diff DEPENDENCIES/generic/include/boost/graph/betweenness_centrality.hpp @ 16:2665513ce2d3

Add boost headers

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/DEPENDENCIES/generic/include/boost/graph/betweenness_centrality.hpp	Tue Aug 05 11:11:38 2014 +0100
@@ -0,0 +1,616 @@
+// Copyright 2004 The Trustees of Indiana University.
+
+// Distributed under the Boost Software License, Version 1.0.
+// (See accompanying file LICENSE_1_0.txt or copy at
+// http://www.boost.org/LICENSE_1_0.txt)
+
+//  Authors: Douglas Gregor
+//           Andrew Lumsdaine
+#ifndef BOOST_GRAPH_BRANDES_BETWEENNESS_CENTRALITY_HPP
+#define BOOST_GRAPH_BRANDES_BETWEENNESS_CENTRALITY_HPP
+
+#include <stack>
+#include <vector>
+#include <boost/graph/overloading.hpp>
+#include <boost/graph/dijkstra_shortest_paths.hpp>
+#include <boost/graph/breadth_first_search.hpp>
+#include <boost/graph/relax.hpp>
+#include <boost/graph/graph_traits.hpp>
+#include <boost/tuple/tuple.hpp>
+#include <boost/type_traits/is_convertible.hpp>
+#include <boost/type_traits/is_same.hpp>
+#include <boost/mpl/if.hpp>
+#include <boost/property_map/property_map.hpp>
+#include <boost/graph/named_function_params.hpp>
+#include <algorithm>
+
+namespace boost {
+
+namespace detail { namespace graph {
+
+  /**
+   * Customized visitor passed to Dijkstra's algorithm by Brandes'
+   * betweenness centrality algorithm. This visitor is responsible for
+   * keeping track of the order in which vertices are discovered, the
+   * predecessors on the shortest path(s) to a vertex, and the number
+   * of shortest paths.
+   */
+  template<typename Graph, typename WeightMap, typename IncomingMap,
+           typename DistanceMap, typename PathCountMap>
+  struct brandes_dijkstra_visitor : public bfs_visitor<>
+  {
+    typedef typename graph_traits<Graph>::vertex_descriptor vertex_descriptor;
+    typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor;
+
+    brandes_dijkstra_visitor(std::stack<vertex_descriptor>& ordered_vertices,
+                             WeightMap weight,
+                             IncomingMap incoming,
+                             DistanceMap distance,
+                             PathCountMap path_count)
+      : ordered_vertices(ordered_vertices), weight(weight), 
+        incoming(incoming), distance(distance),
+        path_count(path_count)
+    { }
+
+    /**
+     * Whenever an edge e = (v, w) is relaxed, the incoming edge list
+     * for w is set to {(v, w)} and the shortest path count of w is set to
+     * the number of paths that reach {v}.
+     */
+    void edge_relaxed(edge_descriptor e, const Graph& g) 
+    { 
+      vertex_descriptor v = source(e, g), w = target(e, g);
+      incoming[w].clear();
+      incoming[w].push_back(e);
+      put(path_count, w, get(path_count, v));
+    }
+
+    /**
+     * If an edge e = (v, w) was not relaxed, it may still be the case
+     * that we've found more equally-short paths, so include {(v, w)} in the
+     * incoming edges of w and add all of the shortest paths to v to the
+     * shortest path count of w.
+     */
+    void edge_not_relaxed(edge_descriptor e, const Graph& g) 
+    {
+      typedef typename property_traits<WeightMap>::value_type weight_type;
+      typedef typename property_traits<DistanceMap>::value_type distance_type;
+      vertex_descriptor v = source(e, g), w = target(e, g);
+      distance_type d_v = get(distance, v), d_w = get(distance, w);
+      weight_type w_e = get(weight, e);
+
+      closed_plus<distance_type> combine;
+      if (d_w == combine(d_v, w_e)) {
+        put(path_count, w, get(path_count, w) + get(path_count, v));
+        incoming[w].push_back(e);
+      }
+    }
+
+    /// Keep track of vertices as they are reached
+    void examine_vertex(vertex_descriptor w, const Graph&) 
+    { 
+      ordered_vertices.push(w);
+    }
+
+  private:
+    std::stack<vertex_descriptor>& ordered_vertices;
+    WeightMap weight;
+    IncomingMap incoming;
+    DistanceMap distance;
+    PathCountMap path_count;
+  };
+
+  /**
+   * Function object that calls Dijkstra's shortest paths algorithm
+   * using the Dijkstra visitor for the Brandes betweenness centrality
+   * algorithm.
