Abstract
This paper generalizes Freeman’s geodesic centrality measures for betweenness on undirected graphs to the more general directed case. Four steps are taken. The point centrality measure is first generalized for directed graphs. Second, a unique maximally centralized graph is defined for directed graphs, holding constant the numbers of points with reciprocatable (incoming and outgoing) versus only unreciprocatable (outgoing only or incoming only) arcs, and focusing the measure on the maximally central arrangement of arcs within these constraints. Alternatively, one may simply normalize on the number of arcs. This enables the third step of defining the relative behveenness centralities of a point, independent of the number of points. This normalization step for directed centrality measures removes Gould’s objection that centrality measures for directed graphs are not interpretable because they lack a standard for maximality. The relative directed centrality converges with Freeman’s betweenness measure in the case of undirected graphs with no isolates. The fourth step is to define the measures of this concept of graph centralization in terms of the dominance of the most central point.
Highlights
Betweenness centrality (Freeman 1977, 1979, 1980) is a fundamental measurement concept for the analysis of social networks
The present study identifies a unit of measurement and a uniquely maximal centralized graph for the directed case, and so provides for a proper generalization of betweenness centrality to directed graphs
Normalizing the point centrality measure establishes comparability between graphs of different sizes but of differing degrees of symmetry, and prevents the measure of relative centrality in graphs with very few reciprocated relations from being deflated because of their lack of symmetry. This step in normalizing the directed centrality measure removes Gould’s (1987) objection that centrality measures for directed graphs are uninterpretable because they lack a standard for maximality
Summary
Betweenness centrality (Freeman 1977, 1979, 1980) is a fundamental measurement concept for the analysis of social networks. The recent book by Hage and Harary (1991) demonstrates some of its many descriptive and predictive uses. It was originally defined, only for undirected graphs. This constitutes a rather severe limit on its potential utility for directed (nonsymmetric) graphs and social networks. Gould (1987) argues that measures of betweenness centrality are possible for the directed case, but that owing to lack of a unit of measurement and of a unique definition of the maximally centralized graph for the directed case, the measure remains uninterpretable. The present study identifies a unit of measurement and a uniquely maximal centralized graph for the directed case, and so provides for a proper generalization of betweenness centrality to directed graphs
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