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- <title>Parallel BGL Betweenness Centrality</title>
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- <div class="document" id="logo-betweenness-centrality">
- <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>
- <!-- Copyright (C) 2004-2009 The Trustees of Indiana University.
- Use, modification and distribution is subject to 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) -->
- <pre class="literal-block">
- // named parameter versions
- template<typename Graph, typename Param, typename Tag, typename Rest>
- void
- brandes_betweenness_centrality(const Graph& g,
- const bgl_named_params<Param,Tag,Rest>& params);
- template<typename Graph, typename CentralityMap>
- void
- brandes_betweenness_centrality(const Graph& g, CentralityMap centrality);
- template<typename Graph, typename CentralityMap, typename EdgeCentralityMap>
- void
- brandes_betweenness_centrality(const Graph& g, CentralityMap centrality,
- EdgeCentralityMap edge_centrality_map);
- // non-named parameter versions
- template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
- typename IncomingMap, typename DistanceMap, typename DependencyMap,
- typename PathCountMap, typename VertexIndexMap, typename Buffer>
- void
- brandes_betweenness_centrality(const Graph& g,
- CentralityMap centrality,
- EdgeCentralityMap edge_centrality_map,
- IncomingMap incoming,
- DistanceMap distance,
- DependencyMap dependency,
- PathCountMap path_count,
- VertexIndexMap vertex_index,
- Buffer sources,
- typename property_traits<DistanceMap>::value_type delta);
- template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
- typename IncomingMap, typename DistanceMap, typename DependencyMap,
- typename PathCountMap, typename VertexIndexMap, typename WeightMap,
- typename Buffer>
- void
- brandes_betweenness_centrality(const Graph& g,
- CentralityMap centrality,
- EdgeCentralityMap edge_centrality_map,
- IncomingMap incoming,
- DistanceMap distance,
- DependencyMap dependency,
- PathCountMap path_count,
- VertexIndexMap vertex_index,
- Buffer sources,
- typename property_traits<WeightMap>::value_type delta,
- WeightMap weight_map);
- // helper functions
- template<typename Graph, typename CentralityMap>
- typename property_traits<CentralityMap>::value_type
- central_point_dominance(const Graph& g, CentralityMap centrality);
- </pre>
- <p>The <tt class="docutils literal"><span class="pre">brandes_betweenness_centrality()</span></tt> function computes the
- betweenness centrality of the vertices and edges in a graph. The
- method of calculating betweenness centrality in <em>O(V)</em> space is due to
- Brandes <a class="citation-reference" href="#brandes01" id="id1">[Brandes01]</a>. The algorithm itself is a modification of
- Brandes algorithm by Edmonds <a class="citation-reference" href="#edmonds09" id="id2">[Edmonds09]</a>.</p>
- <div class="contents topic" id="contents">
- <p class="topic-title first">Contents</p>
- <ul class="simple">
- <li><a class="reference internal" href="#where-defined" id="id3">Where Defined</a></li>
- <li><a class="reference internal" href="#parameters" id="id4">Parameters</a></li>
- <li><a class="reference internal" href="#complexity" id="id5">Complexity</a></li>
- <li><a class="reference internal" href="#algorithm-description" id="id6">Algorithm Description</a></li>
- <li><a class="reference internal" href="#bibliography" id="id7">Bibliography</a></li>
- </ul>
- </div>
- <div class="section" id="where-defined">
- <h1><a class="toc-backref" href="#id3">Where Defined</a></h1>
- <p><<tt class="docutils literal"><span class="pre">boost/graph/distributed/betweenness_centrality.hpp</span></tt>></p>
- <p>also accessible from</p>
- <p><<tt class="docutils literal"><span class="pre">boost/graph/betweenness_centrality.hpp</span></tt>></p>
- </div>
- <div class="section" id="parameters">
- <h1><a class="toc-backref" href="#id4">Parameters</a></h1>
- <dl class="docutils">
- <dt>IN: <tt class="docutils literal"><span class="pre">const</span> <span class="pre">Graph&</span> <span class="pre">g</span></tt></dt>
- <dd>The graph type must be a model of <a class="reference external" href="DistributedGraph.html">Distributed Graph</a>. The graph
- 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
- edges in <tt class="docutils literal"><span class="pre">g</span></tt> will result in undefined behavior.</dd>
- <dt>IN: <tt class="docutils literal"><span class="pre">CentralityMap</span> <span class="pre">centrality</span></tt></dt>
