Relational Data Partitioning using EvolutionaryGame Theory

Lawrence O. Hall, Fellow, IEEE, Alireza ChakeriDepartment of Computer Science and Engineering

University of South FloridaTampa, Florida

Email: hall@cse.usf.edu, chakeri@mail.usf.edu

Abstract—This paper presents a new approach for relationaldata partitioning using the notion of dominant sets. A dominantset is a subset of data points satisfying the constraints of internalhomogeneity and external in-homogeneity, i.e. a cluster. However,since any subset of a dominant set cannot be a dominant set itself,dominant sets tend to be compact sets. Hence, in this paper, wepresent a novel approach to enumerate well distributed clusterswhere the number of clusters need not be known. When thenumber of clusters is known, in order to search the solution spaceappropriately, after finding each dominant set, data points arepartitioned into two disjoint subsets of data points using spectralgraph image segmentation methods to enumerate the other welldistributed dominant sets. For the latter case, we introduce a newhierarchical approach for relational data partitioning using a newclass of evolutionary game theory dynamics called InImDynamicswhich is very fast and linear, in computational time, with thenumber of data points. In this regard, at each level of theproposed hierarchy, Dunn’s index is used to find the appropriatenumber of clusters. Then the objects are partitioned based on theprojected number of clusters using game theoretic relations. Thesame method is applied to each partition to extract its underlyingstructure. Although the resulting clusters exist in their equivalentpartitions, they may not be clusters of the entire data. Hence, theyare checked for being an actual cluster and if they are not, theyare extended to an existing cluster of the data. The approach canalso be used to assign unseen data to existing clusters, as well.

I. INTRODUCTION

Clustering is an important unsupervised learning approachand widely used in pattern recognition, data mining and otherrelated fields. Consider a set of n objects O = {o1, o2, ..., on}.There are two possible types of data, numerical data andrelational data. While in numerical data, objects are repre-sented by their numeric features, in relational data only simi-larity (dissimilarity) between each pair of objects is available.Specifically, a numerical object data oi containing p features(dimensions) is represented by a vector {xi1, xi2, ..., xip} ⊂<p, where each xij provides a feature value. On the otherhand, relational data consists n2 similarities (dissimilarities)between pairs of objects in O. The relational data are usuallyrepresented by an n × n similarity matrix A = (aij) (ordissimilarity matrix D = (dij)), such that aij (dij) is thesimilarity (dissimilarity) between object i and object j. Onecan convert an numerical data X into a dissimilarity matrixD using a distance function. For instance, dij can be the Eu-clidean distance between objects i and j, i.e. dij = ||xi−xj ||.Hence, a relational clustering algorithm can be applied to bothtypes of data. However, there are relational data sets that do

not come from a numerical object data. Consequently, one canonly use relational data clustering methods in these situations.

A classical approach to relational data clustering is to usethe concepts and approaches from graph theory [1]. In thisregard, the data points are mapped to the vertices of a weightededge graph such that the edge weights represent the similaritiesbetween objects of the corresponding nodes. Hence, theseapproaches map clustering problems to graph theoretic onesin which powerful algorithms have been developed includingminimum spanning tree [2], and minimum cut [3], [4]. Re-cently in [5], the notion of a dominant set was introducedas a definition of a cluster arising from the generalizationof maximal cliques in weighted edge graphs. The underlyingassumption is that a cluster can be seen as a group of objectshaving more similarity to one another than objects not in thecluster. The two most important features of this approach arethat it provides a measure of a cluster cohesiveness as well asa degree of belonging of every vertex to each cluster.

A dominant set is a subset of data points satisfying the con-straint of internal homogeneity and external in-homogeneity.In [5], a very significant relation between dominant sets andthe standard quadratic optimization problem was established.In particular, it was proved that dominant sets are strict localsolutions of the corresponding quadratic problem. In addition,it is well known from evolutionary game theory that strict localsolutions are in one-to-one correspondence with evolutionarystable strategies (ESSs) [6] of the game that has the similaritymatrix of the data points as its payoff. In this regard, replicatordynamics, a class of evolutionary game-theoretic algorithmsinspired by Darwinian selection processes [6], [7] providea very efficient tool to find dominant sets. However, sinceevolutionary dynamics will find one equilibrium depending onthe initial state, an enumeration method of dominant sets isneeded. A simple approach is to use a peeling-off strategy [5],where a dominant set is detected, and then all of its verticesis removed from the graph. Repeating this procedure in theremaining data, we obtain a partitioning of the data whereeach dominant set corresponds to a cluster. However, since thegraph is changing during every step of this method, one caneasily argue that the resulted dominant sets may not exist in theoriginal graph. To overcome the problem, in [8] and [9], twomethods were introduced to enumerate the equilibriums of anysymmetric game. In [8], after extraction of ESS, the graph ischanged is such a way that the located ESS becomes unstableunder the dynamics without affecting the remaining ESSs. In[9], a tabu-set method was developed using InImDynamics978-1-4799-4518-4/14/$31.00 c© 2014 IEEE

