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<h1>Source code for networkx.linalg.laplacianmatrix</h1><div class="highlight"><pre>
<span class="sd">"""</span>
<span class="sd">Laplacian matrix of graphs.</span>
<span class="sd">"""</span>
<span class="c"># Copyright (C) 2004-2013 by</span>
<span class="c"># Aric Hagberg <hagberg@lanl.gov></span>
<span class="c"># Dan Schult <dschult@colgate.edu></span>
<span class="c"># Pieter Swart <swart@lanl.gov></span>
<span class="c"># All rights reserved.</span>
<span class="c"># BSD license.</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="kn">as</span> <span class="nn">nx</span>
<span class="kn">from</span> <span class="nn">networkx.utils</span> <span class="kn">import</span> <span class="n">require</span><span class="p">,</span> <span class="n">not_implemented_for</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">'Aric Hagberg <aric.hagberg@gmail.com>'</span><span class="p">,</span>
<span class="s">'Pieter Swart (swart@lanl.gov)'</span><span class="p">,</span>
<span class="s">'Dan Schult (dschult@colgate.edu)'</span><span class="p">,</span>
<span class="s">'Alejandro Weinstein <alejandro.weinstein@gmail.com>'</span><span class="p">])</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">'laplacian_matrix'</span><span class="p">,</span>
<span class="s">'normalized_laplacian_matrix'</span><span class="p">,</span>
<span class="s">'directed_laplacian_matrix'</span><span class="p">]</span>
<span class="nd">@require</span><span class="p">(</span><span class="s">'numpy'</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s">'directed'</span><span class="p">)</span>
<div class="viewcode-block" id="laplacian_matrix"><a class="viewcode-back" href="../../../reference/generated/networkx.linalg.laplacianmatrix.laplacian_matrix.html#networkx.linalg.laplacianmatrix.laplacian_matrix">[docs]</a><span class="k">def</span> <span class="nf">laplacian_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodelist</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s">'weight'</span><span class="p">):</span>
<span class="sd">"""Return the Laplacian matrix of G.</span>
<span class="sd"> The graph Laplacian is the matrix L = D - A, where</span>
<span class="sd"> A is the adjacency matrix and D is the diagonal matrix of node degrees.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A NetworkX graph</span>
<span class="sd"> nodelist : list, optional</span>
<span class="sd"> The rows and columns are ordered according to the nodes in nodelist.</span>
<span class="sd"> If nodelist is None, then the ordering is produced by G.nodes().</span>
<span class="sd"> weight : string or None, optional (default='weight')</span>
<span class="sd"> The edge data key used to compute each value in the matrix.</span>
<span class="sd"> If None, then each edge has weight 1.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> L : NumPy matrix</span>
<span class="sd"> The Laplacian matrix of G.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> For MultiGraph/MultiDiGraph, the edges weights are summed.</span>
<span class="sd"> See to_numpy_matrix for other options.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> to_numpy_matrix</span>
<span class="sd"> normalized_laplacian_matrix</span>
<span class="sd"> """</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="k">if</span> <span class="n">nodelist</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">nodelist</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">()</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_multigraph</span><span class="p">():</span>
<span class="c"># this isn't the fastest way to do this...</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">to_numpy_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">,</span><span class="n">nodelist</span><span class="o">=</span><span class="n">nodelist</span><span class="p">,</span><span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">))</span>
<span class="n">I</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">I</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">D</span> <span class="o">-</span> <span class="n">A</span>
<span class="k">else</span><span class="p">:</span>
<span class="c"># Graph or DiGraph, this is faster than above</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodelist</span><span class="p">)</span>
<span class="n">index</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">n</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">nodelist</span><span class="p">)</span> <span class="p">)</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">))</span>
<span class="k">for</span> <span class="n">ui</span><span class="p">,</span><span class="n">u</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">nodelist</span><span class="p">):</span>
<span class="n">totalwt</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">v</span><span class="p">,</span><span class="n">d</span> <span class="ow">in</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="n">items</span><span class="p">():</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">vi</span> <span class="o">=</span> <span class="n">index</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">wt</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">L</span><span class="p">[</span><span class="n">ui</span><span class="p">,</span><span class="n">vi</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">wt</span>
<span class="n">totalwt</span> <span class="o">+=</span> <span class="n">wt</span>
<span class="n">L</span><span class="p">[</span><span class="n">ui</span><span class="p">,</span><span class="n">ui</span><span class="p">]</span> <span class="o">=</span> <span class="n">totalwt</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">asmatrix</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
</div>
<span class="nd">@require</span><span class="p">(</span><span class="s">'numpy'</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s">'directed'</span><span class="p">)</span>
