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mp1: all done except 1.1

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Claudio Maggioni 2020-09-27 15:55:37 +02:00
parent 7355b27cd3
commit b5cc7065de
4 changed files with 450 additions and 39 deletions
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4
mp1/files_data/pagerank1.m
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@ -35,7 +35,7 @@ new_x = x;
old_norm = -1; old_norm = -1;
no = +Inf; no = +Inf;
while old_norm == -1 || old_norm > no while (old_norm == -1 || old_norm > no) && no > 1e-8
x = new_x; x = new_x;
old_norm = no; old_norm = no;
new_x = G * x + e * (z * x); new_x = G * x + e * (z * x);
@ -44,6 +44,8 @@ while old_norm == -1 || old_norm > no
no = norm(new_x - x, 2); no = norm(new_x - x, 2);
end end
display(it);
% ------------------------------------------------------------------------- % -------------------------------------------------------------------------
% Normalize so that sum(x) == 1. % Normalize so that sum(x) == 1.

13
mp1/files_data/pagerank2.m
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@ -32,28 +32,25 @@ x = e/n;
% Check if B will be a singular matrix. If so, change it % Check if B will be a singular matrix. If so, change it
alpha = 1; alpha = 1;
while rcond(A - alpha * I) <= eps while rcond(A - alpha * I) < eps
alpha = alpha + 0.01; alpha = alpha + 0.01;
end end
display(alpha);
B = (A - alpha * I);
old_x = zeros(n, 1); old_x = zeros(n, 1);
old_norm = -1; old_norm = -1;
no = +Inf; no = +Inf;
it = 0; it = 0;
while old_norm == -1 || old_norm > no while (old_norm == -1 || old_norm > no) && no > 1e-8
old_norm = no; old_norm = no;
old_x = x; old_x = x;
x = B \ x; x = B \ x;
x = x/norm(x, 2); x = x/sum(x);
it = it + 1; it = it + 1;
no = norm(x - old_x, 2); no = norm(x - old_x, 1);
end end
x = old_x; x = abs(old_x);
display(it); display(it);
% ------------------------------------------------------------------------- % -------------------------------------------------------------------------