DOI 10.1109/CIDM.2014.7008656

[10]–[12] in order to heuristically search the solution space.

On the other hand, dominant sets tend to be a compact setof vertices. This is because any subset of vertices of a dominantset cannot be a dominant set itself. In other words, based onthe evolutionary game theory terminology, any subset of nonzero elements of an ESS can not be even a Nash equilibrium(and consequently be an ESS). This property may cause a (big)cluster to be subdivided into several dominant sets. Hence, wepropose an approach to efficiently search the solution space inorder to find the desired number of well distributed dominantsets. In this regard, the vector containing the payoffs of everyvertex playing against the equilibrium is partitioned into twosubsets and the partition that does not contain the equilibriumvertices is used for detecting the other dominant sets. Also,when the number of clusters is not provided, we propose anefficient approach to extract the appropriate number of clustersusing Dunn’s index and then partition the objects based on theclusters obtained using game theory relations. Then the samemethod is applied to each of the partitions obtained. Althoughthe clusters in each partition are clusters of their equivalentsub graph, they may not be clusters of the entire graph. Hencethey are checked for being a cluster of the entire graph and ifthey are not, they extended to be a cluster of the entire graphusing InImDyn.

II. PRELIMINARIES

This section contains some background on dominant setsand evolutionary game theory, which is needed to develop thepresented approach.

A. Dominant Sets

A classical approach to relational data clustering is to mapthe objects to the vertices of a weighted edge graph suchthat edge weights are the similarities between objects of thecorresponding nodes. Let’s consider an undirected weightededge graph G = (V,A) , where V = {v1, . . . , vn} is the setof vertices and A = (aij) is the similarity matrix in which aijis the edge weight between vertices vi and vj . Recently in [5],the notion of a dominant set was introduced as a definition of acluster. In [5], it was proved that in an unweighted edge graph,dominant sets are equal to strictly maximal cliques. In thissubsection, the fundamentals of the dominant set frameworkare presented briefly.

Let S ⊆ V be a subset of vertices. The average weighteddegree of vi ∈ V with regard to S is defined as

awdegS (vi) =1

|S|∑vj∈S

aij (1)

Also, if vj /∈ S, then a measure which reflects the relativesimilarity between nodes vj and vi with regard to the averagesimilarity between node vi and its neighbors in S is definedas

φS (vi, vj) = aij − awdegS(vi) (2)

As a result, the weight of vi ∈ S with regard to S is defined

as

wS(vi) =

1 if |S| = 1∑vj∈S\{vi}

φS\{vi} (vj , vi) wS\{i}(vj) otherwise

(3)

while the total weight of S is defined as follows

W (S) =∑vi∈S

wS(vi) (4)

Consequently, in [5], the following definition of the domi-nant set representing the notion of a cluster in a weighted edgegraph was presented.

Definition 1: A non empty subset of vertices S ⊆ Vsuch that W (T ) > 0 for any non-empty subset T ⊆ S, isa dominant set if it has the two following conditions:

1) wS (vi) > 0, for all vi ∈ S.2) wS∪{vi} (vi) < 0, for all vi /∈ S.

It can be seen that conditions 1 and 2 correspond to thetwo main characteristics of a cluster, i.e. internal homogeneityand external in-homogeneity, respectively.

