ceph/src/boost/libs/graph_parallel/doc/html/betweenness_centrality.html

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  12 <h1 class="title"><a class="reference external" href="http://www.osl.iu.edu/research/pbgl"><img align="middle" alt="Parallel BGL" class="align-middle" src="pbgl-logo.png" /></a> Betweenness Centrality</h1>
  13
  14 <!-- Copyright (C) 2004-2009 The Trustees of Indiana University.
  15 Use, modification and distribution is subject to the Boost Software
  16 License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
  17 http://www.boost.org/LICENSE_1_0.txt) -->
  18 <pre class="literal-block">
  19 // named parameter versions
  20 template&lt;typename Graph, typename Param, typename Tag, typename Rest&gt;
  21 void
  22 brandes_betweenness_centrality(const Graph&amp; g,
  23                                const bgl_named_params&lt;Param,Tag,Rest&gt;&amp; params);
  24
  25 template&lt;typename Graph, typename CentralityMap&gt;
  26 void
  27 brandes_betweenness_centrality(const Graph&amp; g, CentralityMap centrality);
  28
  29 template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap&gt;
  30 void
  31 brandes_betweenness_centrality(const Graph&amp; g, CentralityMap centrality,
  32                                EdgeCentralityMap edge_centrality_map);
  33
  34 // non-named parameter versions
  35 template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap,
  36          typename IncomingMap, typename DistanceMap, typename DependencyMap,
  37          typename PathCountMap, typename VertexIndexMap, typename Buffer&gt;
  38 void
  39 brandes_betweenness_centrality(const Graph&amp; g,
  40                                CentralityMap centrality,
  41                                EdgeCentralityMap edge_centrality_map,
  42                                IncomingMap incoming,
  43                                DistanceMap distance,
  44                                DependencyMap dependency,
  45                                PathCountMap path_count,
  46                                VertexIndexMap vertex_index,
  47                                Buffer sources,
  48                                typename property_traits&lt;DistanceMap&gt;::value_type delta);
  49
  50 template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap,
  51          typename IncomingMap, typename DistanceMap, typename DependencyMap,
  52          typename PathCountMap, typename VertexIndexMap, typename WeightMap,
  53          typename Buffer&gt;
  54 void
  55 brandes_betweenness_centrality(const Graph&amp; g,
  56                                CentralityMap centrality,
  57                                EdgeCentralityMap edge_centrality_map,
  58                                IncomingMap incoming,
  59                                DistanceMap distance,
  60                                DependencyMap dependency,
  61                                PathCountMap path_count,
  62                                VertexIndexMap vertex_index,
  63                                Buffer sources,
  64                                typename property_traits&lt;WeightMap&gt;::value_type delta,
  65                                WeightMap weight_map);
  66
  67 // helper functions
  68 template&lt;typename Graph, typename CentralityMap&gt;
  69 typename property_traits&lt;CentralityMap&gt;::value_type
  70 central_point_dominance(const Graph&amp; g, CentralityMap centrality);
  71 </pre>
  72 <p>The <tt class="docutils literal"><span class="pre">brandes_betweenness_centrality()</span></tt> function computes the
  73 betweenness centrality of the vertices and edges in a graph.  The
  74 method of calculating betweenness centrality in <em>O(V)</em> space is due to
  75 Brandes <a class="citation-reference" href="#brandes01" id="id1">[Brandes01]</a>.  The algorithm itself is a modification of
  76 Brandes algorithm by Edmonds <a class="citation-reference" href="#edmonds09" id="id2">[Edmonds09]</a>.</p>
  77 <div class="contents topic" id="contents">
  78 <p class="topic-title first">Contents</p>
  79 <ul class="simple">
  80 <li><a class="reference internal" href="#where-defined" id="id3">Where Defined</a></li>
  81 <li><a class="reference internal" href="#parameters" id="id4">Parameters</a></li>
  82 <li><a class="reference internal" href="#complexity" id="id5">Complexity</a></li>
  83 <li><a class="reference internal" href="#algorithm-description" id="id6">Algorithm Description</a></li>
  84 <li><a class="reference internal" href="#bibliography" id="id7">Bibliography</a></li>
  85 </ul>
  86 </div>
  87 <div class="section" id="where-defined">
  88 <h1><a class="toc-backref" href="#id3">Where Defined</a></h1>
  89 <p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/distributed/betweenness_centrality.hpp</span></tt>&gt;</p>
