License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.07071.10
URN: urn:nbn:de:0030-drops-10693
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2007/1069/
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Serra Capizzano, Stefano

Google Pageranking Problem: The Model and the Analysis

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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:DagSemProc.07071.10,
  author =	{Serra Capizzano, Stefano},
  title =	{{Google Pageranking Problem: The Model and the Analysis}},
  booktitle =	{Web Information Retrieval and Linear Algebra Algorithms},
  pages =	{1--34},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7071},
  editor =	{Andreas Frommer and Michael W. Mahoney and Daniel B. Szyld},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2007/1069},
  URN =		{urn:nbn:de:0030-drops-10693},
  doi =		{10.4230/DagSemProc.07071.10},
  annote =	{Keywords: Google matrix, rank-one perturbation, Jordan canonical form, extrapolation formulae.}
}

Keywords: Google matrix, rank-one perturbation, Jordan canonical form, extrapolation formulae.
Collection: 07071 - Web Information Retrieval and Linear Algebra Algorithms
Issue Date: 2007
Date of publication: 28.06.2007

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