Now consider the following quadratic program

maxx

f (x) = xTA x

subject to x ∈ ∆(5)

where ∆ = {x ∈ Rn : x ≥ 0,∑xi = 1} is the

standard simplex, and A is an n × n symmetric similaritymatrix of graph G. Pavan et al [5] established a one toone correspondence between dominant sets and strict localmaximizers of a quadratic form over the standard simplex.Particularly, if S ⊆ V is a dominant set, then its characteristicvector xS , which is defined as

xSi =

{wS(vi)W (S) if vi ∈ S

0 otherwise(6)

is a strict local solution of program (5) with A as beingsimilarity matrix of the weighted edge graph G. Conversely, ifx is a strict local solution of program (5), then its support S =σ(x), where σ (x) = {i ∈ {1, . . . , n} |xi > 0} , will be adominant set.

B. Game Theoretic Viewpoint of Dominant Sets

A novel connection between finding dominant sets usingquadratic programming and evolutionary game theory wasestablished in [5], [13]–[16]. In this subsection, the basicdefinition of evolutionary game theory is briefly reviewed.

A strategic game consists of a set of players, their strate-gies, and payoffs available for all combinations of players’strategies. In evolutionary game theory, the players are re-peatedly drawn from a random infinite population to play thegame. Let us consider a large population of individuals whoare randomly matched to play a symmetric two-player game(doubly symmetric game) G = (S,A). Each player playsa certain strategy from strategy set S = {1, 2, . . . , n} withA = (aij) the n× n symmetric payoff matrix, which assigns

a payoff aij to a player playing the strategy i against a playerplaying strategy j.

The state of the population can be represented by an n-dimensional vector x = (x1, x2, . . . , xn), where xi corre-sponds to the fraction of individuals in the population playingstrategy i ∈ S, such that Σn

i=1xi = 1. This set is formallyidentical to the set of mixed strategies, and consequently itbelongs to ∆. Hence, the expected payoff received for a playerusing mixed strategy y against an opponent playing mixedstrategy x is u(y, x) = yTAx.

The best response for a player is a choice that maxi-mizes the payoff of that player given the actions taken bythe other players. Consequently, the best responses againsta mixed strategy x are the set of mixed strategies β (x) ={y ∈ ∆ |u(y, x) = maxzu(z, x)}. A mixed strategy x isa symmetric Nash equilibrium if and only if it is the bestresponse to itself, i.e. u(y, x) ≤ u(x, x), ∀y ∈ ∆. This impliesthat for all i ∈ σ(x), where σ(x) = {i ∈ S|xi > 0} is calledthe support of mixed strategy x, u(ei, x) = u(x, x). In otherwords, the payoff for every strategy in the support of x is thesame, while all strategies outside the support of x earn a payoffless than or equal to u(x, x) which is called the equilibriumpayoff. It easy to see that all KKT points of the problem (5)with matrix A are the same as the Nash equilibriums of thecorresponding game, and vice versa.

Now assume that the whole population plays mixed strat-egy x and a small group of mutants enter this population andadopt a different mixed strategy y. A mixed strategy x is calledan Evolutionary Stable Strategy (ESS) if and only if:

1) It is a Nash equilibrium, i.e. u(y, x) ≤ u(x, x),∀y ∈∆ (equilibrium condition).

2) If u(y, x) = u(x, x) , then u(y, y) < u(x, y) (stabilitycondition).

Now, in [13], a relation between ESS of two player gameswith payoff matrix A and the quadratic program (5) wasestablished, such that x is the ESS of a two player game withpayoff matrix A if and only if it is a strict local solution ofproblem (5). As a result, based on section II.A, one can findthe ESS of a game with payoff matrix A to find the dominantsets of a graph with similarity matrix A.

On the other hand, replicator dynamics, a class of evo-lutionary game theory dynamics, provide us a very efficienttool to find the strict local solutions. It was proved that in [6],the asymptotically stable points of the replicator dynamics arein one-to-one correspondence of the strict local maximizer ofproblem (5). In other words, x ∈ ∆ is asymptotically stableunder replicator dynamics if and only if x ∈ ∆ is a strictlocal maximizer of problem (5), and if and only if x ∈ ∆ isan ESS for the equivalent symmetric game [6]. Specifically,it was proved that replicator dynamics converge to a Nashequilibrium if the initial population state starts from the interiorof ∆ represented by int(∆) = {x ∈ ∆|xi > 0,∀i}.

