Current-flow betweenness (Brandes and Fleischer 2005), also known as random-walk betweenness (Newman 2005), treats the network as an electrical circuit with the edge weights as conductances. One unit of current is sent between every pair \(s, t\), and the score is the average current that flows through the node: $$CFB(v) = \frac{2}{(n-1)(n-2)} \sum_{s < t} \frac{1}{2} \sum_{u} w_{vu} \left| p^{(st)}_v - p^{(st)}_u \right|,$$ where \(p^{(st)}\) are the node potentials of that pair, taken from the pseudoinverse of the Laplacian, and the inner sum is set to zero for \(v = s\) and \(v = t\).
Arguments
- x
Network input accepted by
centrality.- ...
Further arguments to
centrality, such asdirectedandnormalized.
Details
The measure is defined for connected undirected networks. On a
disconnected network every score is NA, with a
cograph_undefined_measure warning.
On a directed network the Laplacian is built from the asymmetric weight
matrix, and directed = FALSE gives the undirected reading. The
potentials and the currents come from the same weights, and
weighted = FALSE gives every edge conductance one.
The fixed factor \(2/((n-1)(n-2))\) is the normalization of
networkx::current_flow_betweenness_centrality().
References
Brandes, U., & Fleischer, D. (2005). Centrality measures based on current flow. In STACS 2005, Lecture Notes in Computer Science, 3404 (pp. 533-544). Springer. doi:10.1007/978-3-540-31856-9_44 .
Newman, M. E. J. (2005). A measure of betweenness centrality based on random walks. Social Networks, 27(1), 39-54. doi:10.1016/j.socnet.2004.11.009 .