1// Copyright 2004 The Trustees of Indiana University.
2
3// Distributed under the Boost Software License, Version 1.0.
4// (See accompanying file LICENSE_1_0.txt or copy at
5// http://www.boost.org/LICENSE_1_0.txt)
6
7// Authors: Douglas Gregor
8// Andrew Lumsdaine
9#ifndef BOOST_GRAPH_BRANDES_BETWEENNESS_CENTRALITY_HPP
10#define BOOST_GRAPH_BRANDES_BETWEENNESS_CENTRALITY_HPP
11
12#include <stack>
13#include <vector>
14#include <boost/graph/overloading.hpp>
15#include <boost/graph/dijkstra_shortest_paths.hpp>
16#include <boost/graph/breadth_first_search.hpp>
17#include <boost/graph/relax.hpp>
18#include <boost/graph/graph_traits.hpp>
19#include <boost/tuple/tuple.hpp>
20#include <boost/type_traits/is_convertible.hpp>
21#include <boost/type_traits/is_same.hpp>
22#include <boost/mpl/if.hpp>
23#include <boost/property_map/property_map.hpp>
24#include <boost/graph/named_function_params.hpp>
25#include <algorithm>
26
27namespace boost {
28
29namespace detail { namespace graph {
30
31 /**
32 * Customized visitor passed to Dijkstra's algorithm by Brandes'
33 * betweenness centrality algorithm. This visitor is responsible for
34 * keeping track of the order in which vertices are discovered, the
35 * predecessors on the shortest path(s) to a vertex, and the number
36 * of shortest paths.
37 */
38 template<typename Graph, typename WeightMap, typename IncomingMap,
39 typename DistanceMap, typename PathCountMap>
40 struct brandes_dijkstra_visitor : public bfs_visitor<>
41 {
42 typedef typename graph_traits<Graph>::vertex_descriptor vertex_descriptor;
43 typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor;
44
45 brandes_dijkstra_visitor(std::stack<vertex_descriptor>& ordered_vertices,
46 WeightMap weight,
47 IncomingMap incoming,
48 DistanceMap distance,
49 PathCountMap path_count)
50 : ordered_vertices(ordered_vertices), weight(weight),
51 incoming(incoming), distance(distance),
52 path_count(path_count)
53 { }
54
55 /**
56 * Whenever an edge e = (v, w) is relaxed, the incoming edge list
57 * for w is set to {(v, w)} and the shortest path count of w is set to
58 * the number of paths that reach {v}.
59 */
60 void edge_relaxed(edge_descriptor e, const Graph& g)
61 {
62 vertex_descriptor v = source(e, g), w = target(e, g);
63 incoming[w].clear();
64 incoming[w].push_back(e);
65 put(path_count, w, get(path_count, v));
66 }
67
68 /**
69 * If an edge e = (v, w) was not relaxed, it may still be the case
70 * that we've found more equally-short paths, so include {(v, w)} in the
71 * incoming edges of w and add all of the shortest paths to v to the
72 * shortest path count of w.
73 */
74 void edge_not_relaxed(edge_descriptor e, const Graph& g)
75 {
76 typedef typename property_traits<WeightMap>::value_type weight_type;
77 typedef typename property_traits<DistanceMap>::value_type distance_type;
78 vertex_descriptor v = source(e, g), w = target(e, g);
79 distance_type d_v = get(distance, v), d_w = get(distance, w);
80 weight_type w_e = get(weight, e);
81
82 closed_plus<distance_type> combine;
83 if (d_w == combine(d_v, w_e)) {
84 put(path_count, w, get(path_count, w) + get(path_count, v));
85 incoming[w].push_back(e);
86 }
87 }
88
89 /// Keep track of vertices as they are reached
90 void examine_vertex(vertex_descriptor w, const Graph&)
91 {
92 ordered_vertices.push(w);
93 }
94
95 private:
96 std::stack<vertex_descriptor>& ordered_vertices;
97 WeightMap weight;
98 IncomingMap incoming;
99 DistanceMap distance;
100 PathCountMap path_count;
101 };
102
103 /**
104 * Function object that calls Dijkstra's shortest paths algorithm
105 * using the Dijkstra visitor for the Brandes betweenness centrality
106 * algorithm.
