code: def betweenness_centrality(
      G,
      k=None,
      normalized=True,
      weight=None,
      endpoints=False,
      seed=None,
  ):
      r"""Compute the shortest-path betweenness centrality for nodes.
  
      Betweenness centrality of a node $v$ is the sum of the
      fraction of all-pairs shortest paths that pass through $v$
  
      .. math::
  
         c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}
  
      where $V$ is the set of nodes, $\sigma(s, t)$ is the number of
      shortest $(s, t)$-paths,  and $\sigma(s, t|v)$ is the number of
      those paths  passing through some  node $v$ other than $s, t$.
      If $s = t$, $\sigma(s, t) = 1$, and if $v \in {s, t}$,
      $\sigma(s, t|v) = 0$ [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-1)(n-2))`
        for graphs, and `1/((n-1)(n-2))` 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.
  
      endpoints : bool, optional
        If True include the endpoints in the shortest path counts.
  
      seed : integer, random_state, or None (default)
          Indicator of random number generation state.
          See :ref:`Randomness<randomness>`.
          Note that this is only used if k is not None.
  
      Returns
      -------
      nodes : dictionary
         Dictionary of nodes with betweenness centrality as the value.
  
      See Also
      --------
      edge_betweenness_centrality
      load_centrality
  
      Notes
      -----
      The algorithm is from Ulrik Brandes [1]_.
      See [4]_ for the original first published version and [2]_ for details on
      algorithms for variations and related metrics.
  
      For approximate betweenness calculations set k=#samples to use
      k nodes ("pivots") to estimate the betweenness values. For an estimate
      of the number of pivots needed see [3]_.
  
      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 total number of paths between source and target is counted
      differently for directed and undirected graphs. Directed paths
      are easy to count. Undirected paths are tricky: should a path
      from "u" to "v" count as 1 undirected path or as 2 directed paths?
  
      For betweenness_centrality we report the number of undirected
      paths when G is undirected.
  
      For betweenness_centrality_subset the reporting is different.
      If the source and target subsets are the same, then we want
      to count undirected paths. But if the source and target subsets
      differ -- for example, if sources is {0} and targets is {1},
      then we are only counting the paths in one direction. They are
      undirected paths but we are counting them in a directed way.
      To count them as undirected paths, each should count as half a path.
  
      This algorithm is not guaranteed to be correct if edge weights
      are floating point numbers. As a workaround you can use integer
      numbers by multiplying the relevant edge attributes by a convenient
      constant factor (eg 100) and converting to integers.
  
      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
      .. [3] Ulrik Brandes and Christian Pich:
         Centrality Estimation in Large Networks.
         International Journal of Bifurcation and Chaos 17(7):2303-2318, 2007.
         https://dx.doi.org/10.1142/S0218127407018403
      .. [4] Linton C. Freeman:
         A set of measures of centrality based on betweenness.
         Sociometry 40: 35–41, 1977
         https://doi.org/10.2307/3033543
      """
      betweenness = dict.fromkeys(G, 0.0)  # b[v]=0 for v in G
      if k is None:
          nodes = G
      else:
          nodes = seed.sample(list(G.nodes()), k)
      for s in nodes:
          # single source shortest paths
          if weight is None:  # use BFS
              S, P, sigma, _ = _single_source_shortest_path_basic(G, s)
          else:  # use Dijkstra's algorithm
              S, P, sigma, _ = _single_source_dijkstra_path_basic(
                  G, s, weight
              )
          # accumulation
          if endpoints:
              betweenness, _ = _accumulate_endpoints(
                  betweenness, S, P, sigma, s
              )
          else:
              betweenness, _ = _accumulate_basic(
                  betweenness, S, P, sigma, s
              )
      # rescaling
      betweenness = _rescale(
          betweenness,
          len(G),
          normalized=normalized,
          directed=G.is_directed(),
          k=k,
          endpoints=endpoints,
      )
      return betweenness
  
file_path: networkx/networkx/algorithms/centrality/betweenness.py
imports: from collections import deque
  from heapq import heappop, heappush
  from itertools import count
  import networkx
  from networkx.algorithms.shortest_paths.weighted import _weight_function
  from networkx.utils import py_random_state
  from networkx.utils.decorators import not_implemented_for
label_desc: Use image node_id 1 for calling a global function with example usage: betweenness_centrality(G, k, normalized, weight, endpoints, seed) and returns: betweenness