<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>closeness_centrality — NetworkX 1.8.1 documentation</title>
<link rel="stylesheet" href="../../_static/networkx.css" type="text/css" />
<link rel="stylesheet" href="../../_static/pygments.css" type="text/css" />
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: '../../',
VERSION: '1.8.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: false
};
</script>
<script type="text/javascript" src="../../_static/jquery.js"></script>
<script type="text/javascript" src="../../_static/underscore.js"></script>
<script type="text/javascript" src="../../_static/doctools.js"></script>
<link rel="search" type="application/opensearchdescription+xml"
title="Search within NetworkX 1.8.1 documentation"
href="../../_static/opensearch.xml"/>
<link rel="top" title="NetworkX 1.8.1 documentation" href="../../index.html" />
<link rel="up" title="Centrality" href="../algorithms.centrality.html" />
<link rel="next" title="betweenness_centrality" href="networkx.algorithms.centrality.betweenness_centrality.html" />
<link rel="prev" title="out_degree_centrality" href="networkx.algorithms.centrality.out_degree_centrality.html" />
</head>
<body>
<div style="color: black;background-color: white; font-size: 3.2em; text-align: left; padding: 15px 10px 10px 15px">
NetworkX
</div>
<div class="related">
<h3>Navigation</h3>
<ul>
<li class="right" style="margin-right: 10px">
<a href="../../genindex.html" title="General Index"
accesskey="I">index</a></li>
<li class="right" >
<a href="../../py-modindex.html" title="Python Module Index"
>modules</a> |</li>
<li class="right" >
<a href="networkx.algorithms.centrality.betweenness_centrality.html" title="betweenness_centrality"
accesskey="N">next</a> |</li>
<li class="right" >
<a href="networkx.algorithms.centrality.out_degree_centrality.html" title="out_degree_centrality"
accesskey="P">previous</a> |</li>
<li><a href="http://networkx.github.com/">NetworkX Home </a> | </li>
<li><a href="http://networkx.github.com/documentation.html">Documentation </a>| </li>
<li><a href="http://networkx.github.com/download.html">Download </a> | </li>
<li><a href="http://github.com/networkx">Developer (Github)</a></li>
<li><a href="../index.html" >Reference</a> »</li>
<li><a href="../pdf_reference.html" >Reference</a> »</li>
<li><a href="../algorithms.html" >Algorithms</a> »</li>
<li><a href="../algorithms.centrality.html" accesskey="U">Centrality</a> »</li>
</ul>
</div>
<div class="sphinxsidebar">
<div class="sphinxsidebarwrapper">
<h4>Previous topic</h4>
<p class="topless"><a href="networkx.algorithms.centrality.out_degree_centrality.html"
title="previous chapter">out_degree_centrality</a></p>
<h4>Next topic</h4>
<p class="topless"><a href="networkx.algorithms.centrality.betweenness_centrality.html"
title="next chapter">betweenness_centrality</a></p>
<div id="searchbox" style="display: none">
<h3>Quick search</h3>
<form class="search" action="../../search.html" method="get">
<input type="text" name="q" />
<input type="submit" value="Go" />
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
<p class="searchtip" style="font-size: 90%">
Enter search terms or a module, class or function name.
</p>
</div>
<script type="text/javascript">$('#searchbox').show(0);</script>
</div>
</div>
<div class="document">
<div class="documentwrapper">
<div class="bodywrapper">
<div class="body">
<div class="section" id="closeness-centrality">
<h1>closeness_centrality<a class="headerlink" href="#closeness-centrality" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="networkx.algorithms.centrality.closeness_centrality">
<tt class="descname">closeness_centrality</tt><big>(</big><em>G</em>, <em>u=None</em>, <em>distance=None</em>, <em>normalized=True</em><big>)</big><a class="headerlink" href="#networkx.algorithms.centrality.closeness_centrality" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute closeness centrality for nodes.</p>
<p>Closeness centrality <a class="reference internal" href="#r157">[R157]</a> of a node <span class="math">u</span> is the reciprocal of the
sum of the shortest path distances from <span class="math">u</span> to all <span class="math">n-1</span> other nodes.
