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<div class="section" id="collaboration-weighted-projected-graph">
<h1>collaboration_weighted_projected_graph<a class="headerlink" href="#collaboration-weighted-projected-graph" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="networkx.algorithms.bipartite.projection.collaboration_weighted_projected_graph">
<tt class="descname">collaboration_weighted_projected_graph</tt><big>(</big><em>B</em>, <em>nodes</em><big>)</big><a class="reference internal" href="../../_modules/networkx/algorithms/bipartite/projection.html#collaboration_weighted_projected_graph"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#networkx.algorithms.bipartite.projection.collaboration_weighted_projected_graph" title="Permalink to this definition">¶</a></dt>
<dd><p>Newman’s weighted projection of B onto one of its node sets.</p>
<p>The collaboration weighted projection is the projection of the
bipartite network B onto the specified nodes with weights assigned
using Newman’s collaboration model <a class="reference internal" href="#r147">[R147]</a>:</p>
<div class="math">
<p><span class="math">w_{v,u} = \sum_k \frac{\delta_{v}^{w} \delta_{w}^{k}}{k_w - 1}</span></p>
</div><p>where <span class="math">v</span> and <span class="math">u</span> are nodes from the same bipartite node set,
and <span class="math">w</span> is a node of the opposite node set.
The value <span class="math">k_w</span> is the degree of node <span class="math">w</span> in the bipartite
network and <span class="math">\delta_{v}^{w}</span> is 1 if node <span class="math">v</span> is
linked to node <span class="math">w</span> in the original bipartite graph or 0 otherwise.</p>
<p>The nodes retain their attributes and are connected in the resulting
graph if have an edge to a common node in the original bipartite
graph.</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>B</strong> : NetworkX graph</p>
<blockquote>
<div><p>The input graph should be bipartite.</p>
</div></blockquote>
<p><strong>nodes</strong> : list or iterable</p>
<blockquote>
<div><p>Nodes to project onto (the “bottom” nodes).</p>
</div></blockquote>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>Graph</strong> : NetworkX graph</p>
<blockquote class="last">
<div><p>A graph that is the projection onto the given nodes.</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"><tt class="xref py py-obj docutils literal"><span class="pre">is_bipartite</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">is_bipartite_node_set</span></tt>, <a class="reference external" href="http://docs.python.org/library/sets.html#sets" title="(in Python v2.7)"><tt class="xref py py-obj docutils literal"><span class="pre">sets</span></tt></a>, <a class="reference internal" href="networkx.algorithms.bipartite.projection.weighted_projected_graph.html#networkx.algorithms.bipartite.projection.weighted_projected_graph" title="networkx.algorithms.bipartite.projection.weighted_projected_graph"><tt class="xref py py-obj docutils literal"><span class="pre">weighted_projected_graph</span></tt></a>, <a class="reference internal" href="networkx.algorithms.bipartite.projection.overlap_weighted_projected_graph.html#networkx.algorithms.bipartite.projection.overlap_weighted_projected_graph" title="networkx.algorithms.bipartite.projection.overlap_weighted_projected_graph"><tt class="xref py py-obj docutils literal"><span class="pre">overlap_weighted_projected_graph</span></tt></a>, <a class="reference internal" href="networkx.algorithms.bipartite.projection.generic_weighted_projected_graph.html#networkx.algorithms.bipartite.projection.generic_weighted_projected_graph" title="networkx.algorithms.bipartite.projection.generic_weighted_projected_graph"><tt class="xref py py-obj docutils literal"><span class="pre">generic_weighted_projected_graph</span></tt></a>, <a class="reference internal" href="networkx.algorithms.bipartite.projection.projected_graph.html#networkx.algorithms.bipartite.projection.projected_graph" title="networkx.algorithms.bipartite.projection.projected_graph"><tt class="xref py py-obj docutils literal"><span class="pre">projected_graph</span></tt></a></p>
</div>
<p class="rubric">Notes</p>
<p>No attempt is made to verify that the input graph B is bipartite.
The graph and node properties are (shallow) copied to the projected graph.</p>
<p class="rubric">References</p>
<table class="docutils citation" frame="void" id="r147" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[R147]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> Scientific collaboration networks: II.
Shortest paths, weighted networks, and centrality,
M. E. J. Newman, Phys. Rev. E 64, 016132 (2001).</td></tr>
</tbody>
</table>
<p class="rubric">Examples</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">networkx.algorithms</span> <span class="kn">import</span> <span class="n">bipartite</span>
<span class="gp">>>> </span><span class="n">B</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">path_graph</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">B</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span> <span class="o">=</span> <span class="n">bipartite</span><span class="o">.</span><span class="n">collaboration_weighted_projected_graph</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="gp">>>> </span><span class="k">print</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">())</span>
<span class="go">[0, 2, 4, 5]</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">edge</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span> <span class="k">print</span><span class="p">(</span><span class="n">edge</span><span class="p">)</span>
<span class="gp">... </span>
<span class="go">(0, 2, {'weight': 0.5})</span>
<span class="go">(0, 5, {'weight': 0.5})</span>
<span class="go">(2, 4, {'weight': 1.0})</span>
<span class="go">(2, 5, {'weight': 0.5})</span>
</pre></div>
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