+   */
+  template<typename WeightMap>
+  struct brandes_dijkstra_shortest_paths
+  {
+    brandes_dijkstra_shortest_paths(WeightMap weight_map) 
+      : weight_map(weight_map) { }
+
+    template<typename Graph, typename IncomingMap, typename DistanceMap, 
+             typename PathCountMap, typename VertexIndexMap>
+    void 
+    operator()(Graph& g, 
+               typename graph_traits<Graph>::vertex_descriptor s,
+               std::stack<typename graph_traits<Graph>::vertex_descriptor>& ov,
+               IncomingMap incoming,
+               DistanceMap distance,
+               PathCountMap path_count,
+               VertexIndexMap vertex_index)
+    {
+      typedef brandes_dijkstra_visitor<Graph, WeightMap, IncomingMap, 
+                                       DistanceMap, PathCountMap> visitor_type;
+      visitor_type visitor(ov, weight_map, incoming, distance, path_count);
+
+      dijkstra_shortest_paths(g, s, 
+                              boost::weight_map(weight_map)
+                              .vertex_index_map(vertex_index)
+                              .distance_map(distance)
+                              .visitor(visitor));
+    }
+
+  private:
+    WeightMap weight_map;
+  };
+
+  /**
+   * Function object that invokes breadth-first search for the
+   * unweighted form of the Brandes betweenness centrality algorithm.
+   */
+  struct brandes_unweighted_shortest_paths
+  {
+    /**
+     * Customized visitor passed to breadth-first search, which
+     * records predecessor and the number of shortest paths to each
+     * vertex.
+     */
+    template<typename Graph, typename IncomingMap, typename DistanceMap, 
+             typename PathCountMap>
+    struct visitor_type : public bfs_visitor<>
+    {
+      typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor;
+      typedef typename graph_traits<Graph>::vertex_descriptor 
+        vertex_descriptor;
+      
+      visitor_type(IncomingMap incoming, DistanceMap distance, 
+                   PathCountMap path_count, 
+                   std::stack<vertex_descriptor>& ordered_vertices)
+        : incoming(incoming), distance(distance), 
+          path_count(path_count), ordered_vertices(ordered_vertices) { }
+
+      /// Keep track of vertices as they are reached
+      void examine_vertex(vertex_descriptor v, Graph&)
+      {
+        ordered_vertices.push(v);
+      }
+
+      /**
+       * Whenever an edge e = (v, w) is labelled a tree edge, the
+       * incoming edge list for w is set to {(v, w)} and the shortest
+       * path count of w is set to the number of paths that reach {v}.
+       */
+      void tree_edge(edge_descriptor e, Graph& g)
+      {
+        vertex_descriptor v = source(e, g);
+        vertex_descriptor w = target(e, g);
+        put(distance, w, get(distance, v) + 1);
+        
+        put(path_count, w, get(path_count, v));
+        incoming[w].push_back(e);
+      }
+
+      /**
+       * If an edge e = (v, w) is not a tree edge, it may still be the
+       * case that we've found more equally-short paths, so include (v, w)
+       * in the incoming edge list of w and add all of the shortest
+       * paths to v to the shortest path count of w.
