Google Pageranking Problem: The Model and the Analysis

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.