From 24a1ec711bbe6c376389ae8605df650f3bf1b25f Mon Sep 17 00:00:00 2001 From: Iva Miholic Date: Mon, 7 Jul 2014 18:38:02 +0200 Subject: [PATCH] updejt --- izvornik.pdf | Bin 0 -> 840014 bytes text_source/fer.cls | 381 +++++++++++++++++++ text_source/figs/034.png | Bin 0 -> 13105 bytes text_source/figs/abg1000.png | Bin 0 -> 21401 bytes text_source/figs/abg1000_2.png | Bin 0 -> 17634 bytes text_source/figs/abg500v.png | Bin 0 -> 16199 bytes text_source/figs/alters2.png | Bin 0 -> 35263 bytes text_source/figs/jutarnji1.png | Bin 0 -> 15542 bytes text_source/figs/log_mali.png | Bin 0 -> 19694 bytes text_source/figs/sig.png | Bin 0 -> 17479 bytes text_source/figs/y0.png | Bin 0 -> 9959 bytes text_source/izvornik.pdf | Bin 0 -> 840014 bytes text_source/literatura.bib | 124 +++++++ text_source/thesis.tex | 646 +++++++++++++++++++++++++++++++++ thesis.pdf | Bin 0 -> 1257641 bytes 15 files changed, 1151 insertions(+) create mode 100644 izvornik.pdf create mode 100644 text_source/fer.cls create 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Podešavanje oznaka kod listi (1. razina je crtica, 2. puni krug) +\renewcommand{\labelitemi}{\textendash} +\renewcommand{\labelitemii}{\textbullet} + +% http://www.tex.ac.uk/cgi-bin/texfaq2html?label=seccntfmt +% dodaje tocku nakon section broja ali ne i nakon chapter! +% za chapter se koristi jednostavno http://theoval.cmp.uea.ac.uk/~nlct/latex/thesis/node10.html +\renewcommand*{\@seccntformat}[1]{% + \csname the#1\endcsname.\quad +} + +% http://stackoverflow.com/questions/2426963/modifying-latex-table-of-contents-to-add-a-period-after-chapter-table-figure-numb +% Dodaje u toc tocke +\let \savenumberline \numberline +\def \numberline#1{\savenumberline{#1.}} + +\renewcommand\theequation{\ifnum \c@chapter>\z@ \thechapter.\fi \@arabic\c@equation} +\renewcommand\thefigure{\ifnum \c@chapter>\z@ \thechapter.\fi \@arabic\c@figure} +\renewcommand\thetable{\ifnum \c@chapter>\z@ \thechapter.\fi \@arabic\c@table} + +% Rimski brojevi stranica za zahvalu i tablicu sadržaja. +\renewcommand{\thepage}{\roman{page}} + +\if@radseminar\else% +% Ispis napomene o umetanju izvornika. +\newcommand{\izvornik}{% + \newpage + \thispagestyle{empty} + \vspace*{\fill} + \hfil \textsl{Umjesto ove stranice umetnite izvornik Va\v{s}eg rada.} \par \hfil + \hfil \textsl{Da bi ste uklonili ovu stranicu obri\v{s}ite naredbu \texttt{\textbackslash izvornik}.} \hfil + \vspace*{\fill} +} +\fi + +% Dodana zahvala +\newif\if@dodanazahvala \global\@dodanazahvalafalse +\if@radseminar\else% +% Naredba za dodavanje zahvale. +\newcommand{\zahvala}[1]{% + \newpage + \setcounter{page}{3} + \vspace*{\fill} + \hfil {\itshape #1}\hfil + \vspace*{\fill} + \@dodanazahvalatrue +} +\fi + +% TOC +\renewcommand\tableofcontents{% + \if@dodanazahvala\else \setcounter{page}{4} \fi + \if@radseminar \setcounter{page}{2} \fi + \chapter*{\contentsname + \@mkboth{% + \MakeUppercase\contentsname}{\MakeUppercase\contentsname}}% + \@starttoc{toc}% +} + +\newboolean{atappendix} +\setboolean{atappendix}{false} +\newboolean{secondpart} +\setboolean{secondpart}{false} + +\renewcommand{\appendix}{ + \setboolean{atappendix}{true} + \setcounter{chapter}{0} + \renewcommand{\thechapter}{\Alph{chapter}} +} + +\renewcommand{\@makechapterhead}[1]{% +\ifthenelse{\boolean{atappendix}}{% + \vspace*{50\p@}% + {\setlength{\parskip}{0em} \parindent \z@ \raggedright \normalfont + \interlinepenalty\@M + \ifnum \c@secnumdepth >\m@ne + \Huge\bfseries Appendix \thechapter \\ #1 + \par\nobreak + \fi + \nobreak + \vskip 40\p@ + } +} +{ + \ifthenelse{\boolean{secondpart}} + { + } + { + \setcounter{page}{1} + \renewcommand{\thepage}{\arabic{page}} + \setboolean{secondpart}{true} + } + + \vspace*{50\p@}% + {\setlength{\parskip}{0em} \parindent \z@ \raggedright \normalfont + \interlinepenalty\@M + \ifnum \c@secnumdepth >\m@ne + \Huge\bfseries \thechapter.