on the design of utility functions in community formation games

On the Design of Utility Functions in Community Formation Game Xiaohu Zhu∗† , Jun Wu‡ , Chongjun Wang∗† , Junyuan Xie∗† Key Laboratory for Novel Software Technology(Nanjing University) China † Department of Computer Science and Technology, Nanjing University, China ‡ School of Computer and Information Science, Hohai University, China wendyneil.zhu@gmail.com,{iip, chjwang, jyxie}@nju.edu.cn ∗ State Abstract—To understand the overlapping community structure in complex networks, a great deal of methods have been proposed by researchers from different areas. Community formation game is a game-theoretical view for this problem. In this article, we develop an new concept named social distance to describe a natural phenomenon in real world reflecting the strength of connections between different individuals. Some improvements have been given to acquire better performance of detecting the overlapping community structure. We illustrate them by conducting a series of experiments both on real-world and benchmark networks. Keywords-game theory; overlapping community structure; utility function I. I NTRODUCTION Individuals in various complex networks formed relatively intense communities with different ways and structures. Palla et al. studied the statistical properties of community structure in the large networks [1]. They pointed out that community was universal and evolving dynamically. In many research areas, similar property emerged. For example, the protein networks in biology research, the propagation of epidemic in medicine research, and the design of large scale computer networks in computer science were tightly related with community structures. Researchers from different areas were interested in this kind of property and the research for finding community structures in complex networks lasted for years. Nowadays scientists continued the formers’ work, and introduced thoughts and tools from mathematics, physics, computer science, which pushed detecting community structure to one of the most hot directions in the research of complex networks. As research on community detection problems went further, some new problems emerged, especially the overlap of communities. In real world, the overlapping community structure actually was a typical characteristic of various networks. We were surrounded by social networks, such as family, working and classmates networks. In research circle, one could be in different circles if he or she was doing interdisciplinary research. And in the online social networks, people also could choose many communities to enter in. All these examples told us that the universality of overlapping community structure in natural world and human world. Recently, a bunch of solutions for overlapping community detection problem have been proposed, such as clique percolation algorithm [2]–[4], line graph transformation [5]– [8], local expansion [9] [10], model based algorithm and game theoretic strategy [11] [12]. And measurement for the evaluation of the performance of different algorithms have also been proposed. Both could equip the toolbox of human being searching for the truth of real world. Then we could use these methods to simulate real networks. In the process of modeling the true complex systems, we use the information of simulating to build more proper systems. However, there exist many unsatisfiable aspects in this area. Previous works often lost the precision of the description of the problems. In the work of Chen et al., they designed a neat game theoretical framework for overlapping detection problem [11]. Following their work, Alvari et al. proposed new utility functions trying to dig out more useful insights of community formation [12]. Arora et al. used classic methods in theoretic computer science to make community detection rigorous and also gave a systematic analysis [13]. In this paper, we use game theoretic framework for community detection, and try to understand community formation game. Recently, there are only two papers focusing on this kind of game. Chen et al. designed a gain function named personal modularity function based on Newman’s modularity function. Alvari et al. made some new utility functions based on similarity. We give a new utility function and make improvement for the former utility functions. In experiments, we have found some thing deeper than what we expected. Our paper is organized as follows. In section II, we borrow some results of potential game [14]. In section III, the framework of community formation game has been formally proposed. We introduce the new utility function, social distance function, and use the game framework to design a community detection algorithm in section IV. In section V, we show the improvements in experiments under real world networks and benchmark graphs and discovery of a