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Flow Betweenness Centrality


Unnormalized flow betweenness
Freeman et al. define the raw or unnormalized flow betweenness of a vertex, v ∈ V(G) as:
Flow Betweenness Centrality
where f(i, j, G) is the maximum flow from i to j within G (under the assumption of infinite vertex capacities, finite edge capacities, and non-simultaneity of pairwise flows). Intuitively, unnormalized flow betweenness is simply the total maximum flow (aggregated across all pairs of third parties) mediated by v.

Normalized flow betweenness
Normalize the raw flow betweenness by the total maximum flow among third parties (including v); this leads to the following normalized flow betweenness measure:
Flow Betweenness Centrality

Variant by Koschutzki et al.
Flow Betweenness Centrality
where 0/0 flow ratios are treated as 0 (as in shortest-path betweenness).


  • FREEMAN, L. C.; BORGATTI, S.P.; and WHITE, D.R. (1991). “Centrality in Valued Graphs: A Measure of Betweenness Based on Network Flow.” Social Networks, 13(2), 141-154. DOI: 10.1016/0378-8733(91)90017-N Publisher web site Endnote RIS file
  • Koschutzki, D.; Lehmann, K.A.; Peeters, L.; Richter, S.; Tenfelde-Podehl, D.; Zlotowski, O. (2005). "Centrality Indices." In BRANDES, U. & ERLEBACH, T. 2005. Network Analysis: Methodological Foundations, U.S. Government Printing Office.


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