<!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>navigable_small_world_graph — 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="Graph generators" href="../generators.html" /> <link rel="next" title="kl_connected_subgraph" href="networkx.generators.hybrid.kl_connected_subgraph.html" /> <link rel="prev" title="waxman_graph" href="networkx.generators.geometric.waxman_graph.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.generators.hybrid.kl_connected_subgraph.html" title="kl_connected_subgraph" accesskey="N">next</a> |</li> <li class="right" > <a href="networkx.generators.geometric.waxman_graph.html" title="waxman_graph" 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="../generators.html" accesskey="U">Graph generators</a> »</li> </ul> </div> <div class="sphinxsidebar"> <div class="sphinxsidebarwrapper"> <h4>Previous topic</h4> <p class="topless"><a href="networkx.generators.geometric.waxman_graph.html" title="previous chapter">waxman_graph</a></p> <h4>Next topic</h4> <p class="topless"><a href="networkx.generators.hybrid.kl_connected_subgraph.html" title="next chapter">kl_connected_subgraph</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="navigable-small-world-graph"> <h1>navigable_small_world_graph<a class="headerlink" href="#navigable-small-world-graph" title="Permalink to this headline">¶</a></h1> <dl class="function"> <dt id="networkx.generators.geometric.navigable_small_world_graph"> <tt class="descname">navigable_small_world_graph</tt><big>(</big><em>n</em>, <em>p=1</em>, <em>q=1</em>, <em>r=2</em>, <em>dim=2</em>, <em>seed=None</em><big>)</big><a class="reference internal" href="../../_modules/networkx/generators/geometric.html#navigable_small_world_graph"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#networkx.generators.geometric.navigable_small_world_graph" title="Permalink to this definition">¶</a></dt> <dd><p>Return a navigable small-world graph.</p> <p>A navigable small-world graph is a directed grid with additional long-range connections that are chosen randomly. From <a class="reference internal" href="#r265">[R265]</a>:</p> <p>Begin with a set of nodes that are identified with the set of lattice points in an <span class="math">n \times n</span> square, <span class="math">{(i,j): i\in {1,2,\ldots,n}, j\in {1,2,\ldots,n}}</span> and define the lattice distance between two nodes <span class="math">(i,j)</span> and <span class="math">(k,l)</span> to be the number of “lattice steps” separating them: <span class="math">d((i,j),(k,l)) = |k-i|+|l-j|</span>.</p> <p>For a universal constant <span class="math">p</span>, the node <span class="math">u</span> has a directed edge to every other node within lattice distance <span class="math">p</span> (local contacts) .</p> <p>For universal constants <span class="math">q\ge 0</span> and <span class="math">r\ge 0</span> construct directed edges from <span class="math">u</span> to <span class="math">q</span> other nodes (long-range contacts) using independent random trials; the i’th directed edge from <span class="math">u</span> has endpoint <span class="math">v</span> with probability proportional to <span class="math">d(u,v)^{-r}</span>.</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>n</strong> : int</p> <blockquote> <div><p>The number of nodes.</p> </div></blockquote> <p><strong>p</strong> : int</p> <blockquote> <div><p>The diameter of short range connections. Each node is connected to every other node within lattice distance p.</p> </div></blockquote> <p><strong>q</strong> : int</p> <blockquote> <div><p>The number of long-range connections for each node.</p> </div></blockquote> <p><strong>r</strong> : float</p> <blockquote> <div><p>Exponent for decaying probability of connections. The probability of connecting to a node at lattice distance d is 1/d^r.</p> </div></blockquote> <p><strong>dim</strong> : int</p> <blockquote> <div><p>Dimension of grid</p> </div></blockquote> <p><strong>seed</strong> : int, optional</p> <blockquote class="last"> <div><p>Seed for random number generator (default=None).</p> </div></blockquote> </td> </tr> </tbody> </table> <p class="rubric">References</p> <table class="docutils citation" frame="void" id="r265" rules="none"> <colgroup><col class="label" /><col /></colgroup> <tbody valign="top"> <tr><td class="label">[R265]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> J. Kleinberg. The small-world phenomenon: An algorithmic perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000.</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.generators.hybrid.kl_connected_subgraph.html" title="kl_connected_subgraph" >next</a> |</li> <li class="right" > <a href="networkx.generators.geometric.waxman_graph.html" title="waxman_graph" >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="../generators.html" >Graph generators</a> »</li> </ul> </div> <div class="footer"> © Copyright 2013, NetworkX Developers. 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