# Flow Betweenness Centrality

#### Definition

**Unnormalized flow betweenness**

Freeman et al. define the raw or unnormalized flow betweenness of a vertex, v ∈ V(G) as: 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:

**Variant by Koschutzki et al.**

where 0/0 flow ratios are treated as 0 (as in shortest-path betweenness).

#### References

- 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
- 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.