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<h1>Source code for networkx.algorithms.bipartite.cluster</h1><div class="highlight"><pre>
<span class="c">#-*- coding: utf-8 -*-</span>
<span class="c"># Copyright (C) 2011 by </span>
<span class="c"># Jordi Torrents <jtorrents@milnou.net></span>
<span class="c"># Aric Hagberg <hagberg@lanl.gov></span>
<span class="c"># All rights reserved.</span>
<span class="c"># BSD license.</span>
<span class="kn">import</span> <span class="nn">itertools</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="kn">as</span> <span class="nn">nx</span>
<span class="n">__author__</span> <span class="o">=</span> <span class="s">"""</span><span class="se">\n</span><span class="s">"""</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s">'Jordi Torrents <jtorrents@milnou.net>'</span><span class="p">,</span>
<span class="s">'Aric Hagberg (hagberg@lanl.gov)'</span><span class="p">])</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span> <span class="s">'clustering'</span><span class="p">,</span>
<span class="s">'average_clustering'</span><span class="p">,</span>
<span class="s">'latapy_clustering'</span><span class="p">,</span>
<span class="s">'robins_alexander_clustering'</span><span class="p">]</span>
<span class="c"># functions for computing clustering of pairs</span>
<span class="k">def</span> <span class="nf">cc_dot</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span><span class="n">nv</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nu</span> <span class="o">&</span> <span class="n">nv</span><span class="p">))</span><span class="o">/</span><span class="nb">len</span><span class="p">(</span><span class="n">nu</span> <span class="o">|</span> <span class="n">nv</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">cc_max</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span><span class="n">nv</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nu</span> <span class="o">&</span> <span class="n">nv</span><span class="p">))</span><span class="o">/</span><span class="nb">max</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nu</span><span class="p">),</span><span class="nb">len</span><span class="p">(</span><span class="n">nv</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">cc_min</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span><span class="n">nv</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nu</span> <span class="o">&</span> <span class="n">nv</span><span class="p">))</span><span class="o">/</span><span class="nb">min</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nu</span><span class="p">),</span><span class="nb">len</span><span class="p">(</span><span class="n">nv</span><span class="p">))</span>
<span class="n">modes</span><span class="o">=</span><span class="p">{</span><span class="s">'dot'</span><span class="p">:</span><span class="n">cc_dot</span><span class="p">,</span>
<span class="s">'min'</span><span class="p">:</span><span class="n">cc_min</span><span class="p">,</span>
<span class="s">'max'</span><span class="p">:</span><span class="n">cc_max</span><span class="p">}</span>
<div class="viewcode-block" id="latapy_clustering"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.bipartite.cluster.latapy_clustering.html#networkx.algorithms.bipartite.cluster.latapy_clustering">[docs]</a><span class="k">def</span> <span class="nf">latapy_clustering</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodes</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">'dot'</span><span class="p">):</span>
<span class="sd">r"""Compute a bipartite clustering coefficient for nodes.</span>
<span class="sd"> The bipartie clustering coefficient is a measure of local density</span>
<span class="sd"> of connections defined as [1]_:</span>
<span class="sd"> </span>
<span class="sd"> .. math::</span>
<span class="sd"> c_u = \frac{\sum_{v \in N(N(v))} c_{uv} }{|N(N(u))|}</span>
<span class="sd"> where `N(N(u))` are the second order neighbors of `u` in `G` excluding `u`, </span>
<span class="sd"> and `c_{uv}` is the pairwise clustering coefficient between nodes </span>
<span class="sd"> `u` and `v`.</span>
<span class="sd"> The mode selects the function for `c_{uv}` which can be:</span>
<span class="sd"> `dot`: </span>
<span class="sd"> .. math::</span>
<span class="sd"> c_{uv}=\frac{|N(u)\cap N(v)|}{|N(u) \cup N(v)|}</span>
<span class="sd"> `min`: </span>
<span class="sd"> .. math::</span>
<span class="sd"> c_{uv}=\frac{|N(u)\cap N(v)|}{min(|N(u)|,|N(v)|)}</span>
<span class="sd"> `max`: </span>
<span class="sd"> .. math::</span>
<span class="sd"> c_{uv}=\frac{|N(u)\cap N(v)|}{max(|N(u)|,|N(v)|)}</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A bipartite graph</span>
<span class="sd"> nodes : list or iterable (optional)</span>
<span class="sd"> Compute bipartite clustering for these nodes. The default </span>
<span class="sd"> is all nodes in G.</span>
<span class="sd"> mode : string</span>
<span class="sd"> The pariwise bipartite clustering method to be used in the computation.</span>
<span class="sd"> It must be "dot", "max", or "min". </span>
<span class="sd"> </span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> clustering : dictionary</span>
<span class="sd"> A dictionary keyed by node with the clustering coefficient value.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import bipartite</span>