  90 <p>also accessible from</p>
  91 <p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/betweenness_centrality.hpp</span></tt>&gt;</p>
  92 </div>
  93 <div class="section" id="parameters">
  94 <h1><a class="toc-backref" href="#id4">Parameters</a></h1>
  95 <dl class="docutils">
  96 <dt>IN:  <tt class="docutils literal"><span class="pre">const</span> <span class="pre">Graph&amp;</span> <span class="pre">g</span></tt></dt>
  97 <dd>The graph type must be a model of <a class="reference external" href="DistributedGraph.html">Distributed Graph</a>.  The graph
  98 type must also model the <a class="reference external" href="http://www.boost.org/libs/graph/doc/IncidenceGraph.html">Incidence Graph</a> concept.  0-weighted
  99 edges in <tt class="docutils literal"><span class="pre">g</span></tt> will result in undefined behavior.</dd>
 100 <dt>IN: <tt class="docutils literal"><span class="pre">CentralityMap</span> <span class="pre">centrality</span></tt></dt>
 101 <dd><p class="first">A centrality map may be supplied to the algorithm, if not supplied a
 102 <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no vertex centrality
 103 information will be recorded.  The <tt class="docutils literal"><span class="pre">CentralityMap</span></tt> type must be a
 104 <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  The key type must be the graph's
 105 vertex descriptor type.</p>
 106 <p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
 107 </dd>
 108 <dt>IN:  <tt class="docutils literal"><span class="pre">EdgeCentralityMap</span> <span class="pre">edge_centrality_map</span></tt></dt>
 109 <dd><p class="first">An edge centrality map may be supplied to the algorithm, if not
 110 supplied a <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no edge
 111 centrality information will be recorded.  The <tt class="docutils literal"><span class="pre">EdgeCentralityMap</span></tt>
 112 type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  The key type must be
 113 the graph's vertex descriptor type.</p>
 114 <p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
 115 </dd>
 116 <dt>IN:  <tt class="docutils literal"><span class="pre">IncomingMap</span> <span class="pre">incoming</span></tt></dt>
 117 <dd><p class="first">The incoming map contains the incoming edges to a vertex that are
 118 part of shortest paths to that vertex.  The <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> type
 119 must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  Its key type and value type
 120 must both be the graph's vertex descriptor type.</p>
 121 <dl class="last docutils">
 122 <dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
 123 <dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of <tt class="docutils literal"><span class="pre">std::vector</span></tt> of the graph's vertex
 124 descriptor type.</dd>
 125 </dl>
 126 </dd>
 127 <dt>IN:  <tt class="docutils literal"><span class="pre">DistanceMap</span> <span class="pre">distance</span></tt></dt>
 128 <dd><p class="first">The distance map records the distance to vertices during the
 129 shortest paths portion of the algorithm.  The <tt class="docutils literal"><span class="pre">DistanceMap</span></tt> type
 130 must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  Its key type must be the
 131 graph's vertex descriptor type.</p>
 132 <dl class="last docutils">
 133 <dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
 134 <dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the value type of the <tt class="docutils literal"><span class="pre">CentralityMap</span></tt>.</dd>
 135 </dl>
 136 </dd>
 137 <dt>IN: <tt class="docutils literal"><span class="pre">DependencyMap</span> <span class="pre">dependency</span></tt></dt>
 138 <dd><p class="first">The dependency map records the dependency of each vertex during the
 139 centrality calculation portion of the algorithm.  The
 140 <tt class="docutils literal"><span class="pre">DependencyMap</span></tt> type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  Its
 141 key type must be the graph's vertex descriptor type.</p>
 142 <dl class="last docutils">
 143 <dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
 144 <dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the value type of the <tt class="docutils literal"><span class="pre">CentralityMap</span></tt>.</dd>
 145 </dl>
 146 </dd>
 147 </dl>
 148 <p>IN:  <tt class="docutils literal"><span class="pre">PathCountMap</span> <span class="pre">path_count</span></tt></p>
 149 <blockquote>
 150 <p>The path count map records the number of shortest paths each vertex
 151 is on during the centrality calculation portion of the algorithm.
 152 The <tt class="docutils literal"><span class="pre">PathCountMap</span></tt> type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.
