text: """
  Betweenness centrality measures for subsets of nodes.
  """
  #    Copyright (C) 2004-2015 by 
  #    Aric Hagberg <hagberg@lanl.gov>
  #    Dan Schult <dschult@colgate.edu>
  #    Pieter Swart <swart@lanl.gov>
  #    All rights reserved.
  #    BSD license.
  __author__ = """Aric Hagberg (hagberg@lanl.gov)"""
  
  __all__ = ['betweenness_centrality_subset',
             'edge_betweenness_centrality_subset',
             'betweenness_centrality_source']
  
  import networkx as nx
  
  from networkx.algorithms.centrality.betweenness import\
      _single_source_dijkstra_path_basic as dijkstra
  from networkx.algorithms.centrality.betweenness import\
      _single_source_shortest_path_basic as shortest_path
  
  
  def betweenness_centrality_subset(G,sources,targets,
                                    normalized=False,
                                    weight=None):
      r"""Compute betweenness centrality for a subset of nodes.
  
      .. math::
  
         c_B(v) =\sum_{s\in S, t \in T} \frac{\sigma(s, t|v)}{\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|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
  
      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-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  
        If None, all edge weights are considered equal.
        Otherwise holds the name of the edge attribute used as weight.
  
      Returns
      -------
      nodes : dictionary
         Dictionary of nodes with betweenness centrality as the value.
  
      See Also
      --------
      edge_betweenness_centrality
      load_centrality
  
      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 betweenness_centrality() and is designed to make
      betweenness_centrality(G) be the same as
      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.
         http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
      .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness 
         Centrality and their Generic Computation. 
         Social Networks 30(2):136-145, 2008.
         http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf
      """
      b=dict.fromkeys(G,0.0) # b[v]=0 for v in G
      for s in sources:
          # single source shortest paths
          if weight is None:  # use BFS
              S,P,sigma=shortest_path(G,s)
          else:  # use Dijkstra's algorithm
              S,P,sigma=dijkstra(G,s,weight)
          b=_accumulate_subset(b,S,P,sigma,s,targets)
      b=_rescale(b,len(G),normalized=normalized,directed=G.is_directed())
      return b
  
  
  def edge_betweenness_centrality_subset(G,sources,targets,
                                         normalized=False,
                                         weight=None):
      r"""Compute betweenness centrality for edges for a subset of nodes.
  
      .. math::
  
         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  
        If None, all edge weights are considered equal.
        Otherwise holds the name of the edge attribute used as weight.
  
      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.
         http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
      .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness 
         Centrality and their Generic Computation. 
         Social Networks 30(2):136-145, 2008.
         http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf
  
      """
  
      b=dict.fromkeys(G,0.0) # b[v]=0 for v in G
      b.update(dict.fromkeys(G.edges(),0.0)) # b[e] for e in G.edges()
      for s in sources:
          # single source shortest paths
          if weight is None:  # use BFS
              S,P,sigma=shortest_path(G,s)
          else:  # use Dijkstra's algorithm
              S,P,sigma=dijkstra(G,s,weight)
          b=_accumulate_edges_subset(b,S,P,sigma,s,targets)
      for n in G: # remove nodes to only return edges 
          del b[n]
      b=_rescale_e(b,len(G),normalized=normalized,directed=G.is_directed())
      return b
  
  # obsolete name
  def betweenness_centrality_source(G,normalized=True,weight=None,sources=None):
      if sources is None:
          sources=G.nodes()
      targets=G.nodes()
      return betweenness_centrality_subset(G,sources,targets,normalized,weight)
  
  
  def _accumulate_subset(betweenness,S,P,sigma,s,targets):
      delta=dict.fromkeys(S,0)
      target_set=set(targets)
      while S:
          w=S.pop()
          for v in P[w]:
              if w in target_set:
                  delta[v]+=(sigma[v]/sigma[w])*(1.0+delta[w])
              else:
                  delta[v]+=delta[w]/len(P[w])
          if w != s:
              betweenness[w]+=delta[w]
      return betweenness
  
  def _accumulate_edges_subset(betweenness,S,P,sigma,s,targets):
      delta=dict.fromkeys(S,0)
      target_set=set(targets)
      while S:
          w=S.pop()
          for v in P[w]:
              if w in target_set:
                  c=(sigma[v]/sigma[w])*(1.0+delta[w])
              else:
                  c=delta[w]/len(P[w])
              if (v,w) not in betweenness:
                  betweenness[(w,v)]+=c
              else:
                  betweenness[(v,w)]+=c
              delta[v]+=c
          if w != s:
              betweenness[w]+=delta[w]
      return betweenness
  
  
  
  
  def _rescale(betweenness,n,normalized,directed=False):
      if normalized is True:
          if n <=2:
              scale=None  # no normalization b=0 for all nodes
          else:
              scale=1.0/((n-1)*(n-2))
      else: # rescale by 2 for undirected graphs
          if not directed:
              scale=1.0/2.0
          else:
              scale=None
      if scale is not None:
          for v in betweenness:
              betweenness[v] *= scale
      return betweenness
  
  def _rescale_e(betweenness,n,normalized,directed=False):
      if normalized is True:
          if n <=1:
              scale=None  # no normalization b=0 for all nodes
          else:
              scale=1.0/(n*(n-1))
      else: # rescale by 2 for undirected graphs
          if not directed:
              scale=1.0/2.0
          else:
              scale=None
      if scale is not None:
          for v in betweenness:
              betweenness[v] *= scale
      return betweenness
  