 when quoting this document, please refer to the following
URN: urn:nbn:de:0030-drops-10693
URL:

### Google Pageranking Problem: The Model and the Analysis

 pdf-format:

### 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.

### BibTeX - Entry

```@InProceedings{serracapizzano:DSP:2007:1069,
author =	{Stefano Serra Capizzano},
title =	{Google Pageranking Problem: The Model and the Analysis},
booktitle =	{Web Information Retrieval and Linear Algebra Algorithms},
year =	{2007},
editor =	{Andreas Frommer and Michael W. Mahoney and Daniel B. Szyld},
number =	{07071},
series =	{Dagstuhl Seminar Proceedings},
ISSN =	{1862-4405},
publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
DROPS-Home | Fulltext Search | Imprint | Privacy 