Questions tagged [page-rank]

For questions about Google PageRank algorithm and other similar algorithms.

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How does Pagerank iteratively calculate an eigenvector?

I understand that Pagerank works by finding the eigenvector $\lambda$ to the eigenvalue $\epsilon=1$ of a Markov matrix $A \in \mathbb{R}^{n \times n}$. And, as far as I know, the iterative algorithm ...
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Help with a google rank for each site in the network.

Consider the following web sites on the internet. Each node represents a website and the arrows indicate hyperlinks between the sites. Find the most important website in this network. In this case we ...
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34 views

Find a Google rank for each site in the following network. [closed]

Consider the following web sites on the internet. Each node represents a website and the arrows indicate hyperlinks between the sites. Find the most important website in this network. Let’s define ...
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Testing from which of two jars my sample set has come from and with what significance? What if the significance is a function of the success events?

Assume I have two jars, each having two sets of balls, black and white. The balls have different weights, and for the white (success draw) balls, the heavier weight makes a more significant success if ...
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73 views

PageRank sensitive calculation

𝐀 is the adj. matrix of a directed graph 𝐺. 𝐱 is the PageRank vector we calculate for a given 𝛼 (e.g. 0.85). 𝐶 is a subset of pages of 𝐺, for which we change some of the outgoing links. By 𝐱̃ ...
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39 views

Rank of a jacobian matrix

I have the following problem: Let $x,v\in\mathbb{R}^n$ be strictly stochastic vectors ($x_i>0; \sum_{i=1}^n x_i=1$). Let $R\in\mathbb{R}^n\times\mathbb{R}^{n^2}$ a matrix with stochastic columns ...
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59 views

Topic sensitive page rank [closed]

I have a question regarding the topic sensitive PageRank algorithm. The picture shows how the transition matrix A is modified in order to make the normal PageRank algorithm topic sensitive. Why don'...
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34 views

Computing PageRank Vector

I have been given the adjacency matrix $$A = \begin{pmatrix} 0 & 1 & 0&1&1 \\ 1 & 0 &1&1& 0\\ 0& 1 &0&0&1 \\1&1&0&0&1 \\0&1&1&...
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28 views

Show that the vector of ones, e, is an eigenvector of the Google matrix transpose

In the Google matrix where $$G=\alpha A+(1-\alpha)\frac{1}{n}ee^T$$ and $e$ is a vector of ones, how do I show that $e$ is the eigenvector of $G^T$ corresponding to the eigenvalue of 1 I need to ...
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61 views

Show that columns of PageRank matrix sum to 1

Where the matrix A is replaced with $$G=\alpha A+(1-\alpha)\frac{1}{n}ee^T$$ Is it a sufficient condition a matrix is stochastic if the largest eigenvalue is 1? Or that in this case since A is ...
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73 views

Calculating PageRank of a Google Matrix

Let $A$ the adjacency matrix of a Web Digraph, with $\{0,1\}$ entries. For sake of clarity, we assume that the matrix is irreducible and without full-zero rows (i.e. no leaf nodes in the graph) . ...
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171 views

Perron Frobenius Theorem modified

On this site I found a modified version of Perron Frobenius Theorem Perron-Frobenius Theorem: If M is a positive, column stochastic matrix, then: 1 is an eigenvalue of multiplicity one. 1 is ...
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102 views

Connections between the randomness of the normal distribution and Textrank?

In a TED speech on 8:40 the mathematician said that: This algorithm uses the laws of mathematical randomness to determine automatically the most relevant web pages, in the same way as we used ...
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What is the relationship between the fluid in PageRank and web traffic?

PageRank can be interpreted as the amount of an imaginary fluid that collects at different nodes in a web graph. I would like to know what the relationship is between this imaginary fluid and actual ...
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45 views

Ranking participants based on tiers and totals

I am trying to rank participants based on sets of data that I have. The data used is for a competition. In this competition, you can participate in X amount of events, at the end of the event you ...
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303 views

Prove that the dimension of the eigenspace corresponding to the eigenvalue $\lambda=1$ of $H$ is at least the number of the clusters..

There are lots of ’islands’ in the world-wide-web, meaning clusters of websites that are not connected to other parts of the world wide web via hyperlinks. Let $H$ denote the column stochastic ...
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270 views

Initial distribution of a Markov Chain (Power Model)

So, I was trying to model the PageRank algorithm based on the information of an article, and it said that in order to implement the power method, I needed the distribution of the process given by $$\...
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43 views

Calculate PageRank for small web

Calculate PageRank for: A links to B, B links to C and C links to B and C where the damping factor $\beta=0.8$ I have: $M=\begin{bmatrix} 0&0&\frac{1}{2} \\ 1&0&\frac{1}{2} \\ 0&...
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88 views

PageRank algorithm for grid graph

I am currently studying the PageRank algorithm. To find the ranks i know you have two options: Compute the result of a large linear system Apply the surfer concept (like Markov chains) I have this ...
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112 views

Why is the matrix $(I - A)$ theoretically singular?

I've the following Matlab code to compute the eigenvector using the inverse iteration (or power) method: A = p * G * D + delta; x = (I − A) \ e; x = x / sum(x); ...
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798 views

Understanding PageRank as an eigenvalue problem

In the book "A first course in numerical methods" by U. Arscher and C. Greif, chapter 8 on "Eingenvalues and singular values", example 8.1, we have: Given a network linkage graph ...
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187 views

Rewriting the simplified google algorithm in linear algebra form

I have the expression for the rank ($x_{i}$) of a page $i$ in an internet with $n$ sites, each site contains $n_{i}$ links to other sites and is linked to by the pages $L_{i}\subset\{1,\dots,n\}$. The ...