interesting phenomenon that the distance shorting in the process of community formation is a constant for individuals. Section VI concludes our work. II. F UNDAMENTAL THEORY Game theory is a strong tool for modeling and analysis, and has been applied for research of human action in vast areas. At the beginning of the development of game theory, it mainly have been used for economics study to model the behaviors of enterprises and markets. Now game theory is some bit of generalized tool for social science, politics, and psychology researches. Recently, scientists from theoretic computer science focus on the computation aspects of game theory, and have developed a novel research area, named algorithmic game theory. By using game theory we can model social network problems naturally, and easily generalize into other complex networks. Complex networks always forms in a decentralized way, the individuals in networks interact with each other on the basis of their strategies. Therefore the two areas have a intimate relations. For example, the subject in evolution game theory is similar to the formation of complex networks, finding influential individuals also could be considered as compute the solution to cooperative game. We introduce some theoretic conclusion, the potential game. The pure strategic Nash equilibrium always exits in potential game and best response dynamic must converge. All these properties are helpful to modeling community formation process. Potential game was first proposed by Monderer and Shapley [14]. Definition II.1. (Potential game)A game G = (N, A, u) is called a potential game, if there exists a function P : A 7→ R such that, for all i ∈ N , all a−i ∈ A−i and ai , a′i ∈ Ai ui (ai , a−i ) − ui (a′i , a−i ) = P (ai , a−i ) − P (a′i , a−i ). Theorem II.2. [15] Every (finite) potential game has at least one pure strategic Nash equilibrium. Proof: Let a∗ = arg maxa∈A P (a), then for any other action profile a′ P (a∗ ) ≥ P (a′ ). Therefore by definition of the potential function for any player i, switching from a∗ to a′ , we have ui (a∗ ) ≥ ui (a′ ). III. C OMMUNITY FORMATION GAME A. Overlapping community detection problem A network, or equivalent speaking, a graph G is formed by nodes and the connections(or edges) between them, sometimes each connection could be assigned a cost. Community detection problem is to find proper partition of the nodes under some rule. It is common accepted by researchers that connections inner a community are denser than those between communities. In overlapping community detection problem, nodes in the network being studied could be contained in multiple communities. Although interdisciplinary community of scientists have being working on it over many years, it still remains a hard task to give a well-defined definition for it. There are different aspects of community structures, such as local definition, global definition and Figure 1. an example of overlapping community structure similarity based definition. We give a formal definition as follows: Given a network or graph, G = (V, E), V is a set of n nodes. E is a set of m edges. We call G dense graph, if m = O(n2 ) and sparse graph, if m = O(n). We could represent this graph with a n × n adjacency matrix A. Adjacency matrix can be used for both unweighted graph and weighted graph. Every element Aij = 1 of Aij if there is a edge between i and j, otherwise Aij = 0 In weighted graph, we can use cij to represent the strength of the connection between i and j. Overlapping community structure, in brief, is a set of clusters of nodes in a network. These clusters may be overlapped on some nodes, i.e., they have same nodes in common, in figure 1 the green nodes are the overlapping nodes. We define the overlapping community structure as follows: Definition III.1 (Overlapping community structure). Overlapping community structure C = {c1 , c2 , · · · , ck }where ci (i ∈ [k]) is the cluster of nodes in G. Each node i in graph G could be in one or more ci (i ∈ [k]), i.e., ∀i, j ∈ [k], |ci ∩ cj | ≥ 0. B. Community formation game In this subsection we define a game for the formation of community in a network. Given a network, we acquire the interactions of individuals. We consider every node in the network as a agent with rationality, who always is trying to maximize his/her utility by entering or leaving communities. Then we could get the equilibrium as the final solution to the problem. The framework for community formation game is natural. In real world, when people enter some community, they could get some benefit from this action. At the same time, they may pay some fee for entering. With this idea, it is natural to design a utility function for every individual. This function says the total payoff of an individual, both gain and loss. By this way, we fix each individual a gain function and a loss function. During formation, they choose