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  <head>
    <title>Boost Graph Library: Brandes' Betweenness Centrality</title>
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    <IMG SRC="../../../boost.png" 
     ALT="C++ Boost" width="277" height="86"> 
<h1><img src="figs/python.gif" alt="(Python)"/><tt>brandes_betweenness_centrality</tt></h1>

    <p>
    <pre>
<em>// named parameter versions</em>
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_map);

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

<em>// non-named parameter versions</em>
template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap,
         typename IncomingMap, typename DistanceMap, typename DependencyMap, 
         typename PathCountMap, typename VertexIndexMap&gt;
void 
brandes_betweenness_centrality(const Graph&amp; g, CentralityMap centrality_map,
                               EdgeCentralityMap edge_centrality,
                               IncomingMap incoming,
                               DistanceMap distance, DependencyMap dependency,
                               PathCountMap path_count, 
                               VertexIndexMap vertex_index);

template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap,
         typename IncomingMap, typename DistanceMap, typename DependencyMap, 
         typename PathCountMap, typename VertexIndexMap, typename WeightMap&gt;    
void 
brandes_betweenness_centrality(const Graph&amp; g, CentralityMap centrality_map,
                               EdgeCentralityMap edge_centrality,
                               IncomingMap incoming, 
                               DistanceMap distance,  DependencyMap dependency,
                               PathCountMap path_count,      
                               VertexIndexMap vertex_index,
                               WeightMap weight_map);

<em>// helper functions</em>
template&lt;typename Graph, typename CentralityMap&gt;
void 
relative_betweenness_centrality(const Graph&amp; g, CentralityMap centrality_map);

template&lt;typename Graph, typename CentralityMap&gt;
typename property_traits&lt;CentralityMap&gt;::value_type
central_point_dominance(const Graph&amp; g, CentralityMap centrality_map);
    </pre>

<p>This algorithm&nbsp;[<a href="bibliography.html#brandes01">54</a>]
computes the <em>betweenness centrality</em>&nbsp;[<a
href="bibliography.html#freeman77">55</a>,<a
href="bibliography.html#anthonisse71">56</a>] of each vertex or each
edge in the graph. The betweenness centrality of a vertex <em>v</em>
is defined by

<p><img src="figs/betweenness_centrality.gif">,

<p>where <img src="figs/sigma_st.gif"> is the number of shortest paths from
vertex <em>s</em> to vertex <em>t</em> and <img src="figs/sigma_stv.gif">
is the number of shortest paths from vertex <em>s</em> to vertex
<em>t</em> that pass through vertex <em>v</em>.

<!-- \sum_{s \neq v \neq t}\frac{\sigma_{st}(v)}{\sigma_{st}} -->

<p>The edge betweenness centrality indicates for each edge the
betweenness centrality that was contributed to the target(s) of the
edge (plural for undirected graphs). Similar to (vertex) betweenness
centrality, edge betweenness centrality can be used to determine the
edges through which most shortest paths must pass. A single invocation
of this algorithm can compute either the vertex or edge centrality (or
both).</p>

<p>This algorithm can operate either on weighted graphs (if a suitable
edge weight map is supplied) or unweighted graphs (if no edge weight
map is supplied). The result is the absolute betweenness centrality;
to convert to the relative betweenness centrality, which scales each
absolute centrality by <img src="figs/rel_betweenness_centrality.gif">
(where <em>n</em> is the number of vertices in the graph), use
<tt>relative_betweenness_centrality</tt>. Given the relative
betweenness centrality, one can compute the <em>central point
dominance</em>&nbsp;[<a href="bibliography.html#freeman77">55</a>], which is a measure of the maximum "betweenness" of any
point in the graph: it will be 0 for complete graphs and
1 for "wheel" graphs (in which there is a central vertex that all
paths include; see <a href="#Fig1">Fig. 1</a>). Let <img src="figs/v_star.gif"> be the vertex with the largest relative betweenness centrality; then, the central point dominance is defined as:

<p><img src="figs/central_point_dominance.gif">

<!-- C_B' = \frac{\sum_{v \in V} C_B(v^*) - C_B'(v)}{n-1} -->

<p><a name="Fig1">
    <table border="1">
      <thead>
        <tr>
          <th>Fig. 1: A wheel graph, where every path travels through the central node. <br>The central point dominance of this graph is 1.</td>
        </tr>
      </thead>
      <tbody>
        <tr>
          <td align="center"><img src="figs/wheel_graph.gif"></td>
        </tr>
      </tbody>
    </table>

<h3>Where Defined</h3>
<a href="../../../boost/graph/betweenness_centrality.hpp"><tt>boost/graph/betweenness_centrality.hpp</tt></a>

<h3>Parameters</h3>
IN: <tt>const Graph&amp; g</tt>
<blockquote>
  The graph object on which the algorithm will be applied.  The type
  <tt>Graph</tt> must be a model of <a
  href="VertexListGraph.html">Vertex List Graph</a> and <a
  href="IncidenceGraph.html">Incidence Graph</a>. When an edge
  centrality map is supplied, it must also model <a
  href="EdgeListGraph.html">Edge List Graph</a>.<br>

<b>Python</b>: The parameter is named <tt>graph</tt>.
</blockquote>
    
UTIL: <tt>IncomingMap incoming</tt> 
<blockquote>
  This property map records the set of edges incoming to each vertex that comprise a shortest path from a particular source vertex through this vertex, and is used internally by the algorithm.The <tt>IncomingMap</tt> type must be a <a
  href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property
  Map</a> whose key type is the same as the vertex descriptor type of
  the graph and whose value type is a Sequence (e.g., an
  <tt>std::vector</tt>) containing edge descriptors.<br>

  <b>Default:</b> <a
  href="../../property_map/doc/iterator_property_map.html">
  <tt>iterator_property_map</tt></a> created from a
  <tt>std::vector</tt> of <tt>std::vector&lt;Edge&gt;</tt>, where
  <tt>Edge</tt> is the edge descriptor type of the graph.<br>

  <b>Python</b>: Unsupported parameter.
</blockquote>

UTIL: <tt>DistanceMap distance_map</tt> 
<blockquote>
  The shortest path weight from each source vertex <tt>s</tt> to each
  vertex in the graph <tt>g</tt> is recorded in this property map, but
  the result is only used internally. The type <tt>DistanceMap</tt>
  must be a model of <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>. The vertex descriptor type of the graph needs to
  be usable as the key type of the distance map. The value type of the
  distance map is the element type of a <a
  href="./Monoid.html">Monoid</a>.<br>

  <b>Default:</b> <a
  href="../../property_map/doc/iterator_property_map.html">
  <tt>iterator_property_map</tt></a> created from a
  <tt>std::vector</tt> of the <tt>WeightMap</tt>'s value type (or the
  <tt>vertices_size_type</tt> of the graph when no weight map exists)
  of size <tt>num_vertices(g)</tt> and using the <tt>vertex_index</tt> for
  the index map.<br>

  <b>Python</b>: Unsupported parameter.
</blockquote>

UTIL: <tt>DependencyMap dependency</tt> 
<blockquote>
  Property map used internally to accumulate partial betweenness
  centrality results. The type <tt>DependencyMap</tt> must be a model
  of <a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>. The vertex descriptor type of the graph needs to
  be usable as the key type of the dependency map. The value type of
  the dependency map must be compatible with the value type of the
  centrality map.<br>

  <b>Default:</b> <a
  href="../../property_map/doc/iterator_property_map.html">
  <tt>iterator_property_map</tt></a> created from a
  <tt>std::vector</tt> of the <tt>CentralityMap</tt>'s value type of
  size <tt>num_vertices(g)</tt> and using the <tt>vertex_index</tt>
  for the index map.<br>

  <b>Python</b>: Unsupported parameter.
</blockquote>

UTIL: <tt>PathCountMap path_count</tt> 
<blockquote>
  Property map used internally to accumulate the number of paths that
  pass through each particular vertex. The type <tt>PathCountMap</tt>
  must be a model of <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>. The vertex descriptor type of the graph needs to
  be usable as the key type of the dependency map. The value type of
  the dependency map must be an integral type large enough to store
  the number of paths in the graph.<br>