- <dd><p class="first">A centrality map may be supplied to the algorithm, if not supplied a
- <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no vertex centrality
- information will be recorded. The <tt class="docutils literal"><span class="pre">CentralityMap</span></tt> type must be a
- <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>. The key type must be the graph's
- vertex descriptor type.</p>
- <p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
- </dd>
- <dt>IN: <tt class="docutils literal"><span class="pre">EdgeCentralityMap</span> <span class="pre">edge_centrality_map</span></tt></dt>
- <dd><p class="first">An edge centrality map may be supplied to the algorithm, if not
- supplied a <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no edge
- centrality information will be recorded. The <tt class="docutils literal"><span class="pre">EdgeCentralityMap</span></tt>
- type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>. The key type must be
- the graph's vertex descriptor type.</p>
- <p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
- </dd>
- <dt>IN: <tt class="docutils literal"><span class="pre">IncomingMap</span> <span class="pre">incoming</span></tt></dt>
- <dd><p class="first">The incoming map contains the incoming edges to a vertex that are
- part of shortest paths to that vertex. The <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> type
- must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>. Its key type and value type
- must both be the graph's vertex descriptor type.</p>
- <dl class="last docutils">
- <dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
- <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
- descriptor type.</dd>
- </dl>
- </dd>
- <dt>IN: <tt class="docutils literal"><span class="pre">DistanceMap</span> <span class="pre">distance</span></tt></dt>
- <dd><p class="first">The distance map records the distance to vertices during the
- shortest paths portion of the algorithm. The <tt class="docutils literal"><span class="pre">DistanceMap</span></tt> type
- must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>. Its key type must be the
- graph's vertex descriptor type.</p>
- <dl class="last docutils">
- <dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
- <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>
- </dl>
- </dd>
- <dt>IN: <tt class="docutils literal"><span class="pre">DependencyMap</span> <span class="pre">dependency</span></tt></dt>
- <dd><p class="first">The dependency map records the dependency of each vertex during the
- centrality calculation portion of the algorithm. The
- <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
- key type must be the graph's vertex descriptor type.</p>
- <dl class="last docutils">
- <dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
- <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>
- </dl>
- </dd>
- </dl>
- <p>IN: <tt class="docutils literal"><span class="pre">PathCountMap</span> <span class="pre">path_count</span></tt></p>
- <blockquote>
- <p>The path count map records the number of shortest paths each vertex
- is on during the centrality calculation portion of the algorithm.
- 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>.
- Its key type must be the graph's vertex descriptor type.</p>
- <dl class="docutils">
- <dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
- <dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the graph's degree size type.</dd>
- </dl>
- </blockquote>
- <dl class="docutils">
- <dt>IN: <tt class="docutils literal"><span class="pre">VertexIndexMap</span> <span class="pre">vertex_index</span></tt></dt>
- <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
- descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt> and whose value type is an
- integral type. The property map should map from vertices to their
- (local) indices in the range <em>[0, num_vertices(g))</em>.</p>
- <p class="last"><strong>Default</strong>: <tt class="docutils literal"><span class="pre">get(vertex_index,</span> <span class="pre">g)</span></tt></p>
- </dd>
- <dt>IN: <tt class="docutils literal"><span class="pre">WeightMap</span> <span class="pre">weight_map</span></tt></dt>
- <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
- descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt>. If not supplied the betweenness
- centrality calculation will be unweighted.</dd>
- <dt>IN: <tt class="docutils literal"><span class="pre">Buffer</span> <span class="pre">sources</span></tt></dt>
- <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