their actions based on their own utility functions. Definition III.2 (Community formation game). Give a underlying graph G = (V, E), V is the set of players(nodes), E is the set of edges, where n = |V |, m = |E|. Without loss of generality, we consider this network is unweighted and undirected. And we use agent or nodes freely to mean the same thing, players in the game. Each agent choose the communities he or she intending to enter ing as his/her strategy. All possible communities could be represented as [k] = {1, 2, · · · , k}, where k ≪ n. This conforms to real situation. For each agent, there is a utility function comprised by a gain function, gi (·) and a loss function, li (·). For a function∑ set {f1 , f2 , . . . , fn } defined on an agent, we define f (·) = i∈[n] fi (·). In the definition, we almost state the framework for community formation game. We shall give some basic concepts as follows. Definition III.3 (Strategy space). The strategy space for agent i is defined as the subset of community set, containing the communities this agent intends to enter, i.e. all subsets of [k]. We use Ci ⊂ [k] to represent the strategy of agent i. This is also just the agent’s label set of communities. We allow Ci = ∅ to mean that i doesn’t belong to any community. Definition III.4 (Strategy profile). C = (C1 , C2 , . . . , Cn ) is defined as a strategy profile, actually could be considered as a vector of all agents’ label sets. Each agent i has a gain function gi (·) and a loss function li (·). Then utility function for i maps C to a real number. Definition III.5 (Utility Function). Let the community label set exclude community i be C−i , and (C−i , Ci′ ) be the strategy profile in which node i’s community label set changed into Ci′ , for each agent agent i, We define utility function ui (C) = gi (C) − li (C). Then we could give the best response strategy for agent i as follows: gi (C−i , Ci′ ) − li (C−i , Ci′ ). arg max ′ Ci ⊂[k] Definition III.6 (Pure strategic Nash equilibrium). Given a graph G, the strategy profile C forms a (pure) Nash equilibrium for a community formation game if all agent are using their own best response strategy, i.e., ∀i ∈ [n] and Ci′ ̸= Ci , ui (C−i , Ci′ ) ≤ ui (C−i , Ci ). In short, in Nash equilibrium, every agent can’t improve their utility function by unilaterally alter their strategy. In real world setting, we could consider this state as a situation everyone is satisfied. People could enter multiple communities forming overlapping community structure. However, this kind of a game even may not have a Nash equilibrium. Therefore we should use some restriction on the utility functions to guarantee the existence of Nash equilibria. In potential games, we define a potential function on the strategy profile of all agents. In a word, this function strictly decreases with the same quantity of the change of the utility of a agent changing his/her strategy to make improvement. Last section we give a result of potential game, that Nash equilibrium always exists in a finite potential game. And the dynamic process as a agent sequentially change his/her strategy to achieve better response will converge to a Nash equilibrium. Definition III.7 (Local linearity). A function set {fi (·) : 1 ≤ i ≤ n} is locally linear with linear factor ρ if for each strategy profile C and agent i’s every strategy Ci′ , the following result holds. ∀i ∈ [n], fi (C−i , Ci′ ) − fi (C) = ρ(f (C−i , Ci′ ) − f (C)). Theorem III.8. Let {gi (·) : i ∈ [n]} and {li (·) : i ∈ [n]} be the gain function and loss function of community formation game. If {gi (·)} and {li (·)} are local linear with linear factor ρg and ρl , then community formation game is a potential game. Proof: Define potential function as follows: Φ(C) = ρl · l(C) − ρg · g(C). Consider that agent i changes his/her strategy from Ci to Ci′ . By the definition of locally linearity and utility functions, we get Φ(C) − Φ(C−i ) = ui (C−i , Ci′ ) − ui (C). Therefore community formation game is a potential game. Now we could alter our attention to the computing of Nash equilibrium. As we know, computing Nash equilibrium is a hard problem in general setting. C. The Fundamental Problem Now we focus on the design of utility function. Recently, main results for this are some variations of classic measurements, such as modularity and similarity. In this framework, we could compare the efficiency between different measurements. While we design utility functions, the most important thing is to guarantee the existence of equilibrium of the game, i.e. we should construct locally linear functions. The key factor of community formation game is the proper utility function. Chen et al. proposed in 2010 personalized modularity function based on Newman’s preceding work [11]. And Alvari et al. gave some new insight by using the concept of structural equivalence [12]. 