and powerful proper-ties such as: it makes no assumption on the underlying data representation, it makes noassumption on the structure of the affinity matrix, being it able to work with asymmet-ric and even negative similarity functions alike; it does not require a priori knowledgeon the number of clusters; able to solve one-class clustering problems (figure groundseparation); it allows extracting overlapping clusters [14]; it generalizes naturally to hy-pergraph clustering problems, in which the clustering game is played by more than twoplayers [11]. The dominant sets formulation, using replicator equation, have proven tobe an effective tool for graph-based clustering and have found applications in a varietyof different domains, such as bioinformatics [5], computer vision [15], image process-ing [18],[2], group detection [8][17][16], security and video surveillance [7], [1], [12],[6] etc. Very recently, Eyasu et al. , in their two consecutive works [10, 21], general-ized the dominant set formulation. The generalized version, using replicator dynamicsas a solver, can be applied in different fields of studies such as in computer vision,biomedical analysis, human behavior and social network analysis and others.The notion of clustering using replicator equation has also be used in [4] where thecore idea is to combine an effective diffusion process, based on iteratively approach-ing evolutionary stable strategies. Similarly, Xingwei et al. used it for dense neighborselection for affinity learning with diffusion on tensor product graph. Few of the mostefficient and most recent computer vision applications that uses replicator equationsinclude: Retrieval[19] where replicator is used to different size dense neighbors whichimproves the quality of affinity propagation; tracking [12] which uses replicator as asolver; interactive image segmentation where the segmentation is guided by the userprovided information [21]; large scale image geo-localization [22]Morteza showed, in his adaptive trajectory analysis of replicator dynamics, howmuch effective replicator dynamics is for data clustering and structure identification [3].Extraction of dense subgraphs using game dynamics is done in [9], and in [20] it hasbeen shown that replicator have no difficulty in extracting a few natural clusters fromheavy background noise. Moreover, Eyasu et al. proposed effective replicator basedalgorithm for simultaneous clustering and outlier detections [23]. eferenceseferences