+       */
+      void non_tree_edge(edge_descriptor e, Graph& g)
+      {
+        vertex_descriptor v = source(e, g);
+        vertex_descriptor w = target(e, g);
+        if (get(distance, w) == get(distance, v) + 1) {
+          put(path_count, w, get(path_count, w) + get(path_count, v));
+          incoming[w].push_back(e);
+        }
+      }
+
+    private:
+      IncomingMap incoming;
+      DistanceMap distance;
+      PathCountMap path_count;
+      std::stack<vertex_descriptor>& ordered_vertices;
+    };
+
+    template<typename Graph, typename IncomingMap, typename DistanceMap, 
+             typename PathCountMap, typename VertexIndexMap>
+    void 
+    operator()(Graph& g, 
+               typename graph_traits<Graph>::vertex_descriptor s,
+               std::stack<typename graph_traits<Graph>::vertex_descriptor>& ov,
+               IncomingMap incoming,
+               DistanceMap distance,
+               PathCountMap path_count,
+               VertexIndexMap vertex_index)
+    {
+      typedef typename graph_traits<Graph>::vertex_descriptor
+        vertex_descriptor;
+
+      visitor_type<Graph, IncomingMap, DistanceMap, PathCountMap>
+        visitor(incoming, distance, path_count, ov);
+      
+      std::vector<default_color_type> 
+        colors(num_vertices(g), color_traits<default_color_type>::white());
+      boost::queue<vertex_descriptor> Q;
+      breadth_first_visit(g, s, Q, visitor, 
+                          make_iterator_property_map(colors.begin(), 
+                                                     vertex_index));
+    }
+  };
+
+  // When the edge centrality map is a dummy property map, no
+  // initialization is needed.
+  template<typename Iter>
+  inline void 
+  init_centrality_map(std::pair<Iter, Iter>, dummy_property_map) { }
+
+  // When we have a real edge centrality map, initialize all of the
+  // centralities to zero.
+  template<typename Iter, typename Centrality>
+  void 
+  init_centrality_map(std::pair<Iter, Iter> keys, Centrality centrality_map)
+  {
+    typedef typename property_traits<Centrality>::value_type 
+      centrality_type;
+    while (keys.first != keys.second) {
+      put(centrality_map, *keys.first, centrality_type(0));
+      ++keys.first;
+    }
+  }
+
+  // When the edge centrality map is a dummy property map, no update
+  // is performed.
+  template<typename Key, typename T>
+  inline void 
+  update_centrality(dummy_property_map, const Key&, const T&) { }
+
+  // When we have a real edge centrality map, add the value to the map
+  template<typename CentralityMap, typename Key, typename T>
+  inline void 
+  update_centrality(CentralityMap centrality_map, Key k, const T& x)
+  { put(centrality_map, k, get(centrality_map, k) + x); }
+
+  template<typename Iter>
+  inline void 
+  divide_centrality_by_two(std::pair<Iter, Iter>, dummy_property_map) {}
+
+  template<typename Iter, typename CentralityMap>
+  inline void
+  divide_centrality_by_two(std::pair<Iter, Iter> keys, 
+                           CentralityMap centrality_map)
+  {
+    typename property_traits<CentralityMap>::value_type two(2);
+    while (keys.first != keys.second) {
+      put(centrality_map, *keys.first, get(centrality_map, *keys.first) / two);
+      ++keys.first;
+    }
+  }
+
+  template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
+           typename IncomingMap, typename DistanceMap, 
+           typename DependencyMap, typename PathCountMap,
+           typename VertexIndexMap, typename ShortestPaths>
+  void 
+  brandes_betweenness_centrality_impl(const Graph& g, 
+                                      CentralityMap centrality,     // C_B
+                                      EdgeCentralityMap edge_centrality_map,
+                                      IncomingMap incoming, // P
+                                      DistanceMap distance,         // d
+                                      DependencyMap dependency,     // delta
+                                      PathCountMap path_count,      // sigma
+                                      VertexIndexMap vertex_index,
+                                      ShortestPaths shortest_paths)
+  {
+    typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator;
+    typedef typename graph_traits<Graph>::vertex_descriptor vertex_descriptor;
+
+    // Initialize centrality
+    init_centrality_map(vertices(g), centrality);
+    init_centrality_map(edges(g), edge_centrality_map);
+
+    std::stack<vertex_descriptor> ordered_vertices;
+    vertex_iterator s, s_end;
+    for (boost::tie(s, s_end) = vertices(g); s != s_end; ++s) {
+      // Initialize for this iteration
+      vertex_iterator w, w_end;
+      for (boost::tie(w, w_end) = vertices(g); w != w_end; ++w) {
+        incoming[*w].clear();
+        put(path_count, *w, 0);
+        put(dependency, *w, 0);
+      }
+      put(path_count, *s, 1);
+      
+      // Execute the shortest paths algorithm. This will be either
+      // Dijkstra's algorithm or a customized breadth-first search,
+      // depending on whether the graph is weighted or unweighted.
