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\begin{document}


\setassignment
\setduedate{Thursday, 8 October 2020, 12:00 AM}

\serieheader{Numerical Computing}{2020}{Student: Claudio Maggioni}{Discussed with: --}{Solution for Project 1}{}
\newline

\assignmentpolicy
The purpose of this assignment\footnote{This document is originally based on a SIAM book chapter from \textsl{Numerical Computing with Matlab} from  Clever B. Moler.} is to learn the importance of numerical linear algebra algorithms to solve fundamental  linear algebra problems that occur in search engines.



\section{Page-Rank Algorithm }

\subsection{Theory [20 points]}

\subsubsection{What assumptions should be made to guarantee convergence of the power method?}
The first assumption to make is that the biggest eigenvalue in terms of absolute values should (let's name it $\lambda_1$)
be strictly greater than all other eigenvectors, so:

$$|\lambda_1| < |\Lambda_i| \forall i \in \{2..n\}$$

Also, the eigenvector \textit{guess} from which the power iteration starts must have a component in the direction of $x_i$, the eigenvector for the eigenvalue $\lambda_1$ from before.

\subsubsection{What is a shift and invert approach?}

The shift and invert approach is a variant of the power method that may significantly increase the rate of convergence where some application of the vanilla method require large numbers of iterations. This improvement is achieved by taking the input matrix $A$ and deriving a matrix $B$ defined as:

\[B = (A - \alpha I)^{-1}\]

where $\alpha$ is an arbitrary constant that must be chosen wisely in order to increase the rate of convergence. Since the eigenvalues $u_i$ of B can be derived from the eigenvalues $\lambda_i$ of A, namely:

$$u_i = \frac{1}{\lambda_i - \alpha}$$

the rate of convergence of the power method on B is:

$$\left|\frac{u_2}{u_1}\right| = \left|\frac{\frac1{\lambda_2 - \alpha}}{\frac1{\lambda_1 - \alpha}}\right| = \left|\frac{\lambda_1 - \alpha}{\lambda_2 - \alpha}\right|$$

By choosing $\alpha$ close to $\lambda_1$, the convergence is sped up. To further increase the rate of convergence (up to a cubic rate), a new $\alpha$, and thus a new $B$, may be chosen for every iteration.

\subsubsection{What is the difference in cost of a single iteration of the power method, compared to the inverse iteration?}

Inverse iteration is generally more expensive than a regular application of the power method, due to the overhead caused by the intermediate matrix B. One must either recompute B every time $\alpha$ changes, which is rather expensive due to the inverse operation in the definition of B, or one must solve the matrix equation $(A - \alpha I)v_k = v_{k-1}$ in every iteration.

\subsubsection{What is a Rayleigh quotient and how can it be used for eigenvalue computations?}

The Railegh quotient is an effective way to either compute the corresponding eigenvalue of an eigenvector or the corresponding eigenvalue approximation of an eigenvector approximation. I.e., if $x$ is an eigenvector, then:

\[\lambda = \mu(x) = \frac{x^T A x}{x^T x}\]

is the corresponding eigenvalue, while if $x$ is an eigenvector approximation, for example found through some iterations of the power method, then $\lambda$ is the closest possible approximation to the corresponding eigenvalue in a least-square sense.

\subsection{Other webgraphs [10 points]}

The provided PageRank MATLAB implementation was run 3 times on the starting websites \texttt{http://atelier.inf.usi.ch/~maggicl}, \texttt{https://www.iisbadoni.edu.it}, and \texttt{https://www.usi.ch}, with results listed respectively in Figure \ref{fig:run1}, Figure \ref{fig:run2} and Figure \ref{fig:run3}.

One patten that emerges on the first and third execution is the presence of 1s in the main diagonal. This indicates that several pages found have a link to themselves. Another interesting pattern, this time observable in all executions, is the presence of contiguous rectangular regions filled with 1s, especially along the main diagonal. This may be due to the presence of pages belonging to the same website, thus having a common layout and perhaps linking to a common set of internal (when near to the main diagonal) or external pages.

\begin{figure}[h]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width = \textwidth]{run1spy}
\caption{Spy plot of connectivity matrix}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width = \textwidth]{run1rank}
\caption{Page rank bar graph}
\end{subfigure}
\begin{subfigure}{\textwidth}
\begin{verbatim}