107 */
108 template<typename WeightMap>
109 struct brandes_dijkstra_shortest_paths
110 {
111 brandes_dijkstra_shortest_paths(WeightMap weight_map)
112 : weight_map(weight_map) { }
113
114 template<typename Graph, typename IncomingMap, typename DistanceMap,
115 typename PathCountMap, typename VertexIndexMap>
116 void
117 operator()(Graph& g,
118 typename graph_traits<Graph>::vertex_descriptor s,
119 std::stack<typename graph_traits<Graph>::vertex_descriptor>& ov,
120 IncomingMap incoming,
121 DistanceMap distance,
122 PathCountMap path_count,
123 VertexIndexMap vertex_index)
124 {
125 typedef brandes_dijkstra_visitor<Graph, WeightMap, IncomingMap,
126 DistanceMap, PathCountMap> visitor_type;
127 visitor_type visitor(ov, weight_map, incoming, distance, path_count);
128
129 dijkstra_shortest_paths(g, s,
130 boost::weight_map(weight_map)
131 .vertex_index_map(vertex_index)
132 .distance_map(distance)
133 .visitor(visitor));
134 }
135
136 private:
137 WeightMap weight_map;
138 };
139
140 /**
141 * Function object that invokes breadth-first search for the
142 * unweighted form of the Brandes betweenness centrality algorithm.
143 */
144 struct brandes_unweighted_shortest_paths
145 {
146 /**
147 * Customized visitor passed to breadth-first search, which
148 * records predecessor and the number of shortest paths to each
149 * vertex.
150 */
151 template<typename Graph, typename IncomingMap, typename DistanceMap,
152 typename PathCountMap>
153 struct visitor_type : public bfs_visitor<>
154 {
155 typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor;
156 typedef typename graph_traits<Graph>::vertex_descriptor
157 vertex_descriptor;
158
159 visitor_type(IncomingMap incoming, DistanceMap distance,
160 PathCountMap path_count,
161 std::stack<vertex_descriptor>& ordered_vertices)
162 : incoming(incoming), distance(distance),
163 path_count(path_count), ordered_vertices(ordered_vertices) { }
164
165 /// Keep track of vertices as they are reached
166 void examine_vertex(vertex_descriptor v, Graph&)
167 {
168 ordered_vertices.push(v);
169 }
170
171 /**
172 * Whenever an edge e = (v, w) is labelled a tree edge, the
173 * incoming edge list for w is set to {(v, w)} and the shortest
174 * path count of w is set to the number of paths that reach {v}.
175 */
176 void tree_edge(edge_descriptor e, Graph& g)
177 {
178 vertex_descriptor v = source(e, g);
179 vertex_descriptor w = target(e, g);
180 put(distance, w, get(distance, v) + 1);
181
182 put(path_count, w, get(path_count, v));
183 incoming[w].push_back(e);
184 }
185
186 /**
187 * If an edge e = (v, w) is not a tree edge, it may still be the
188 * case that we've found more equally-short paths, so include (v, w)
189 * in the incoming edge list of w and add all of the shortest
190 * paths to v to the shortest path count of w.
