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wikipedia_principal_eigenvector.html
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<p class="first admonition-title">Note</p>
<p class="last">Click <a class="reference internal" href="#sphx-glr-download-auto-examples-applications-wikipedia-principal-eigenvector-py"><span class="std std-ref">here</span></a> to download the full example code or run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="wikipedia-principal-eigenvector">
<span id="sphx-glr-auto-examples-applications-wikipedia-principal-eigenvector-py"></span><h1>Wikipedia principal eigenvector<a class="headerlink" href="#wikipedia-principal-eigenvector" title="Permalink to this headline">¶</a></h1>
<p>A classical way to assert the relative importance of vertices in a
graph is to compute the principal eigenvector of the adjacency matrix
so as to assign to each vertex the values of the components of the first
eigenvector as a centrality score:</p>
<blockquote>
<div><a class="reference external" href="https://en.wikipedia.org/wiki/Eigenvector_centrality">https://en.wikipedia.org/wiki/Eigenvector_centrality</a></div></blockquote>
<p>On the graph of webpages and links those values are called the PageRank
scores by Google.</p>
<p>The goal of this example is to analyze the graph of links inside
wikipedia articles to rank articles by relative importance according to
this eigenvector centrality.</p>
<p>The traditional way to compute the principal eigenvector is to use the
power iteration method:</p>
<blockquote>
<div><a class="reference external" href="https://en.wikipedia.org/wiki/Power_iteration">https://en.wikipedia.org/wiki/Power_iteration</a></div></blockquote>
<p>Here the computation is achieved thanks to Martinsson’s Randomized SVD
algorithm implemented in scikit-learn.</p>
<p>The graph data is fetched from the DBpedia dumps. DBpedia is an extraction
of the latent structured data of the Wikipedia content.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="c1"># Author: Olivier Grisel <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">print_function</span>
<span class="kn">from</span> <span class="nn">bz2</span> <span class="kn">import</span> <span class="n">BZ2File</span>
<span class="kn">import</span> <span class="nn">os</span>
<span class="kn">from</span> <span class="nn">datetime</span> <span class="kn">import</span> <span class="n">datetime</span>
<span class="kn">from</span> <span class="nn">pprint</span> <span class="kn">import</span> <span class="n">pprint</span>
<span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <span class="n">time</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">sparse</span>
<span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">randomized_svd</span>
<span class="kn">from</span> <span class="nn">sklearn.externals.joblib</span> <span class="kn">import</span> <span class="n">Memory</span>
<span class="kn">from</span> <span class="nn">sklearn.externals.six.moves.urllib.request</span> <span class="kn">import</span> <span class="n">urlopen</span>
<span class="kn">from</span> <span class="nn">sklearn.externals.six</span> <span class="kn">import</span> <span class="n">iteritems</span>
<span class="k">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="c1"># #############################################################################</span>
<span class="c1"># Where to download the data, if not already on disk</span>
<span class="n">redirects_url</span> <span class="o">=</span> <span class="s2">"http://downloads.dbpedia.org/3.5.1/en/redirects_en.nt.bz2"</span>
<span class="n">redirects_filename</span> <span class="o">=</span> <span class="n">redirects_url</span><span class="o">.</span><span class="n">rsplit</span><span class="p">(</span><span class="s2">"/"</span><span class="p">,</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
<span class="n">page_links_url</span> <span class="o">=</span> <span class="s2">"http://downloads.dbpedia.org/3.5.1/en/page_links_en.nt.bz2"</span>
<span class="n">page_links_filename</span> <span class="o">=</span> <span class="n">page_links_url</span><span class="o">.</span><span class="n">rsplit</span><span class="p">(</span><span class="s2">"/"</span><span class="p">,</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
<span class="n">resources</span> <span class="o">=</span> <span class="p">[</span>
<span class="p">(</span><span class="n">redirects_url</span><span class="p">,</span> <span class="n">redirects_filename</span><span class="p">),</span>
<span class="p">(</span><span class="n">page_links_url</span><span class="p">,</span> <span class="n">page_links_filename</span><span class="p">),</span>
<span class="p">]</span>
<span class="k">for</span> <span class="n">url</span><span class="p">,</span> <span class="n">filename</span> <span class="ow">in</span> <span class="n">resources</span><span class="p">:</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">filename</span><span class="p">):</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Downloading data from '</span><span class="si">%s</span><span class="s2">', please wait..."</span> <span class="o">%</span> <span class="n">url</span><span class="p">)</span>
<span class="n">opener</span> <span class="o">=</span> <span class="n">urlopen</span><span class="p">(</span><span class="n">url</span><span class="p">)</span>
<span class="nb">open</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="s1">'wb'</span><span class="p">)</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">opener</span><span class="o">.</span><span class="n">read</span><span class="p">())</span>
<span class="k">print</span><span class="p">()</span>
<span class="c1"># #############################################################################</span>
<span class="c1"># Loading the redirect files</span>
<span class="n">memory</span> <span class="o">=</span> <span class="n">Memory</span><span class="p">(</span><span class="n">cachedir</span><span class="o">=</span><span class="s2">"."</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">index</span><span class="p">(</span><span class="n">redirects</span><span class="p">,</span> <span class="n">index_map</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="sd">"""Find the index of an article name after redirect resolution"""</span>