<span class="sd"> >>> G = nx.path_graph(4) # path graphs are bipartite</span>
<span class="sd"> >>> c = bipartite.clustering(G) </span>
<span class="sd"> >>> c[0]</span>
<span class="sd"> 0.5</span>
<span class="sd"> >>> c = bipartite.clustering(G,mode='min') </span>
<span class="sd"> >>> c[0]</span>
<span class="sd"> 1.0</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> robins_alexander_clustering</span>
<span class="sd"> square_clustering</span>
<span class="sd"> average_clustering</span>
<span class="sd"> </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008).</span>
<span class="sd"> Basic notions for the analysis of large two-mode networks. </span>
<span class="sd"> Social Networks 30(1), 31--48.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">algorithms</span><span class="o">.</span><span class="n">bipartite</span><span class="o">.</span><span class="n">is_bipartite</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">"Graph is not bipartite"</span><span class="p">)</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">cc_func</span> <span class="o">=</span> <span class="n">modes</span><span class="p">[</span><span class="n">mode</span><span class="p">]</span>
<span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span>\
<span class="s">"Mode for bipartite clustering must be: dot, min or max"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">nodes</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="n">G</span>
<span class="n">ccs</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
<span class="n">cc</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">nbrs2</span><span class="o">=</span><span class="nb">set</span><span class="p">([</span><span class="n">u</span> <span class="k">for</span> <span class="n">nbr</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">nbr</span><span class="p">]])</span><span class="o">-</span><span class="nb">set</span><span class="p">([</span><span class="n">v</span><span class="p">])</span>
<span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">nbrs2</span><span class="p">:</span>
<span class="n">cc</span> <span class="o">+=</span> <span class="n">cc_func</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">]),</span><span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">v</span><span class="p">]))</span>
<span class="k">if</span> <span class="n">cc</span> <span class="o">></span> <span class="mf">0.0</span><span class="p">:</span> <span class="c"># len(nbrs2)>0</span>
<span class="n">cc</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nbrs2</span><span class="p">)</span>
<span class="n">ccs</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">cc</span>
<span class="k">return</span> <span class="n">ccs</span>
</div>
<span class="n">clustering</span> <span class="o">=</span> <span class="n">latapy_clustering</span>
<div class="viewcode-block" id="average_clustering"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.bipartite.cluster.average_clustering.html#networkx.algorithms.bipartite.cluster.average_clustering">[docs]</a><span class="k">def</span> <span class="nf">average_clustering</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodes</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">'dot'</span><span class="p">):</span>
<span class="sd">r"""Compute the average bipartite clustering coefficient.</span>
<span class="sd"> A clustering coefficient for the whole graph is the average, </span>
<span class="sd"> .. math::</span>
<span class="sd"> C = \frac{1}{n}\sum_{v \in G} c_v,</span>
<span class="sd"> </span>
<span class="sd"> where `n` is the number of nodes in `G`.</span>
<span class="sd"> Similar measures for the two bipartite sets can be defined [1]_</span>
<span class="sd"> </span>
<span class="sd"> .. math::</span>
<span class="sd"> C_X = \frac{1}{|X|}\sum_{v \in X} c_v,</span>
<span class="sd"> </span>
<span class="sd"> where `X` is a bipartite set of `G`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> a bipartite graph</span>
<span class="sd"> nodes : list or iterable, optional</span>
<span class="sd"> A container of nodes to use in computing the average. </span>
<span class="sd"> The nodes should be either the entire graph (the default) or one of the </span>
<span class="sd"> bipartite sets.</span>
<span class="sd"> mode : string</span>
<span class="sd"> The pariwise bipartite clustering method. </span>
<span class="sd"> It must be "dot", "max", or "min" </span>
<span class="sd"> </span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> clustering : float</span>
<span class="sd"> The average bipartite clustering for the given set of nodes or the </span>
<span class="sd"> entire graph if no nodes are specified.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import bipartite</span>
<span class="sd"> >>> G=nx.star_graph(3) # star graphs are bipartite</span>
<span class="sd"> >>> bipartite.average_clustering(G) </span>
<span class="sd"> 0.75</span>
<span class="sd"> >>> X,Y=bipartite.sets(G)</span>
<span class="sd"> >>> bipartite.average_clustering(G,X) </span>
<span class="sd"> 0.0</span>
<span class="sd"> >>> bipartite.average_clustering(G,Y) </span>
<span class="sd"> 1.0</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> clustering</span>
<span class="sd"> </span>