 153 Its key type must be the graph's vertex descriptor type.</p>
 154 <dl class="docutils">
 155 <dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
 156 <dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the graph's degree size type.</dd>
 157 </dl>
 158 </blockquote>
 159 <dl class="docutils">
 160 <dt>IN:  <tt class="docutils literal"><span class="pre">VertexIndexMap</span> <span class="pre">vertex_index</span></tt></dt>
 161 <dd><p class="first">A model of <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose key type is the vertex
 162 descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt> and whose value type is an
 163 integral type. The property map should map from vertices to their
 164 (local) indices in the range <em>[0, num_vertices(g))</em>.</p>
 165 <p class="last"><strong>Default</strong>: <tt class="docutils literal"><span class="pre">get(vertex_index,</span> <span class="pre">g)</span></tt></p>
 166 </dd>
 167 <dt>IN:  <tt class="docutils literal"><span class="pre">WeightMap</span> <span class="pre">weight_map</span></tt></dt>
 168 <dd>A model of <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose key type is the edge
 169 descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt>.  If not supplied the betweenness
 170 centrality calculation will be unweighted.</dd>
 171 <dt>IN: <tt class="docutils literal"><span class="pre">Buffer</span> <span class="pre">sources</span></tt></dt>
 172 <dd><p class="first">A model of <a class="reference external" href="http://www.boost.org/libs/graph/doc/Buffer.html">Buffer</a> containing the starting vertices for the
 173 algorithm.  If <tt class="docutils literal"><span class="pre">sources</span></tt> is empty a complete betweenness
 174 centrality calculation using all vertices in <tt class="docutils literal"><span class="pre">g</span></tt> will be
 175 performed.  The value type of the Buffer must be the graph's vertex
 176 descriptor type.</p>
 177 <p class="last"><strong>Default</strong>: An empty <tt class="docutils literal"><span class="pre">boost::queue</span></tt> of int.</p>
 178 </dd>
 179 </dl>
 180 </div>
 181 <div class="section" id="complexity">
 182 <h1><a class="toc-backref" href="#id5">Complexity</a></h1>
 183 <p>Computing the shortest paths, counting them, and computing the
 184 contribution to the centrality map is <em>O(V log V)</em>.  Calculating exact
 185 betweenness centrality requires counting the shortest paths from all
 186 vertices in <tt class="docutils literal"><span class="pre">g</span></tt>, thus exact betweenness centrality is <em>O(V^2 log
 187 V)</em>.</p>
 188 </div>
 189 <div class="section" id="algorithm-description">
 190 <h1><a class="toc-backref" href="#id6">Algorithm Description</a></h1>
 191 <p>For the vertices in <tt class="docutils literal"><span class="pre">sources</span></tt> (or all vertices in <tt class="docutils literal"><span class="pre">g</span></tt> when
 192 <tt class="docutils literal"><span class="pre">sources</span></tt> is empty) the algorithm first calls a customized
 193 implementation of <a class="reference external" href="dijkstra_shortest_paths.html">delta_stepping_shortest_paths</a> which maintains a
 194 shortest path tree using an <tt class="docutils literal"><span class="pre">IncomingMap</span></tt>.  The <tt class="docutils literal"><span class="pre">IncomingMap</span></tt>
 195 contains the source of all incoming edges on shortest paths.</p>
 196 <p>The <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> defines the shortest path DAG at the target of the
 197 edges in the shortest paths tree.  In the bidirectional case edge
 198 flags could be used to translate the shortest paths information to the
 199 source of the edges.  Setting edge flags during the shortest path
 200 computation rather than using an <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> would result in
 201 adding an <em>O(V)</em> factor to the inner loop of the shortest paths
 202 computation to account for having to clear edge flags when a new
 203 shortest path is found.  This would increase the complexity of the
 204 algorithm.  Asymptotically, the current implementation is better,
 205 however using edge flags in the bidirectional case would reduce the
 206 number of supersteps required by the depth of the shortest paths DAG
 207 for each vertex.  Currently an <tt class="docutils literal"><span class="pre">outgoing</span></tt> map is explicitly
 208 constructed by simply reversing the edges in the incoming map.  Once
 209 the <tt class="docutils literal"><span class="pre">outgoing</span></tt> map is constructed it is traversed in dependency
 210 order from the source of the shortest paths calculation in order to
 211 compute path counts.  Once path counts are computed the shortest paths
 212 DAG is again traversed in dependency order from the source to
 213 calculate the dependency and centrality of each vertex.</p>
 214 <p>The algorithm is complete when the centrality has been computed from
 215 all vertices in <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
 216 </div>
 217 <div class="section" id="bibliography">
 218 <h1><a class="toc-backref" href="#id7">Bibliography</a></h1>
 219 <table class="docutils citation" frame="void" id="brandes01" rules="none">
 220 <colgroup><col class="label" /><col /></colgroup>
 221 <tbody valign="top">
 222 <tr><td class="label"><a class="fn-backref" href="#id1">[Brandes01]</a></td><td>Ulrik Brandes.  A Faster Algorithm for Betweenness
 223 Centrality. In the Journal of Mathematical Sociology, volume 25
 224 number 2, pages 163--177, 2001.</td></tr>
 225 </tbody>
 226 </table>
 227 <table class="docutils citation" frame="void" id="edmonds09" rules="none">
 228 <colgroup><col class="label" /><col /></colgroup>
 229 <tbody valign="top">
 230 <tr><td class="label"><a class="fn-backref" href="#id2">[Edmonds09]</a></td><td>Nick Edmonds, Torsten Hoefler, and Andrew Lumsdaine.
 231 A Space-Efficient Parallel Algorithm for Computing Betweenness
 232 Centrality in Sparse Networks.  Indiana University tech report.
 233 2009.</td></tr>
 234 </tbody>
 235 </table>
 236 <hr class="docutils" />
 237 <p>Copyright (C) 2009 The Trustees of Indiana University.</p>
 238 <p>Authors: Nick Edmonds and Andrew Lumsdaine</p>
 239 </div>
 240 </div>
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