191 */
192 void non_tree_edge(edge_descriptor e, Graph& g)
193 {
194 vertex_descriptor v = source(e, g);
195 vertex_descriptor w = target(e, g);
196 if (get(distance, w) == get(distance, v) + 1) {
197 put(path_count, w, get(path_count, w) + get(path_count, v));
198 incoming[w].push_back(e);
199 }
200 }
201
202 private:
203 IncomingMap incoming;
204 DistanceMap distance;
205 PathCountMap path_count;
206 std::stack<vertex_descriptor>& ordered_vertices;
207 };
208
209 template<typename Graph, typename IncomingMap, typename DistanceMap,
210 typename PathCountMap, typename VertexIndexMap>
211 void
212 operator()(Graph& g,
213 typename graph_traits<Graph>::vertex_descriptor s,
214 std::stack<typename graph_traits<Graph>::vertex_descriptor>& ov,
215 IncomingMap incoming,
216 DistanceMap distance,
217 PathCountMap path_count,
218 VertexIndexMap vertex_index)
219 {
220 typedef typename graph_traits<Graph>::vertex_descriptor
221 vertex_descriptor;
222
223 visitor_type<Graph, IncomingMap, DistanceMap, PathCountMap>
224 visitor(incoming, distance, path_count, ov);
225
226 std::vector<default_color_type>
227 colors(num_vertices(g), color_traits<default_color_type>::white());
228 boost::queue<vertex_descriptor> Q;
229 breadth_first_visit(g, s, Q, visitor,
230 make_iterator_property_map(colors.begin(),
231 vertex_index));
232 }
233 };
234
235 // When the edge centrality map is a dummy property map, no
236 // initialization is needed.
237 template<typename Iter>
238 inline void
239 init_centrality_map(std::pair<Iter, Iter>, dummy_property_map) { }
240
241 // When we have a real edge centrality map, initialize all of the
242 // centralities to zero.
243 template<typename Iter, typename Centrality>
244 void
245 init_centrality_map(std::pair<Iter, Iter> keys, Centrality centrality_map)
246 {
247 typedef typename property_traits<Centrality>::value_type
248 centrality_type;
249 while (keys.first != keys.second) {
250 put(centrality_map, *keys.first, centrality_type(0));
251 ++keys.first;
252 }
253 }
254
255 // When the edge centrality map is a dummy property map, no update
256 // is performed.
257 template<typename Key, typename T>
258 inline void
259 update_centrality(dummy_property_map, const Key&, const T&) { }
260
261 // When we have a real edge centrality map, add the value to the map
262 template<typename CentralityMap, typename Key, typename T>
263 inline void
264 update_centrality(CentralityMap centrality_map, Key k, const T& x)
265 { put(centrality_map, k, get(centrality_map, k) + x); }
266
267 template<typename Iter>
268 inline void
269 divide_centrality_by_two(std::pair<Iter, Iter>, dummy_property_map) {}
270
271 template<typename Iter, typename CentralityMap>
272 inline void
273 divide_centrality_by_two(std::pair<Iter, Iter> keys,
274 CentralityMap centrality_map)
275 {
276 typename property_traits<CentralityMap>::value_type two(2);
277 while (keys.first != keys.second) {
278 put(centrality_map, *keys.first, get(centrality_map, *keys.first) / two);
279 ++keys.first;
280 }
281 }
282
283 template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
284 typename IncomingMap, typename DistanceMap,
285 typename DependencyMap, typename PathCountMap,
286 typename VertexIndexMap, typename ShortestPaths>
287 void
288 brandes_betweenness_centrality_impl(const Graph& g,
289 CentralityMap centrality, // C_B
290 EdgeCentralityMap edge_centrality_map,
291 IncomingMap incoming, // P
292 DistanceMap distance, // d
293 DependencyMap dependency, // delta
294 PathCountMap path_count, // sigma
295 VertexIndexMap vertex_index,
296 ShortestPaths shortest_paths)
297 {
298 typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator;
299 typedef typename graph_traits<Graph>::vertex_descriptor vertex_descriptor;
300
301 // Initialize centrality
302 init_centrality_map(vertices(g), centrality);
303 init_centrality_map(edges(g), edge_centrality_map);
304
305 std::stack<vertex_descriptor> ordered_vertices;
306 vertex_iterator s, s_end;
307 for (boost::tie(s, s_end) = vertices(g); s != s_end; ++s) {
308 // Initialize for this iteration
309 vertex_iterator w, w_end;
310 for (boost::tie(w, w_end) = vertices(g); w != w_end; ++w) {
311 incoming[*w].clear();
312 put(path_count, *w, 0);
313 put(dependency, *w, 0);
314 }
315 put(path_count, *s, 1);
316
317 // Execute the shortest paths algorithm. This will be either
318 // Dijkstra's algorithm or a customized breadth-first search,
319 // depending on whether the graph is weighted or unweighted.