+      shortest_paths(g, *s, ordered_vertices, incoming, distance,
+                     path_count, vertex_index);
+      
+      while (!ordered_vertices.empty()) {
+        vertex_descriptor w = ordered_vertices.top();
+        ordered_vertices.pop();
+        
+        typedef typename property_traits<IncomingMap>::value_type
+          incoming_type;
+        typedef typename incoming_type::iterator incoming_iterator;
+        typedef typename property_traits<DependencyMap>::value_type 
+          dependency_type;
+        
+        for (incoming_iterator vw = incoming[w].begin();
+             vw != incoming[w].end(); ++vw) {
+          vertex_descriptor v = source(*vw, g);
+          dependency_type factor = dependency_type(get(path_count, v))
+            / dependency_type(get(path_count, w));
+          factor *= (dependency_type(1) + get(dependency, w));
+          put(dependency, v, get(dependency, v) + factor);
+          update_centrality(edge_centrality_map, *vw, factor);
+        }
+        
+        if (w != *s) {
+          update_centrality(centrality, w, get(dependency, w));
+        }
+      }
+    }
+
+    typedef typename graph_traits<Graph>::directed_category directed_category;
+    const bool is_undirected = 
+      is_convertible<directed_category*, undirected_tag*>::value;
+    if (is_undirected) {
+      divide_centrality_by_two(vertices(g), centrality);
+      divide_centrality_by_two(edges(g), edge_centrality_map);
+    }
+  }
+
+} } // end namespace detail::graph
+
+template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
+         typename IncomingMap, typename DistanceMap, 
+         typename DependencyMap, typename PathCountMap, 
+         typename VertexIndexMap>
+void 
+brandes_betweenness_centrality(const Graph& g, 
+                               CentralityMap centrality,     // C_B
+                               EdgeCentralityMap edge_centrality_map,
+                               IncomingMap incoming, // P
+                               DistanceMap distance,         // d
+                               DependencyMap dependency,     // delta
+                               PathCountMap path_count,      // sigma
+                               VertexIndexMap vertex_index
+                               BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
+{
+  detail::graph::brandes_unweighted_shortest_paths shortest_paths;
+
+  detail::graph::brandes_betweenness_centrality_impl(g, centrality, 
+                                                     edge_centrality_map,
+                                                     incoming, distance,
+                                                     dependency, path_count,
+                                                     vertex_index, 
+                                                     shortest_paths);
+}
+
+template<typename Graph, typename CentralityMap, typename EdgeCentralityMap, 
+         typename IncomingMap, typename DistanceMap, 
+         typename DependencyMap, typename PathCountMap, 
+         typename VertexIndexMap, typename WeightMap>    
+void 
+brandes_betweenness_centrality(const Graph& g, 
+                               CentralityMap centrality,     // C_B
+                               EdgeCentralityMap edge_centrality_map,
+                               IncomingMap incoming, // P
+                               DistanceMap distance,         // d
+                               DependencyMap dependency,     // delta
+                               PathCountMap path_count,      // sigma
+                               VertexIndexMap vertex_index,
+                               WeightMap weight_map
+                               BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
+{
+  detail::graph::brandes_dijkstra_shortest_paths<WeightMap>
+    shortest_paths(weight_map);
+
+  detail::graph::brandes_betweenness_centrality_impl(g, centrality, 
+                                                     edge_centrality_map,
+                                                     incoming, distance,
+                                                     dependency, path_count,
+                                                     vertex_index, 
+                                                     shortest_paths);
