networkx.algorithms.centrality.edge_current_flow_betweenness_centrality

edge_current_flow_betweenness_centrality(G, normalized=True, weight=None, dtype=<class 'float'>, solver='full')[source]

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 2.

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=None)) – Key for edge data used as the edge weight. If None, then use 1 as each edge weight.

  • dtype (data type (default=float)) – Default data type for internal matrices. Set to np.float32 for lower memory consumption.

  • solver (string (default=’full’)) – 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 of edge tuples with betweenness centrality as the value.

Return type

dictionary

Raises

NetworkXError – The algorithm does not support DiGraphs. If the input graph is an instance of DiGraph class, NetworkXError is raised.

See also

betweenness_centrality(), edge_betweenness_centrality(), current_flow_betweenness_centrality()

Notes

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

1

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://algo.uni-konstanz.de/publications/bf-cmbcf-05.pdf

2

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