Com_Limit - arXiv:physics/0607100 v2 14 Jul 2006 Resolution...

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Unformatted text preview: arXiv:physics/0607100 v2 14 Jul 2006 Resolution limit in community detection Santo Fortunato 1, 2, 3 and Marc Barth´ elemy 1,4 1 School of Informatics and Biocomplexity Center, Indiana University, Eigenmann Hall, 1900 East Tenth Street, Bloomington IN 47406 2 Fakult¨at f¨ur Physik, Universit¨at Bielefeld, D-33501 Bielefeld, Germany 3 Complex Networks Lagrange Laboratory (CNLL), ISI Foundation, Torino, Italy 4 CEA-Centre d’Etudes de Bruy` eres-le-Chˆatel, D´ epartement de Physique Th´ eorique et Appliqu´ ee BP12, 91680 Bruy` eres-Le-Chˆatel, France (Dated: July 18, 2006) Detecting community structure is fundamental to clarify the link between structure and function in complex networks and is used for practical applications in many disciplines. A successful method relies on the optimization of a quantity called modularity [Newman and Girvan, Phys. Rev. E 69 , 026113 (2004)], which is a quality index of a partition of a network into communities. We find that modularity optimization may fail to identify modules smaller than a scale which depends on the total number L of links of the network and on the degree of interconnectedness of the modules, even in cases where modules are unambiguously defined. The probability that a module conceals well-defined substructures is the highest if the number of links internal to the module is of the order of √ 2 L or smaller. We discuss the practical consequences of this result by analyzing partitions obtained through modularity optimization in artificial and real networks. PACS numbers: 89.75.-k, 89.75.Hc, 05.40 -a, 89.75.Fb, 87.23.Ge Keywords: Networks, community structure, modularity I. INTRODUCTION Community detection in complex networks has at- tracted a lot of attention in the last years (for a re- view see [1, 2]). The main reason is that complex net- works [3, 4, 5, 6, 7] are made of a large number of nodes and that so far most of the quantitative investigations were focusing on statistical properties disregarding the roles played by specific subgraphs. Detecting commu- nities (or modules ) can then be a way to identify rele- vant substructures that may also correspond to impor- tant functions. In the case of the World Wide Web, for instance, communities are sets of Web pages dealing with the same topic [8]. Relevant community structures were also found in social networks [9, 10, 11], biochemical net- works [12, 13, 14], the Internet [15], food webs [16], and in networks of sexual contacts [17]. Loosely speaking a community is a subgraph of a net- work whose nodes are more tightly connected with each other than with nodes outside the subgraph. A decisive advance in community detection was made by Newman and Girvan [18], who introduced a quantitative measure for the quality of a partition of a network into commu- nities, the so-called modularity . This measure essentially compares the number of links inside a given module with the expected value for a randomized graph of the same size and degree sequence. If one takes modularity as thesize and degree sequence....
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This note was uploaded on 05/20/2011 for the course CAP 5515 taught by Professor Ungor during the Spring '08 term at University of Florida.

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Com_Limit - arXiv:physics/0607100 v2 14 Jul 2006 Resolution...

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