edge_current_flow_betweenness_centrality¶
- edge_current_flow_betweenness_centrality(G, normalized=True, weight='weight', dtype=<type 'float'>, solver='lu')¶
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 [R148].
Parameters : G : graph
A NetworkX graph
normalized : bool, 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.
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
time [R147], where 
is the time needed to compute the
inverse Laplacian. For a full matrix this is 
but using
sparse methods you can achieve 
where 
is the
Laplacian matrix condition number.The space required is
is the width of the sparse
Laplacian matrix. Worse case is 
for 
.If the edges have a ‘weight’ attribute they will be used as weights in this algorithm. Unspecified weights are set to 1.
References
[R147] (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 [R148] (1, 2) A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 39-54 (2005).