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# edge_betweenness_centrality_subset

edge_betweenness_centrality_subset(G, sources, targets, normalized=False, weight=None)

  Compute betweenness centrality for edges for a subset of nodes.

  \[c_B(v) =\sum_{s\in S,t \in T} \frac{\sigma(s, t|e)}{\sigma(s, t)}\]

  where \(S\) is the set of sources, \(T\) is the set of targets, \(\sigma(s, t)\) is the number of shortest \((s, t)\)-paths, and \(\sigma(s, t|e)\) is the number of those paths passing through edge \(e\) [2].

  Parameters:
    G graph

      A networkx graph.

    sources: list of nodes

      Nodes to use as sources for shortest paths in betweenness

    targets: list of nodes

      Nodes to use as targets for shortest paths in betweenness

    normalized bool, optional

      If True the betweenness values are normalized by `2/(n(n-1))` for graphs, and `1/(n(n-1))` for directed graphs where `n` is the number of nodes in G.

    weight None or string, optional (default=None)

      If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. Weights are used to calculate weighted shortest paths, so they are interpreted as distances.

  Returns:
    edges dictionary

      Dictionary of edges with Betweenness centrality as the value.

  See also

  `betweenness_centrality`
  `edge_load`

  Notes

  The basic algorithm is from [1].

  For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes.

  The normalization might seem a little strange but it is the same as in edge_betweenness_centrality() and is designed to make edge_betweenness_centrality(G) be the same as edge_betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).

  References

  [1]

  Ulrik Brandes, A Faster Algorithm for Betweenness Centrality. Journal of Mathematical Sociology 25(2):163-177, 2001. https://doi.org/10.1080/0022250X.2001.9990249

  [2]

  Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. https://doi.org/10.1016/j.socnet.2007.11.001