+}
+
+namespace detail { namespace graph {
+  template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
+           typename WeightMap, typename VertexIndexMap>
+  void 
+  brandes_betweenness_centrality_dispatch2(const Graph& g,
+                                           CentralityMap centrality,
+                                           EdgeCentralityMap edge_centrality_map,
+                                           WeightMap weight_map,
+                                           VertexIndexMap vertex_index)
+  {
+    typedef typename graph_traits<Graph>::degree_size_type degree_size_type;
+    typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor;
+    typedef typename mpl::if_c<(is_same<CentralityMap, 
+                                        dummy_property_map>::value),
+                                         EdgeCentralityMap, 
+                               CentralityMap>::type a_centrality_map;
+    typedef typename property_traits<a_centrality_map>::value_type 
+      centrality_type;
+
+    typename graph_traits<Graph>::vertices_size_type V = num_vertices(g);
+    
+    std::vector<std::vector<edge_descriptor> > incoming(V);
+    std::vector<centrality_type> distance(V);
+    std::vector<centrality_type> dependency(V);
+    std::vector<degree_size_type> path_count(V);
+
+    brandes_betweenness_centrality(
+      g, centrality, edge_centrality_map,
+      make_iterator_property_map(incoming.begin(), vertex_index),
+      make_iterator_property_map(distance.begin(), vertex_index),
+      make_iterator_property_map(dependency.begin(), vertex_index),
+      make_iterator_property_map(path_count.begin(), vertex_index),
+      vertex_index,
+      weight_map);
+  }
+  
+
+  template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
+           typename VertexIndexMap>
+  void 
+  brandes_betweenness_centrality_dispatch2(const Graph& g,
+                                           CentralityMap centrality,
+                                           EdgeCentralityMap edge_centrality_map,
+                                           VertexIndexMap vertex_index)
+  {
+    typedef typename graph_traits<Graph>::degree_size_type degree_size_type;
+    typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor;
+    typedef typename mpl::if_c<(is_same<CentralityMap, 
+                                        dummy_property_map>::value),
+                                         EdgeCentralityMap, 
+                               CentralityMap>::type a_centrality_map;
+    typedef typename property_traits<a_centrality_map>::value_type 
+      centrality_type;
+
+    typename graph_traits<Graph>::vertices_size_type V = num_vertices(g);
+    
+    std::vector<std::vector<edge_descriptor> > incoming(V);
+    std::vector<centrality_type> distance(V);
+    std::vector<centrality_type> dependency(V);
+    std::vector<degree_size_type> path_count(V);
+
+    brandes_betweenness_centrality(
+      g, centrality, edge_centrality_map,
+      make_iterator_property_map(incoming.begin(), vertex_index),
+      make_iterator_property_map(distance.begin(), vertex_index),
+      make_iterator_property_map(dependency.begin(), vertex_index),
+      make_iterator_property_map(path_count.begin(), vertex_index),
+      vertex_index);
+  }
+
+  template<typename WeightMap>
+  struct brandes_betweenness_centrality_dispatch1
+  {
+    template<typename Graph, typename CentralityMap, 
+             typename EdgeCentralityMap, typename VertexIndexMap>
+    static void 
+    run(const Graph& g, CentralityMap centrality, 
+        EdgeCentralityMap edge_centrality_map, VertexIndexMap vertex_index,
+        WeightMap weight_map)
+    {
+      brandes_betweenness_centrality_dispatch2(g, centrality, edge_centrality_map,
+                                               weight_map, vertex_index);
+    }
+  };
+
+  template<>
+  struct brandes_betweenness_centrality_dispatch1<param_not_found>
+  {
+    template<typename Graph, typename CentralityMap, 
+             typename EdgeCentralityMap, typename VertexIndexMap>
+    static void 
+    run(const Graph& g, CentralityMap centrality, 
+        EdgeCentralityMap edge_centrality_map, VertexIndexMap vertex_index,
+        param_not_found)
