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% vim: set ts=4 sw=4 et tw=80:
\documentclass[unicode,11pt,a4paper,oneside,numbers=endperiod,openany]{scrartcl}
\usepackage{graphicx}
\usepackage{subcaption}
\usepackage{amsmath}
\input{assignment.sty}
\begin{document}
\setassignment
\setduedate{Thursday, 8 October 2020, 12:00 AM}
\serieheader{Numerical Computing}{2020}{Student: Claudio Maggioni}{Discussed with: --}{Solution for Project 1}{}
ewline
\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{Show that the order of convergence of the power method is linear,
and state what the asymptotic error constant is.}
First of all, we show the the sequence of vectors computed by power iteration
indeed converges to $\lambda_1$ or the biggest eigenvector (we assume we name
eigenvectors in decreasing order of magnitude, with $|\lambda_1| > |\lambda_i|$
for $i \in 2..n$).
We can express the seed for the eigenvector (i.e. the initial value of $v$ of
the power iteration) as a linear combination of eigenvalues:
\[v_0 = \sum_{i=1}^n a_i x_i\]
We can then express the result of the n-th power method as
\[v_n = \gamma A v_{n-1} = A^n v_0 = \sum_{i=1}^n \gamma a_i \lambda_i^n x_i =
\lambda_1^n \sum_{i=1}^n \gamma a_i \left( \frac{\lambda_i}{\lambda_1} \right)^n x_i =
\gamma a_1 \lambda_1^n x_1 + \lambda_1^n \sum_{i=2}^n \gamma a_i
\left(\frac{\lambda_i}{\lambda_1}\right)^n x_i \]
Here, $\gamma$ is just a normalization term to make $||v_n|| = 1$. $v_n$ clearly
converges to $x_1$ since all the terms in the $\sum_{i=2}^n$ contain
$\frac{\lambda_i}{\lambda_1}$, which is always less than 0 if $i > 1$ for the
sorting of eigenvalues we did before. Therefore, these terms to the power of n
converge to 0, and $\gamma$ will cancel out $a_1 \lambda_1^k$ due to the
normalization, thus making the sequence converge to $\lambda_1$.
To see if the sequence converges linearly we use the definitions of rate of
convergence:
\[\lim_{n \to \infty}\frac{|x_{n+1} - \lambda_1|}{|x_n - \lambda_1|^1} = \mu\]
If this limit has a finite solution then the sequence converges linearly with
rate $\mu$.
\[\lim_{n \to \infty}\frac{\left| a_1 \lambda_1^{n+1} x_1 + \lambda_1^{n+1}
\sum_{i=2}^n a_i \left(\frac{\lambda_i}{\lambda_1}\right)^{n+1}
x_i - \beta_{n+1} x_1\right|}
{\left| a_1 \lambda_1^n x_1 + \lambda_1^n \sum_{i=2}^n a_i
\left(\frac{\lambda_i}{\lambda_1}\right)^n x_i - \beta_n x_1\right|^1} = \mu\]
To simplify calculations, we consider the sequence without the normalization
factor $\gamma$ that will converge to a denormalized version of $x_1$, named
$\beta x_1$. We can then simplify the $a_1\lambda_1^{i}x_1$ terms in the
sequences with $\beta_{i} x_1$ since $\beta_i$ can be set freely.
Now we consider that if $|\lambda_2| > |\lambda_i| \forall i \in 3..n$ (since we
sorted the eigenvalues), then
$\left(\frac{\lambda_i}{\lambda_1}\right)^n$ for $i > 2$ will always converge faster to
0 than $\left(\frac{\lambda_2}{\lambda_1}\right)^n$ thus all terms other than
$i=2$ can be ignored in the limit computation. Therefore, the limit has finite
solution and the convergence rate
is
\[\mu = \frac{\lambda_2}{\lambda_1}\]
\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.
Finally, we can always observe a line starting from the top-left of G and ending in its bottom-left, running along a steep path slighly going right. This may be a side effect of the way pages are discovered and numbered: if new pages are continuously discovered, these pages will be added at the end of U and a corresponding vertical strip on 1s will appear in the bottomest non-colored region of G. This continues until $n$ unique pages are visited and the line reaches the bottom edge of the connectivity matrix. The steepness of the line thus formed depends on the amount of new pages discovered in each of the first iterations of the \texttt{surfer(...)} function.