However, in this paper, we use a new class of dynamiccalled InImDyn (Infection and Immunization Dynamics) whichcan overcome some limitations of the replicator dynamics [11],[12]. Every fixed point of this dynamic is a Nash equilibriumand vice versa. Specifically, any trajectory starting in ∆ con-verges to a Nash equilibrium; in other words, the limits points

of this dynamic are Nash equilibriums (also, there is a oneto one correspondence between ESSs and the asymptoticallystable points of InImDyn). In addition, from a computationalviewpoint, every iteration in InImDyn is linear in the numberof strategies enumerated compared to the quadratic one inreplicator dynamics [10].

III. EFFICIENT DOMINANT SET PARTITIONING FORRELATIONAL DATA

To use the dominant set clustering method when numericalobject data X is provided, one converts the obtained dissim-ilarity matrix D to a similarity matrix A. As a result, it iscommon to use the following transformation

aij = exp(−d2ij/σ2) (7)

where σ is a positive real number which effects the rate ofdecrease rate of the similarity values. Although we expect aclustering method to be invariant to the changes in similaritymatrices caused by different σ’s and yield similar clusteringresults, but the dominant set clustering is not robust. Theresults of the clustering vary widely with different σ’s.

In clustering problems, two possible cases happen regard-ing the number of clusters in a data set. The first is when thenumber of clusters is known in advance (we call it ”knownnumber of clusters”) which is more common in numericaldata than relational data. The second situation is when weneed to determine the appropriate number of clusters (we callit ”unknown number of clusters situation”). In this paper, wetackle both situations using evolutionary game theory relations.

Now, to lay the theoretical groundwork for our algorithm,we further analyze the properties of the Nash equilibriumsand InImDyn. Let’s denote GB as the subgraph of G inducedby the set of vertices in B ⊂ S and x as the expansion ofmixed strategy x of a subgraph GB to G by adding zeroentries to x for vertices that do not belong in GB . If x isa Nash equilibrium of the subgraph GB , it is easy to judgewhether x is a Nash equilibrium of G [17]. In this regard, ifall other vertices outside GB playing against x earn payoffsnot larger than equilibrium payoff u(x, x), then x is a Nashequilibrium of the entire graph G. Hence, if there is a vertexi such that it earns a higher payoff playing against x thanequilibrium payoff u(x, x), x is no longer a Nash equilibriumof the whole graph G. We use the above analysis to develop ourproposed algorithm. Specifically, after finding each equilibriumof a subgraph, the algorithm checks whether the obtainedequilibrium is an equilibrium of the whole graph. If it is not,the obtained equilibria of the subgraph is used as the initialpoint for the InImDyn to run on the whole graph. Now, theimportant feature of InImDyn, as stated in subsection II.B,that makes it appropriate for our approach is that there isa one to one correspondence between all of its fixed pointsand Nash equilibriums. In other words, in spite of replicatordynamics where the trajectory needs to be started in the interiorof the simplex to converge to a Nash equilibrium, the InImDynprovides convergence to a Nash equilibria starting from a pointin the simplex.

A. Known number of clusters

For the situation in which the number of clusters c isprovided, using the notion of dominant sets is limited to a

data set that has a number of equilibriums greater than orequal the provided number of clusters. However, this limitationis generally satisfied for most data sets (This is because ofthe fact that recently in [18], it was proved that in randomtwo player games in which each player has the same numberof strategies |S|, the mean number of Nash equilibriums isexp(|S| [B + O( log|S|

|S| )]) where B ≈ 0.281644). Hence, anaive approach is to enumerate dominant sets and consider thefirst c dominant sets created. Particularly, after finding eachdominant set, the peeling-off strategy is employed with thedifference that if the obtained equilibrium of the subgraph isnot the equilibrium of the whole graph, it is used as the initialpoint for the InImDyn to run on the whole graph [9]. But, aswe show in the experimental section, the obtained dominantsets might not be well distributed and they may belong tothe same clusters. Hence, we propose an approach to extractthe informative dominant sets and then efficiently assign theremaining objects to them. Here, the informative dominant setsmean highly distributed ones, i.e. their supports are mutuallydisjoint sets if possible.