<span class="n">k</span> <span class="o">=</span> <span class="n">redirects</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
<span class="k">return</span> <span class="n">index_map</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">index_map</span><span class="p">))</span>
<span class="n">DBPEDIA_RESOURCE_PREFIX_LEN</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="s2">"http://dbpedia.org/resource/"</span><span class="p">)</span>
<span class="n">SHORTNAME_SLICE</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="n">DBPEDIA_RESOURCE_PREFIX_LEN</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">short_name</span><span class="p">(</span><span class="n">nt_uri</span><span class="p">):</span>
<span class="sd">"""Remove the < and > URI markers and the common URI prefix"""</span>
<span class="k">return</span> <span class="n">nt_uri</span><span class="p">[</span><span class="n">SHORTNAME_SLICE</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">get_redirects</span><span class="p">(</span><span class="n">redirects_filename</span><span class="p">):</span>
<span class="sd">"""Parse the redirections and build a transitively closed map out of it"""</span>
<span class="n">redirects</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Parsing the NT redirect file"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">l</span><span class="p">,</span> <span class="n">line</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">BZ2File</span><span class="p">(</span><span class="n">redirects_filename</span><span class="p">)):</span>
<span class="n">split</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">split</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">4</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"ignoring malformed line: "</span> <span class="o">+</span> <span class="n">line</span><span class="p">)</span>
<span class="k">continue</span>
<span class="n">redirects</span><span class="p">[</span><span class="n">short_name</span><span class="p">(</span><span class="n">split</span><span class="p">[</span><span class="mi">0</span><span class="p">])]</span> <span class="o">=</span> <span class="n">short_name</span><span class="p">(</span><span class="n">split</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
<span class="k">if</span> <span class="n">l</span> <span class="o">%</span> <span class="mi">1000000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"[</span><span class="si">%s</span><span class="s2">] line: </span><span class="si">%08d</span><span class="s2">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">datetime</span><span class="o">.</span><span class="n">now</span><span class="p">()</span><span class="o">.</span><span class="n">isoformat</span><span class="p">(),</span> <span class="n">l</span><span class="p">))</span>
<span class="c1"># compute the transitive closure</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Computing the transitive closure of the redirect relation"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">l</span><span class="p">,</span> <span class="n">source</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">redirects</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
<span class="n">transitive_target</span> <span class="o">=</span> <span class="bp">None</span>
<span class="n">target</span> <span class="o">=</span> <span class="n">redirects</span><span class="p">[</span><span class="n">source</span><span class="p">]</span>
<span class="n">seen</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">source</span><span class="p">])</span>
<span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
<span class="n">transitive_target</span> <span class="o">=</span> <span class="n">target</span>
<span class="n">target</span> <span class="o">=</span> <span class="n">redirects</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">target</span><span class="p">)</span>
<span class="k">if</span> <span class="n">target</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">target</span> <span class="ow">in</span> <span class="n">seen</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">seen</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">target</span><span class="p">)</span>
<span class="n">redirects</span><span class="p">[</span><span class="n">source</span><span class="p">]</span> <span class="o">=</span> <span class="n">transitive_target</span>
<span class="k">if</span> <span class="n">l</span> <span class="o">%</span> <span class="mi">1000000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"[</span><span class="si">%s</span><span class="s2">] line: </span><span class="si">%08d</span><span class="s2">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">datetime</span><span class="o">.</span><span class="n">now</span><span class="p">()</span><span class="o">.</span><span class="n">isoformat</span><span class="p">(),</span> <span class="n">l</span><span class="p">))</span>
<span class="k">return</span> <span class="n">redirects</span>
<span class="c1"># disabling joblib as the pickling of large dicts seems much too slow</span>
<span class="c1">#@memory.cache</span>
<span class="k">def</span> <span class="nf">get_adjacency_matrix</span><span class="p">(</span><span class="n">redirects_filename</span><span class="p">,</span> <span class="n">page_links_filename</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">"""Extract the adjacency graph as a scipy sparse matrix</span>
<span class="sd"> Redirects are resolved first.</span>
<span class="sd"> Returns X, the scipy sparse adjacency matrix, redirects as python</span>
<span class="sd"> dict from article names to article names and index_map a python dict</span>
<span class="sd"> from article names to python int (article indexes).</span>
<span class="sd"> """</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Computing the redirect map"</span><span class="p">)</span>