\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 1 https://creativecommons.org/licenses/by-sa/3.0
204 0.0406 8 1 https://forum.gitlab.com
82 0.01897 https://www.mediawiki.org
81 0.01887 4 https://wikimediafoundation.org
87 0.0150 6 1 https://docs.gitea.io
78 0.01454 9 https://www.mediawiki.org/wiki/Special:MyLanguage/
How_to_contribute
77 0.0132 https://foundation.wikimedia.org/wiki/Privacy_policy
217 0.0127 8 https://bugs.archlinux.org
80 0.01157 6 https://foundation.wikimedia.org/wiki/Cookie_statement
215 0.0114 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 1 https://twitter.com/mozilla
63 0.02485 1 https://twitter.com/firefox
68 0.02032 1 https://www.instagram.com/firefox
412 0.0164 1 https://www.instagram.com/mozilla
62 0.0080 1 https://github.com/mozilla/kitsune
81 0.00700 2 https://www.apple.com
384 0.0064 5 1 https://www.xfinity.com/privacy/policy/dns
4 0.0064 0 https:
377 0.0059 1 https://abouthome-snippets-service.readthedocs.io/en/
latest/data_collection.html
393 0.0059 1 https://www.adjust.com/terms/privacy-policy
410 0.0057 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 0.07414 1 https://www.instagram.com/usiuniversity
53 0.03246 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 0.01556 2 https://www.swissuniversities.ch
424 0.0144 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 0.01070 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]}
The following ETH organization are following for the near cliques along the diagonal of the connectivity matrix in \texttt{eth500.mat}. The clique approximate position on the diagonal is indicated throgh the ranges in parenthesis.
\begin{itemize}
\item \texttt{baug.ethz.ch} (74-100)
\item \texttt{mat.ethz.ch} (114-129)
\item \texttt{mavt.ethz.ch} (164-182)
\item \texttt{biol.ethz.ch} (198-216)
\item \texttt{chab.ethz.ch} (221-236)
\item \texttt{math.ethz.ch} (264-278)
\item \texttt{erdw.ethz.ch} (321-337)
\item \texttt{usys.ethz.ch} (358-373)
\item \texttt{mtec.ethz.ch} (396-416)
\item \texttt{gess.ethz.ch} (436-462)
\end{itemize}
\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:
\[x'\approx\begin{bmatrix}
0.4771\\
0.2630\\
0.3747\\
0.4905\\
0.4013\\
0.4013\\
\end{bmatrix}\]
To obtain more easily interpretable PageRank values, we can require that the sum
of all PageRanks should be one. Once we normalize in this way, the result is:
\[x=\frac{x'}{\sum_{i=1}^{6}x_i}\approx\begin{bmatrix}
0.1981\\
0.1092\\
0.1556\\
0.2037\\
0.1667\\
0.1667\\
\end{bmatrix}\]
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]}
\subsubsection{Create \texttt{pagerank1(G)} by modifying \texttt{pagerank.m} to
use the power method instead of solving thesparse linear system. What is an
appropriate test for terminating the power iteration?}
The MATLAB solution to this question can be found on
\texttt{files\_data/pagerank1.m}.
An appropriate termination condition is to stop iterating when the sequence of
solution vectors stops converging. This can be achieved by checking when, for an
iteration n
\[||x_n - x_{n-1}|| < ||x_{n-1} - x_{n-2}||\]
stops holding.
Since this condition might be too aggressive for a resonably approximated
solution, a cutoff value for the norm (say $10^{-8}$) can also be introduced.
\subsubsection{Create \texttt{pagerank2(G)} by modifying \texttt{pagerank.m}
to use the inverse iteration. Set $\alpha$ equal to 0.8, 0.9 and 1 and comment on
the different number of iterations they take until convergence. Also, what
should be done in the unlikely event that the backslash operation involves a
division by zero?}
The MATLAB solution to this question can be found on
\texttt{files\_data/pagerank1.m}.
The termination condition for the previous exercise is used also in this
implementation.
Iterations are minimal with $\alpha = 1$. For example, for the graph in Figure
5 in the assignment, we have 5 iterations for $\alpha = 1$, 8 iterations for
$\alpha=0.9$, and 11 iterations for $\alpha=0.8$.
To avoid divisions by 0 or near-0 numbers we check the reciprocal condition
number of the matrix $A - \alpha * I$. If this number is below \texttt{eps},
$\alpha$ must be changed. My implementation simply increases $\alpha$ in increments
of $10^{-2}$ until a condition number higher than \texttt{eps} is found.