~#1 + \par\nobreak + \fi + \nobreak + \vskip 40\p@ + } +} +} + +\renewcommand{\@makeschapterhead}[1]{% +\ifthenelse{\boolean{atappendix}}{% + \vspace*{50\p@}% + {\setlength{\parskip}{0em} \parindent \z@ \raggedright + \normalfont + \interlinepenalty\@Mdef + \Huge\scshape Appendix \\ #1\par + \nobreak + \vskip 40\p@ + } +} +{% + \vspace*{50\p@}% + {\setlength{\parskip}{0em} \parindent \z@ \raggedright + \normalfont + \interlinepenalty\@M + \Huge\scshape #1\par + \nobreak + \vskip 40\p@ + } +} +} + +\if@radseminar% + \def\voditelj#1{\gdef\@voditelj{#1}} + \def\@voditelj{\@latex@warning@no@line{Voditelj nije naveden. + Koristi \noexpand\voditelj za definiranje}} +\else% + \def\thesisnumber#1{\gdef\@thesisnumber{#1}} + \def\@thesisnumber{\@latex@warning@no@line{Broj rada nije definiran. + Koristi \noexpand\thesisnumber za definiranje}} +\fi + +\if@titlepage + \renewcommand\maketitle{% + \begin{titlepage}% + \let\footnotesize\small + \let\footnoterule\relax + + \begin{center} + {\large\sffamily UNIVERSITY OF ZAGREB} + + {\large\bfseries\sffamily FACULTY OF ELECTRICAL ENGINEERING AND COMPUTING} + \end{center} + \vfill + \begin{center}% + \if@radzavrsni% + {\large\sffamily BACHELOR THESIS No. ~\@thesisnumber \par}% + \else \if@raddiplomski% + {\large\sffamily DIPLOMSKI RAD br.~\@thesisnumber \par}% + \else {\large\sffamily\bfseries SEMINAR \par}% + \fi\fi + \if@radseminar \vskip 3em% + \else \vskip 1em% + \fi + {\huge \bfseries\sffamily \@title \par}% + \vskip 1em% + {\large\sffamily\lineskip .75em% + \begin{tabular}[t]{c}% + \if@radseminar% + {\itshape \@author} \\ + Voditelj: {\itshape \@voditelj} + \else + \@author + \fi + \end{tabular}\par}% + \end{center}\par + \vfill + \begin{center} + {\sffamily\large Zagreb, + \today% + } + \end{center} + \end{titlepage}% + \setcounter{footnote}{0}% + \global\let\maketitle\relax +% \global\let\@thanks\@empty +% \global\let\@author\@empty + \global\let\@date\@empty +% \global\let\@title\@empty +% \global\let\title\relax +% \global\let\author\relax + \global\let\date\relax + \global\let\and\relax + } +\fi + +\newdimen\listindentation +\setlength{\listindentation}{3 em} + +% Podešavanje izemize okoline +% Zahtjeva enumitem paket +\renewenvironment{itemize}% +{ + \setlength{\parskip}{0em} + \advance\@itemdepth\@ne + \edef\@itemitem{labelitem\romannumeral\@itemdepth}% + + \begin{list}{\csname\@itemitem\endcsname}{\setlength{\leftmargin}{\listindentation}\setlength{\itemsep}{0 em}\setlength{\topsep}{0 em}} +}% +{\end{list}} + +% Naredba \engl +\newcommand{\engl}[1]{(engl.~\emph{#1})} + +% Sažetak na hrvatskom +\newenvironment{sazetak} +{ +\newpage +\vspace*{\fill} +\thispagestyle{empty} +\begin{center} + {\bf Širenje informacija u društvenim mrežama} +\end{center} +\hspace*{\fill} {\bf Sa\v{z}etak} \hspace*{\fill} \par +\vspace*{25pt} +} +{ +\vspace*{\fill} +} + +% Ključne riječi na hrvatskom +\newcommand{\kljucnerijeci}[1] +{ +\vspace{15pt} +\noindent \textbf{Klju\v{c}ne rije\v{c}i:} #1 +} + +% Ključne riječi na engleskom +\newcommand{\keywords}[1] +{ +\vspace{15pt} +\noindent \textbf{Keywords:} #1 +} + +% Sažetak na engleskom +\def\engtitle#1{\gdef\@engtitle{#1}} +\def\@engtitle{\@latex@warning@no@line{Engleski naziv rada nije definiran. +Koristi \noexpand\engtitle