Measures of betweenness in non-symmetric networks

doi:10.1016/0378-8733(87)90023-2

Social Networks

Volume 9, Issue 3, September 1987, Pages 277-282

https://doi.org/10.1016/0378-8733(87)90023-2 Get rights and content

Abstract

Betweenness has commonly been regarded as a centrality measure that can only be employed with symmetric networks. This paper shows that computation of absolute partial betweenness scores is unproblematic for non-symmetric networks as well, as long as ordered pairs of points are considered. The maximum absolute score is also derived, making it possible to compute “relative” betweenness scores which can be compared across non-symmetric networks.

References (5)

Ronald S. Burt
Models of network structure
Annual Review of Sociology
(1980)
Linton C. Freeman
Centrality in social networks I: Conceptual clarification
Social Networks
(1979)

There are more references available in the full text version of this article.

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^☆: I would like to thank Roberto M. Fernandez, Linton C. Freeman, and Harrison C. White for their helpful comments and suggestions. This material is based upon work supported under a National Science Foundation Graduate Fellowship. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation.

^∗∗: Department of Sociology, University of Arizona, Tucson, AZ 85721, U.S.A.

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Measures of betweenness in non-symmetric networks☆

Abstract

Models of network structure

Annual Review of Sociology

Centrality in social networks I: Conceptual clarification

Social Networks