\subsubsection{Use your functions \texttt{pagerank1.m} and \texttt{pagerank2.m}
(set $\alpha = 1$) to compute the PageRanks of the six-node example presented in
Figure 1. Make sure you get the correct result from each of your three
functions.}
Here are the results for \texttt{pagerank1.m}:
\begin{verbatim}
Using power method implementation
page-rank in out url
1 0.3210 2 2 alpha
6 0.2007 2 1 sigma
2 0.1705 1 2 beta
4 0.1368 2 1 delta
3 0.1066 1 3 gamma
5 0.0643 1 0 rho
\end{verbatim}
And here are the results for \texttt{pagerank2.m}:
\begin{verbatim}
Using inverse iteration implementation
page-rank in out url
1 0.3210 2 2 alpha
6 0.2007 2 1 sigma
2 0.1705 1 2 beta
4 0.1368 2 1 delta
3 0.1066 1 3 gamma
5 0.0643 1 0 rho
\end{verbatim}
\subsubsection{Use your functions \texttt{pagerank1.m} and \texttt{pagerank2.m}
(set $\alpha = 1$) to compute the PageRanks of the six-node example presented in
Figure 5. Discuss the differences between the results obtained now and the ones
obtained in question 4.}
Here are the results for \texttt{pagerank1.m}:
\begin{verbatim}
Using power method implementation
page-rank in out url
4 0.2037 2 1 delta
1 0.1981 1 2 alpha
5 0.1667 1 1 rho
6 0.1667 1 1 sigma
3 0.1556 2 1 gamma
2 0.1092 1 2 beta
\end{verbatim}
And here are the results for \texttt{pagerank2.m}:
\begin{verbatim}
Using inverse iteration implementation
page-rank in out url
4 0.2037 2 1 delta
1 0.1981 1 2 alpha
5 0.1667 1 1 rho
6 0.1667 1 1 sigma
3 0.1556 2 1 gamma
2 0.1092 1 2 beta
\end{verbatim}
Both results are almost identical, with differences in the order of $10^{-8}$
due to the approximation introduced by the termination conditions. This results
are also approximately equal to the ones obtained manually once the normalization process
by the sum of components is applied.
\subsubsection{Use your functions \texttt{pagerank1.m} and \texttt{pagerank2.m}
(set $\alpha = 1$) to compute the PageRanks of the three selected graphs from
exercise 2. Report on the convergence of e.g. the power iteration for these
subgraphs and summarize the advantage of the power method implemented in
\texttt{pagerank2.m} against the original implementation in
\texttt{pagerank.m}.}
In the following outputs, \texttt{it = <number>} represents the number of
iterations required to compute the PageRanks within the termination criteria
described before.
Here are the results for \texttt{pagerank1.m} for starting website
\texttt{http://atelier.inf.usi.ch/\~{}maggicl}:
\begin{verbatim}
Using power method implementation
it = 73
page-rank in out url
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
216 0.0114 38 8 https://wiki.archlinux.org
218 0.0114 38 6 https://security.archlinux.org
428 0.0107 9 1 https://www.dnb.de/kataloghilfe
219 0.0102 38 7 https://aur.archlinux.org
359 0.0098 9 1 https://creativecommons.org/publicdomain/zero/1.0
366 0.0092 27 21 https://archive.org
446 0.0089 24 5 https://foundation.wikimedia.org/wiki/Terms_of_Use
83 0.0079 78 0 https:\/\/schema.org
85 0.0074 77 0 https:\/\/www.wikimedia.org\/static\/images
\/wmf-hor-googpub.png
181 0.0066 8 2 https://gitlab.com
95 0.0062 2 1 https://www.enable-javascript.com
113 0.0061 13 1 https://www.britannica.com/topic/polenta
429 0.0058 8 1 https://www.dnb.de/EN/Home/home_node.html
432 0.0058 8 1 https://www.dnb.de/expertensuche
417 0.0058 8 1 https://www.dnb.de/DE/Home/home_node.html
379 0.0057 24 2 https://blog.archive.org
213 0.0057 32 7 https://www.archlinux.org
99 0.0056 3 1 https://www.usi.ch/it
19 0.0051 4 1 https://creativecommons.org/licenses/by-nc-sa/4.0
214 0.0050 31 5 https://www.archlinux.org/packages
220 0.0050 31 6 https://www.archlinux.org/download
\end{verbatim}
Here are the results for \texttt{pagerank2.m} for starting website
\texttt{http://atelier.inf.usi.ch/\~{}maggicl}:
\begin{verbatim}
Using inverse iteration implementation
it = 7
page-rank in out url