+    {
+      brandes_betweenness_centrality_dispatch2(g, centrality, edge_centrality_map,
+                                               vertex_index);
+    }
+  };
+
+  template <typename T>
+  struct is_bgl_named_params {
+    BOOST_STATIC_CONSTANT(bool, value = false);
+  };
+
+  template <typename Param, typename Tag, typename Rest>
+  struct is_bgl_named_params<bgl_named_params<Param, Tag, Rest> > {
+    BOOST_STATIC_CONSTANT(bool, value = true);
+  };
+
+} } // end namespace detail::graph
+
+template<typename Graph, typename Param, typename Tag, typename Rest>
+void 
+brandes_betweenness_centrality(const Graph& g, 
+                               const bgl_named_params<Param,Tag,Rest>& params
+                               BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
+{
+  typedef bgl_named_params<Param,Tag,Rest> named_params;
+
+  typedef typename get_param_type<edge_weight_t, named_params>::type ew;
+  detail::graph::brandes_betweenness_centrality_dispatch1<ew>::run(
+    g, 
+    choose_param(get_param(params, vertex_centrality), 
+                 dummy_property_map()),
+    choose_param(get_param(params, edge_centrality), 
+                 dummy_property_map()),
+    choose_const_pmap(get_param(params, vertex_index), g, vertex_index),
+    get_param(params, edge_weight));
+}
+
+// disable_if is required to work around problem with MSVC 7.1 (it seems to not
+// get partial ordering getween this overload and the previous one correct)
+template<typename Graph, typename CentralityMap>
+typename disable_if<detail::graph::is_bgl_named_params<CentralityMap>,
+                    void>::type
+brandes_betweenness_centrality(const Graph& g, CentralityMap centrality
+                               BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
+{
+  detail::graph::brandes_betweenness_centrality_dispatch2(
+    g, centrality, dummy_property_map(), get(vertex_index, g));
+}
+
+template<typename Graph, typename CentralityMap, typename EdgeCentralityMap>
+void 
+brandes_betweenness_centrality(const Graph& g, CentralityMap centrality,
+                               EdgeCentralityMap edge_centrality_map
+                               BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
+{
+  detail::graph::brandes_betweenness_centrality_dispatch2(
+    g, centrality, edge_centrality_map, get(vertex_index, g));
+}
+
+/**
+ * Converts "absolute" betweenness centrality (as computed by the
+ * brandes_betweenness_centrality algorithm) in the centrality map
+ * into "relative" centrality. The result is placed back into the
+ * given centrality map.
+ */
+template<typename Graph, typename CentralityMap>
+void 
+relative_betweenness_centrality(const Graph& g, CentralityMap centrality)
+{
+  typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator;
+  typedef typename property_traits<CentralityMap>::value_type centrality_type;
+
+  typename graph_traits<Graph>::vertices_size_type n = num_vertices(g);
+  centrality_type factor = centrality_type(2)/centrality_type(n*n - 3*n + 2);
+  vertex_iterator v, v_end;
+  for (boost::tie(v, v_end) = vertices(g); v != v_end; ++v) {
+    put(centrality, *v, factor * get(centrality, *v));
+  }
+}
+
+// Compute the central point dominance of a graph.
+template<typename Graph, typename CentralityMap>
+typename property_traits<CentralityMap>::value_type
+central_point_dominance(const Graph& g, CentralityMap centrality
+                        BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
+{
+  using std::max;
+
+  typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator;
+  typedef typename property_traits<CentralityMap>::value_type centrality_type;
+
+  typename graph_traits<Graph>::vertices_size_type n = num_vertices(g);
+
+  // Find max centrality
+  centrality_type max_centrality(0);
+  vertex_iterator v, v_end;
+  for (boost::tie(v, v_end) = vertices(g); v != v_end; ++v) {
+    max_centrality = (max)(max_centrality, get(centrality, *v));
+  }
+
+  // Compute central point dominance
+  centrality_type sum(0);
+  for (boost::tie(v, v_end) = vertices(g); v != v_end; ++v) {
+    sum += (max_centrality - get(centrality, *v));
+  }
+  return sum/(n-1);
+}
+
+} // end namespace boost
+
+#endif // BOOST_GRAPH_BRANDES_BETWEENNESS_CENTRALITY_HPP