za definiranje}} +\renewenvironment{abstract} +{ +\vspace*{\fill} +\thispagestyle{empty} +\begin{center} + {\bf \@engtitle} +\end{center} +\hspace*{\fill} {\bf Abstract} \hspace*{\fill} \par +\vspace*{25pt} +} +{ +\vspace*{\fill} +} + +\endinput diff --git a/text_source/figs/034.png b/text_source/figs/034.png new file mode 100644 index 0000000000000000000000000000000000000000..01ede246f2045b90d4a14b3083fe5342571463a7 GIT binary patch literal 13105 zcmd6O2UJwqwryc-6SM(P5UMXCh)57Yl7In4BuSPmk|d+#P=Ed4kmeu?g*qaB zTT~8(+7$wS>3-P3dwQfXMp-{h}#6_L9&?SphmAhzZ-=T> zT2!`ORDn!kt3Rhig`OIkh+W z3xc!cLPbuty6!%?K98OpV7hxMeT1__YpOAezHEilsa)SY(mujK(oOs1E>yt<3JRY& zZ(@%NNm+>SSx1rz6{O9;MOKlkXJF8v%az-|HE26L6S*}LY1P1%KRuUG9#~#3;tmiARy`WFhSjf6T{wpds_dUYt*2IbQm4eliNZbGJUPmyzvi-< z-?Sy18(kPIw*ETuLE6PUr7QWJM34FsoN3S_|Z!Dl;q@O`Hef0lAU4ZWxX?tLJvOug28twYd8#gd3n)X 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information item is authority propagated. The question remains, how much larger should the value of $\alpha$ (or $\beta$) be in order to characterize information item as peer- (or authority- ) propagated and how to verify that this result is due to strong evidence in the data. + +In order to verify that this kind of conclusion is based on strong evidence in the data, \cite{akm-icsn-kdd08} and \cite{authority} proposed randomization test called \emph{time-shuffle test}. + +Let $D$ be the subset of $V \cup A$ that contains all nodes (peer or authority) that eventually become active. The time-shuffle test permutes the activation times of the nodes in $D$. The randomized version of $D$ made in this way contains the same nodes as $D$ (those that eventually become active) with different attributes $t$. Denote by $D'$ randomized version of $D$ with permuted timestamp attributes $t$. + +The input for the estimation algorithm consist of network $H$ and set of nodes $D$, denoted by $\langle H, D\rangle$. Let $\alpha(D')$ and $\beta(D')$ be the estimates of peer and authority coefficients for input $\langle H, D'\rangle$. Let $\alpha$ and $\beta$ denote the maximum-likelihood estimation of the peer and authority coefficients for the original input $\langle H, D\rangle$. Let $\mathcal{D}$ be the set of all possible randomized versions of set $D$ via the time-shuffle test. + +The \emph{strength of peer influence}s $S_{\alpha}$ is defined as the fraction of randomized datasets $D' \in \mathcal{D}$ for which $\alpha > \alpha(D')$, thus +\begin{equation} +S_{\alpha} = P_{D' \in \mathcal{D}} (\alpha > \alpha(D')). +\end{equation} + +The \emph{strength of authority influence} $S_{\beta}$ is defined similarly, as the fraction of randomized datasets $D' \in \mathcal{D}$ for which $\beta > \beta(D')$, thus +\begin{equation} +S_{\beta} = P_{D' \in \mathcal{D}} (\beta > \beta(D')). +\end{equation} + + Both $S_{\alpha}$ and $S_{\beta}$ take values in $[0, 1]$. The large value of strength of peer influence indicates stronger evidence of peer influence in the data. Similarly, larger value of $\beta$ indicates stronger authority influence in the data. The proof for correctnes of decision making based on thus defined strengths for the similar model parametrized with $2$ parameters can be found in \cite{akm-icsn-kdd08}. + +\chapter{Experimental results and analysis} + +The model estimation was performed on the Referendum Dataset. Firstly, the estimation was performed on the restricted graph of registered users without the nodes that never become active. After that, the estimation was preformed on the complete graph of both