1) Modularity aspect: Newman and Girvan presented the concept of modularity describing the effectiveness of community partition [16]. Modularity is a function defined on the partition of a graph. Let G be a graph, every node i in it belongs to a community ci , then we can define characteristic function δ(ci , cj ) = 1 iff ci = cj , otherwise δ(ci , cj ). We define modularity function Q of a partition as follows: di dj 1 ∑ (Aij − )δ(ci , cj ). Q= 2m ij 2m where A is the adjacency matrix of G, di and dj are degrees of i and j respectively. Q sums over all pairs of nodes in G, but only the pairs of nodes in same community. Actually, it is the number of edges in same communities subtracting the expected number of edges in a corresponding same degree distribution random network. The edges in random graph appears with a given probability [17]. We often call this kind of model configuration model. The probability that two nodes with degree di and dj are connected is exactly ki kj /2m, and the edges are generated independently. Chen et al. proposed personalized modularity function. Definition III.9 (Personalized modularity function). For agent i, define his/her personalized modularity function as follows: di dj 1 ∑ (Aij δ̂(i, j) − · |Ci ∩ Cj |). Qi (C) = 2m 2m of edges, n is the number of nodes, di is the degree of i. We define neighbor node similarity as follows: { di dj    ωij (1 − 2m ), Aij = 1, , ωij ≥ 0,   ωij , Aij = 0. sij = { dni dj  Aij = 1,  4m ,  ωij = 0.   − di dj , A = 0. , 4m ij When i and j have some nodes as their neighbors in common, i.e., ωij ≥ 1, and they are directly linked, sij acquires the maximum value. If they don’t share common neighbors and the neighbors don’t have connections, then sij is the smallest. This definition is reasonable for it combines the structural and connectedness similarities of this graph but lacks theoretical rigorous proof. We can assure the game has an equilibrium. D. Computing Nash equilibria of CFG In game theory, many results are non-constructive proofs. Now people alternate their notation to computing equilibrium in game theory. For if we don’t have efficient algorithms for it, our study just stays at the beginning. Algorithms for searching for Nash equilibria almost are non-polynomial. Theorem III.11. [11] If there is a community formation game, in which the gain function and the loss function are both locally linear. then to compute the best response for a single agent and a Nash equilibrium are NP-hard. j∈[n] where δ̂(i, j) = 1 if |Ci ∩ Cj | ≥ 1, otherwise δ̂(i, j) = 0. A is the adjacency matrix of G. 2) Similarity aspect: Alvari et al. proposed a similarity based gain function. The measurement with the concept of structural equivalency describes the idea that two nodes even not directed connected, if they share same neighbors they are structural equivalent. They gave the utility function as follows: n 1 ∑ gi (C) = sij δij . 2m j=1,j̸=i 1 (|Ci | − 1). m where C represents the strategy profile of all agents’, i.e., a set of all belonging labels of agents m is the number of G, n is the number of nodes or agents, sij is the similarity between node i and j. If |Ci ∩ Cj | ≥ 1, δij = 1, otherwise δij = 0. And |Ci | is the number of communities which agent i belongs to. In their work, a neighbors similarity measurement has been designed to describe the similarity between two nodes. li = Definition III.10 (Neighbor node similarity). Let ωij = |Γ(i) ∩ Γ(j)|, where Γ(i) is the neighbor set of node i. Assume that A is the adjacency matrix of G, m is the number As a result, we try to find approximated optimal solution. Actually in real world, individuals don’t always give a best response. Therefore here we consider that agents will choose a strategy in a restricted strategic space according to others’ strategies. In our configuration, individuals could take actions from four choices. • Join. Agent i joins in a new community, add a new label into Ci . • Leave. Agent i leaves a community, remove the corresponding label in Ci . • Switch. Agent i switches one community to another, rewrite the label. • Stay the same. Agent i does nothing. In a restricted strategy space, equilibrium is a state in which no agent intends to alter his/her strategy. This kind of equilibrium is often called local equilibrium [18]. We represent the local strategy space of agent i with ls(Ci ), the community label set after all these possible actions. We define local equilibrium as follows: Definition III.12 (Local equilibrium). Given strategy profile C = (C1 , C2 , · · · , Cn ), we call C forms a local equilibrium if all agents are using local best strategy of their own, i.e., ∀i and Ci′ ∈ ls(Ci ), ui (C−i , Ci′ ) ≤ ui (C−i , Ci ). In local equilibrium, the utility function of each agent reaches a local optimal, but not the global optimal in Nash equilibrium. We can prove computing local optimal is polynomial time solvable for we only need to enumerate