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mp1/template.pdf
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462
mp1/template.tex
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@ -1,3 +1,4 @@
% vim: set ts=4 sw=4 et tw=80:
\documentclass[unicode,11pt,a4paper,oneside,numbers=endperiod,openany]{scrartcl} \documentclass[unicode,11pt,a4paper,oneside,numbers=endperiod,openany]{scrartcl}
\usepackage{graphicx} \usepackage{graphicx}
\usepackage{subcaption} \usepackage{subcaption}
@ -10,7 +11,7 @@
\setduedate{Thursday, 8 October 2020, 12:00 AM} \setduedate{Thursday, 8 October 2020, 12:00 AM}
\serieheader{Numerical Computing}{2020}{Student: Claudio Maggioni}{Discussed with: --}{Solution for Project 1}{} \serieheader{Numerical Computing}{2020}{Student: Claudio Maggioni}{Discussed with: --}{Solution for Project 1}{}
\newline ewline
\assignmentpolicy \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. 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.
@ -82,17 +83,17 @@ Finally, we can always observe a line starting from the top-left of G and ending
\begin{subfigure}{\textwidth} \begin{subfigure}{\textwidth}
\begin{verbatim} \begin{verbatim}
360 0.0869 31 1 https://creativecommons.org/licenses/by-sa/3.0 360 0.0869 1 https://creativecommons.org/licenses/by-sa/3.0
204 0.0406 8 1 https://forum.gitlab.com 204 0.0406 8 1 https://forum.gitlab.com
82 0.0189 117 18 https://www.mediawiki.org 82 0.01897 https://www.mediawiki.org
81 0.0188 117 4 https://wikimediafoundation.org 81 0.01887 4 https://wikimediafoundation.org
87 0.0150 6 1 https://docs.gitea.io 87 0.0150 6 1 https://docs.gitea.io
78 0.0145 114 9 https://www.mediawiki.org/wiki/Special:MyLanguage/ 78 0.01454 9 https://www.mediawiki.org/wiki/Special:MyLanguage/
How_to_contribute How_to_contribute
77 0.0132 77 13 https://foundation.wikimedia.org/wiki/Privacy_policy 77 0.0132 https://foundation.wikimedia.org/wiki/Privacy_policy
217 0.0127 40 8 https://bugs.archlinux.org 217 0.0127 8 https://bugs.archlinux.org
80 0.0115 107 6 https://foundation.wikimedia.org/wiki/Cookie_statement 80 0.01157 6 https://foundation.wikimedia.org/wiki/Cookie_statement
215 0.0114 38 5 https://bbs.archlinux.org 215 0.0114 5 https://bbs.archlinux.org
\end{verbatim} \end{verbatim}
\caption{Top 10 webpages with highest PageRank} \caption{Top 10 webpages with highest PageRank}
\end{subfigure} \end{subfigure}
@ -115,18 +116,18 @@ Finally, we can always observe a line starting from the top-left of G and ending
\begin{subfigure}{\textwidth} \begin{subfigure}{\textwidth}
\begin{verbatim} \begin{verbatim}
411 0.0249 42 1 https://twitter.com/mozilla 411 0.0249 1 https://twitter.com/mozilla
63 0.0248 145 1 https://twitter.com/firefox 63 0.02485 1 https://twitter.com/firefox
68 0.0203 142 1 https://www.instagram.com/firefox 68 0.02032 1 https://www.instagram.com/firefox
412 0.0164 37 1 https://www.instagram.com/mozilla 412 0.0164 1 https://www.instagram.com/mozilla
62 0.0080 21 1 https://github.com/mozilla/kitsune 62 0.0080 1 https://github.com/mozilla/kitsune
81 0.0070 110 2 https://www.apple.com 81 0.00700 2 https://www.apple.com
384 0.0064 5 1 https://www.xfinity.com/privacy/policy/dns 384 0.0064 5 1 https://www.xfinity.com/privacy/policy/dns
4 0.0064 32 0 https: 4 0.0064 0 https:
377 0.0059 19 1 https://abouthome-snippets-service.readthedocs.io/en/ 377 0.0059 1 https://abouthome-snippets-service.readthedocs.io/en/
latest/data_collection.html latest/data_collection.html
393 0.0059 19 1 https://www.adjust.com/terms/privacy-policy 393 0.0059 1 https://www.adjust.com/terms/privacy-policy
410 0.0057 16 1 https://wiki.mozilla.org/Firefox/Data_Collection 410 0.0057 1 https://wiki.mozilla.org/Firefox/Data_Collection
\end{verbatim} \end{verbatim}
\caption{Top 10 webpages with highest PageRank} \caption{Top 10 webpages with highest PageRank}
\end{subfigure} \end{subfigure}
@ -149,16 +150,16 @@ Finally, we can always observe a line starting from the top-left of G and ending
\begin{subfigure}{\textwidth} \begin{subfigure}{\textwidth}
\begin{verbatim} \begin{verbatim}
55 0.0741 354 1 https://www.instagram.com/usiuniversity 55 0.07414 1 https://www.instagram.com/usiuniversity
53 0.0324 366 3 https://www.facebook.com/usiuniversity 53 0.03246 3 https://www.facebook.com/usiuniversity
299 0.0248 6 1 https://twitter.com/usi_en 299 0.0248 6 1 https://twitter.com/usi_en
329 0.0243 8 1 https://www.facebook.com/USIeLab 329 0.0243 8 1 https://www.facebook.com/USIeLab
308 0.0156 7 3 https://www.facebook.com/USIFinancialCommunication 308 0.0156 7 3 https://www.facebook.com/USIFinancialCommunication
60 0.0155 316 2 https://www.swissuniversities.ch 60 0.01556 2 https://www.swissuniversities.ch
424 0.0144 96 1 https://it.bul.sbu.usi.ch 424 0.0144 1 https://it.bul.sbu.usi.ch
330 0.0123 6 4 https://www.facebook.com/USI.ITDxC 330 0.0123 6 4 https://www.facebook.com/USI.ITDxC
320 0.0122 7 1 https://www.facebook.com/usiimeg 320 0.0122 7 1 https://www.facebook.com/usiimeg
56 0.0107 320 0 https://www.youtube.com/usiuniversity 56 0.01070 0 https://www.youtube.com/usiuniversity
\end{verbatim} \end{verbatim}
\caption{Top 10 webpages with highest PageRank} \caption{Top 10 webpages with highest PageRank}
\end{subfigure} \end{subfigure}
@ -214,7 +215,7 @@ First we compute the matrix A, finding:
We then find the eigenvectors and eigenvalues of A through MATLAB, finding that the solution of $A x = 1 x$ is: 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} \[x'\approx\begin{bmatrix}
0.4771\\ 0.4771\\
0.2630\\ 0.2630\\
0.3747\\ 0.3747\\
@ -223,6 +224,18 @@ We then find the eigenvectors and eigenvalues of A through MATLAB, finding that
0.4013\\ 0.4013\\
\end{bmatrix}\] \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. 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 $p \to 1$.} \subsubsection{Describe what happens with this example to both the definition of PageRank and the computation done by pagerank in the limit $p \to 1$.}
@ -233,5 +246,404 @@ In the computation, increasing $p$ decreases $\delta$ (which represents the prob
\subsection{PageRanks by solving a sparse linear system [50 points]} \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
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\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} \end{document}