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<div class="section" id="betweenness-centrality">
<h1>betweenness_centrality<a class="headerlink" href="#betweenness-centrality" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="networkx.algorithms.centrality.betweenness_centrality">
<tt class="descname">betweenness_centrality</tt><big>(</big><em>G</em>, <em>k=None</em>, <em>normalized=True</em>, <em>weight=None</em>, <em>endpoints=False</em>, <em>seed=None</em><big>)</big><a class="headerlink" href="#networkx.algorithms.centrality.betweenness_centrality" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute the shortest-path betweenness centrality for nodes.</p>
<p>Betweenness centrality of a node <span class="math">v</span> is the sum of the
fraction of all-pairs shortest paths that pass through <span class="math">v</span>:</p>
<div class="math">
<p><span class="math">c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}</span></p>
</div><p>where <span class="math">V</span> is the set of nodes, <span class="math">\sigma(s, t)</span> is the number of
shortest <span class="math">(s, t)</span>-paths, and <span class="math">\sigma(s, t|v)</span> is the number of those
paths passing through some node <span class="math">v</span> other than <span class="math">s, t</span>.
If <span class="math">s = t</span>, <span class="math">\sigma(s, t) = 1</span>, and if <span class="math">v \in {s, t}</span>,
<span class="math">\sigma(s, t|v) = 0</span> <a class="reference internal" href="#r155">[R155]</a>.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>G</strong> : graph</p>
<blockquote>
<div><p>A NetworkX graph</p>
</div></blockquote>
<p><strong>k</strong> : int, optional (default=None)</p>
<blockquote>
<div><p>If k is not None use k node samples to estimate betweenness.
The value of k <= n where n is the number of nodes in the graph.
Higher values give better approximation.</p>
</div></blockquote>
<p><strong>normalized</strong> : bool, optional</p>
<blockquote>
<div><p>If True the betweenness values are normalized by <span class="math">2/((n-1)(n-2))</span>
for graphs, and <span class="math">1/((n-1)(n-2))</span> for directed graphs where <span class="math">n</span>
is the number of nodes in G.</p>
</div></blockquote>
<p><strong>weight</strong> : None or string, optional</p>
<blockquote>
<div><p>If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.</p>
</div></blockquote>
<p><strong>endpoints</strong> : bool, optional</p>
<blockquote>
<div><p>If True include the endpoints in the shortest path counts.</p>
</div></blockquote>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>nodes</strong> : dictionary</p>
<blockquote class="last">
<div><p>Dictionary of nodes with betweenness centrality as the value.</p>
</div></blockquote>
</td>
</tr>
</tbody>
</table>
<div class="admonition-see-also admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><a class="reference internal" href="networkx.algorithms.centrality.edge_betweenness_centrality.html#networkx.algorithms.centrality.edge_betweenness_centrality" title="networkx.algorithms.centrality.edge_betweenness_centrality"><tt class="xref py py-obj docutils literal"><span class="pre">edge_betweenness_centrality</span></tt></a>, <a class="reference internal" href="networkx.algorithms.centrality.load_centrality.html#networkx.algorithms.centrality.load_centrality" title="networkx.algorithms.centrality.load_centrality"><tt class="xref py py-obj docutils literal"><span class="pre">load_centrality</span></tt></a></p>
</div>
<p class="rubric">Notes</p>
<p>The algorithm is from Ulrik Brandes <a class="reference internal" href="#r154">[R154]</a>.
See <a class="reference internal" href="#r155">[R155]</a> for details on algorithms for variations and related metrics.</p>
<p>For approximate betweenness calculations set k=#samples to use
k nodes (“pivots”) to estimate the betweenness values. For an estimate
of the number of pivots needed see <a class="reference internal" href="#r156">[R156]</a>.</p>
<p>For weighted graphs the edge weights must be greater than zero.
Zero edge weights can produce an infinite number of equal length
paths between pairs of nodes.</p>
<p class="rubric">References</p>
<table class="docutils citation" frame="void" id="r154" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[R154]</td><td><em>(<a class="fn-backref" href="#id2">1</a>, <a class="fn-backref" href="#id5">2</a>)</em> A Faster Algorithm for Betweenness Centrality.
Ulrik Brandes,
Journal of Mathematical Sociology 25(2):163-177, 2001.
<a class="reference external" href="http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf">http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf</a></td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="r155" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[R155]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id3">2</a>, <a class="fn-backref" href="#id6">3</a>)</em> Ulrik Brandes: On Variants of Shortest-Path Betweenness
Centrality and their Generic Computation.
Social Networks 30(2):136-145, 2008.
<a class="reference external" href="http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf">http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf</a></td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="r156" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[R156]</td><td><em>(<a class="fn-backref" href="#id4">1</a>, <a class="fn-backref" href="#id7">2</a>)</em> Ulrik Brandes and Christian Pich:
Centrality Estimation in Large Networks.
International Journal of Bifurcation and Chaos 17(7):2303-2318, 2007.
<a class="reference external" href="http://www.inf.uni-konstanz.de/algo/publications/bp-celn-06.pdf">http://www.inf.uni-konstanz.de/algo/publications/bp-celn-06.pdf</a></td></tr>
</tbody>
</table>
</dd></dl>
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