<!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>directed_laplacian_matrix — 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="Linear algebra" href="../linalg.html" />
<link rel="next" title="laplacian_spectrum" href="networkx.linalg.spectrum.laplacian_spectrum.html" />
<link rel="prev" title="normalized_laplacian_matrix" href="networkx.linalg.laplacianmatrix.normalized_laplacian_matrix.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.linalg.spectrum.laplacian_spectrum.html" title="laplacian_spectrum"
accesskey="N">next</a> |</li>
<li class="right" >
<a href="networkx.linalg.laplacianmatrix.normalized_laplacian_matrix.html" title="normalized_laplacian_matrix"
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="../linalg.html" accesskey="U">Linear algebra</a> »</li>
</ul>
</div>
<div class="sphinxsidebar">
<div class="sphinxsidebarwrapper">
<h4>Previous topic</h4>
<p class="topless"><a href="networkx.linalg.laplacianmatrix.normalized_laplacian_matrix.html"
title="previous chapter">normalized_laplacian_matrix</a></p>
<h4>Next topic</h4>
<p class="topless"><a href="networkx.linalg.spectrum.laplacian_spectrum.html"
title="next chapter">laplacian_spectrum</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="directed-laplacian-matrix">
<h1>directed_laplacian_matrix<a class="headerlink" href="#directed-laplacian-matrix" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="networkx.linalg.laplacianmatrix.directed_laplacian_matrix">
<tt class="descname">directed_laplacian_matrix</tt><big>(</big><em>G</em>, <em>nodelist=None</em>, <em>weight='weight'</em>, <em>walk_type=None</em>, <em>alpha=0.95</em><big>)</big><a class="reference internal" href="../../_modules/networkx/linalg/laplacianmatrix.html#directed_laplacian_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#networkx.linalg.laplacianmatrix.directed_laplacian_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the directed Laplacian matrix of G.</p>
<p>The graph directed Laplacian is the matrix</p>
<div class="math">
<p><span class="math">L = I - (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} ) / 2</span></p>
</div><p>where <span class="math">I</span> is the identity matrix, <span class="math">P</span> is the transition matrix of the
graph, and <span class="math">\Phi</span> a matrix with the Perron vector of <span class="math">P</span> in the diagonal and
zeros elsewhere.</p>
<p>Depending on the value of walk_type, <span class="math">P</span> can be the transition matrix
induced by a random walk, a lazy random walk, or a random walk with
teleportation (PageRank).</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> : DiGraph</p>
<blockquote>
<div><p>A NetworkX graph</p>
</div></blockquote>
<p><strong>nodelist</strong> : list, optional</p>
<blockquote>
<div><p>The rows and columns are ordered according to the nodes in nodelist.
If nodelist is None, then the ordering is produced by G.nodes().</p>
</div></blockquote>
<p><strong>weight</strong> : string or None, optional (default=’weight’)</p>
<blockquote>
<div><p>The edge data key used to compute each value in the matrix.
If None, then each edge has weight 1.</p>
</div></blockquote>
<p><strong>walk_type</strong> : string or None, optional (default=None)</p>
<blockquote>
<div><p>If None, <span class="math">P</span> is selected depending on the properties of the
graph. Otherwise is one of ‘random’, ‘lazy’, or ‘pagerank’</p>
</div></blockquote>
<p><strong>alpha</strong> : real</p>
<blockquote>
<div><p>(1 - alpha) is the teleportation probability used with pagerank</p>
</div></blockquote>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>L</strong> : NumPy array</p>
<blockquote>
<div><p>Normalized Laplacian of G.</p>
</div></blockquote>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>NetworkXError</strong> :</p>
<blockquote>
<div><p>If NumPy cannot be imported</p>
</div></blockquote>
<p><strong>NetworkXNotImplemnted</strong> :</p>
<blockquote class="last">
<div><p>If G is not a DiGraph</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.linalg.laplacianmatrix.laplacian_matrix.html#networkx.linalg.laplacianmatrix.laplacian_matrix" title="networkx.linalg.laplacianmatrix.laplacian_matrix"><tt class="xref py py-obj docutils literal"><span class="pre">laplacian_matrix</span></tt></a></p>
</div>
<p class="rubric">Notes</p>
<p>Only implemented for DiGraphs</p>
<p class="rubric">References</p>
<table class="docutils citation" frame="void" id="r293" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id1">[R293]</a></td><td>Fan Chung (2005).
Laplacians and the Cheeger inequality for directed graphs.
Annals of Combinatorics, 9(1), 2005</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.linalg.spectrum.laplacian_spectrum.html" title="laplacian_spectrum"
>next</a> |</li>
<li class="right" >
<a href="networkx.linalg.laplacianmatrix.normalized_laplacian_matrix.html" title="normalized_laplacian_matrix"
>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="../linalg.html" >Linear algebra</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>