registered users and their friends. + +The complete procedure was implemented in Python. Although the Referendum Dataset cannot be currently provided, reusable parts of the implementation can be found in \cite{code}. For the work with GML \cite{gml} format and the graph structure, \emph{igraph} library \cite{igraph} was used and for the minimization step, implementation of BFGS algorithm from the \emph{SciPy} library \cite{scipy} was used. The $1^{st}$ order derivatives as derived in \eqref{dera}, \eqref{derb}, \eqref{derg} were given to the procedure directly while the $2^{nd}$ order derivatives in the form of Hessian matrix were approximated internally. The initial values of $\alpha$, $\beta$ and $\gamma$ were taken uniformly at random from the range $[0, 1]$. For all experiments, a step size of $1$ minute was taken and the data from the first $5$ days of application being online was analyzed. + +\section{About the values of $A(x,y)$} +Let's examine values of $A(x,y)$ with respect to $x$ for some fixed values of $y$ and compare them to the activation frequency for the time when all authorities were inactive, \emph{i.e.} when external influence was not present. The differences between these frequencies may indicate the possibility of external influence. + + For a fixed $y$, the value of $A(x,y)$ stands for the frequency of activation for nodes with $x$ active neighbours at the time while exactly $y$ authorities were active, \emph{i.e.} the activation has happened between the activation of $y^{th}$ and $(y+1)^{th}$ activated authority. Value of $A(x,0)$ is thus calculated for nodes that became active before the activation of the $1^{st}$ activated authority and the value of $A(x,3)$ is calculated for the nodes activated between the activation of $3^{rd}$ activated and $4^{th}$ activated authority. + + The graphs of $A(x,y)$ for the fixed value of $ 0 \le y \leq 11$ on the Referendum Dataset have the shape similar to Figure \ref{jutarnji1}. However, there is one exception and it is shown on the Figure \ref{034}. For the reference, $A(x, 0)$, frequency of activation before authorities were present is shown on the Figure \ref{y0}. +\begin{figure} +\begin{subfigure}{\textwidth} +\centering +\includegraphics[scale=0.4]{figs/y0.png} +\caption{Values of $A(x, y)$ for $y=0$.} +\label{y0} +\end{subfigure}% +\par\bigskip +\begin{subfigure}{\textwidth} +\centering +\includegraphics[scale=0.4]{figs/jutarnji1.png} +\caption{Values of $A(x, y)$ for $y=3$.} +\label{jutarnji1} +\end{subfigure}\\[1ex] +\begin{subfigure}{\textwidth} +\centering +\includegraphics[scale=0.4]{figs/034.png} +\caption{Values of $A(x, y)$ for $y=7$.} +\label{034} +\end{subfigure} +\caption{Values of $A(x,y)$ for fixed $y \in \{0, 3, 7\}$ with respect to number of activations with $x$ neighbours active calculated for the Referendum Dataset.} +\label{fig:test} +\end{figure} + +In the case of information propagation where only social influence is present, graph of activation frequency with respect to the number of activated neighbours $x$ would look similar to the Figure \ref{y0}. That is, activation frequency for nodes that have at least one activated neighbour will be greater than the number of activations for nodes with $0$ neighbours active. A certain sign of authority presence is unexpected frequency of activation for nodes that have $0$ activated neighbours. The Figure \ref{jutarnji1} is an example of that kind of activity where frequency of activation with $0$ neighbours active exceed the frequency of activation with $1$ friend