In order to find desired number of well distributed equi-libriums, we developed a method to search the solution spaceappropriately. In this regard, after finding each equilibrium, thevector containing payoffs for each vertex playing against theequilibrium strategy is computed and then it is discretized inorder to remove the vertices close to the equilibrium vertices.In fact, we are doing this in the hope of removing the verticesin the basin of attraction of the obtained equilibrium. One canuse several discretization criteria that are used in spectral graphmethods for image segmentation [4], [19] such as median cut,jump cut and ratio cut [20]. In this paper, we use the jumpcut method. For jump cut, we sort the payoff vector entries indecending order and find the largest difference between twoentries in the sequence. The entries before these two entries,which have high payoffs, are grouped as close vertices to thevertices in the equilibrium. It is easy to see that vertices in theequilibrium should be included in this group since they havethe highest payoff value. By doing this, close vertices to theones in the cluster as well as the vertices in the cluster willnot be considered to find the next equilibrium.

The proposed approach is summarized in Algorithm 1. Ittakes a graph G = (S,A) (S is the set of vertices and A isthe similarity matrix) as input and returns a set N containingdesired number c clusters of G and two sets of vertices thatare explained below. The enumeration algorithm uses InImDyn(function InImDyn in line 6) that is initialized to the slightlyperturbed barycenter of ∆S\T (function Barycenter in line5). Also, a tabu-set T is used to explore the solution spacemore efficiently. After finding each equilibrium, its support andthose comparable vertices obtained by the jump cut method areadded to T (function JumpCut in line 11), and then InImDynrun on the remaining objects, i.e on the subgame obtainedby strategies S \ T . If the resulting equilibrium is not anequilibrium of the entire game, it is used as the initial point forInImDyn to run again on the whole graph G. This procedurerepeats until c clusters are enumerated or T becomes equal toS. In the latter case, one can use a more strict condition forconstructing T such as selecting the vertices whose payoffs arebigger than the average plus the variance of all vertices playingagainst the equilibrium. The algorithm also gives us two more

sets. Let’s assume that there are c clusters represented by theset of equilibriums N = {n1, n2, ..., nc}. This set partitionsthe vertices into two subsets Z = {j ∈ S|j ∈ σ(ni),∃i} andZ = {j ∈ S|j /∈ σ(ni),∀i} that the set of objects that belongto the support of at least one equilibrium and do not belongto any equilibrium, respectively.

Algorithm 1. Equilibrium Enumeration

1) Input: Graph G = (S,A), and c as the number of clusters.2) Output: N a set containing at most c Nash equilibriums

of G, Z and Z.3) Initialize N = ∅, T = ∅ and counter = 14) while counter ≤ c or T 6= S5) b = Barycenter(GS\T )6) x = InImDyn(GS\T , b)7) if there exists i ∈ S such that u(ei, x) > u(x, x)8) y = InImDyn(GS , x)9) else

10) y = x11) T = T ∪ JumpCut(y)12) Add y to N , and counter = counter + 113) Partition the vertices into Z = {j ∈ S|j ∈ σ(ni),∃i} and

Z = {j ∈ S|j /∈ σ(ni),∀i} induced by N

Based on Algorithm 1, every mixed strategy Nash equilib-rium x provides a cluster in which its support represents theobjects belong to that cluster. In addition, the correspondingequilibrium payoff u(x, x) shows the cohesiveness (compact-ness) of the cluster. Now, consider the mapping C : S →{1, ..., c} that assigns each object a class label. To partitionthe data, we propose the following relation to assign object ito the appropriate partition

C(i) =

argmax

j

u(ei,nj)u(nj ,nj)

if i ∈ Z

argmaxj

nj(i) if i ∈ Z(8)

where C(i) represents the partition assigned to object i. Theintuition behind this relation is that every object i in Z isassigned to the partition in which it gains higher relativesimilarity with regard to the cluster cohesiveness. In otherwords, although it is necessary to gain high payoff playingagainst the equilibrium strategy, but it also depends on theequilibrium payoff. Also, every object i in Z belongs tothe partition that has highest degree of participation. Thepartitioning approach is summarized in Algorithm 2.

