current_flow_betweenness_centrality

current_flow_betweenness_centrality(G, normalized=True, weight=None, dtype=<class 'float'>, solver='full')

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 [2].

Parameters:

Ggraph

A NetworkX graph

normalizedbool, 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.

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.

dtypedata type (float)

Default data type for internal matrices. Set to np.float32 for lower memory consumption.

solverstring (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:

nodesdictionary

Dictionary of nodes with betweenness centrality as the value.

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 [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. https://doi.org/10.1007/978-3-540-31856-9_44

[2]

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