all possible join, leave and switch actions. to two reasons, one is to guarantee the locally linearity of utility function and the other is to match the real situation. Because when a network forms, the nodes in it often have only temporary information about his/her situation. IV. S OCIAL DISTANCE FUNCTION In this section, we introduce our new utility function. There exist path between nodes in networks. Using these path, we could define distances. For example, distance between two nodes often is defined as the length of shortest path between them. In real world, when some people forms a community, they actually distance themselves from those not in the same community. We design the following social distance function to describe this idea. Distance can be considered as the strength between two nodes. Figure 2. A. Social distance before joining the community Shortest path in graph theory is a sequence of alternate edges and nodes connecting two nodes. Searching for shortest paths between nodes in graph has many applications. We notice that individuals often seems to prefer to be closer to the member of the same community than those of different communities. Thence, we model this phenomenon by letting individuals reduce their distance between the nodes within the same community. We shall give the formal definition. Definition IV.1 (Original distance). Original distance dij between nodes i and j is defined as the length of shortest path between them. As the network forms communities, for some node i, other nodes are divided into inner and out the community of i. Intuitively, when some nodes form a community, they distance themselves from other nodes. We define social distance function as follows: Definition IV.2 (Social distance function). For each node in a community, we define social distance function of i: 1 ∑ ρi (C) = ((dij − ω(j)) · δij + dij (1 − δij )). 2m j̸=i where δij = 1 iff i and j have some community in common, i.e. Ci ∩ Cj ̸= ∅, δij = 0 iff i and j don’t share same community, i.e. Ci ∩ Cj = ∅. ω(j) is defined as a function of j to represent the reducing value of gain function. Social distance function sums the distances of i and nodes inner community and out community. Then it could properly describe the distance dynamic process between nodes as community structure forms in networks. When one node play the game, first he/she judges the actions’ profits by the change of social distance function and selects the best response strategy. One thing to point out here is that only the nodes in the community influenced by agent i’s action changed the distances to i. We make this restriction due Figure 3. after joining the community As figure2 shows, graph G has six nodes A, B, C, D, E and F , where the cost of edges are all 1. By definition, we could get the social distances of all nodes. In figure3, assuming the red nodes on the right have formed a community. Node C chooses to join this community. Then we compute the social distances as following tableI. Table I COMPUTING EXAMPLE OF SOCIAL DISTANCE FUNCTION Node A B C D E F Social distance(before) ρ 1.1 1.1 0.8 0.8 1.1 1.1 Social distance(after) ρ′ 1.1 1.1 0.8 − 0.1(ω(D) − ω(E) − ω(F )) 0.8 − 0.1(ω(D)) 1.1 − 0.1(ω(E)) 1.1 − 0.1(ω(F )) Now we give the concrete definition of community formation game based on social distance function Definition IV.3 (Gain function). Agent i’s gain function gi (C) is defined as follows: gi (C) = ρi (C). where C is current strategy profile. Definition IV.4 (Loss function). Agent i’s loss function li (C) is defined as follows: li (C) = 1 (|Ci | − 1). m Now we give the proof of the local linearity of social distance functions. By the definition of local linearity, function set {fi (·) : 1 ≤ i ≤ n} is locally linear with linear factor c it is sufficient to show that for each strategy profile C and agent i’s each strategy Ci′ , the following holds. Algorithm 1 Social distance based community formation game Require: Graph, G = (V, E). Ensure: Community structure of this graph, C. 1: Initialize each nodes as a single community. 2: Initialize a set C to represent the final result. 3: Compute all pairs shortest paths 4: repeat 5: Randomly pick an agent who has been seen less. 6: The chosen agent chooses best operation from four possible actions,join, leave,switch or do nothing. 7: until reach the local equilibrium ∀i ∈ [n], fi (C−i , Ci′ ) − fi (C) = c(f (C−i , Ci′ ) − f (C)). Lemma IV.5. Let fi be ρi (·) and f be ∑ ρi (·), then i∈[n] {fi (·) : 1 ≤ i ≤ n} is locally linear. Proof: Without loss of generality, let agent i change his/her strategy from Ci to Ci′ . • i joins community S, ∑ then the social distance of i ω(i). By the definition of changes with ∆i = i∈S social distance, for global function f changes with ∑ ω(i). ∆=2 i∈S • • i leaves ∑ S, then the social distance changes with ∆i = ω(i) and the global function changes with ∆ = − i∈S ∑ ω(i). 