active. + +The information propagation dynamic after activation of authority $y=7$ is suspected to be primarily pressured by internal (social influence) what is further approved with Figure \ref{034}. It is important to notice that, as time passes and more nodes become active, the probability of a node having $0$ active neighbours decreases, even for the nodes that become active pressured by external influence. Although the authority $y = 7$ becomes active relatively late, the frequency of activation related to authorities that activate after $y = 7$ form a shape similar to \ref{jutarnji1}. The suspicion of information about the application being propagated mostly pressured by external influence will be confirmed with a model. + + +\section{Estimating peer and authority coefficients} +Estimation of $\alpha$, $\beta$ and $\gamma$ for the Restricted Referendum dataset gives $\alpha = 0.16432$, $\beta = 0.41826 $ and $\gamma = -8.84820$. The strength of peer and authority influence was obtained by the time-shuffle test based on $1000$ randomized instances of the input graph and the strengths $S_{\alpha} = 0.996$, $S_{\beta} = 0.633$ were obtained. Frequency histograms of estimated $\alpha(D')$ and $\beta(D')$ for randomized datasets $D' \in \mathcal{D}$ used in this test are shown on the Figure \ref{hist_full}. +\begin{figure}[htp] +\centering +\includegraphics[scale=0.6]{figs/abg1000.png} +\caption{Frequency histograms of estimated values of $\alpha$ (blue) and $\beta$ (green) obtained with original version of the model for the Restricted Referendum Dataset with time-shuffle test based on $1000$ randomized instances.} +\label{hist_full} +\end{figure} + +The estimation algorithm was ran for the Complete Referendum Dataset and the values $\alpha=1.44990$, $\beta=-1.02360$ and $\gamma=-13.52627$ were obtained. The negative value of $\beta$ may lead to conclusion the model, along with its assumptions, is not suitable for the Complete Referendum Dataset. + +The maximum likelihood function for the model was constructed by examining the events of an inactive node 'trying' to activate independently for all time steps and all inactive nodes at that time. To reason about the negative value of $\beta$, the average frequency of activation and average number of trials (observed events) for the duration of time intervals while $y$ authorities were active, along with the corresponding success rate is given in the Table \ref{T1} for the Restricted and in Table \ref{T2} for the Complete Referendum Dataset. Since the frequencies of activation and trials depend on the length of the observed time interval, they were given in the tables as average frequency per time step for every observed time interval. The success rate is observed, since the maximum likelihood estimator tries to fit the distribution model to similarly defined success rate. + +\begin{table}[htp] +\small +%0 0.77791 12549.02 0.062 +%1 9.00000 11792.91 0.763 +%2 7.34400 11058.83 0.664 +%3 3.91023 7636.39 0.512 +%4 4.52381 6764.81 0.669 +%5 3.74797 6443.38 0.582 +\begin{subtable}{\textwidth} +\centering +\begin{tabular}{|c|c|c|c|c|c|c|} + \hline + \rule[-1ex]{0pt}{2.5ex} $y$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} successes & $0.77791$ & $9.00000$ & $7.34400$ & $3.91023$ & $4.52381$ & $3.74797$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} trials & $12549.02$ & $11792.91$ & $11058.83$ & $7636.39$ & $6764.81$ & $6443.38$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} success rate ($10 ^ {-3}$)& $0.062$ & $0.763$ & $0.664$ & $0.512$ & $0.669$ & $0.582$ \\ + \hline + \hline +%6 1.26634 5156.96 0.246 +%7 0.89345 3001.87 0.298 +%8 0.67893 1882.61 0.361 +%9 1.41053 1664.66 0.847 +%10 3.40826 1434.86 2.375 +%11 1.62857 879.45 1.852 + \rule[-1ex]{0pt}{2.5ex} $y$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} successes & $ 1.26634$ & $0.89345$ & $0.67893$ & $1.41053$ & $3.40826$ & $1.62857$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} trials & $5156.96$ & $3001.87$ & $1882.61$ & $1664.66$ & $1434.86$ & $879.45$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} success rate ($10 ^ {-3}$)& $0.246$ & $0.298$ & $0.361$ & $0.847$ & $2.375$ & $1.852$ \\ + \hline + \end{tabular} + \caption{Restricted Referendum Dataset.} + \label{T1} + \end{subtable} +\\[1ex] +\begin{subtable}{\textwidth} +\centering +%0 0.77791 1697339.72 0.458 +%1 9.00000 1694518.91 5.311 +%2 7.34400 1693784.83 4.336 +%3 3.91023 1690362.39 2.313 +%4 4.52381 1689490.81 2.678 +%5 3.74797 1689169.38 2.219 +\begin{tabular}{|c|c|c|c|c|c|c|} + \hline + \rule[-1ex]{0pt}{2.5ex} $y$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} successes & $0.77791$ & $9.00000$ & $7.34400$ & $3.91023$ & $4.52381$ & $3.74797$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} trials & $1697340$ & $1694519$ & $1693785$ & $1690362$ & $1689491$ & $1689169$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} success rate ($10 ^ {-6}$)& $0.458$ & $5.311$ & $4.336$ & $2.313$ & $2.678$ & $2.219$ \\ + \hline + \hline + \rule[-1ex]{0pt}{2.5ex} $y$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ \\ + \hline +%6 1.26634 1687883 0.750 +%7 0.89344 1685728 0.530 +%8 0.67893 1684609 0.403 +%9 1.41053 1684391 0.837 +%10 3.40816 1684161 2.024 +%11 1.62857 1639898 0.993 + \rule[-1ex]{0pt}{2.5ex} successes & $ 1.26634$ & $0.89345$ & $0.67893$ & $1.41053$ & $3.40826$ & $1.62857$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} trials & $1687883$ & $1685728$ & $1684609$ & $1684391$ & $1684161$ & $1639898$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex} success rate ($10 ^ {-6}$)& $0.750$ & $0.530$ & $0.403$ & $0.837$ & $2.024$ & $0.993$ \\ + \hline + \end{tabular} + \caption{Complete referendum dataset.} + \label{T2} +\end{subtable} + \caption{Average frequencies of observed successful activations per time step, average frequencies of trials per time step and success rates for the time period when $y$ authorities were active on the Restricted and Complete Referendum Datasets.} +\label{tab} +\end{table} + +It is important to notice that average frequency of successful events for each time interval stays the same for both Restricted and Complete Referendum Dataset. Since at each time step only inactive nodes are observed and conducted in trials, it is expected for the averaged frequency of trials to decrease with time as the observed set of nodes shrinks in size. The smaller averaged number of trials for the events observed in late time intervals for the Restricted Referendum Dataset, further induces the growth of success rate as the number of active authorities increases. + +Since there are $99.25\%$ of nodes in the Complete Referendum Dataset that never become active, the number of trials stays roughly the same for the observed time intervals, as seen in the Table \ref{T2}. Therefore the success rate is roughly proportional to the number of successful activations for every interval, \emph{i.e.