360    0.0869    31     1    https://creativecommons.org/licenses/by-sa/3.0
204    0.0406    8      1    https://forum.gitlab.com
82     0.0189    117    18   https://www.mediawiki.org
81     0.0188    117    4    https://wikimediafoundation.org
87     0.0150    6      1    https://docs.gitea.io
78     0.0145    114    9    https://www.mediawiki.org/wiki/Special:MyLanguage/
                                 How_to_contribute
77     0.0132    77     13   https://foundation.wikimedia.org/wiki/Privacy_policy
217    0.0127    40     8    https://bugs.archlinux.org
80     0.0115    107    6    https://foundation.wikimedia.org/wiki/Cookie_statement
215    0.0114    38     5    https://bbs.archlinux.org
\end{verbatim}
\caption{Top 10 webpages with highest PageRank}
\end{subfigure}
\label{fig:run1}
\caption{Results of first PageRank calculation (for starting website \texttt{http://atelier.inf.usi.ch/~maggicl/})}
\end{figure}

\begin{figure}[h]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width = \textwidth]{run2spy}
\caption{Spy plot of connectivity matrix}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width = \textwidth]{run2rank}
\caption{Page rank bar graph}
\end{subfigure}
\begin{subfigure}{\textwidth}
\begin{verbatim}

411    0.0249    42     1    https://twitter.com/mozilla
63     0.0248    145    1    https://twitter.com/firefox
68     0.0203    142    1    https://www.instagram.com/firefox
412    0.0164    37     1    https://www.instagram.com/mozilla
62     0.0080    21     1    https://github.com/mozilla/kitsune
81     0.0070    110    2    https://www.apple.com
384    0.0064    5      1    https://www.xfinity.com/privacy/policy/dns
4      0.0064    32     0    https:
377    0.0059    19     1    https://abouthome-snippets-service.readthedocs.io/en/
                                 latest/data_collection.html
393    0.0059    19     1    https://www.adjust.com/terms/privacy-policy
410    0.0057    16     1    https://wiki.mozilla.org/Firefox/Data_Collection
\end{verbatim}
\caption{Top 10 webpages with highest PageRank}
\end{subfigure}
\label{fig:run2}
\caption{Results of second PageRank calculation (for starting website \texttt{https://www.iisbadoni.edu.it/})}
\end{figure}

\begin{figure}[h]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width = \textwidth]{run3spy}
\caption{Spy plot of connectivity matrix}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width = \textwidth]{run3rank}
\caption{Page rank bar graph}
\end{subfigure}
\begin{subfigure}{\textwidth}
\begin{verbatim}

55     0.0741    354    1    https://www.instagram.com/usiuniversity
53     0.0324    366    3    https://www.facebook.com/usiuniversity
299    0.0248    6      1    https://twitter.com/usi_en
329    0.0243    8      1    https://www.facebook.com/USIeLab
308    0.0156    7      3    https://www.facebook.com/USIFinancialCommunication
60     0.0155    316    2    https://www.swissuniversities.ch
424    0.0144    96     1    https://it.bul.sbu.usi.ch
330    0.0123    6      4    https://www.facebook.com/USI.ITDxC
320    0.0122    7      1    https://www.facebook.com/usiimeg
56     0.0107    320    0    https://www.youtube.com/usiuniversity
\end{verbatim}
\caption{Top 10 webpages with highest PageRank}
\end{subfigure}
\label{fig:run3}
\caption{Results of third PageRank calculation (for starting website \texttt{https://www.usi.ch/})}
\end{figure}

\subsection{Connectivity matrix and subcliques [10 points]}

\subsection{Connectivity matrix and disjoint subgraphs [10 points]}

\subsubsection{What is the connectivity matrix G (w.r.t figure 5)?}

The connectivity matrix G, with U being defined as $\{"alpha", "beta", "gamma", "delta", "rho", "sigma"\}$ is:

\[G = \begin{bmatrix}
	0&0&0&1&0&0\\
	1&0&0&0&0&0\\
	1&1&0&0&0&0\\
	0&1&1&0&0&0\\
	0&0&0&0&0&1\\
	0&0&0&0&1&0\\
\end{bmatrix}\]

\subsubsection{What are the PageRanks if the hyperlink transition probability $p$ is the default value 0.85?}

First we compute the matrix A, finding:

\[A = \frac1{40} \begin{bmatrix}
	1 &1 &1 &35&1 &1 \\
	18&1 &1 &1 &1 &1 \\
	18&18&1 &1 &1 &1 \\
	1 &18&35&1 &1 &1 \\
	1 &1 &1 &1 &1 &35 \\
	1 &1 &1 &1 &35&1 \\
\end{bmatrix}\]

We then find the eigenvectors and eigenvalues of A through MATLAB, finding that the solution of $A x = 1 x$ is:

\[x\approx\begin{bmatrix}
	0.4771\\
    	0.2630\\
    	0.3747\\
    	0.4905\\
    	0.4013\\
    	0.4013\\
\end{bmatrix}\]

Thus the pageranks are the components of vector $x$, w.r.t. the order given in U.

\subsubsection{Describe what happens with this example to both the definition of PageRank and the computation done by pagerank in the limit $p \to 1$.}

If $p$ is closer to 1, then the probability a web user will visit a certain page randomly decreases, thus giving more weight in the computation of PageRank to the links between one page and another.

In the computation, increasing $p$ decreases $\delta$ (which represents the probability of a user randomly visiting a page), eventually making it 0 when $p$ is 1.

\subsection{PageRanks by solving a sparse linear system [50 points]}


\end{document}
