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  <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 &#8220;lattice steps&#8221; 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&#8217;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>

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