1<?xml version="1.0" encoding="utf-8" ?> 2<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> 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. 15Use, modification and distribution is subject to the Boost Software 16License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at 17http://www.boost.org/LICENSE_1_0.txt) --> 18<pre class="literal-block"> 19// named parameter versions 20template<typename Graph, typename Param, typename Tag, typename Rest> 21void 22brandes_betweenness_centrality(const Graph& g, 23 const bgl_named_params<Param,Tag,Rest>& params); 24 25template<typename Graph, typename CentralityMap> 26void 27brandes_betweenness_centrality(const Graph& g, CentralityMap centrality); 28 29template<typename Graph, typename CentralityMap, typename EdgeCentralityMap> 30void 31brandes_betweenness_centrality(const Graph& g, CentralityMap centrality, 32 EdgeCentralityMap edge_centrality_map); 33 34// non-named parameter versions 35template<typename Graph, typename CentralityMap, typename EdgeCentralityMap, 36 typename IncomingMap, typename DistanceMap, typename DependencyMap, 37 typename PathCountMap, typename VertexIndexMap, typename Buffer> 38void 39brandes_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 50template<typename Graph, typename CentralityMap, typename EdgeCentralityMap, 51 typename IncomingMap, typename DistanceMap, typename DependencyMap, 52 typename PathCountMap, typename VertexIndexMap, typename WeightMap, 53 typename Buffer> 54void 55brandes_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 68template<typename Graph, typename CentralityMap> 69typename property_traits<CentralityMap>::value_type 70central_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 73betweenness centrality of the vertices and edges in a graph. The 74method of calculating betweenness centrality in <em>O(V)</em> space is due to 75Brandes <a class="citation-reference" href="#brandes01" id="id1">[Brandes01]</a>. The algorithm itself is a modification of 76Brandes 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 98type must also model the <a class="reference external" href="http://www.boost.org/libs/graph/doc/IncidenceGraph.html">Incidence Graph</a> concept. 0-weighted 99edges 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 103information 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 105vertex 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 110supplied a <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no edge 111centrality information will be recorded. The <tt class="docutils literal"><span class="pre">EdgeCentralityMap</span></tt> 112type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>. The key type must be 113the 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 118part of shortest paths to that vertex. The <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> type 119must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>. Its key type and value type 120must 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 124descriptor 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 129shortest paths portion of the algorithm. The <tt class="docutils literal"><span class="pre">DistanceMap</span></tt> type 130must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>. Its key type must be the 131graph'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 139centrality 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 141key 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 151is on during the centrality calculation portion of the algorithm. 152The <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>. 153Its 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 162descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt> and whose value type is an 163integral 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 169descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt>. If not supplied the betweenness 170centrality 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 173algorithm. If <tt class="docutils literal"><span class="pre">sources</span></tt> is empty a complete betweenness 174centrality calculation using all vertices in <tt class="docutils literal"><span class="pre">g</span></tt> will be 175performed. The value type of the Buffer must be the graph's vertex 176descriptor 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 184contribution to the centrality map is <em>O(V log V)</em>. Calculating exact 185betweenness centrality requires counting the shortest paths from all 186vertices in <tt class="docutils literal"><span class="pre">g</span></tt>, thus exact betweenness centrality is <em>O(V^2 log 187V)</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 193implementation of <a class="reference external" href="dijkstra_shortest_paths.html">delta_stepping_shortest_paths</a> which maintains a 194shortest 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> 195contains 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 197edges in the shortest paths tree. In the bidirectional case edge 198flags could be used to translate the shortest paths information to the 199source of the edges. Setting edge flags during the shortest path 200computation rather than using an <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> would result in 201adding an <em>O(V)</em> factor to the inner loop of the shortest paths 202computation to account for having to clear edge flags when a new 203shortest path is found. This would increase the complexity of the 204algorithm. Asymptotically, the current implementation is better, 205however using edge flags in the bidirectional case would reduce the 206number of supersteps required by the depth of the shortest paths DAG 207for each vertex. Currently an <tt class="docutils literal"><span class="pre">outgoing</span></tt> map is explicitly 208constructed by simply reversing the edges in the incoming map. Once 209the <tt class="docutils literal"><span class="pre">outgoing</span></tt> map is constructed it is traversed in dependency 210order from the source of the shortest paths calculation in order to 211compute path counts. Once path counts are computed the shortest paths 212DAG is again traversed in dependency order from the source to 213calculate the dependency and centrality of each vertex.</p> 214<p>The algorithm is complete when the centrality has been computed from 215all 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 223Centrality. In the Journal of Mathematical Sociology, volume 25 224number 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. 231A Space-Efficient Parallel Algorithm for Computing Betweenness 232Centrality in Sparse Networks. Indiana University tech report. 2332009.</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" /> 243Generated on: 2009-05-31 00:21 UTC. 244Generated 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. 245 246</div> 247</body> 248</html> 249
