[ceph.git] / ceph / src / boost / libs / graph_parallel / doc / html / betweenness_centrality.html

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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>
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	3	<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
	4	<head>
	5	<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
	6	<meta name="generator" content="Docutils 0.6: http://docutils.sourceforge.net/" />
	7	<title>Parallel BGL Betweenness Centrality</title>
	8	<link rel="stylesheet" href="../../../../rst.css" type="text/css" />
	9	</head>
	10	<body>
	11	<div class="document" id="logo-betweenness-centrality">
	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<typename Graph, typename Param, typename Tag, typename Rest>
	21	void
	22	brandes_betweenness_centrality(const Graph& g,
	23	const bgl_named_params<Param,Tag,Rest>& params);
	24
	25	template<typename Graph, typename CentralityMap>
	26	void
	27	brandes_betweenness_centrality(const Graph& g, CentralityMap centrality);
	28
	29	template<typename Graph, typename CentralityMap, typename EdgeCentralityMap>
	30	void
	31	brandes_betweenness_centrality(const Graph& g, CentralityMap centrality,
	32	EdgeCentralityMap edge_centrality_map);
	33
	34	// non-named parameter versions
	35	template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
	36	typename IncomingMap, typename DistanceMap, typename DependencyMap,
	37	typename PathCountMap, typename VertexIndexMap, typename Buffer>
	38	void
	39	brandes_betweenness_centrality(const Graph& 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<DistanceMap>::value_type delta);
	49
	50	template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
	51	typename IncomingMap, typename DistanceMap, typename DependencyMap,
	52	typename PathCountMap, typename VertexIndexMap, typename WeightMap,
	53	typename Buffer>
	54	void
	55	brandes_betweenness_centrality(const Graph& 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<WeightMap>::value_type delta,
65	WeightMap weight_map);
66
67	// helper functions
68	template<typename Graph, typename CentralityMap>
69	typename property_traits<CentralityMap>::value_type
70	central_point_dominance(const Graph& 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><<tt class="docutils literal"><span class="pre">boost/graph/distributed/betweenness_centrality.hpp</span></tt>></p>
90	<p>also accessible from</p>
91	<p><<tt class="docutils literal"><span class="pre">boost/graph/betweenness_centrality.hpp</span></tt>></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&</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>
241	<div class="footer">
242	<hr class="footer" />
243	Generated on: 2009-05-31 00:21 UTC.
244	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.
245
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