Why Eigen values of adajcency matrix are actually the sentence scores in Textrank
<p>Here is the route for TextRank:</p>

<ol>
<li>Document to be summarized expressed as tf-idf matrix</li>
<li>(tf-idf matrix)*(tf-idf matrix).Transpose = Adjacency matrix of some graph whose vertices are
actually the sentences of above document</li>
<li>Page rank is applied on this graph -> returns PR values of each sentence</li>
</ol>

<p>Now, <strong>this PR values are actually Eigen values of that adjacency matrix</strong><br>
What is the physical meaning or intuition behind this.? </p>

<p><strong>Why Eigen values are actually the ranks ?</strong></p>

<p>Here is the link for Page Rank:
<a href="http://www.cs.princeton.edu/~chazelle/courses/BIB/pagerank.htm" rel="nofollow">http://www.cs.princeton.edu/~chazelle/courses/BIB/pagerank.htm</a></p>

<p>Here is an extract from above page: <br>
<strong>PageRank or PR(A) can be calculated using a simple iterative algorithm, and corresponds to the principal eigenvector of the normalized link matrix of the web.</strong></p>

<p>Link for TextRank:
<a href="https://joshbohde.com/blog/document-summarization" rel="nofollow">https://joshbohde.com/blog/document-summarization</a></p>

<p>To begin with, your question is a bit mistaken. The eignevalues are <em>not</em> the scores. Rather, the <em>entries of the stationary eigenvector</em> are the scores.</p>

<p>Textrank works on a <a href="https://web.eecs.umich.edu/~mihalcea/papers/mihalcea.emnlp04.pdf" rel="nofollow">graphical approach to words</a>. It has a number of variations, but they have the following common steps:</p>

<ol>
<li><p>Create a weighted graph where the vertices are entities (words or sentences), and the weights are the transition probabilities between entities.</p></li>
<li><p>Find the <a href="https://en.wikipedia.org/wiki/Stochastic_matrix" rel="nofollow">stochastic matrix</a> associated with the graph, and score each entity according to its stationary distribution.</p></li>
</ol>

<p>In this case, the graph is built as follows. First, a matrix is built where the rows are sentences and the columns are words. The entries of the matrix are specified by TF-IDF. To find the similarity between sentences, the normalized matrix is multiplied by its transform. This is because, for each two sentences and a word, there is a similarity between the sentences based on the product of the TF-IDF of the word in each sentence, and we need to sum up over all words. If you think about it a bit, summing up the products is exactly what matrix multiplication by the transpose does.</p>

<p>So now we have a stochastic matrix <em>P</em> which can be interpreted as the probability of transition from sentence <em>i</em> to sentence <em>j</em>. The score is the stationary distribution <em>x</em>, which means that</p>

<p><em>P x = x = 1 x</em>.</p>

<p>This means that <em>x</em> is the eigenvector associated with the eigenvalue 1. By the <a href="https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem" rel="nofollow">Perron-Frobenius Theorem</a>, this eigenvector exists under some mild conditions, and 1 is the largest eigenvalue. This last part is basically Pagerank.</p>

python nlp eigenvector pagerank summarization