2 i∈S ∑ ∑ ω(j), ω(i)− i switches from S to T , then ∆i = ∑ i∈T ∑ j∈S ω(j)). ω(i) − for global function f ∆ = 2( i∈T j∈S Therefore c = 1/2. Theorem IV.6 (The Existence of Nash Equilibrium). Let gain function be gi (C) and loss function be li (C), then community formation game based on social distance function has at least a Nash equilibrium. Proof: By theoremIII.8 and LemmaIV.5, it is trivial that 1 (|Ci | − 1) is locally linear with linear factor 1, then li = m it is a potential game, by TheoremII.2 this game has at least a Nash equilibrium. Now we could guarantee the existence of Nash equilibrium in community formation game based on social distance function. Based on the discussion of local equilibrium, we give a community detection algorithm as follows. B. Algorithm for computing a Nash equilibrium Here we give an algorithm to acquire the local equilibrium of community formation game. In section III-D, we introduce the strategy for computing equilibrium. We need to search result not in global solution space but in local strategy space. Three gain func1 mod tions ρi , ρi + Qmod i (ωj = c) and ρi + Qi (ωj = dj ) are used in this algorithm framework. The corresponding algorithms are SDGAME, SDMODGAME(invariant) and SDMODGAME(variant), where invariant means the ωi is a constant and variant means that it is a function determined by i. For example, ωi could be arbitrary function of the degree of i, i.e., ωi = (di ). V. E XPERIMENTS AND D ISCUSSIONS We conduct experiments on both real world networks and artificial networks to quantitatively analyze the performance of algorithms. By experiments, we discover some phenomenons called little attention in research before and assure that the concept of distance is useful to analysis of networks. A. Real World Networks Two real world networks have been used for testing the algorithm, the dolphin network and Zachary’s karate club network. Karate club network describes a social network about the relationship situation of a karate club by Zachary’s two years observation [19]. This network divides into two non-overlapping communities. Dolphin network [20] is about the relationship between 62 dolphins in Doubtful Sound. This network studied by Lusseau et al. during seven years, demonstrating the community structure in animal relation networks. The results for both networks are showed in Figure4 and Figure5. The white nodes in the two figures are the overlapping nodes found by our SDMODGAME algorithm. B. Artificial Networks In this section we compare the performance of different algorithms in LFR benchmark. LFR was proposed by Lancichinetti et al. [21] for evaluating the overlapping community detection algorithms. Table II gives some introduction to the parameters of LFR graph generating model. Figure 4. result for Zachary karate club network Figure 6. Performance compare about this concept. The social distance reflecting the strength of connections in another aspects is of great importance for each individual. More work should be done to disclose information underneath the topology relation of complex networks. VI. C ONCLUSION Figure 5. Result for Dolphin network Table II LFR Parameter N maxk mu t1 t2 minc maxc om on DataSet1 1000 15 0.1 2 1 20 100 2 0-500 PARAMETERS DataSet2 1000 15 0.3 2 1 20 100 2 0-500 DataSet3 1000 15 0.1 2 1 20 100 2-8 100 DataSet4 1000 15 0.3 2 1 20 100 2-8 100 C. Discussion The performance of our algorithm is evaluated by the normalized mutual information(NMI). Two series of data are generated, with different µ, 0.1 and 0.3. SDMODGAME, NGGAME, MODGAME are compared in datasets with parameter overlapping membership ranging from 2 to 8. We set ω = 0.01 for invariant and ω = dj for variant. In the right two figures6, when µ chooses a greater value, our method gives an improvement on the the accuracy of the overlapping community structure result. In the second line of figure6, invariant SDMODGAME and variant SDMODGAME are compared. It is showed that the invariant ω = 0.01 is better than ω = dj . The result of our experiments shows that the concept of social distance, compared with the ordinary distance, is a more proper notion for real scene. However, more properties remain unknown The goal of complex network analysis is to understand the world we live and to find the underlying truth. Community detection is a fundamental tools. Although we have plenty methods for overlapping community detection, we even don’t have a precise image on the formation of community structure of networks. By using rigorous tools such as game theory, we could define formally the process of the formation of communities. Guaranteed by the results of game theory, we find more insightful results. Our work focus on the new concept named social distance, which naturally describes the situation in real world. By experiments, we find that if we combine the information of structure of connection