320 shortest_paths(g, *s, ordered_vertices, incoming, distance,
321 path_count, vertex_index);
322
323 while (!ordered_vertices.empty()) {
324 vertex_descriptor w = ordered_vertices.top();
325 ordered_vertices.pop();
326
327 typedef typename property_traits<IncomingMap>::value_type
328 incoming_type;
329 typedef typename incoming_type::iterator incoming_iterator;
330 typedef typename property_traits<DependencyMap>::value_type
331 dependency_type;
332
333 for (incoming_iterator vw = incoming[w].begin();
334 vw != incoming[w].end(); ++vw) {
335 vertex_descriptor v = source(*vw, g);
336 dependency_type factor = dependency_type(get(path_count, v))
337 / dependency_type(get(path_count, w));
338 factor *= (dependency_type(1) + get(dependency, w));
339 put(dependency, v, get(dependency, v) + factor);
340 update_centrality(edge_centrality_map, *vw, factor);
341 }
342
343 if (w != *s) {
344 update_centrality(centrality, w, get(dependency, w));
345 }
346 }
347 }
348
349 typedef typename graph_traits<Graph>::directed_category directed_category;
350 const bool is_undirected =
351 is_convertible<directed_category*, undirected_tag*>::value;
352 if (is_undirected) {
353 divide_centrality_by_two(vertices(g), centrality);
354 divide_centrality_by_two(edges(g), edge_centrality_map);
355 }
356 }
357
358} } // end namespace detail::graph
359
360template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
361 typename IncomingMap, typename DistanceMap,
362 typename DependencyMap, typename PathCountMap,
363 typename VertexIndexMap>
364void
365brandes_betweenness_centrality(const Graph& g,
366 CentralityMap centrality, // C_B
367 EdgeCentralityMap edge_centrality_map,
368 IncomingMap incoming, // P
369 DistanceMap distance, // d
370 DependencyMap dependency, // delta
371 PathCountMap path_count, // sigma
372 VertexIndexMap vertex_index
373 BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
374{
375 detail::graph::brandes_unweighted_shortest_paths shortest_paths;
376
377 detail::graph::brandes_betweenness_centrality_impl(g, centrality,
378 edge_centrality_map,
379 incoming, distance,
380 dependency, path_count,
381 vertex_index,
382 shortest_paths);
383}
384
385template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
386 typename IncomingMap, typename DistanceMap,
387 typename DependencyMap, typename PathCountMap,
388 typename VertexIndexMap, typename WeightMap>
389void
390brandes_betweenness_centrality(const Graph& g,
391 CentralityMap centrality, // C_B
392 EdgeCentralityMap edge_centrality_map,
393 IncomingMap incoming, // P
394 DistanceMap distance, // d
395 DependencyMap dependency, // delta
396 PathCountMap path_count, // sigma
397 VertexIndexMap vertex_index,
398 WeightMap weight_map
399 BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
400{
401 detail::graph::brandes_dijkstra_shortest_paths<WeightMap>
402 shortest_paths(weight_map);
403
404 detail::graph::brandes_betweenness_centrality_impl(g, centrality,
405 edge_centrality_map,
406 incoming, distance,
407 dependency, path_count,
408 vertex_index,
409 shortest_paths);
410}
411
412namespace detail { namespace graph {
413 template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
414 typename WeightMap, typename VertexIndexMap>
415 void
416 brandes_betweenness_centrality_dispatch2(const Graph& g,
417 CentralityMap centrality,
418 EdgeCentralityMap edge_centrality_map,
419 WeightMap weight_map,
420 VertexIndexMap vertex_index)
421 {
422 typedef typename graph_traits<Graph>::degree_size_type degree_size_type;
423 typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor;
424 typedef typename mpl::if_c<(is_same<CentralityMap,