Algorithm 2. Dominant Sets Partitioning

1) Input: Graph G = (S,A), the set of clusters N ={n1, ..., nc}, Z and Z

2) Initialize the vector C of size n3) for each i ∈ {1, ..., n}4) if i ∈ Z5) C(i) = argmaxj nj(i);6) else if i ∈ Z7) C(i) = argmaxj

u(ei,nj)

u(nj ,nj);

8) Output: The vector C representing the partition number ofeach object

B. Unknown number of clusters

There are several approaches to determine the appropriatenumber of clusters in a data set [21]. In this paper, Dunn’s

(a) (b)

(c) (d)

Fig. 1: (a) The similarity matrix of three Gaussian clouds data set, sorted by the class, represented by an image where each pixelshows the scaled similarity value between two objects; (b) Three dominant sets, represented by ’×’, ’o’ and ’+’, were extractednaively; (c) Three dominant sets, represented by ’×’, ’o’ and ’+’, were extracted based on Algorithm 1; (d) Partitioning datapoints into the obtained dominant sets in (c) using Algorithm 2.

(a) (b) (c)

Fig. 2: (a) Sorted payoffs against the equilibrium belonging to the largest cluster (450 examples); (b) Sorted payoffs against theequilibrium belonging to the second largest cluster (100 examples); (c) Sorted payoffs against the equilibrium belonging to thesmallest cluster (50 examples).

index is used to identify sets of clusters that are compact,with a small variance between members of the clusters, andwell separated as compared to the within cluster variance.We develop a hierarchical clustering method that employsalgorithms similar to Algorithm 1 and 2 to enumerate theequilibriums of the graph using Dunn’s index.

First, Dunn’s index is used to determine the appropriatenumber of clusters in the first level of the hierarchy usingAlgorithm 1, and data is partitioned using Algorithm 2. Then,each partition is considered as an independent data set andthe same routine is applied, with the difference that clustersin each partition are extended to be clusters of G (if theyare not already a clusters of G). This hierarchical equilibriumenumeration continues until the desired level of hierarchies is

reached.

The following summerizes the proposed approach in threesteps. To simplify, we assume that the minimum (min) andmaximum (max) number of clusters to use Dunn’s index isthe same for all levels. The min and max values are definedapriori by the user.

1) Apply the Algorithm 1 for all number of clustersbetween min and max.

2) Find the best number of cluster using Dunn’s index.3) Apply Algorithm 2 and go to step 4 if the number of

levels is not reached, otherwise terminate.4) Do step 1 for each partition in step 3.

The first evidence which is congruent with our experimen-

(a) (b)

(c) (d)

Fig. 3: (a) Three dominant sets, represented by ’×’, ’o’ and ’+’, were extracted based on Algorithm 1; (b) Partitioning datapoints to the obtained dominant sets in (a) using Algorithm 2; (c) Partitioning the data points to the two equilibriums in thelevel 1 hierarchy using ’×’, ’o’ and ’+’; (d) Partitioning the data points to the four equilibriums in the level 2 hierarchy using’×’, ’o’, ’∗’ and ’+’.

tal results is that the clusters obtained in a level generally willbe obtained in the following levels. This is because of thefact that equilibriums obtained by InImDyn are asymptoticallystable points. Also, the clusters in each partition are usuallyobtained in decreasing order of size which can be an advantageof our method. In addition, for the first levels, equilibriumsobtained in each subgraph are usually the equilibriums of theentire graph, hence there is no need to extend them to anequilibrium of the whole graph.

IV. EXPERIMENTAL RESULTS

In order to test the effectiveness of the proposed algorithm,we present a number of examples that illustrate its variousfacets in both numerical and relational data sets. All thealgorithms were written in m code for Matlab and run on amachine equipped with 2 Intel 2 GHz CPUs and 8 GB RAM.For numerical data sets, we set σ in (7) equal to the mean ofthe similarity values.

A. Three Gaussian Clouds

This example contains 600 object vectors X ⊂ <2. Thedata were drawn from three Gaussian distributions containing450, 100 and 50 examples. Figure 1(a) depicts the simi-larity matrix obtained by (7) as an image. Each pixel ofthe image displays scaled similarity value of two objectsbetween [0, 255]. White pixels represent high similarity, whereblack pixels represent low similarity. The cluster tendency isshown by the number of white blocks along the diagonalof the image. Hence, the image shows that there are three

separable clusters represented by high similarities around themain diagonal. Figure 1(b) shows the situation where the firstthree equilibriums are considered as the desired three clusters.It can be easily seen that they are not distributed and they allbelong to the same cluster. However, the result of the proposedapproach in Algorithm 1 is shown in Figure 1(c). In this case,the obtained three equilibriums are well dispersed among thewhole data, and each of them represents a true visualizedcluster. As one can see there are some data points which donot belong to any clusters. Hence, Figure 1(d) shows the resultof applying the Algorithm 2 to completely partition the data.

