A measure of betweenness centrality based on random walks

Betweenness is a measure of the centrality of a node in a network, and is normally calculated as the fraction of shortest paths between node pairs that pass through the node of interest. Betweenness is, in some sense, a measure of the influence a node has over the spread of information through the network. By counting only shortest paths, however, the conventional definition implicitly assumes that information spreads only along those shortest paths. Here, we propose a betweenness measure that relaxes this assumption, including contributions from essentially all paths between nodes, not just the shortest, although it still gives more weight to short paths. The measure is based on random walks, counting how often a node is traversed by a random walk between two other nodes. We show how our measure can be calculated using matrix methods, and give some examples of its application to particular networks.

[1]  John F. Padgett,et al.  Robust Action and the Rise of the Medici, 1400-1434 , 1993, American Journal of Sociology.

[2]  Adrian Mindel,et al.  Recent advances: Sexually transmitted infections , 1998 .

[3]  John Scott.  What is social network analysis , 2010 .

[4]  P. Bonacich.  Power and Centrality: A Family of Measures , 1987, American Journal of Sociology.

[5]  M. Zelen,et al.  Rethinking centrality: Methods and examples☆ , 1989 .

[6]  D. Watts,et al.  An Experimental Study of Search in Global Social Networks , 2003, Science.

[7]  John Scott.  Social Network Analysis , 1988 .

[8]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[9]  M. Newman,et al.  Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  J. Kertész,et al.  Random walks on complex networks with inhomogeneous impact. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Heiko Rieger,et al.  Stability of shortest paths in complex networks with random edge weights. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Heiko Rieger,et al.  Random walks on complex networks. , 2004, Physical review letters.

[14]  R. Rothenberg,et al.  Risk network structure in the early epidemic phase of HIV transmission in Colorado Springs , 2002, Sexually transmitted infections.

[15]  Stanley Milgram,et al.  An Experimental Study of the Small World Problem , 1969 .

[16]  Alexandre Arenas,et al.  Optimal network topologies for local search with congestion , 2002, Physical review letters.

[17]  Sharon L. Milgram,et al.  The Small World Problem , 1967 .

[18]  S. Wasserman,et al.  Social Network Analysis: Computer Programs , 1994 .

[19]  I. Denham,et al.  Sexually Transmitted Infections , 2013 .

[20]  K. Goh,et al.  Betweenness centrality correlation in social networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  M. Newman,et al.  Why social networks are different from other types of networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  L. Freeman,et al.  Centrality in valued graphs: A measure of betweenness based on network flow , 1991 .

[23]  L. Freeman.  Centrality in social networks conceptual clarification , 1978 .

[24]  Leonard M. Freeman,et al.  A set of measures of centrality based upon betweenness , 1977 .

[25]  David K. Smith.  Network Flows: Theory, Algorithms, and Applications , 1994 .

[26]  U. Brandes.  A faster algorithm for betweenness centrality , 2001 .