425 dummy_property_map>::value),
426 EdgeCentralityMap,
427 CentralityMap>::type a_centrality_map;
428 typedef typename property_traits<a_centrality_map>::value_type
429 centrality_type;
430
431 typename graph_traits<Graph>::vertices_size_type V = num_vertices(g);
432
433 std::vector<std::vector<edge_descriptor> > incoming(V);
434 std::vector<centrality_type> distance(V);
435 std::vector<centrality_type> dependency(V);
436 std::vector<degree_size_type> path_count(V);
437
438 brandes_betweenness_centrality(
439 g, centrality, edge_centrality_map,
440 make_iterator_property_map(incoming.begin(), vertex_index),
441 make_iterator_property_map(distance.begin(), vertex_index),
442 make_iterator_property_map(dependency.begin(), vertex_index),
443 make_iterator_property_map(path_count.begin(), vertex_index),
444 vertex_index,
445 weight_map);
446 }
447
448
449 template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
450 typename VertexIndexMap>
451 void
452 brandes_betweenness_centrality_dispatch2(const Graph& g,
453 CentralityMap centrality,
454 EdgeCentralityMap edge_centrality_map,
455 VertexIndexMap vertex_index)
456 {
457 typedef typename graph_traits<Graph>::degree_size_type degree_size_type;
458 typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor;
459 typedef typename mpl::if_c<(is_same<CentralityMap,
460 dummy_property_map>::value),
461 EdgeCentralityMap,
462 CentralityMap>::type a_centrality_map;
463 typedef typename property_traits<a_centrality_map>::value_type
464 centrality_type;
465
466 typename graph_traits<Graph>::vertices_size_type V = num_vertices(g);
467
468 std::vector<std::vector<edge_descriptor> > incoming(V);
469 std::vector<centrality_type> distance(V);
470 std::vector<centrality_type> dependency(V);
471 std::vector<degree_size_type> path_count(V);
472
473 brandes_betweenness_centrality(
474 g, centrality, edge_centrality_map,
475 make_iterator_property_map(incoming.begin(), vertex_index),
476 make_iterator_property_map(distance.begin(), vertex_index),
477 make_iterator_property_map(dependency.begin(), vertex_index),
478 make_iterator_property_map(path_count.begin(), vertex_index),
479 vertex_index);
480 }
481
482 template<typename WeightMap>
483 struct brandes_betweenness_centrality_dispatch1
484 {
485 template<typename Graph, typename CentralityMap,
486 typename EdgeCentralityMap, typename VertexIndexMap>
487 static void
488 run(const Graph& g, CentralityMap centrality,
489 EdgeCentralityMap edge_centrality_map, VertexIndexMap vertex_index,
490 WeightMap weight_map)
491 {
492 brandes_betweenness_centrality_dispatch2(g, centrality, edge_centrality_map,
493 weight_map, vertex_index);
494 }
495 };
496
497 template<>
498 struct brandes_betweenness_centrality_dispatch1<param_not_found>
499 {
500 template<typename Graph, typename CentralityMap,
501 typename EdgeCentralityMap, typename VertexIndexMap>
502 static void
503 run(const Graph& g, CentralityMap centrality,
504 EdgeCentralityMap edge_centrality_map, VertexIndexMap vertex_index,
505 param_not_found)
506 {
507 brandes_betweenness_centrality_dispatch2(g, centrality, edge_centrality_map,
508 vertex_index);
509 }
510 };
511
512 template <typename T>
513 struct is_bgl_named_params {
514 BOOST_STATIC_CONSTANT(bool, value = false);
515 };
516
517 template <typename Param, typename Tag, typename Rest>
518 struct is_bgl_named_params<bgl_named_params<Param, Tag, Rest> > {
519 BOOST_STATIC_CONSTANT(bool, value = true);
520 };
521
522} } // end namespace detail::graph
523
524template<typename Graph, typename Param, typename Tag, typename Rest>
525void
526brandes_betweenness_centrality(const Graph& g,
