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).

Returns :

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).