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
216 0.0114 38 8 https://wiki.archlinux.org
218 0.0114 38 6 https://security.archlinux.org
428 0.0107 9 1 https://www.dnb.de/kataloghilfe
219 0.0102 38 7 https://aur.archlinux.org
359 0.0098 9 1 https://creativecommons.org/publicdomain/zero/1.0
366 0.0092 27 21 https://archive.org
446 0.0089 24 5 https://foundation.wikimedia.org/wiki/Terms_of_Use
83 0.0079 78 0 https:\/\/schema.org
85 0.0074 77 0 https:\/\/www.wikimedia.org\/static\/images
\/wmf-hor-googpub.png
181 0.0066 8 2 https://gitlab.com
95 0.0062 2 1 https://www.enable-javascript.com
113 0.0061 13 1 https://www.britannica.com/topic/polenta
429 0.0058 8 1 https://www.dnb.de/EN/Home/home_node.html
432 0.0058 8 1 https://www.dnb.de/expertensuche
417 0.0058 8 1 https://www.dnb.de/DE/Home/home_node.html
379 0.0057 24 2 https://blog.archive.org
213 0.0057 32 7 https://www.archlinux.org
99 0.0056 3 1 https://www.usi.ch/it
19 0.0051 4 1 https://creativecommons.org/licenses/by-nc-sa/4.0
220 0.0050 31 6 https://www.archlinux.org/download
214 0.0050 31 5 https://www.archlinux.org/packages
\end{verbatim}
Here are the results for \texttt{pagerank1.m} for starting website
\texttt{https://www.iisbadoni.edu.it/}:
\begin{verbatim}
Using power method implementation
it = 75
page-rank in out url
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
400 0.0057 15 1 https://yandex.ru/legal/confidential
396 0.0057 15 1 https://github.com/mozilla-mobile/firefox-ios
/blob/master/Docs/MMA.md
5 0.0056 31 0 https://ssl
3 0.0054 36 0 https://www.iisbadoni.edu.it/sites/default/files
/favicon.ico
6 0.0054 36 0 https://www.iisbadoni.edu.it/sites/default/files
/logo.png
208 0.0054 159 0 https://schema.org
74 0.0052 178 5 https://foundation.mozilla.org
72 0.0052 33 32 https://www.mozilla.org/privacy/websites/#cookies
23 0.0051 2 1 https://www.iisbadoni.edu.it/mad
300 0.0051 157 0 https://accounts.firefox.com
\end{verbatim}
Here are the results for \texttt{pagerank2.m} for starting website
\texttt{https://www.iisbadoni.edu.it}:
\begin{verbatim}
Using inverse iteration implementation
it = 7
page-rank in out url
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:
393 0.0059 19 1 https://www.adjust.com/terms/privacy-policy
377 0.0059 19 1 https://abouthome-snippets-service.readthedocs.io
/en/latest/data_collection.html
410 0.0057 16 1 https://wiki.mozilla.org/Firefox/Data_Collection
400 0.0057 15 1 https://yandex.ru/legal/confidential
396 0.0057 15 1 https://github.com/mozilla-mobile/firefox-ios
/blob/master/Docs/MMA.md
5 0.0056 31 0 https://ssl
3 0.0054 36 0 https://www.iisbadoni.edu.it/sites/default/files
/favicon.ico
6 0.0054 36 0 https://www.iisbadoni.edu.it/sites/default/files
/logo.png
208 0.0054 159 0 https://schema.org
74 0.0052 178 5 https://foundation.mozilla.org
72 0.0052 33 32 https://www.mozilla.org/privacy/websites/#cookies
23 0.0051 2 1 https://www.iisbadoni.edu.it/mad
300 0.0051 157 0 https://accounts.firefox.com
\end{verbatim}
Here are the results for \texttt{pagerank1.m} for starting website
\texttt{https://www.usi.ch}:
\begin{verbatim}
Using power method implementation
it = 66
page-rank in out url
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
5 0.0096 317 71 https://usi.ch
62 0.0090 319 18 https://search.usi.ch
337 0.0087 7 1 https://twitter.com/usisoftware
63 0.0080 303 19 https://desk.usi.ch
130 0.0077 25 0 https://www.swissuniversities.ch/it
54 0.0072 208 0 https://twitter.com/USI_university
323 0.0066 9 5 https://www.facebook.com/usiorientamento
150 0.0062 12 1 https://www.innosuisse.ch/inno/it/home.html
248 0.0061 10 1 https://www.facebook.com/usimdfc
106 0.0060 132 8 https://newsletter.usi.ch/archive/en
135 0.0057 201 0 https://schema.org
326 0.0057 6 1 https://www.facebook.com/usialloggimendrisio
322 0.0055 6 1 https://www.facebook.com/USImem
366 0.0054 6 1 https://www.instagram.com/usi_ics_lugano
