1.4/auto_examples/applications/wikipedia_principal_eigenvector.html



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              <ul>
<li><a class="reference internal" href="#">Wikipedia principal eigenvector</a><ul>
<li><a class="reference internal" href="#download-data-if-not-already-on-disk">Download data, if not already on disk</a></li>
<li><a class="reference internal" href="#loading-the-redirect-files">Loading the redirect files</a></li>
<li><a class="reference internal" href="#computing-the-adjacency-matrix">Computing the Adjacency matrix</a></li>
<li><a class="reference internal" href="#computing-principal-singular-vector-using-randomized-svd">Computing Principal Singular Vector using Randomized SVD</a></li>
<li><a class="reference internal" href="#computing-centrality-scores">Computing Centrality scores</a></li>
</ul>
</li>
</ul>

            </div>
      </div>
    </div>
    <div id="sk-page-content-wrapper">
      <div class="sk-page-content container-fluid body px-md-3" role="main">
        
  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p><a class="reference internal" href="#sphx-glr-download-auto-examples-applications-wikipedia-principal-eigenvector-py"><span class="std std-ref">Go to the end</span></a>
to download the full example code or to run this example in your browser via JupyterLite or Binder</p>
</div>
<section class="sphx-glr-example-title" 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="Link to this heading">¶</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><p><a class="reference external" href="https://en.wikipedia.org/wiki/Eigenvector_centrality">https://en.wikipedia.org/wiki/Eigenvector_centrality</a></p>
</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><p><a class="reference external" href="https://en.wikipedia.org/wiki/Power_iteration">https://en.wikipedia.org/wiki/Power_iteration</a></p>
</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 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Author: Olivier Grisel &lt;olivier.grisel@ensta.org&gt;</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="kn">import</span> <span class="nn">os</span>
<span class="kn">from</span> <span class="nn">bz2</span> <span class="kn">import</span> <a href="https://docs.python.org/3/library/bz2.html#bz2.BZ2File" title="bz2.BZ2File" class="sphx-glr-backref-module-bz2 sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">BZ2File</span></a>
<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> <a href="https://docs.python.org/3/library/pprint.html#pprint.pprint" title="pprint.pprint" class="sphx-glr-backref-module-pprint sphx-glr-backref-type-py-function"><span class="n">pprint</span></a>
<span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a>
<span class="kn">from</span> <span class="nn">urllib.request</span> <span class="kn">import</span> <a href="https://docs.python.org/3/library/urllib.request.html#urllib.request.urlopen" title="urllib.request.urlopen" class="sphx-glr-backref-module-urllib-request sphx-glr-backref-type-py-function"><span class="n">urlopen</span></a>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">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>
</pre></div>
</div>
<section id="download-data-if-not-already-on-disk">
<h2>Download data, if not already on disk<a class="headerlink" href="#download-data-if-not-already-on-disk" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">redirects_url</span> <span class="o">=</span> <span class="s2">&quot;http://downloads.dbpedia.org/3.5.1/en/redirects_en.nt.bz2&quot;</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">&quot;/&quot;</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">&quot;http://downloads.dbpedia.org/3.5.1/en/page_links_en.nt.bz2&quot;</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">&quot;/&quot;</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> <a href="https://docs.python.org/3/library/os.path.html#os.path.exists" title="os.path.exists" class="sphx-glr-backref-module-os-path sphx-glr-backref-type-py-function"><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span></a><span class="p">(</span><span class="n">filename</span><span class="p">):</span>
        <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Downloading data from &#39;</span><span class="si">%s</span><span class="s2">&#39;, please wait...&quot;</span> <span class="o">%</span> <span class="n">url</span><span class="p">)</span>
        <span class="n">opener</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/urllib.request.html#urllib.request.urlopen" title="urllib.request.urlopen" class="sphx-glr-backref-module-urllib-request sphx-glr-backref-type-py-function"><span class="n">urlopen</span></a><span class="p">(</span><span class="n">url</span><span class="p">)</span>
        <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="s2">&quot;wb&quot;</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
            <span class="n">f</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="nb">print</span><span class="p">()</span>
</pre></div>
</div>
</section>
<section id="loading-the-redirect-files">
<h2>Loading the redirect files<a class="headerlink" href="#loading-the-redirect-files" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></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="w">    </span><span class="sd">&quot;&quot;&quot;Find the index of an article name after redirect resolution&quot;&quot;&quot;</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">&quot;http://dbpedia.org/resource/&quot;</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="w">    </span><span class="sd">&quot;&quot;&quot;Remove the &lt; and &gt; URI markers and the common URI prefix&quot;&quot;&quot;</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="w">    </span><span class="sd">&quot;&quot;&quot;Parse the redirections and build a transitively closed map out of it&quot;&quot;&quot;</span>
    <span class="n">redirects</span> <span class="o">=</span> <span class="p">{}</span>
    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Parsing the NT redirect file&quot;</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><a href="https://docs.python.org/3/library/bz2.html#bz2.BZ2File" title="bz2.BZ2File" class="sphx-glr-backref-module-bz2 sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">BZ2File</span></a><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="nb">print</span><span class="p">(</span><span class="s2">&quot;ignoring malformed line: &quot;</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="nb">print</span><span class="p">(</span><span class="s2">&quot;[</span><span class="si">%s</span><span class="s2">] line: </span><span class="si">%08d</span><span class="s2">&quot;</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Computing the transitive closure of the redirect relation&quot;</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="kc">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="p">{</span><span class="n">source</span><span class="p">}</span>
        <span class="k">while</span> <span class="kc">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="kc">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="nb">print</span><span class="p">(</span><span class="s2">&quot;[</span><span class="si">%s</span><span class="s2">] line: </span><span class="si">%08d</span><span class="s2">&quot;</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>
</pre></div>
</div>
</section>
<section id="computing-the-adjacency-matrix">
<h2>Computing the Adjacency matrix<a class="headerlink" href="#computing-the-adjacency-matrix" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></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="kc">None</span><span class="p">):</span>
<span class="w">    </span><span class="sd">&quot;&quot;&quot;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">    &quot;&quot;&quot;</span>