<span class="sd"> Notes </span>
<span class="sd"> -----</span>
<span class="sd"> The container of nodes passed to this function must contain all of the nodes</span>
<span class="sd"> in one of the bipartite sets ("top" or "bottom") in order to compute </span>
<span class="sd"> the correct average bipartite clustering coefficients.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008).</span>
<span class="sd"> Basic notions for the analysis of large two-mode networks. </span>
<span class="sd"> Social Networks 30(1), 31--48.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">nodes</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">nodes</span><span class="o">=</span><span class="n">G</span>
<span class="n">ccs</span><span class="o">=</span><span class="n">latapy_clustering</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodes</span><span class="o">=</span><span class="n">nodes</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="n">mode</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">ccs</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">))</span><span class="o">/</span><span class="nb">len</span><span class="p">(</span><span class="n">nodes</span><span class="p">)</span>
</div>
<div class="viewcode-block" id="robins_alexander_clustering"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.bipartite.cluster.robins_alexander_clustering.html#networkx.algorithms.bipartite.cluster.robins_alexander_clustering">[docs]</a><span class="k">def</span> <span class="nf">robins_alexander_clustering</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="sd">r"""Compute the bipartite clustering of G.</span>
<span class="sd"> Robins and Alexander [1]_ defined bipartite clustering coefficient as</span>
<span class="sd"> four times the number of four cycles `C_4` divided by the number of</span>
<span class="sd"> three paths `L_3` in a bipartite graph:</span>
<span class="sd"> .. math::</span>
<span class="sd"> CC_4 = \frac{4 * C_4}{L_3}</span>
<span class="sd"> </span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> a bipartite graph</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> clustering : float</span>
<span class="sd"> The Robins and Alexander bipartite clustering for the input graph.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import bipartite</span>
<span class="sd"> >>> G = nx.davis_southern_women_graph()</span>
<span class="sd"> >>> print(round(bipartite.robins_alexander_clustering(G), 3))</span>
<span class="sd"> 0.468</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> latapy_clustering</span>
<span class="sd"> square_clustering</span>
<span class="sd"> </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Robins, G. and M. Alexander (2004). Small worlds among interlocking </span>
<span class="sd"> directors: Network structure and distance in bipartite graphs. </span>
<span class="sd"> Computational & Mathematical Organization Theory 10(1), 69–94.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o"><</span> <span class="mi">4</span> <span class="ow">or</span> <span class="n">G</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="o"><</span> <span class="mi">3</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="n">L_3</span> <span class="o">=</span> <span class="n">_threepaths</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">if</span> <span class="n">L_3</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="n">C_4</span> <span class="o">=</span> <span class="n">_four_cycles</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">return</span> <span class="p">(</span><span class="mf">4.</span> <span class="o">*</span> <span class="n">C_4</span><span class="p">)</span> <span class="o">/</span> <span class="n">L_3</span>
</div>
<span class="k">def</span> <span class="nf">_four_cycles</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">cycles</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">v</span><span class="p">],</span> <span class="mi">2</span><span class="p">):</span>
<span class="n">cycles</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">((</span><span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">])</span> <span class="o">&</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">w</span><span class="p">]))</span> <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="n">v</span><span class="p">]))</span>
<span class="k">return</span> <span class="n">cycles</span> <span class="o">/</span> <span class="mi">4</span>
<span class="k">def</span> <span class="nf">_threepaths</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">paths</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">v</span><span class="p">]:</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">])</span> <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="n">v</span><span class="p">]):</span>
<span class="n">paths</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">w</span><span class="p">])</span> <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="p">]))</span>
<span class="c"># Divide by two because we count each three path twice</span>
<span class="c"># one for each possible starting point</span>
<span class="k">return</span> <span class="n">paths</span> <span class="o">/</span> <span class="mi">2</span>
</pre></div>
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