File: betweenness_centrality.hpp

package info (click to toggle)
boost1.62 1.62.0+dfsg-4
  • links: PTS, VCS
  • area: main
  • in suites: stretch
  • size: 686,420 kB
  • sloc: cpp: 2,609,004; xml: 972,558; ansic: 53,674; python: 32,437; sh: 8,829; asm: 3,071; cs: 2,121; makefile: 964; perl: 859; yacc: 472; php: 132; ruby: 94; f90: 55; sql: 13; csh: 6
file content (616 lines) | stat: -rw-r--r-- 24,666 bytes parent folder | download | duplicates (13)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
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