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Com_LocalShell - PHYSICAL REVIEW E 72 046108 2005 Local...

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Local method for detecting communities James P. Bagrow 1 and Erik M. Bollt 2,1 1 Department of Physics, Clarkson University, Potsdam, New York 13699-5820, USA 2 Department of Math and Computer Science, Clarkson University, Potsdam, New York 13699-5815, USA Received 24 December 2004; published 10 October 2005 We propose a method of community detection that is computationally inexpensive and possesses physical significance to a member of a social network. This method is unlike many divisive and agglomerative tech- niques and is local in the sense that a community can be detected within a network without requiring knowl- edge of the entire network. A global application of this method is also introduced. Several artificial and real-world networks, including the famous Zachary karate club, are analyzed. DOI: 10.1103/PhysRevE.72.046108 PACS number s : 89.75.Hc, 05.10. a, 87.23.Ge, 89.20.Hh I. INTRODUCTION It has been shown in the past that many interesting sys- tems can be represented as networks composed of vertices and edges 1–4 . Such systems include the Internet 5 , so- cial and friendship networks 6 , food webs 7 , and citation networks 8,9 . For example, a social network may represent people as vertices and edges linking vertices when those people are on a first-name basis. A topic of current interest in the area of networks has been the idea of communities and their detection. A commu- nity could be loosely described as a collection of vertices within a graph that are densely connected amongst them- selves while being loosely connected to the rest of the graph 10–12 . Many networks exhibit such a community structure and this motivates our work. This description, however, is somewhat vague and open to interpretation. This leads to the possibility that different techniques for detecting these com- munities may lead to slightly different yet equally valid re- sults. We emphasize this variation in Sec. II D. Several techniques have been proposed to detect commu- nity structure inside of a network. The recent and highly successful betweenness centrality algorithm due to Newman and Girvan 13–15 performs well within a variety of net- works but it is costly to compute O n 2 m on a graph with n vertices and m edges 15 . More importantly, while be- tweenness centrality has been shown to be a useful quantity for detecting community structure, it is knowledge not usu- ally attainable to a vertex within the graph . In this paper we ask, if a person were to move to a new town, what actions would he or she take to see what com- munity or communities they belong to? Most community detection methods using hierarchical clustering fall within two categories: divisive and agglomerative 6,15 . Both forms, including those using betweenness and other methods, are global algorithms and do not represent feasible actions that a member of a network could undertake to identify the network’s community structure. The method proposed here may better represent actions that members of a network would undertake to identify their own communities.
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