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:
- 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 (default=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 edge tuples with betweenness centrality as the value.
- Raises:
- NetworkXError
The algorithm does not support DiGraphs. If the input graph is an instance of DiGraph class, NetworkXError is raised.
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).