212 0.0054 12 3 https://www.facebook.com/usimt
7 0.0051 211 32 https://search.usi.ch/it
6 0.0051 204 0 https://www.usi.ch/sites/all/themes/usiclean
/img/bollino-usi.svg
14 0.0051 204 62 https://www.usi.ch/originalnode/342
15 0.0051 204 57 https://www.usi.ch/originalnode/358
16 0.0051 204 62 https://www.usi.ch/originalnode/343
17 0.0051 204 57 https://www.usi.ch/originalnode/344
18 0.0051 204 58 https://www.usi.ch/en/originalnode/12174
20 0.0051 204 60 https://www.usi.ch/originalnode/349
21 0.0051 204 62 https://www.usi.ch/originalnode/8996
22 0.0051 204 60 https://www.usi.ch/originalnode/348
23 0.0051 204 59 https://www.usi.ch/originalnode/351
24 0.0051 204 58 https://www.usi.ch/originalnode/350
25 0.0051 204 61 https://www.usi.ch/originalnode/353
26 0.0051 204 58 https://www.usi.ch/en/originalnode/354
27 0.0051 204 59 https://www.usi.ch/originalnode/8014
61 0.0051 204 0 https://www.usi.ch/sites/all/themes/usiclean
/img/swissuniversities.svg
57 0.0050 188 9 https://newsletter.usi.ch/archive
\end{verbatim}
Here are the results for \texttt{pagerank2.m} for starting website
\texttt{https://www.usi.ch}:
\begin{verbatim}
Using inverse iteration implementation
it = 7
page-rank in out url
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
5 0.0096 317 71 https://usi.ch
62 0.0090 319 18 https://search.usi.ch
337 0.0087 7 1 https://twitter.com/usisoftware
63 0.0080 303 19 https://desk.usi.ch
130 0.0077 25 0 https://www.swissuniversities.ch/it
54 0.0072 208 0 https://twitter.com/USI_university
323 0.0066 9 5 https://www.facebook.com/usiorientamento
150 0.0062 12 1 https://www.innosuisse.ch/inno/it/home.html
248 0.0061 10 1 https://www.facebook.com/usimdfc
106 0.0060 132 8 https://newsletter.usi.ch/archive/en
135 0.0057 201 0 https://schema.org
326 0.0057 6 1 https://www.facebook.com/usialloggimendrisio
322 0.0055 6 1 https://www.facebook.com/USImem
366 0.0054 6 1 https://www.instagram.com/usi_ics_lugano
212 0.0054 12 3 https://www.facebook.com/usimt
7 0.0051 211 32 https://search.usi.ch/it
14 0.0051 204 62 https://www.usi.ch/originalnode/342
27 0.0051 204 59 https://www.usi.ch/originalnode/8014
21 0.0051 204 62 https://www.usi.ch/originalnode/8996
22 0.0051 204 60 https://www.usi.ch/originalnode/348
25 0.0051 204 61 https://www.usi.ch/originalnode/353
61 0.0051 204 0 https://www.usi.ch/sites/all/themes/usiclean
/img/swissuniversities.svg
16 0.0051 204 62 https://www.usi.ch/originalnode/343
23 0.0051 204 59 https://www.usi.ch/originalnode/351
15 0.0051 204 57 https://www.usi.ch/originalnode/358
17 0.0051 204 57 https://www.usi.ch/originalnode/344
24 0.0051 204 58 https://www.usi.ch/originalnode/350
6 0.0051 204 0 https://www.usi.ch/sites/all/themes/usiclean
/img/bollino-usi.svg
20 0.0051 204 60 https://www.usi.ch/originalnode/349
26 0.0051 204 58 https://www.usi.ch/en/originalnode/354
18 0.0051 204 58 https://www.usi.ch/en/originalnode/12174
57 0.0050 188 9 https://newsletter.usi.ch/archive
\end{verbatim}
The potential of the algorithm in \texttt{pagerank2.m} compared to the one in
\texttt{pagerank.m} is that by using inverse iteration, thus theoretically
avoiding a full matrix system solution process, the sparsity of matrix G can be
maintained during the computation thus saving memory costs.
However, since my \texttt{pagerank2.m} implementation is a simplified
implementation of the algorithm and indeed uses system solution
(MATLAB's \texttt{mldivide}), this
theoretical advantages are voided: my implementation is 3 times as long as the
original one and uses approximately the same amount of memory. Advantages over
\texttt{pagerank1.m} (due to the intentionally increased rate of convergence)
are clear: in
all the examples above, the inverse iteration implementation is at least 8 times
faster than the ``vanilla'' power method.
\end{document}
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