Node centrality in weighted networks: Generalizing degree and shortest paths

Abstract Ties often have a strength naturally associated with them that differentiate them from each other. Tie strength has been operationalized as weights. A few network measures have been proposed for weighted networks, including three common measures of node centrality: degree, closeness, and betweenness. However, these generalizations have solely focused on tie weights, and not on the number of ties, which was the central component of the original measures. This paper proposes generalizations that combine both these aspects. We illustrate the benefits of this approach by applying one of them to Freeman’s EIES dataset.

[1]  Ulrik Brandes,et al.  On variants of shortest-path betweenness centrality and their generic computation , 2008, Soc. Networks.

[2]  A. Vespignani,et al.  The architecture of complex weighted networks. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Stephen P. Borgatti,et al.  Centrality and network flow , 2005, Soc. Networks.

[4]  Paul Honigsheim.  Book Review:The Sociology of Georg Simmel. Kurt H. Wolff , 1950 .

[5]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[6]  Leo Katz,et al.  A new status index derived from sociometric analysis , 1953 .

[7]  M. McPherson,et al.  Birds of a Feather: Homophily in Social Networks , 2001 .

[8]  Mark S. Granovetter.  The Strength of Weak Ties , 1973, American Journal of Sociology.

[9]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

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

[11]  Thomas W. Valente.  Network models of the diffusion of innovations , 1996, Comput. Math. Organ. Theory.

[12]  Daniel J. Brass.  Being in the right place: A structural analysis of individual influence in an organization. , 1984 .

[13]  Tore Opsahl,et al.  Prominence and control: the weighted rich-club effect. , 2008, Physical review letters.

[14]  E. R. Peay.  Connectedness in a general model for valued networks , 1980 .

[15]  Steven B. Andrews,et al.  Structural Holes: The Social Structure of Competition , 1995, The SAGE Encyclopedia of Research Design.

[16]  M. Newman.  Erratum: Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality (Physical Review e (2001) 64 (016132)) , 2006 .

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

[18]  A. Ferligoj,et al.  Generalized Blockmodeling: Preface , 2004 .

[19]  Tom A. B. Snijders,et al.  Social Network Analysis , 2011, International Encyclopedia of Statistical Science.

[20]  Morten T. Hansen,et al.  The Search-Transfer Problem: The Role of Weak Ties in Sharing Knowledge across Organization Subunits , 1999 .

[21]  Kathleen M. Carley,et al.  On the robustness of centrality measures under conditions of imperfect data , 2006, Soc. Networks.

[22]  Martin Suter,et al.  Small World , 2002 .

[23]  L. Freeman,et al.  The Development of Social Network Analysis: A Study in the Sociology of Science , 2005 .

[24]  Tore Opsahl,et al.  Clustering in weighted networks , 2009, Soc. Networks.

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

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

[27]  G. Simmel.  The sociology of Georg Simmel , 1950 .

[28]  P. Piccone,et al.  For Sociology , 1974, Telos.

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

[30]  Alessandro Vespignani,et al.  Reaction–diffusion processes and metapopulation models in heterogeneous networks , 2007, cond-mat/0703129.

[31]  David Knoke,et al.  Optimal connections: strength and distance in valued graphs , 2001, Soc. Networks.

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

[33]  M. Newman.  Analysis of weighted networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.