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Unformatted text preview: A Faster Algorithm for Betweenness Centrality * Ulrik Brandes University of Konstanz Department of Computer & Information Science Box D 67, 78457 Konstanz, Germany Ulrik.Brandes@uni-konstanz.de Abstract The betweenness centrality index is essential in the analysis of social networks, but costly to compute. Currently, the fastest known algo- rithms require ( n 3 ) time and ( n 2 ) space, where n is the number of actors in the network. Motivated by the fast-growing need to compute centrality indices on large, yet very sparse, networks, new algorithms for betweenness are introduced in this paper. They require O ( n + m ) space and run in O ( nm ) and O ( nm + n 2 log n ) time on unweighted and weighted networks, respectively, where m is the number of links. Experimen- tal evidence is provided that this substantially increases the range of networks for which centrality analysis is feasible. Keywords: Social networks, betweenness centrality, algorithms. 1 Introduction In social network analysis, graph-theoretic concepts are used to understand and explain social phenomena. A social network consists of a set of actors, who may be arbitrary entities like persons or organizations, and one or more types of relations between them. For a comprehensive overview of methods and applications see Wasserman and Faust (1994) or Scott (1991). * Part of this research was done while with the Department of Computer Science at Brown University. I gratefully acknowledge financial support from the German Academic Exchange Service (DAAD, Hochschulsonderprogramm III). Published in Journal of Mathematical Sociology 25(2):163-177, (2001). 1 An essential tool for the analysis of social networks are centrality indices defined on the vertices of the graph (Bavelas, 1948; Sabidussi, 1966; Freeman, 1979). They are designed to rank the actors according to their position in the network and interpreted as the prominence of actors embedded in a social structure. Many centrality indices are based on shortest paths linking pairs of actors, measuring, e.g., the average distance from other actors, or the ratio of shortest paths an actor lies on. Many network-analytic studies rely at least in part on an evaluation of these indices. With the increasing practicality of electronic data collection and, of course, the advent of the Web, there is a likewise increasing demand for the computation of centrality indices on networks with thousands of ac- tors. Several notions of centrality originating from social network analysis are in use to determine the structural prominence of Web pages (Kleinberg, 1999; Brin et al., 1998; Bharat and Henzinger, 1998). However, there is an ( n 3 ) bottleneck in existing implementations, due to the particularly important betweenness centrality index (Freeman, 1977; Anthonisse, 1971), which makes comparative centrality analyses of networks with more than a few hundred actors prohibitive. As a remedy, network analysts are now suggesting simpler indices, for instance based only on linkages between the...
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