527 const bgl_named_params<Param,Tag,Rest>& params
528 BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
529{
530 typedef bgl_named_params<Param,Tag,Rest> named_params;
531
532 typedef typename get_param_type<edge_weight_t, named_params>::type ew;
533 detail::graph::brandes_betweenness_centrality_dispatch1<ew>::run(
534 g,
535 choose_param(get_param(params, vertex_centrality),
536 dummy_property_map()),
537 choose_param(get_param(params, edge_centrality),
538 dummy_property_map()),
539 choose_const_pmap(get_param(params, vertex_index), g, vertex_index),
540 get_param(params, edge_weight));
541}
542
543// disable_if is required to work around problem with MSVC 7.1 (it seems to not
544// get partial ordering getween this overload and the previous one correct)
545template<typename Graph, typename CentralityMap>
546typename disable_if<detail::graph::is_bgl_named_params<CentralityMap>,
547 void>::type
548brandes_betweenness_centrality(const Graph& g, CentralityMap centrality
549 BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
550{
551 detail::graph::brandes_betweenness_centrality_dispatch2(
552 g, centrality, dummy_property_map(), get(vertex_index, g));
553}
554
555template<typename Graph, typename CentralityMap, typename EdgeCentralityMap>
556void
557brandes_betweenness_centrality(const Graph& g, CentralityMap centrality,
558 EdgeCentralityMap edge_centrality_map
559 BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
560{
561 detail::graph::brandes_betweenness_centrality_dispatch2(
562 g, centrality, edge_centrality_map, get(vertex_index, g));
563}
564
565/**
566 * Converts "absolute" betweenness centrality (as computed by the
567 * brandes_betweenness_centrality algorithm) in the centrality map
568 * into "relative" centrality. The result is placed back into the
569 * given centrality map.
570 */
571template<typename Graph, typename CentralityMap>
572void
573relative_betweenness_centrality(const Graph& g, CentralityMap centrality)
574{
575 typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator;
576 typedef typename property_traits<CentralityMap>::value_type centrality_type;
577
578 typename graph_traits<Graph>::vertices_size_type n = num_vertices(g);
579 centrality_type factor = centrality_type(2)/centrality_type(n*n - 3*n + 2);
580 vertex_iterator v, v_end;
581 for (boost::tie(v, v_end) = vertices(g); v != v_end; ++v) {
582 put(centrality, *v, factor * get(centrality, *v));
583 }
584}
585
586// Compute the central point dominance of a graph.
587template<typename Graph, typename CentralityMap>
588typename property_traits<CentralityMap>::value_type
589central_point_dominance(const Graph& g, CentralityMap centrality
590 BOOST_GRAPH_ENABLE_IF_MODELS_PARM(Graph,vertex_list_graph_tag))
591{
592 using std::max;
593
594 typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator;
595 typedef typename property_traits<CentralityMap>::value_type centrality_type;
596
597 typename graph_traits<Graph>::vertices_size_type n = num_vertices(g);
598
599 // Find max centrality
600 centrality_type max_centrality(0);
601 vertex_iterator v, v_end;
602 for (boost::tie(v, v_end) = vertices(g); v != v_end; ++v) {
603 max_centrality = (max)(max_centrality, get(centrality, *v));
604 }
605
606 // Compute central point dominance
607 centrality_type sum(0);
608 for (boost::tie(v, v_end) = vertices(g); v != v_end; ++v) {
609 sum += (max_centrality - get(centrality, *v));
610 }
611 return sum/(n-1);
612}
613
614} // end namespace boost
615
616#endif // BOOST_GRAPH_BRANDES_BETWEENNESS_CENTRALITY_HPP
617

source code of boost/boost/graph/betweenness_centrality.hpp