    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Computing the redirect map&quot;</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Computing the integer index map&quot;</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><a href="https://docs.python.org/3/library/bz2.html#bz2.BZ2File" title="bz2.BZ2File" class="sphx-glr-backref-module-bz2 sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">BZ2File</span></a><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="nb">print</span><span class="p">(</span><span class="s2">&quot;ignoring malformed line: &quot;</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="nb">print</span><span class="p">(</span><span class="s2">&quot;[</span><span class="si">%s</span><span class="s2">] line: </span><span class="si">%08d</span><span class="s2">&quot;</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="kc">None</span> <span class="ow">and</span> <span class="n">l</span> <span class="o">&gt;=</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Computing the adjacency matrix&quot;</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="scipy.sparse.lil_matrix" class="sphx-glr-backref-module-scipy-sparse sphx-glr-backref-type-py-class sphx-glr-backref-instance"><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><a href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.float32" title="numpy.float32" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-attribute"><span class="n">np</span><span class="o">.</span><span class="n">float32</span></a><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="nb">print</span><span class="p">(</span><span class="s2">&quot;Converting to CSR representation&quot;</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="nb">print</span><span class="p">(</span><span class="s2">&quot;CSR conversion done&quot;</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="p">{</span><span class="n">i</span><span class="p">:</span> <span class="n">name</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">index_map</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span>
</pre></div>
</div>
</section>
<section id="computing-principal-singular-vector-using-randomized-svd">
<h2>Computing Principal Singular Vector using Randomized SVD<a class="headerlink" href="#computing-principal-singular-vector-using-randomized-svd" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Computing the principal singular vectors using randomized_svd&quot;</span><span class="p">)</span>
<span class="n">t0</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><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="nb">print</span><span class="p">(</span><span class="s2">&quot;done in </span><span class="si">%0.3f</span><span class="s2">s&quot;</span> <span class="o">%</span> <span class="p">(</span><a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><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="nb">print</span><span class="p">(</span><span class="s2">&quot;Top wikipedia pages according to principal singular vectors&quot;</span><span class="p">)</span>
<a href="https://docs.python.org/3/library/pprint.html#pprint.pprint" title="pprint.pprint" class="sphx-glr-backref-module-pprint sphx-glr-backref-type-py-function"><span class="n">pprint</span></a><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>
<a href="https://docs.python.org/3/library/pprint.html#pprint.pprint" title="pprint.pprint" class="sphx-glr-backref-module-pprint sphx-glr-backref-type-py-function"><span class="n">pprint</span></a><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>
</pre></div>
</div>
</section>
<section id="computing-centrality-scores">
<h2>Computing Centrality scores<a class="headerlink" href="#computing-centrality-scores" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></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="w">    </span><span class="sd">&quot;&quot;&quot;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 &lt;hagberg@lanl.gov&gt;</span>
<span class="sd">      Dan Schult &lt;dschult@colgate.edu&gt;</span>
<span class="sd">      Pieter Swart &lt;swart@lanl.gov&gt;</span>
<span class="sd">    &quot;&quot;&quot;</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://numpy.org/doc/stable/reference/generated/numpy.asarray.html#numpy.asarray" title="numpy.asarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><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="nb">print</span><span class="p">(</span><span class="s2">&quot;Normalizing the graph&quot;</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="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://numpy.org/doc/stable/reference/generated/numpy.asarray.html#numpy.asarray" title="numpy.asarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">asarray</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.where.html#numpy.where" title="numpy.where" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">where</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.isclose.html#numpy.isclose" title="numpy.isclose" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">isclose</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="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://numpy.org/doc/stable/reference/generated/numpy.full.html#numpy.full" title="numpy.full" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">full</span></a><span class="p">(</span><span class="n">n</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="n">dtype</span><span class="o">=</span><a href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.float32" title="numpy.float32" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-attribute"><span class="n">np</span><span class="o">.</span><span class="n">float32</span></a><span class="p">)</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="nb">print</span><span class="p">(</span><span class="s2">&quot;power iteration #</span><span class="si">%d</span><span class="s2">&quot;</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://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot" title="numpy.dot" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><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="nb">print</span><span class="p">(</span><span class="s2">&quot;error: </span><span class="si">%0.6f</span><span class="s2">&quot;</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">&lt;</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Computing principal eigenvector score using a power iteration method&quot;</span><span class="p">)</span>
<span class="n">t0</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><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="nb">print</span><span class="p">(</span><span class="s2">&quot;done in </span><span class="si">%0.3f</span><span class="s2">s&quot;</span> <span class="o">%</span> <span class="p">(</span><a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>
<a href="https://docs.python.org/3/library/pprint.html#pprint.pprint" title="pprint.pprint" class="sphx-glr-backref-module-pprint sphx-glr-backref-type-py-function"><span class="n">pprint</span></a><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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