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Com_Density - PHYSICAL REVIEW E 77 036109 2008 Quantitative...

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Quantitative function for community detection Zhenping Li, 1,2, * Shihua Zhang, 2,3, * Rui-Sheng Wang, 4 Xiang-Sun Zhang, 2, and Luonan Chen 5,6, 1 Beijing Wuzi University, Beijing 101149, China 2 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China 3 Graduate University of Chinese Academy of Sciences, Beijing 100049, China 4 School of Information, Renmin University of China, Beijing 100872, China 5 Institute of Systems Biology, Shanghai University, Shanghai 200444, China 6 Department of Electrical Engineering and Electronics, Osaka Sangyo University, Osaka 574-8530, Japan Received 6 October 2007; revised manuscript received 2 December 2007; published 10 March 2008 We propose a quantitative function for community partition—i.e., modularity density or D value. We dem- onstrate that this quantitative function is superior to the widely used modularity Q and also prove its equiva- lence with the objective function of the kernel k means. Both theoretical and numerical results show that optimizing the new criterion not only can resolve detailed modules that existing approaches cannot achieve, but also can correctly identify the number of communities. DOI: 10.1103/PhysRevE.77.036109 PACS number s : 89.75.Hc, 87.23.Ge I. INTRODUCTION It has been widely demonstrated in the past that many interesting systems can be represented as networks com- posed of vertices and edges 1 3 . Such systems include the internet, social and friendship networks, food webs, biomo- lecular networks, and citation networks. The prolific progress in the study of complex networks driven by the development of information technology and the increasing availability of huge networked data in the real world have revealed many interesting topological properties, such as small-world prop- erties, power-law degree distributions, and network motifs. A topic of great interest in the area of complex networks is the community structure and its detection. A community could be roughly described as a collection of vertices in a subgraph that are densely connected among themselves while being loosely connected to the vertices outside the subgraph. Since many networks exhibit such a community structure, the characterization and detection of such a com- munity structure have great practical significance. Taking biological molecular networks as an example, dividing pro- tein interaction networks into modular groups provides strong evidence of independent functions and actions for proteins in different subgraphs 3 , 4 . There have been abundant techniques proposed to detect community structure 5 , 6 and fuzzy community structure 7 10 in a network from various fields, but most methods require a definition of community that imposes the limit up to which a group should be considered as a community.
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