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# edge_current_flow_betweenness_centrality¶

edge_current_flow_betweenness_centrality(G, normalized=True, weight='weight', dtype=<type 'float'>, solver='full')

Compute current-flow betweenness centrality for edges.

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 [R188].

Parameters : G : graph A NetworkX graph normalized : bool, optional (default=True) If True the betweenness values are normalized by 2/[(n-1)(n-2)] where n is the number of nodes in G. weight : string or None, optional (default=’weight’) Key for edge data used as the edge weight. If None, then use 1 as each edge weight. dtype: data type (float) Default data type for internal matrices. Set to np.float32 for lower memory consumption. solver: string (default=’lu’) Type of linear solver to use for computing the flow matrix. Options are “full” (uses most memory), “lu” (recommended), and “cg” (uses least memory). nodes : dictionary Dictionary of edge tuples with betweenness centrality as the value.

Notes

Current-flow betweenness can be computed in $$O(I(n-1)+mn \log n)$$ time [R187], 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

 [R187] (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
 [R188] (1, 2) A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 39-54 (2005).