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 betweenness 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.
|Number of pages||12|
|State||Published - Oct 1994|
Bibliographical noteFunding Information:
DRW is endebtedt o Linton C. Freeman for helpful criticismsa nd suggestionsi n completion of this paper, which was stimulated by collaborationsc arried out with him, Vincent Duquenne, Alain De-genne, Thomas Schweizera nd other colleaguesp articipatingi n the Working Groups in Discrete Structure Analysis, sponsored by the Maison des Science de 1’Hommeo f Paris, the Ministry of Research, Maison Suger,a nd the Laboratoired ’AnalyseS econdairee t Methodes Appliques a la Sociologic( LASMAS-CNRS).
ASJC Scopus subject areas
- Sociology and Political Science
- Social Sciences (all)
- Psychology (all)