Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [page-rank]

For questions about Google PageRank algorithm and other similar algorithms.

For questions about Google PageRank algorithm and other similar algorithms.

PageRank is an algorithm assigning numerical values to web pages based on the structure of links between them (i.e., on the webgraph). It can be viewed as a computation of an eigenvector of a special stochastic matrix. It can be understood as a Markov chain where pages are states.

37 questions
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Sensitivity of the second eigenvector (mixing time) in the Google markov transition matrix to changes in $\alpha$

I am reading materials on PageRank seen here: https://arxiv.org/pdf/2207.02296, and here: https://www.stat.uchicago.edu/~lekheng/meetings/mathofranking/ref/langville.pdf. Given the walk matrix $\mathbf{M}$ depicting a Markov chain, some $0 < \alpha…
chibro2
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Does PageRank imply that eigenvalue one exists for any matrix?

I learned from this lecture that for the PageRank algorithm the following equation holds: $$r^{i+1}=L r^{i}$$ I thought when the $r$ vector converges $r^{i+1}=r^{i}$, and hence the equation would become this: $$r=L r$$ which means that $r$ is just…
Lerner Zhang
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Why does PageRank use 0.85 as a standard damping factor?

I understand that for a low damping factor, some nodes will rarely be reached and a high damping factor will slow down the algorithm and cause the random walk to be stuck in 'sinks', and as such a middle ground is preferable. However, is there a…
justworks
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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); taken from the 4th page of this chapter about the pagerank algorithm by Cleve…
user168764
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Graph where PageRank will not converge

I have been stuck on a homework problem for days. Construct a strongly connected graph in which the basic PageRank computation does not converge. I tried everything and still cannot find a solution. My latest attempt is to brute force generate…
mchangun
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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 participate in you get ranked in a Tier (eg from Tier…
Taum
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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: $M=\begin{bmatrix} 0&0&\frac{1}{2} \\ 1&0&\frac{1}{2} \\ 0&1&0 \end{bmatrix}$ and $v=\begin{bmatrix} \frac{1}{3} \\ \frac{1}{3}…
Lauren Bathers
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How page rank relates to the power iteration method

I do not see how pageRank relates to the power method. Since for the pageRank we are looking for the steady stable state (vector) for a Markov (transition) matrix and the matrix has already an eigenvalue equals to one, why multiplication is used…
Nawaf
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Expression of the stationary distribution of a Markov Chain (PageRank)

I want to find the expression for the PageRank of a webpage defined as in the original paper of Sergey and Larry (The Anatomy of a Large-Scale Hypertextual Web Search Engine). Consider a directed graph of $n$ vertices with adjacency matrix $A$.…
René Quijada
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Does there exist a graph that the page ranks of each node are distinct?

Suppose that there is a graph of $n$ nodes ($n>3$), is it possible that every node has a distinct page rank value? the page rank is defined as $R$ while $MR=R$, $M$ is the transition matrix. I have sampled one thousand random graph (10 nodes) and…
Nekomiya Kasane
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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 matrix that describes the probability of going from a…
JustANoob
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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 graph below, is a grid composed of 3x3 nodes. The…
Marc Ortiz
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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 with $n$ nodes (webpages), the importance of a webpage is given by…
user168764
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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 expression is: $$x_{i}=\sum_{j\in…
Thomas Russell
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Geometric interpretation of eigenvector centrality/PageRank

I am trying to achieve a better understanding of the relationship between different uses of eigenvectors, in particular how network applications (eigenvector centrality, PageRank) relate to dimension reduction applications (like principal components…
tvg
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