<span class="n">redirects</span> <span class="o">=</span> <span class="n">get_redirects</span><span class="p">(</span><span class="n">redirects_filename</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Computing the integer index map"</span><span class="p">)</span>
<span class="n">index_map</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">()</span>
<span class="n">links</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
<span class="k">for</span> <span class="n">l</span><span class="p">,</span> <span class="n">line</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">BZ2File</span><span class="p">(</span><span class="n">page_links_filename</span><span class="p">)):</span>
<span class="n">split</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">split</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">4</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"ignoring malformed line: "</span> <span class="o">+</span> <span class="n">line</span><span class="p">)</span>
<span class="k">continue</span>
<span class="n">i</span> <span class="o">=</span> <span class="n">index</span><span class="p">(</span><span class="n">redirects</span><span class="p">,</span> <span class="n">index_map</span><span class="p">,</span> <span class="n">short_name</span><span class="p">(</span><span class="n">split</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
<span class="n">j</span> <span class="o">=</span> <span class="n">index</span><span class="p">(</span><span class="n">redirects</span><span class="p">,</span> <span class="n">index_map</span><span class="p">,</span> <span class="n">short_name</span><span class="p">(</span><span class="n">split</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
<span class="n">links</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
<span class="k">if</span> <span class="n">l</span> <span class="o">%</span> <span class="mi">1000000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"[</span><span class="si">%s</span><span class="s2">] line: </span><span class="si">%08d</span><span class="s2">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">datetime</span><span class="o">.</span><span class="n">now</span><span class="p">()</span><span class="o">.</span><span class="n">isoformat</span><span class="p">(),</span> <span class="n">l</span><span class="p">))</span>
<span class="k">if</span> <span class="n">limit</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">l</span> <span class="o">>=</span> <span class="n">limit</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
<span class="k">break</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Computing the adjacency matrix"</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.lil_matrix.html#scipy.sparse.lil_matrix" title="View documentation for scipy.sparse.lil_matrix"><span class="n">sparse</span><span class="o">.</span><span class="n">lil_matrix</span></a><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">index_map</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">index_map</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">links</span><span class="p">:</span>
<span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="k">del</span> <span class="n">links</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Converting to CSR representation"</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tocsr</span><span class="p">()</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"CSR conversion done"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">X</span><span class="p">,</span> <span class="n">redirects</span><span class="p">,</span> <span class="n">index_map</span>
<span class="c1"># stop after 5M links to make it possible to work in RAM</span>
<span class="n">X</span><span class="p">,</span> <span class="n">redirects</span><span class="p">,</span> <span class="n">index_map</span> <span class="o">=</span> <span class="n">get_adjacency_matrix</span><span class="p">(</span>
<span class="n">redirects_filename</span><span class="p">,</span> <span class="n">page_links_filename</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="mi">5000000</span><span class="p">)</span>
<span class="n">names</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">name</span><span class="p">)</span> <span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">iteritems</span><span class="p">(</span><span class="n">index_map</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Computing the principal singular vectors using randomized_svd"</span><span class="p">)</span>
<span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="n">U</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">V</span> <span class="o">=</span> <span class="n">randomized_svd</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">n_iter</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"done in </span><span class="si">%0.3f</span><span class="s2">s"</span> <span class="o">%</span> <span class="p">(</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>
<span class="c1"># print the names of the wikipedia related strongest components of the</span>
<span class="c1"># principal singular vector which should be similar to the highest eigenvector</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Top wikipedia pages according to principal singular vectors"</span><span class="p">)</span>
<span class="n">pprint</span><span class="p">([</span><span class="n">names</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="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">U</span><span class="o">.</span><span class="n">T</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">argsort</span><span class="p">()[</span><span class="o">-</span><span class="mi">10</span><span class="p">:]])</span>
<span class="n">pprint</span><span class="p">([</span><span class="n">names</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="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">V</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">argsort</span><span class="p">()[</span><span class="o">-</span><span class="mi">10</span><span class="p">:]])</span>
<span class="k">def</span> <span class="nf">centrality_scores</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.85</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-10</span><span class="p">):</span>