} it decreases, thus explaining the negative value of $\beta$. + +Since the conducted analysis has shown the proposed model is not suitable for the given Complete Referendum Dataset, the model will be redefined. + +\section{Redefining the model of external influence} + +The activation of a node without any previously activated neighbours is a pattern attributable to external influence. To make the proposed model suitable for our purpose, the new definition of attribute $y$ will be provided. + +If $y$ in \eqref{Pxy} were to be defined as the frequency of activated nodes at the observed time step whose neighbours haven't been activated before, the estimated probability distribution will be expected to increase with $y$. Additionally, thus defined attribute doesn't necessarily have positive correlation with time, unlike the initial definition of $y$. Hence, the errors caused by decreasing number of trials at the end of the observed time for the Restricted Referendum Dataset and the error caused by too many observed nodes that finally don't activate even though maximum number of authorities are active for the Complete Referendum Dataset, have less impact on the results. + +Estimation of $\alpha$, $\beta$ and $\gamma$ for the Restricted Referendum Dataset with newly defined model of external influence gives $\alpha = 0.16697$, $\beta = 0.92495$ and $\gamma = -8.60957$. The strength of peer and authority influence was again obtained by the time-shuffle test based on $1000$ randomized instances of the input graph and strengths $S_{\alpha} = 0.018$, $S_{\beta} = 0.959$ were obtained. Frequency histograms of estimated $\alpha(D')$ and $\beta(D')$ of randomized datasets $D' \in \mathcal{D}$ used in this test are shown on the Figure \ref{hist_full2}. +\begin{figure}[htp] +\centering +\includegraphics[scale=0.7]{figs/abg1000_2.png} +\caption{Frequency histograms of estimated values of $\alpha$ (blue) and $\beta$ (green) obtained with updated version of the model for the Restricted Referendum Dataset with time-shuffle test based on $1000$ randomized instances.} +\label{hist_full2} +\end{figure} + +The estimation for the Complete Referendum dataset returns $\alpha=0.89215$, $\beta=1.36409$, $\gamma=-15.7230$. The strength of peer and authority influence was again obtained by the time-shuffle test based on $500$ randomized instances of the complete input graph and strengths $S_{\alpha} = 1.000$, $S_{\beta} = 0.998$ were obtained. Frequency histograms of estimated $\alpha(D')$ and $\beta(D')$ of randomized datasets $D' \in \mathcal{D}$ used in this test are shown on the Figure \ref{hist_full3}. +\begin{figure}[htp] +\centering +\includegraphics[scale=0.7]{figs/abg500v.png} +\caption{Frequency histograms of estimated values of $\alpha$ (blue) and $\beta$ (green) obtained with updated version of the model for the Complete Referendum Dataset with time-shuffle test based on $500$ randomized instances.} +\label{hist_full3} +\end{figure} + +\chapter{Conclusion} + +The model for information propagation given in \cite{authority} takes into account that information propagation in social networks can be driven by both social (internal) and authority (external) influence. The proposed model is used for determining whether the information item is mainly propagated driven by internal or external influence. The probability of activation for a node that has $x$ active neighbours when $y$ authorities were active is modeled as logistic regression with $3$ parameters, peer influence coefficient $\alpha$, authority influence coefficient $\beta$ and externality coefficient $\gamma$. The logistic regression model is then estimated using the maximal likelihood estimator. Based on the