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# edge_betweenness_centrality

edge_betweenness_centrality(G, k=None, normalized=True, weight=None, seed=None)

  Compute betweenness centrality for edges.

  Betweenness centrality of an edge \(e\) is the sum of the fraction of all-pairs shortest paths that pass through \(e\)

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

  where \(V\) is the set of nodes, \(\sigma(s, t)\) 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.

    k int, optional (default=None)

      If k is not None use k node samples to estimate betweenness. The value of k <= n where n is the number of nodes in the graph. Higher values give better approximation.

    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.

    seed integer, random_state, or None (default)

      Indicator of random number generation state. See Randomness. Note that this is only used if k is not None.

  Returns:
    edges dictionary

      Dictionary of edges with betweenness centrality as the value.

  See also

  `betweenness_centrality`
  `edge_load`

  Notes

  The algorithm is from Ulrik Brandes [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.

  References

  [1]

  A Faster Algorithm for Betweenness Centrality. Ulrik Brandes, 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