Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

networkx.algorithms.centrality.edge_current_flow_betweenness_centrality_subset

edge_current_flow_betweenness_centrality_subset(G, sources, targets, normalized=True, weight=None, dtype=<class 'float'>, solver='lu')[source]

Compute current-flow betweenness centrality for edges using subsets of 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 [2].

Parameters
Ggraph

A NetworkX graph

sources: list of nodes

Nodes to use as sources for current

targets: list of nodes

Nodes to use as sinks for current

normalizedbool, optional (default=True)

If True the betweenness values are normalized by b=b/(n-1)(n-2) where n is the number of nodes in G.

weightstring or None, optional (default=None)

Key for edge data used as the edge weight. If None, then use 1 as each edge weight. The weight reflects the capacity or the strength of the edge.

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
nodesdict

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 [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).