\section{Introduction}

The PageRank model was proposed by Google in a series of papers to evaluate accurately the most important web-pages from the World Wide Web matching a set of keywords entered by a user. For search engine rankings, the importance of web-pages is computed from the stationary probability vector of the random process of a web surfer who keeps visiting a large set of web-pages connected by hyperlinks. The link structure of the World Wide Web is represented by a directed graph, the so-called web link graph, and its corresponding adjacency matrix $G \in \N^{n \times n}$ where $n$ denotes the number of pages and $G_{ij}$ is nonzero (being 1) only if the \emph{jth} page has a hyperlink pointing to the \emph{ith} page. The transition probability matrix $P \in \R^{n \times n}$ of the random process has entries as described in \ref{eq:transition}.

\begin{equation}\label{eq:transition}
    P(i,j) =
    \begin{cases}
        \displaystyle \frac{1}{\sum_{k=1}^n G_{kj}} & \text{if } G_{i,j} = 0 \\
        0 & \text{otherwise}
    \end{cases}
\end{equation}

\noindent The entire random process needs a unique stationary distribution. To ensure this propriety is satisfied , the transition matrix $P$ is usually modified to be an irreducible stochastic matrix $A$ (called the Google matrix) as follows:

% \noindent To ensure that the random process has a unique stationary distribution and it will not stagnate, the transition matrix P is usually modified to be an irreducible stochastic matrix $A$ (called the Google matrix) as follows

\begin{equation}\label{eq:google}
    A = \alpha \tilde P + (1 - \alpha)v e^T
\end{equation}

\noindent In \ref{eq:google} we have defined a new matrix called $\tilde P = P + vd^T$ where $d \in N^{n \times 1}$ is a binary vector tracing the indices of the damping web pages with no hyperlinks, i.e., $d(i) = 1$ if the \emph{i-th} page has no hyperlink, $v \in \R^{n \times n}$ is a probability vector, $e = [1, 1, ... ,1]^T$ and $0<\alpha<1$, the so-called damping factor that represents the probability in the model that the surfer transfer by clicking a hyperlink rather than other ways. Mathematically, the PageRank model can be formulated as the problem of finding the positive unit eigenvector $x$ (the so-called PageRank vector) such that
\begin{equation}\label{eq:pr}
    Ax = x, \quad \lVert x \rVert = 1, \quad x > 0
\end{equation}
or, equivalently, as the solution of the linear system
\begin{equation}\label{eq:pr2}
    (I - \alpha \tilde P)x = (1 - \alpha)v
\end{equation}

\noindent The authors of the paper \cite{SHEN2022126799} emphasize how in the in the past decade or so, considerable research attention has been devoted to the efficient solution of problems \ref{eq:pr} \ref{eq:pr2}, especially when $n$ is very large. For moderate values of the damping factor, e.g. for $\alpha = 0.85$ as initially suggested by Google for search engine rankings, solution strategies based on the simple Power method have proved to be very effective. However, when $\alpha$ approaches 1, as is required in some applications, the convergence rates of classical stationary iterative methods including the Power method tend to deteriorate sharply, and more robust algorithms need to be used. \vspace*{0.4cm}

\noindent In the reference paper that we are using for this project, the authors focus their attention in the area of PageRank computations with the same network structure but multiple damping factors. For example, in the Random Alpha PageRank model used in the design of anti-spam mechanism \cite{Constantine2009Random}, the rankings corresponding to many different damping factors close to 1 need to be computed simultaneously. They explain that the problem can be expressed mathematically as solving a sequence of linear systems
\begin{equation}\label{eq:pr3}
    (I - \alpha_i \tilde P)x_i = (1 - \alpha_i)v \quad  \alpha_i \in (0, 1) \quad \forall i \in \{1, 2, ..., s\}
\end{equation}
As we know, standard PageRank algorithms applied to \ref{eq:pr3} would solve the $s$ linear systems independently. Although these solutions can be performed in parallel, the process would still demand large computational resources for high dimension problems.
This consideration motived the authors to search novel methods with reduced algorithmic and memory complexity, to afford the solution of larger problems on moderate computing resources. They suggest to write the PageRank problem with multiple damping factors given at once \ref{eq:pr3} as a sequence of shifted linear systems of the form:
\begin{equation}
    \Big(\frac{1}{\alpha_i}I - \tilde P \Big)x^{(i)} = \frac{1 - \alpha_i}{\alpha_i}v \quad \forall i \in \{1, 2, ..., s\} \quad 0 < \alpha_i < 1
\end{equation}
We know from literature that the Shifted Krylov methods may still suffer from slow convergence when the damping factor approaches 1, requiring larger search spaces to converge with satisfactory speed. In \cite{SHEN2022126799} is suggest that, to overcome this problem, we can combine stationary iterative methods and shifted Krylov subspace methods. They derive an implementation of the Power method that solves the PageRank problem with multiple dumpling factors at almost the same computational time of the standard Power method for solving one single system. They also demonstrate that this shifted Power method generates collinear residual vectors. Based on this result, they use the shifted Power iterations to provide smooth initial solutions for running shifted Krylov subspace methods such as \texttt{GMRES}. Besides, they discuss how to apply seed system choosing strategy and extrapolation techniques to further speed up the iterative process.

% As an attempt of a possible remedy in this situation, we present a framework that combines. shifted stationary iterative methods and shifted Krylov subspace methods. In detail, we derive the implementation of the Power method that solves the PageRank problem with multiple damping factors at almost the same computational cost of the standard Power method for solving one single system. Furthermore, we demonstrate that this shifted Power method generates collinear residual vectors. Based on this result, we use the shifted Power iterations to provide smooth initial solutions for running shifted Krylov subspace methods such as GMRES. Besides, we discuss how to apply seed system choosing strategy and extrapolation techniques to further speed up the iterative process.

\subsection{Overview of the classical PageRank problem}
The Power method is considered one of the algorithms of choice for solving either the eigenvalue \ref{eq:pr} or the linear system \ref{eq:pr2} formulation of the PageRank problem, as it was originally used by Google. Power iterations write as
\begin{equation}\label{eq:power}
    x_{(k+1)} = Ax_k =\alpha \tilde P x_{(k)} + (1 - \alpha)v
\end{equation}
The convergence behavior is determined mainly by the ratio between the two largest eigenvalues of A. When $\alpha$ gets closer to $1$, though, the convergence can slow down significantly. \\

\noindent As stated in \cite{SHEN2022126799} The number of iterations required to reduce the initial residual down to a tolerance $\tau$, measured as $\tau = \lVert Ax_k - x_k \rVert = \lVert x_{k+1} - x_k \rVert$ can be estimated as $\frac{\log_{10} \tau}{\log_{10} \alpha}$. The authors provide an example: when $\tau = 10^{-8}$ the Power method requires about 175 steps to converge for $\alpha = 0.9$ but the iteration count rapidly grows to 1833 for $\alpha = 0.99$. Therefore, for values of the damping parameter very close to 1 more robust alternatives to the simple Power algorithm should be used.