betweenness_centrality.hpp

// 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>::edge_iterator edge_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 (tie(s, s_end) = vertices(g); s != s_end; ++s) {
      // Initialize for this iteration
      vertex_iterator w, w_end;
      for (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>::vertex_descriptor vertex_descriptor;
    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>::vertex_descriptor vertex_descriptor;
    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<error_property_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,
        error_property_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 property_value<named_params, edge_weight_t>::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 (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 (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 (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