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 currentflow betweenness centrality for edges.
Currentflow betweenness centrality uses an electrical current model for information spreading in contrast to betweenness centrality which uses shortest paths.
Currentflow betweenness centrality is also known as randomwalk betweenness centrality 2.
 Parameters
G (graph) – A NetworkX graph
normalized (bool, optional (default=True)) – If True the betweenness values are normalized by 2/[(n1)(n2)] 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
Currentflow betweenness can be computed in \(O(I(n1)+mn \log n)\) time 1, where \(I(n1)\) 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. 533544. SpringerVerlag, 2005. http://algo.unikonstanz.de/publications/bfcmbcf05.pdf
 2
A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 3954 (2005).