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# Google Pageranking Problem: The Model and the Analysis

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DagSemProc.07071.10.pdf
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## Cite As

Stefano Serra Capizzano. Google Pageranking Problem: The Model and the Analysis. In Web Information Retrieval and Linear Algebra Algorithms. Dagstuhl Seminar Proceedings, Volume 7071, pp. 1-34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)
https://doi.org/10.4230/DagSemProc.07071.10

## Abstract

Let \$A\$ be a given \$n\$-by-\$n\$ complex matrix with eigenvalues \$lambda ,lambda _{2},ldots ,lambda _{n}\$. Suppose there are nonzero vectors \$% x,yin mathbb{C}^{n}\$ such that \$Ax=lambda x\$, \$y^{ast }A=lambda y^{ast }\$, and \$y^{ast }x=1\$. Let \$vin mathbb{C}^{n}\$ be such that \$v^{ast }x=1\$% , let \$cin mathbb{C}\$, and assume that \$lambda eq clambda _{j}\$ for each \$j=2,ldots ,n\$. Define \$A(c):=cA+(1-c)lambda xv^{ast }\$. The eigenvalues of \$% A(c)\$ are \$lambda ,clambda _{2},ldots ,clambda _{n}\$. Every left eigenvector of \$A(c)\$ corresponding to \$lambda \$ is a scalar multiple of \$% y-z(c)\$, in which the vector \$z(c)\$ is an explicit rational function of \$c\$. If a standard form such as the Jordan canonical form or the Schur triangular form is known for \$A\$, we show how to obtain the corresponding standard form of \$A(c)\$. The web hyper-link matrix \$G(c)\$ used by Google for computing the PageRank is a special case in which \$A\$ is real, nonnegative, and row stochastic (taking into consideration the dangling nodes), \$cin (0,1)\$, \$x\$ is the vector of all ones, and \$v\$ is a positive probability vector. The PageRank vector (the normalized dominant left eigenvector of \$G(c)\$) is therefore an explicit rational function of \$c\$. Extrapolation procedures on the complex field may give a practical and efficient way to compute the PageRank vector when \$c\$ is close to \$1\$. A discussion on the model, on its adherence to reality, and on possible variations is also considered.
##### Keywords
• rank-one perturbation
• Jordan canonical form
• extrapolation formulae.

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