Since the sum of distances depends on the number of nodes in the
graph, closeness is normalized by the sum of minimum possible
distances <span class="math">n-1</span>.</p>
<div class="math">
<p><span class="math">C(u) = \frac{n - 1}{\sum_{v=1}^{n} d(v, u)},</span></p>
</div><p>where <span class="math">d(v, u)</span> is the shortest-path distance between <span class="math">v</span> and <span class="math">u</span>,
and <span class="math">n</span> is the number of nodes in the graph.</p>
<p>Notice that higher values of closeness indicate higher centrality.</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>u</strong> : node, optional</p>
<blockquote>
<div><p>Return only the value for node u</p>
</div></blockquote>
<p><strong>distance</strong> : edge attribute key, optional (default=None)</p>
<blockquote>
<div><p>Use the specified edge attribute as the edge distance in shortest
path calculations</p>
</div></blockquote>
<p><strong>normalized</strong> : bool, optional</p>
<blockquote>
<div><p>If True (default) normalize by the number of nodes in the connected
part of the graph.</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 closeness 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.betweenness_centrality.html#networkx.algorithms.centrality.betweenness_centrality" title="networkx.algorithms.centrality.betweenness_centrality"><tt class="xref py py-obj docutils literal"><span class="pre">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>, <a class="reference internal" href="networkx.algorithms.centrality.eigenvector_centrality.html#networkx.algorithms.centrality.eigenvector_centrality" title="networkx.algorithms.centrality.eigenvector_centrality"><tt class="xref py py-obj docutils literal"><span class="pre">eigenvector_centrality</span></tt></a>, <a class="reference internal" href="networkx.algorithms.centrality.degree_centrality.html#networkx.algorithms.centrality.degree_centrality" title="networkx.algorithms.centrality.degree_centrality"><tt class="xref py py-obj docutils literal"><span class="pre">degree_centrality</span></tt></a></p>
</div>
<p class="rubric">Notes</p>
<p>The closeness centrality is normalized to <span class="math">(n-1)/(|G|-1)</span> where
<span class="math">n</span> is the number of nodes in the connected part of graph
containing the node. If the graph is not completely connected,
this algorithm computes the closeness centrality for each
connected part separately.</p>
<p>If the ‘distance’ keyword is set to an edge attribute key then the
shortest-path length will be computed using Dijkstra’s algorithm with
that edge attribute as the edge weight.</p>
<p class="rubric">References</p>
<table class="docutils citation" frame="void" id="r157" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[R157]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> Freeman, L.C., 1979. Centrality in networks: I.
Conceptual clarification. Social Networks 1, 215–239.
<a class="reference external" href="http://www.soc.ucsb.edu/faculty/friedkin/Syllabi/Soc146/Freeman78.PDF">http://www.soc.ucsb.edu/faculty/friedkin/Syllabi/Soc146/Freeman78.PDF</a></td></tr>
</tbody>
</table>
</dd></dl>
</div>
</div>
</div>
</div>
<div class="clearer"></div>
</div>
<div class="related">
<h3>Navigation</h3>
<ul>
<li class="right" style="margin-right: 10px">
<a href="../../genindex.html" title="General Index"
>index</a></li>
<li class="right" >
<a href="../../py-modindex.html" title="Python Module Index"
>modules</a> |</li>
<li class="right" >
<a href="networkx.algorithms.centrality.betweenness_centrality.html" title="betweenness_centrality"
>next</a> |</li>
<li class="right" >
<a href="networkx.algorithms.centrality.out_degree_centrality.html" title="out_degree_centrality"
>previous</a> |</li>
<li><a href="http://networkx.github.com/">NetworkX Home </a> | </li>
<li><a href="http://networkx.github.com/documentation.html">Documentation </a>| </li>
<li><a href="http://networkx.github.com/download.html">Download </a> | </li>
<li><a href="http://github.com/networkx">Developer (Github)</a></li>
<li><a href="../index.html" >Reference</a> »</li>
<li><a href="../pdf_reference.html" >Reference</a> »</li>
<li><a href="../algorithms.html" >Algorithms</a> »</li>
<li><a href="../algorithms.centrality.html" >Centrality</a> »</li>
</ul>
</div>
<div class="footer">
© Copyright 2013, NetworkX Developers.
Last updated on Oct 23, 2013.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
</div>
</body>
</html>