estimated values of $\alpha$ and $\beta$ and taking into account the results of the randomized test called the time-shuffle test, a decision is made. + +The information propagation analysis for the Referendum Dataset based on the presented model of peer and authority influence has returned estimations for peer and authority coefficients that indicate the model is not suitable for the Referendum Dataset. That is expected since the model assumes the influence of an active node (peer or authority) does not fade away with time which can be easily disproven for the Referendum Dataset. + +If the external influence for an inactive node at time $t$ is measured by the number of nodes that activate at that time and do not have any active neighbours, for estimated peer coefficient $\alpha$ and authority coefficient $\beta$, $0 < \alpha < \beta$ holds on both Restricted and Complete Referendum Dataset. The time shuffle test for the Restricted Referendum Dataset has returned a small value of peer strength and a large value of authority strength. Hence, the information propagation is mainly pressured by external influence. The estimation and the shuffle test for the Complete Referendum Dataset reassures the conclusion is correct. + +\bibliography{literatura} +\bibliographystyle{unsrt} +\pagebreak +\engtitle{Information propagation in online social networks} +\begin{abstract} +In this paper the information propagation model that takes into account both internal (social) and external (authority) influence is presented and revisited. Based on the estimated parameters for the model and the randomization test called time-shuffle test, a decision whether propagation of an information item is mainly peer or authority influence driven can be made. The proposed model and its updated version were used to describe an information item - the existence of the Facebook application referendum2013.hr - as mainly propagated by external (authority) influence. + +\keywords{complex networks, social networks, information propagation, internal and external influence, peer and authority model,time-shuffle test} +\end{abstract} +\begin{sazetak} +U ovom je radu objašnjen model za širenje informacije u društvenoj mreži koji uzima u obzir i unutranji (društveni) i vanjski (medijski) utjecaj na širenje informacije. Na temelju predviđenih parametara modela te statističkog testa, moguće je donošenje odluke o načinu širenja promatrane informacije. Točnije, možemo odrediti da li se promatrana informacija više širila pod društvenim ili medijskim utjecajem. Izloženi model te njegova izmijenjena verzija korišteni su za određivanje prirode utjecaja na širenje informacije o postojanju Facebook aplikacije referendum2013.hr. + +\kljucnerijeci{kompleksne mreže, društvene mreže, širenje informacije, model širenja informacije, društveni utjecaj, medijski utjecaj} +\end{sazetak} + + +\end{document} diff --git a/thesis.pdf b/thesis.pdf new file mode 100644 index 0000000000000000000000000000000000000000..889131a90ab8cbf8755493facdafd39e1703b08f GIT binary patch literal 1257641 zcmeFZcT`hd-yj;KNk@7QO~HrWkrt3HB2Bu0f|P)ONEb*b(wl&QfDn-;U25o|R}twF zdJvQnNRSdufSc$2-tW$=xijm|y?@-bX700-?3|OF?6ZG$|Ju$?Qyo1C8A%1|n_uUK zzER8ZOY{3Wd-AKPQA^!-^>GjI;Fr52&;K8dT1pT6EWq^zzm(pyCjqWHt}eb$U8yxR zsQm+8xIXcw4nf9dPCOWpX4E~upqBDZ1~6W;Ud2;-N7tciaM{yGzk{^LC@9bSu;kpQ=+~pkW}9JAV{6DFD!T#wenTB$RXdk zXi1DMawb6W?DXa6T`iGV&IZ-69n+UUR?&FIBr^KShbS|ba&lf-7I}yJwt_zeX73w& zCX4evz=vs!y>Lkqhv9~$LE!m>H%Z#xcaa0LOTUNuTcuWGoWKkv6lR}=*ku4@b=k)$ 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