The vectors containing the payoffs of every vertex playingagainst the three equilibriums are sorted and plotted in Figure2(a-c). In all three figures, vertices in the equilibriums havethe maximum payoffs, and as a cluster gets smaller thecorresponding maximum value gets smaller. Moreover, thereexists a significant difference in payoffs of some of the verticescomparing to the maximum payoff, that makes the jump cutan appropriate method for discretization purposes.

B. Iris DataSet

The Iris plant data set consists of 50 samples from eachof three species of Iris, i.e., Iris setosa, Iris virginica and Irisversicolor. Four numeric attributes (the length and the widthof the sepals and petals) were measured from each plant.Visualization of the Iris data set indicates that one class iscompletely separable and the other two classes intersect toa degree. Figure 3(a) shows the results of enumerating threeequilibriums using Algorithm 1. Also, Figure 3(b) depicts

(a) (b) (c)

(d) (e)

Fig. 4: (a) The image of the similarity matrix sorted by the class where light color is higher similarity and dark color is lowersimilarity; (b) Two clusters obtained in the first level represented by red solid lines; (c) Two partitions obtained from the first levelrepresented by red dashed lines; (d) Four clusters obtained from the second level represented by red solid lines; (e) Partitioningthe data using the four clusters obtained, where data points are sorted by decreasing size of the clusters in the level to whichthey belong.

assigning clusters to the remaining data points. It is noted thatthe separable class (shown with symbol ”+”) is well separatedon the lower-left corner and the two overlapped classes (”o”and ”×” symbols) are also separated effectively in the upper-right corner.

Moreover, we also applied the unknown number of clustersapproach with 2 levels of hierarchies on this data set. For thefirst level, the Dunn’s index suggests two equilibriums. Theresult is shown in Figure 3(c) representing the two obtainedpartitions. In the second level, Dunn’s index suggests twoequilibriums in each partition, where all of them are eventuallythe equilibriums of the whole data set and hence there isno need to extend them. Figure 3(d) depicts the new fourpartitions. We see here that setosa is split into two clusterswhich does not provide us information as the debate with thisdata is whether there are two or three classes given the features.

C. Bioinformatics Data

The last example is the real world GPD194 [22], [23]relational data set. This data set is not numerical. Rather,dissimilarity relation values were obtained by a fuzzy measureapplied to annotations of 194 human gene products whichappear in the Gene Ontology. The data consists of 21, 87and 86 gene products from Myotubularin, Receptor Precursorand Collagen Alpha Chain protein families, respectively. The

dissimilarity matrix D is converted to the similarity matrix Aas follows

aij = 1− dijmax (D)

(9)

The three protein families are visible in Figure 4(a); theyhave been sorted so that the upper left block is Myotubularin,the middle block is Receptor Precursor, and the lower rightblock is Collagen Alpha Chain. The hierarchical partitioningapproach is applied on this data set. The highest Dunn’s indexvalue in level 1 gives 2 clusters which are shown in Figure 4(b).The partitions induced by the obtained two clusters are shownin Figure 4(c) where the data points are sorted according tothe cluster that they belong to. The results of the second levelof hierarchy are shown in Figures 4(d) and (e). Using Dunn’sindex, each partition contains two clusters which are shown inFigure 4(d). The clusters were obtained in the decreasing orderof size in each partition. In Figure 4(e), for each partition, datapoints are sorted by the decreasing order of size of the clustersto which they belong. We can see that the obtained clusters arewell distributed and represent the whole structure of the data.Also, the clusters in first level are the clusters of the secondlevel as we discussed before.

V. CONCLUSIONS

It is well known that, since dominant sets tend to becompact, there can be many dominant sets in a graph. Also,there can be some data points that are not in the support of anydominant sets. Also, the definition of a dominant set is not arobust one and, consequently, the similarity matrix relies heav-ily on the parameter σ. To overcome these problems, in thispaper an equilibrium enumeration algorithm was introducedusing InImDyn to extract well distributed dominant sets withthe number of clusters either known or unknown. Furthermore,based on game theory relations, a simple and efficient methodwas proposed to put unassigned objects into the appropriateclusters.

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