  <b>Default:</b> <a
  href="../../property_map/doc/iterator_property_map.html">
  <tt>iterator_property_map</tt></a> created from a
  <tt>std::vector</tt> of the <tt>degree_size_type</tt> of the graph of
  size <tt>num_vertices(g)</tt> and using the <tt>vertex_index</tt>
  for the index map.<br>

  <b>Python</b>: Unsupported parameter.
</blockquote>

<h3>Named parameters</h3>
OUT/UTIL: <tt>CentralityMap centrality_map</tt>
<blockquote>
  This property map is used to accumulate the betweenness centrality
  of each vertex, and is the primary output of the algorithm. The type
  <tt>CentralityMap</tt> must be a model of <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>, with the graph's vertex descriptor type as its key
  type. The value type of this property map should be a floating-point
  or rational type.<br>

  <b>Default:</b> a <tt>dummy_property_map</tt>, which requires no
  work to compute and returns no answer.<br>
  <b>Python</b>: The color map must be a <tt>vertex_double_map</tt> for
  the graph.<br>
  <b>Python default</b>: <tt>graph.get_vertex_double_map("centrality")</tt>
</blockquote>

OUT/UTIL: <tt>EdgeCentralityMap edge_centrality_map</tt>
<blockquote>
  This property map is used to accumulate the betweenness centrality
  of each edge, and is a secondary form of output for the
  algorithm. The type <tt>EdgeCentralityMap</tt> must be a model of <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>, with the graph's edge descriptor type as its key
  type. The value type of this property map should be the same as the
  value type of the <tt>CentralityMap</tt> property map.<br>

  <b>Default:</b> a <tt>dummy_property_map</tt>, which requires no
  work to compute and returns no answer.<br>
  <b>Python</b>: The color map must be a <tt>edge_double_map</tt> for
  the graph.<br>
  <b>Python default</b>: <tt>graph.get_edge_double_map("centrality")</tt>
</blockquote>

IN: <tt>vertex_index_map(VertexIndexMap vertex_index)</tt> 
<blockquote>
  This maps each vertex to an integer in the range <tt>[0,
    num_vertices(g))</tt>. This is necessary for efficient updates of the
  heap data structure when an edge is relaxed.  The type
  <tt>VertexIndexMap</tt> must be a model of
  <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
  integer type. The vertex descriptor type of the graph needs to be
  usable as the key type of the map.<br>
  <b>Default:</b> <tt>get(vertex_index, g)</tt>.
    Note: if you use this default, make sure your graph has
    an internal <tt>vertex_index</tt> property. For example,
    <tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does
    not have an internal <tt>vertex_index</tt> property.<br>
  <b>Python</b>: Unsupported parameter.
</blockquote>

IN: <tt>weight_map(WeightMap w_map)</tt>   
<blockquote>
  The weight or ``length'' of each edge in the graph. The weights
  must all be non-negative, and the algorithm will throw a
  <a href="./exception.html#negative_edge"><tt>negative_edge</tt></a> 
  exception is one of the edges is negative.
  The type <tt>WeightMap</tt> must be a model of
  <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
  the graph needs to be usable as the key type for the weight
  map. The value type for this map must be
  the same as the value type of the distance map.<br>
  <b>Default:</b> All edge weights are assumed to be equivalent.
  <b>Python</b>: If supplied, must be an <tt>edge_double_map</tt> for the graph.
</blockquote>

<h3>Complexity</h3> 
The time complexity is <em>O(VE)</em> for unweighted graphs and
<em>O(VE + V(V+E) log V)</em> for weighted graphs. The space complexity
is <em>O(VE)</em>.

    <hr>

<TABLE>
<TR valign=top>
<TD nowrap>Copyright &copy; 2004</TD><TD>
<A HREF="http://www.boost.org/people/doug_gregor.html">Douglas Gregor</A>, Indiana University (dgregor@cs.indiana.edu</A>)<br>
<A HREF="http://www.osl.iu.edu/~lums">Andrew Lumsdaine</A>,
Indiana University (<A
HREF="mailto:lums@osl.iu.edu">lums@osl.iu.edu</A>)
</TD></TR></TABLE>
<!-- Created: Fri Jun  4 16:42:50 EST 2004 -->
<!-- hhmts start -->Last modified: Tue Mar  1 14:14:51 EST 2005 <!-- hhmts end -->
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