<div class="viewcode-block" id="normalized_laplacian_matrix"><a class="viewcode-back" href="../../../reference/generated/networkx.linalg.laplacianmatrix.normalized_laplacian_matrix.html#networkx.linalg.laplacianmatrix.normalized_laplacian_matrix">[docs]</a><span class="k">def</span> <span class="nf">normalized_laplacian_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodelist</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s">'weight'</span><span class="p">):</span>
<span class="sd">r"""Return the normalized Laplacian matrix of G.</span>
<span class="sd"> The normalized graph Laplacian is the matrix</span>
<span class="sd"> .. math::</span>
<span class="sd"> NL = D^{-1/2} L D^{-1/2}</span>
<span class="sd"> where `L` is the graph Laplacian and `D` is the diagonal matrix of</span>
<span class="sd"> node degrees.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A NetworkX graph</span>
<span class="sd"> nodelist : list, optional</span>
<span class="sd"> The rows and columns are ordered according to the nodes in nodelist.</span>
<span class="sd"> If nodelist is None, then the ordering is produced by G.nodes().</span>
<span class="sd"> weight : string or None, optional (default='weight')</span>
<span class="sd"> The edge data key used to compute each value in the matrix.</span>
<span class="sd"> If None, then each edge has weight 1.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> L : NumPy matrix</span>
<span class="sd"> The normalized Laplacian matrix of G.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> For MultiGraph/MultiDiGraph, the edges weights are summed.</span>
<span class="sd"> See to_numpy_matrix for other options.</span>
<span class="sd"> If the Graph contains selfloops, D is defined as diag(sum(A,1)), where A is</span>
<span class="sd"> the adjencency matrix [2]_.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> laplacian_matrix</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Fan Chung-Graham, Spectral Graph Theory,</span>
<span class="sd"> CBMS Regional Conference Series in Mathematics, Number 92, 1997.</span>
<span class="sd"> .. [2] Steve Butler, Interlacing For Weighted Graphs Using The Normalized</span>
<span class="sd"> Laplacian, Electronic Journal of Linear Algebra, Volume 16, pp. 90-98,</span>
<span class="sd"> March 2007.</span>
<span class="sd"> """</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_multigraph</span><span class="p">():</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">laplacian_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodelist</span><span class="o">=</span><span class="n">nodelist</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_selfloops</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">laplacian_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodelist</span><span class="o">=</span><span class="n">nodelist</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">adj_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">D</span><span class="p">)</span> <span class="o">-</span> <span class="n">A</span>
<span class="c"># Handle div by 0. It happens if there are unconnected nodes</span>
<span class="k">with</span> <span class="n">np</span><span class="o">.</span><span class="n">errstate</span><span class="p">(</span><span class="n">divide</span><span class="o">=</span><span class="s">'ignore'</span><span class="p">):</span>
<span class="n">Disqrt</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">D</span><span class="p">))</span>
<span class="n">Disqrt</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">isinf</span><span class="p">(</span><span class="n">Disqrt</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">Ln</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Disqrt</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">L</span><span class="p">,</span><span class="n">Disqrt</span><span class="p">))</span>
<span class="k">return</span> <span class="n">Ln</span>
<span class="c">###############################################################################</span>
<span class="c"># Code based on</span>
<span class="c"># https://bitbucket.org/bedwards/networkx-community/src/370bd69fc02f/networkx/algorithms/community/</span>
</div>
<span class="nd">@require</span><span class="p">(</span><span class="s">'numpy'</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s">'undirected'</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s">'multigraph'</span><span class="p">)</span>
<div class="viewcode-block" id="directed_laplacian_matrix"><a class="viewcode-back" href="../../../reference/generated/networkx.linalg.laplacianmatrix.directed_laplacian_matrix.html#networkx.linalg.laplacianmatrix.directed_laplacian_matrix">[docs]</a><span class="k">def</span> <span class="nf">directed_laplacian_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodelist</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s">'weight'</span><span class="p">,</span>
<span class="n">walk_type</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.95</span><span class="p">):</span>
<span class="sd">r"""Return the directed Laplacian matrix of G.</span>
<span class="sd"> The graph directed Laplacian is the matrix</span>
<span class="sd"> .. math::</span>
<span class="sd"> L = I - (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} ) / 2</span>
<span class="sd"> where `I` is the identity matrix, `P` is the transition matrix of the</span>
<span class="sd"> graph, and `\Phi` a matrix with the Perron vector of `P` in the diagonal and</span>
<span class="sd"> zeros elsewhere.</span>
<span class="sd"> Depending on the value of walk_type, `P` can be the transition matrix</span>
<span class="sd"> induced by a random walk, a lazy random walk, or a random walk with</span>
<span class="sd"> teleportation (PageRank).</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : DiGraph</span>
<span class="sd"> A NetworkX graph</span>
<span class="sd"> nodelist : list, optional</span>
<span class="sd"> The rows and columns are ordered according to the nodes in nodelist.</span>
<span class="sd"> If nodelist is None, then the ordering is produced by G.nodes().</span>
<span class="sd"> weight : string or None, optional (default='weight')</span>
<span class="sd"> The edge data key used to compute each value in the matrix.</span>
<span class="sd"> If None, then each edge has weight 1.</span>
<span class="sd"> walk_type : string or None, optional (default=None)</span>