style and the distance information in networks, better results can be obtained. To put it forward, the influence of social distance kind information makes important affection on the formation of community structures in networks. In the design of social distance function, we even learn that when nodes are forming communities, the shorting distance is a constant, based on our different choices of ωi . Together with the works of Chen et al. and Alvari et al., we conclude that the formation of community structure is mainly dominated by the individuals’ properties and with a local style. With community formation game framework, the detection of overlapping community could be reduced to the study of utility functions. The properties of complex networks are deeply influenced by the individuals in them. The weakness in research is that we can’t have all information. Therefore in our future work, we may design more utility functions with complex combination of different metrics. ACKNOWLEDGMENT The authors would like to thank Yajun Wang and Alvari Hamidreza for the help of algorithm design and implementation and Lancichinetii for generating the benchmark. This Paper is supported by the National Natural Science Foundation of China under Grant No.60503021, No.60721002, No.60875038, No.61105069 and Science and Technology Supporting program of Jiangsu Province under Grant No.BE2011171 and Graduate Innovation Fund of Nanjing University under Grant No. 2011CL07. R EFERENCES [1] G. Palla, T. Vicsek, and A.-L. Barabási, “Community dynamics in social networks,” Fluctuation and Noise Letters, vol. 07, no. 3, pp. 273–287, 2007. [2] G. Palla, I. Derényi, I. Farkas, and T. Vicsek, “Uncovering the overlapping community structure of complex networks in nature and society,” Nature, no. 435, pp. 814–818, 2005. [3] I. Farkas, D. Ábel, G. Palla, and T. Vicsek, “Weighted network modules,” New Journal of Physics, vol. 9, no. 6, p. 180, 2007. [4] J. M. Kumpula, M. Kivelá, K. Kaski, and J. Saramáki, “Sequential algorithm for fast clique percolation,” Physical Review E, vol. 78, no. 2, p. 026109, 2008. [5] Y.-Y. Ahn, J. P. Bagrow, and S. Lehmann, “Link communities reveal multiscale com-plexity in networks,” Nature, no. 466, p. 761C764, 2010. [6] T. Evans and R. Lambiotte, “Line graphs, link partitions and overlapping communities,” Physical Review E, no. 80, p. 016105, 2010. [7] ——, “Line graphs of weighted networks for overlapping communities,” European Physical Journal B, no. 77, p. 265, 2010. [8] Y. Kim and H. Jeong, “The map equation for link community,” Physical Review E, no. 84, p. 026110, 2011. [9] A. Lancichinetti, S. Fortunato, and J. Kertész, “Detecting the overlapping and hier-archical community structure of complex networks,” Journal of Physics, vol. 11, p. 033015, 2009. [10] J. Baumes, M. K. Goldberg, M. S. Krishnamoorthy, M.-I. Malik, and P. Nathan, “Finding communities by clustering a graph into overlapping subgraphs.” in IADIS AC. IADIS, 2005, pp. 97–104. [Online]. Available: http://dblp.unitrier.de/db/conf/iadis/ac2005-1.htmlBaumesGKMP05 [11] W. Chen, Z. Liu, X. Sun, and Y. Wang, “A game-theoretic framework to identify overlapping communities in social networks,” Data Mining and Knowledge Discovery, no. 21, pp. 224–240, 2010. [12] A. Hamidreza, H. Sattar, and H. Ali, “Detecting overlapping communities in social networks by game theory and structural equivalence concept,” Artificial Intelligence and Computational Intelligence, vol. 7003, pp. 620–630, 2011. [13] S. Arora, R. Ge, S. Sachdeva, and G. Schoenebeck, “Finding overlapping communities in social networks: Toward a rigorous approach,” CoRR, vol. abs/1112.1831, 2011. [14] D. Monderer and L. S. Shapley, “Potential games,” Games and Economic Behavior, vol. 14, p. 124C143, 1996. [15] Y. Shoham and K. Leyton-Brown, Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge, UK: Cambridge University Press, 2009. [16] M. Newman, “Modularity and community structure in networks,” Proceedings of the National Academy of Sciences, vol. 103, no. 23, pp. 8577–8582, 2006. [17] P. Erdős and A. Rényi, “On random graphs,” Publicationes Mathematicae, vol. 6, pp. 290–297, 1959. [18] C. Alós-Ferrer and A. Ania, “Local equilibria in economic games,” Economics Letters, vol. 70, no. 2, pp. 165–173, 2001. [19] W. W. Zachary, “An information flow model for conflict and fission in small groups,” Journal of Anthropological Research, no. 33, pp. 452–473, 1977. [20] L. David, S. Karsten, B. O. J., H. Patti, S. Elisabeth, and D. S. M., “The bottlenose dolphin community of Doubtful Sound features a large proportion of longlasting associations,” Behavioral Ecology and Sociobiology, vol. 54, no. 4, pp. 396–405, 2003. [Online]. Available: http://dx.doi.org/10.1007/s00265-003-0651-y [21] A. Lancichinetti and S. Fortunato, “Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities,” Physical Review E, no. 84, p. 016118, 2009.