<span class="sd">"""Power iteration computation of the principal eigenvector</span>
<span class="sd"> This method is also known as Google PageRank and the implementation</span>
<span class="sd"> is based on the one from the NetworkX project (BSD licensed too)</span>
<span class="sd"> with copyrights by:</span>
<span class="sd"> Aric Hagberg <[email protected]></span>
<span class="sd"> Dan Schult <[email protected]></span>
<span class="sd"> Pieter Swart <[email protected]></span>
<span class="sd"> """</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">X</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">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">incoming_counts</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.asarray.html#numpy.asarray" title="View documentation for numpy.asarray"><span class="n">np</span><span class="o">.</span><span class="n">asarray</span></a><span class="p">(</span><span class="n">X</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="n">ravel</span><span class="p">()</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Normalizing the graph"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">incoming_counts</span><span class="o">.</span><span class="n">nonzero</span><span class="p">()[</span><span class="mi">0</span><span class="p">]:</span>
<span class="n">X</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">X</span><span class="o">.</span><span class="n">indptr</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span><span class="n">X</span><span class="o">.</span><span class="n">indptr</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">*=</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="n">incoming_counts</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="n">dangle</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.asarray.html#numpy.asarray" title="View documentation for numpy.asarray"><span class="n">np</span><span class="o">.</span><span class="n">asarray</span></a><span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.where.html#numpy.where" title="View documentation for numpy.where"><span class="n">np</span><span class="o">.</span><span class="n">where</span></a><span class="p">(</span><span class="n">X</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="mf">1.0</span> <span class="o">/</span> <span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
<span class="n">scores</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.ones.html#numpy.ones" title="View documentation for numpy.ones"><span class="n">np</span><span class="o">.</span><span class="n">ones</span></a><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span> <span class="o">/</span> <span class="n">n</span> <span class="c1"># initial guess</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_iter</span><span class="p">):</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"power iteration #</span><span class="si">%d</span><span class="s2">"</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span>
<span class="n">prev_scores</span> <span class="o">=</span> <span class="n">scores</span>
<span class="n">scores</span> <span class="o">=</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">*</span> <span class="p">(</span><span class="n">scores</span> <span class="o">*</span> <span class="n">X</span> <span class="o">+</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html#numpy.dot" title="View documentation for numpy.dot"><span class="n">np</span><span class="o">.</span><span class="n">dot</span></a><span class="p">(</span><span class="n">dangle</span><span class="p">,</span> <span class="n">prev_scores</span><span class="p">))</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">prev_scores</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span>
<span class="c1"># check convergence: normalized l_inf norm</span>
<span class="n">scores_max</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">scores</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="k">if</span> <span class="n">scores_max</span> <span class="o">==</span> <span class="mf">0.0</span><span class="p">:</span>
<span class="n">scores_max</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">err</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">scores</span> <span class="o">-</span> <span class="n">prev_scores</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">/</span> <span class="n">scores_max</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"error: </span><span class="si">%0.6f</span><span class="s2">"</span> <span class="o">%</span> <span class="n">err</span><span class="p">)</span>
<span class="k">if</span> <span class="n">err</span> <span class="o"><</span> <span class="n">n</span> <span class="o">*</span> <span class="n">tol</span><span class="p">:</span>
<span class="k">return</span> <span class="n">scores</span>
<span class="k">return</span> <span class="n">scores</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Computing principal eigenvector score using a power iteration method"</span><span class="p">)</span>
<span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="n">scores</span> <span class="o">=</span> <span class="n">centrality_scores</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-10</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"done in </span><span class="si">%0.3f</span><span class="s2">s"</span> <span class="o">%</span> <span class="p">(</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>
<span class="n">pprint</span><span class="p">([</span><span class="n">names</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="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">scores</span><span class="p">)</span><span class="o">.</span><span class="n">argsort</span><span class="p">()[</span><span class="o">-</span><span class="mi">10</span><span class="p">:]])</span>
</pre></div>
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<p><strong>Total running time of the script:</strong> ( 0 minutes 0.000 seconds)</p>
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