<span class="sd"> If None, `P` is selected depending on the properties of the</span>
<span class="sd"> graph. Otherwise is one of 'random', 'lazy', or 'pagerank'</span>
<span class="sd"> alpha : real</span>
<span class="sd"> (1 - alpha) is the teleportation probability used with pagerank</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> L : NumPy array</span>
<span class="sd"> Normalized Laplacian of G.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If NumPy cannot be imported</span>
<span class="sd"> NetworkXNotImplemnted</span>
<span class="sd"> If G is not a DiGraph</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Only implemented for DiGraphs</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> laplacian_matrix</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Fan Chung (2005).</span>
<span class="sd"> Laplacians and the Cheeger inequality for directed graphs.</span>
<span class="sd"> Annals of Combinatorics, 9(1), 2005</span>
<span class="sd"> """</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="k">if</span> <span class="n">walk_type</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="k">if</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_strongly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">if</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_aperiodic</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">walk_type</span> <span class="o">=</span> <span class="s">"random"</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">walk_type</span> <span class="o">=</span> <span class="s">"lazy"</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">walk_type</span> <span class="o">=</span> <span class="s">"pagerank"</span>
<span class="n">M</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">to_numpy_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodelist</span><span class="o">=</span><span class="n">nodelist</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">shape</span>
<span class="k">if</span> <span class="n">walk_type</span> <span class="ow">in</span> <span class="p">[</span><span class="s">"random"</span><span class="p">,</span> <span class="s">"lazy"</span><span class="p">]:</span>
<span class="n">DI</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diagflat</span><span class="p">(</span><span class="mf">1.0</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="k">if</span> <span class="n">walk_type</span> <span class="o">==</span> <span class="s">"random"</span><span class="p">:</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">DI</span> <span class="o">*</span> <span class="n">M</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">I</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">P</span> <span class="o">=</span> <span class="p">(</span><span class="n">I</span> <span class="o">+</span> <span class="n">DI</span> <span class="o">*</span> <span class="n">M</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
<span class="k">elif</span> <span class="n">walk_type</span> <span class="o">==</span> <span class="s">"pagerank"</span><span class="p">:</span>
<span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="mi">0</span> <span class="o"><</span> <span class="n">alpha</span> <span class="o"><</span> <span class="mi">1</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">'alpha must be between 0 and 1'</span><span class="p">)</span>
<span class="c"># add constant to dangling nodes' row</span>
<span class="n">dangling</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
<span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dangling</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
<span class="n">M</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="n">n</span>
<span class="c"># normalize</span>
<span class="n">M</span> <span class="o">=</span> <span class="n">M</span> <span class="o">/</span> <span class="n">M</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">M</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">)</span> <span class="o">/</span> <span class="n">n</span>
<span class="k">else</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">"walk_type must be random, lazy, or pagerank"</span><span class="p">)</span>
<span class="n">evals</span><span class="p">,</span> <span class="n">evecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="n">index</span> <span class="o">=</span> <span class="n">evals</span><span class="o">.</span><span class="n">argsort</span><span class="p">()[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="c"># index of largest eval,evec</span>
<span class="c"># eigenvector of largest eigenvalue at ind[-1]</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">evecs</span><span class="p">[:,</span><span class="n">index</span><span class="p">])</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span><span class="o">.</span><span class="n">real</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">v</span> <span class="o">/</span> <span class="n">v</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="n">sp</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">sp</span><span class="p">)</span> <span class="o">*</span> <span class="n">P</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">sp</span><span class="p">)</span>
<span class="n">I</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
<span class="k">return</span> <span class="n">I</span> <span class="o">-</span> <span class="p">(</span><span class="n">Q</span> <span class="o">+</span> <span class="n">Q</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">/</span><span class="mf">2.0</span>
<span class="c"># fixture for nose tests</span></div>
<span class="k">def</span> <span class="nf">setup_module</span><span class="p">(</span><span class="n">module</span><span class="p">):</span>
<span class="kn">from</span> <span class="nn">nose</span> <span class="kn">import</span> <span class="n">SkipTest</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">numpy</span>
<span class="k">except</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">SkipTest</span><span class="p">(</span><span class="s">"NumPy not available"</span><span class="p">)</span>
</pre></div>
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