- algorithm. If <tt class="docutils literal"><span class="pre">sources</span></tt> is empty a complete betweenness
- centrality calculation using all vertices in <tt class="docutils literal"><span class="pre">g</span></tt> will be
- performed. The value type of the Buffer must be the graph's vertex
- descriptor type.</p>
- <p class="last"><strong>Default</strong>: An empty <tt class="docutils literal"><span class="pre">boost::queue</span></tt> of int.</p>
- </dd>
- </dl>
- </div>
- <div class="section" id="complexity">
- <h1><a class="toc-backref" href="#id5">Complexity</a></h1>
- <p>Computing the shortest paths, counting them, and computing the
- contribution to the centrality map is <em>O(V log V)</em>. Calculating exact
- betweenness centrality requires counting the shortest paths from all
- vertices in <tt class="docutils literal"><span class="pre">g</span></tt>, thus exact betweenness centrality is <em>O(V^2 log
- V)</em>.</p>
- </div>
- <div class="section" id="algorithm-description">
- <h1><a class="toc-backref" href="#id6">Algorithm Description</a></h1>
- <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
- <tt class="docutils literal"><span class="pre">sources</span></tt> is empty) the algorithm first calls a customized
- implementation of <a class="reference external" href="dijkstra_shortest_paths.html">delta_stepping_shortest_paths</a> which maintains a
- 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>
- contains the source of all incoming edges on shortest paths.</p>
- <p>The <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> defines the shortest path DAG at the target of the
- edges in the shortest paths tree. In the bidirectional case edge
- flags could be used to translate the shortest paths information to the
- source of the edges. Setting edge flags during the shortest path
- computation rather than using an <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> would result in
- adding an <em>O(V)</em> factor to the inner loop of the shortest paths
- computation to account for having to clear edge flags when a new
- shortest path is found. This would increase the complexity of the
- algorithm. Asymptotically, the current implementation is better,
- however using edge flags in the bidirectional case would reduce the
- number of supersteps required by the depth of the shortest paths DAG
- for each vertex. Currently an <tt class="docutils literal"><span class="pre">outgoing</span></tt> map is explicitly
- constructed by simply reversing the edges in the incoming map. Once
- the <tt class="docutils literal"><span class="pre">outgoing</span></tt> map is constructed it is traversed in dependency
- order from the source of the shortest paths calculation in order to
- compute path counts. Once path counts are computed the shortest paths
- DAG is again traversed in dependency order from the source to
- calculate the dependency and centrality of each vertex.</p>
- <p>The algorithm is complete when the centrality has been computed from
- all vertices in <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
- </div>
- <div class="section" id="bibliography">
- <h1><a class="toc-backref" href="#id7">Bibliography</a></h1>
- <table class="docutils citation" frame="void" id="brandes01" rules="none">
- <colgroup><col class="label" /><col /></colgroup>
- <tbody valign="top">
- <tr><td class="label"><a class="fn-backref" href="#id1">[Brandes01]</a></td><td>Ulrik Brandes. A Faster Algorithm for Betweenness
- Centrality. In the Journal of Mathematical Sociology, volume 25
- number 2, pages 163--177, 2001.</td></tr>
- </tbody>
- </table>
- <table class="docutils citation" frame="void" id="edmonds09" rules="none">
- <colgroup><col class="label" /><col /></colgroup>
- <tbody valign="top">
- <tr><td class="label"><a class="fn-backref" href="#id2">[Edmonds09]</a></td><td>Nick Edmonds, Torsten Hoefler, and Andrew Lumsdaine.
- A Space-Efficient Parallel Algorithm for Computing Betweenness
- Centrality in Sparse Networks. Indiana University tech report.
- 2009.</td></tr>
- </tbody>
- </table>
- <hr class="docutils" />
- <p>Copyright (C) 2009 The Trustees of Indiana University.</p>
- <p>Authors: Nick Edmonds and Andrew Lumsdaine</p>
- </div>
- </div>
- <div class="footer">
- <hr class="footer" />
- Generated on: 2009-05-31 00:21 UTC.
- Generated by <a class="reference external" href="http://docutils.sourceforge.net/">Docutils</a> from <a class="reference external" href="http://docutils.sourceforge.net/rst.html">reStructuredText</a> source.
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