On the Design of Utility Functions in Community Formation Game Xiaohu Zhu *† , Jun Wu ‡ , Chongjun Wang *† , Junyuan Xie *† * State 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 Abstract—To understand the overlapping community struc- ture 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. Nowa- days 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 fur- ther, 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 commu- nity 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 de- scription 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 for- mation 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.