mp1: done 2 and 4

mp1/files_data/ex2.m

@@ -0,0 +1,22 @@
+%[U,G] = surfer('https://www.usi.ch',500);
+% pagerank(U,G);
+
+% A = (1/40) * [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];
+A = (1/40) * [
+    0  0  0  40 0  0;
+    20 0  0  0  0  0;
+    20 20 0  0  0  0;
+    0  20 40 0  0  0;
+    0  0  0  0  0 40;
+    0  0  0  0  40  0];
+
+[v,d] = eig(A);
+
+display(d);
+display(v);
+

mp1/files_data/run1rank.txt

@@ -0,0 +1,32 @@
+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
+417	0.0058	8	1	https://www.dnb.de/DE/Home/home_node.html
+429	0.0058	8	1	https://www.dnb.de/EN/Home/home_node.html
+432	0.0058	8	1	https://www.dnb.de/expertensuche
+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

mp1/files_data/run2rank.txt

@@ -0,0 +1,23 @@
+	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	1
+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
+

mp1/files_data/run3rank.txt

@@ -0,0 +1,43 @@
+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
+27	0.0051	204	59	https://www.usi.ch/originalnode/8014
+26	0.0051	204	58	https://www.usi.ch/en/originalnode/354
+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

mp1/template.tex

@@ -1,5 +1,7 @@
 \documentclass[unicode,11pt,a4paper,oneside,numbers=endperiod,openany]{scrartcl}
-
+\usepackage{graphicx}
+\usepackage{subcaption}
+\usepackage{amsmath}
 \input{assignment.sty}
 \begin{document}
 
@@ -57,10 +59,158 @@ is the corresponding eigenvalue, while if $x$ is an eigenvector approximation, f
 
 \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]}