Detect overlapping and hierarchical community structure in networks

arXiv:0810.3093v3 [cs.CY] 3 Nov 2008 APS/123-QED Detect overlapping and hierarchical community structure in networks Huawei Shen 1,2, Xueqi Cheng 1,∗Kai Cai 1, and Mao-Bin Hu 3 1Institute of Computing Technology, Chinese Academy of Sciences, Beijing, P.R. China 2 Graduate University of Chinese Academy of Sciences, Beijing, P.R. China 3 School of Engineering Science, University of Science and Technology of China, Hefei 230026, P.R. China (Dated: November 26, 2024) Clustering and community structure is crucial for many network systems and the related dynamic processes. It has been shown that communities are usually overlapping and hierarchical. However, previous methods investigate these two properties of community structure separately. This pa- per proposes an algorithm (EAGLE) to detect both the overlapping and hierarchical properties of complex community structure together. This algorithm deals with the set of maximal cliques and adopts an agglomerative framework. The quality function of modularity is extended to evaluate the goodness of a cover. The examples of application to real world networks give excellent results. PACS numbers: 89.75.Hc, 05.10.-a, 87.23.Ge, 89.20.Hh I. INTRODUCTION Many complex systems in nature and society can be de- scribed in terms of networks or graphs. Examples include the Internet, the world-wide-web, social and biological systems of various kinds, and many others [1, 2, 3]. In the past decade, the theory of complex network has attracted much attention. Complex networks are usually charac- terized by several distinctive properties: power law de- gree distribution, short path length, clustering and com- munity structure. The problem becomes important be- cause complex system’s dynamics is actually determined by the interaction of many components and the topologi- cal properties of the network will affect the dynamics in a very fundamental way. Therefore, an efficient and sound approach that can capture the topological properties of network is needed. Identifying the community structure is crucial to un- derstand the structural and functional properties of the networks [4, 5, 6]. Many methods have been proposed to identify the community structure of complex networks [7, 8, 9, 10, 11, 12]. One can refer to [13] for reviews. These methods can be roughly classified into two cate- gories in terms of their results, i.e., to form a partition or a cover of the network. The first kind of methods produce a partition, i.e each vertex belongs to one and only one community and is regarded as equally important. Different from classi- cal graph-partition problem, the number of communi- ties and the size of each community are previously un- known. Newman et al. proposed a quality function Q, namely modularity, to evaluate the goodness of a partition [9]. A high value of Q indicates a signifi- cant community structure. Several community detec- tion methods have been proposed by optimizing mod- ularity [11, 14, 15]. Generally, this kind of methods are suitable to understand the entire structure of networks, ∗Electronic address: cxq@ict.ac.cn especially for the networks with a small size. Recently, some authors [17, 18] have pointed out that the optimiza- tion of modularity has a fundamental drawback, i.e. the existence of a resolution limit. The second kind of methods aim to discover the vertex sets (i.e. communities) with a high density of edges. In this case, overlapping is allowed, that is, some vertices may belong to more than one community. Meanwhile, some vertices may be neglected as subordinate vertices. Therefore, these methods result in an incomplete cover of the network. Numerous methods have been proposed, based on k-clique [8], k-dense [25] or other patterns. Un- fortunately, there is no commonly accepted standard to evaluate the goodness of a cover up to now. Compared to the partition methods, this kind of methods are appro- priate to find the cohesive regions in large-scale networks. In real networks, communities are usually overlapping and hierarchical [8, 19, 20, 21]. Overlapping means that some vertices may belong to more than one community. Hierarchical means that communities may be further di- vided into sub-communities. The two kinds of existing methods, as mentioned above, investigate these two phe- nomena separately. The first kind of methods can be used to explore the hierarchical community structure, however, they are unable to deal with overlaps between communities. The second kind of methods can uncover overlapping community structure of networks, but they are incapable of finding the hierarchy of communities. Recently, several authors begin to detect the hierarchical and overlapping community structure [22]. In this paper, a new algorithm EAGLE (agglomera- tivE hierarchicAl clusterinG based on maximaL cliquE) is presented to uncover both hierarchical and overlapping community structure of networks. This algorithm deals with the set of maximal cliques and adopts an agglom- erati

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