Compute current-flow betweenness centrality for nodes.
Current-flow betweenness centrality uses an electrical current model for information spreading in contrast to betweenness centrality which uses shortest paths.
Current-flow betweenness centrality is also known as random-walk betweenness centrality [R167].
Parameters : | G : graph
normalized : bool, optional (default=True)
weight : string or None, optional (default=’weight’)
dtype: data type (float) :
solver: string (default=’lu’) :
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Returns : | nodes : dictionary
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See also
approximate_current_flow_betweenness_centrality, betweenness_centrality, edge_betweenness_centrality, edge_current_flow_betweenness_centrality
Notes
Current-flow betweenness can be computed in O(I(n-1)+mn \log n) time [R166], where I(n-1) is the time needed to compute the inverse Laplacian. For a full matrix this is O(n^3) but using sparse methods you can achieve O(nm{\sqrt k}) where k is the Laplacian matrix condition number.
The space required is O(nw) where `w is the width of the sparse Laplacian matrix. Worse case is w=n for O(n^2).
If the edges have a ‘weight’ attribute they will be used as weights in this algorithm. Unspecified weights are set to 1.
References
[R166] | (1, 2) Centrality Measures Based on Current Flow. Ulrik Brandes and Daniel Fleischer, Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS ‘05). LNCS 3404, pp. 533-544. Springer-Verlag, 2005. http://www.inf.uni-konstanz.de/algo/publications/bf-cmbcf-05.pdf |
[R167] | (1, 2) A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 39-54 (2005). |