boost_1_45_0/libs/graph_parallel/doc/html/betweenness_centrality.html - nest-learning-thermostat/5.0/boost - Git at Google

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 <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
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 <pre class="literal-block">
 // named parameter versions
 template&lt;typename Graph, typename Param, typename Tag, typename Rest&gt;
 void
 brandes_betweenness_centrality(const Graph&amp; g,
                                const bgl_named_params&lt;Param,Tag,Rest&gt;&amp; params);

 template&lt;typename Graph, typename CentralityMap&gt;
 void
 brandes_betweenness_centrality(const Graph&amp; g, CentralityMap centrality);

 template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap&gt;
 void
 brandes_betweenness_centrality(const Graph&amp; g, CentralityMap centrality,
                                EdgeCentralityMap edge_centrality_map);

 // non-named parameter versions
 template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap,
          typename IncomingMap, typename DistanceMap, typename DependencyMap,
          typename PathCountMap, typename VertexIndexMap, typename Buffer&gt;
 void
 brandes_betweenness_centrality(const Graph&amp; g,
                                CentralityMap centrality,
                                EdgeCentralityMap edge_centrality_map,
                                IncomingMap incoming,
                                DistanceMap distance,
                                DependencyMap dependency,
                                PathCountMap path_count,
                                VertexIndexMap vertex_index,
                                Buffer sources,
                                typename property_traits&lt;DistanceMap&gt;::value_type delta);

 template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap,
          typename IncomingMap, typename DistanceMap, typename DependencyMap,
          typename PathCountMap, typename VertexIndexMap, typename WeightMap,
          typename Buffer&gt;
 void
 brandes_betweenness_centrality(const Graph&amp; g,
                                CentralityMap centrality,
                                EdgeCentralityMap edge_centrality_map,
                                IncomingMap incoming,
                                DistanceMap distance,
                                DependencyMap dependency,
                                PathCountMap path_count,
                                VertexIndexMap vertex_index,
                                Buffer sources,
                                typename property_traits&lt;WeightMap&gt;::value_type delta,
                                WeightMap weight_map);

 // helper functions
 template&lt;typename Graph, typename CentralityMap&gt;
 typename property_traits&lt;CentralityMap&gt;::value_type
 central_point_dominance(const Graph&amp; 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>&lt;<tt class="docutils literal"><span class="pre">boost/graph/distributed/betweenness_centrality.hpp</span></tt>&gt;</p>
 <p>also accessible from</p>
 <p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/betweenness_centrality.hpp</span></tt>&gt;</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&amp;</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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	<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>
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	Generated on: 2009-05-31 00:21 UTC.
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