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{screenshot}{5} +\beamer@slide {glas_ponuto}{6} +\beamer@slide {glas_ponuto}{8} +\beamer@slide {Pxy}{11} +\beamer@slide {Pxy}{13} +\beamer@slide {sig}{13} +\beamer@slide {LF}{14} +\beamer@slide {hist_full}{17} +\beamer@slide {T2}{18} +\beamer@slide {tab}{18} +\beamer@slide {Pxy}{19} +\beamer@slide {hist_full2}{20} +\beamer@slide {hist_full3}{20} diff --git a/presentation/prezentacija.synctex.gz b/presentation/prezentacija.synctex.gz new file mode 100644 index 0000000..614fe86 Binary files /dev/null and b/presentation/prezentacija.synctex.gz differ diff --git a/presentation/prezentacija.tex b/presentation/prezentacija.tex new file mode 100644 index 0000000..72505d7 --- /dev/null +++ b/presentation/prezentacija.tex @@ -0,0 +1,329 @@ +\documentclass{beamer} +\usetheme{Antibes} +\usecolortheme{dolphin} + +\setbeamertemplate{footline}[frame number] +\definecolor{g2}{RGB}{25, 0, 51} +\setbeamercolor{title}{fg=g2} +\setbeamercolor{frametitle}{fg=g2} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{subcaption} + +\institute{Department of Electronic Systems and Information Processing + \\ Faculty of Electrical Engineering and Computing \\ University of Zagreb} +\author{Iva Miholić \\ Advisor: Asst. Prof. Mile Šikić} +\title{ Bachelor Thesis No. 3743 \\ Information propagation in online social networks} +\setbeamercovered{transparent} +%\setbeamertemplate{navigation symbols}{} +%\logo{} +\date{9 Jul 2014} +%\subject{} +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +%\begin{frame} +%\tableofcontents +%\end{frame} + +\section{Introduction} +\subsection{Complex networks} +\begin{frame} +\frametitle{Networks} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.2]{figs/alters2.png} +\caption{A network graph of Paul Erd\~os and his collaborators. The nodes represent mathematicians and the edges represent the relationship "wrote a paper with".} +\label{net} +\end{figure} +\end{frame} + +\begin{frame} +\begin{figure}[htp] +\centering +\includegraphics[width=0.7 \textheight]{figs/complex.jpg} +\caption{Comparison of a random and a scale-free network. $p(k)$ stands for the probabilty that a vertex chosen unformly at random has degree $k$.} +\label{net} +\end{figure} +\end{frame} + +\subsection{Information propagation} +\begin{frame} +\begin{itemize} +\item{propagation of the information item amongst the nodes of the social network} +\item{closed world $+$ social influence $=>$ information propagation similar to diffusion} +\item{In the real world, external influence is present. That is visible on the propagation dynamic.} +\item{Model of propagation that takes into account both internal and external influence.} +\end{itemize} +\end{frame} + +\section{Dataset} +\begin{frame} +\begin{figure}[htp] +\centering +\includegraphics[width=\textwidth]{figs/screenshot.png} +\caption{A screenshot of \emph{referendum2013.hr} site taken on 8 Jul 2014} +\label{screenshot} +\end{figure} +\end{frame} + +\begin{frame} +\frametitle{Registration dynamic} +\begin{itemize} +\item{$1.695$ million nodes, $4.461$ million edges} +\item{$11606$ registered users, $11$ recorded articles} +\end{itemize} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.3]{figs/glas_ponuto.png} +\caption{Dynamic of registrations for the first two days of application being active.} +%TODO generiraj sliku +\label{glas_ponuto} +\end{figure} +\end{frame} + +\begin{frame} +\frametitle{Datasets} +\begin{columns} +\column{.5\textwidth} +Restricted Referendum Dataset +\begin{figure} +\includegraphics[scale=0.4]{figs/graphs1.png} +\end{figure} +\column{.5\textwidth} +Complete Referendum Dataset +\begin{figure} +\includegraphics[scale=0.3]{figs/graphs2.png} +\end{figure} +\end{columns} +\end{frame} + +\begin{frame} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.4]{figs/log_mali.png} +\caption{The degree distribution for the network of registered users. $p(k)$ stands for the probabilty that a vertex chosen unformly at random has degree $k$.} +\label{glas_ponuto} +\end{figure} +\end{frame} + +\begin{frame} +\begin{itemize} +\item{\bf{Q: Is the adoption of an information item for the given network mainly internal or external influence driven?}} +\item{information item = information on the existance of the application} +\item{network = Restricted and Complete Referendum Dataset} +\end{itemize} +\end{frame} + +\section{Model} +\subsection{The proces of information propagation} + +\subsection{The structure of the network} +\begin{frame} +\frametitle{The structure of the network} +\begin{itemize} +\item{nodes of social network + external nodes} +\item{every external node can influence every peer node (a node in social network)} +\item{set of external nodes form a complete graph} +\item{peer nodes do not influence the external nodes} +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{Probability of activation} +\begin{equation} +\label{Pxy} + P(x, y; \alpha, \beta, \gamma) = \frac{1}{1 + e ^ {-\alpha \ln(x + 1) - \beta \ln(y + 1) - \gamma}}, +\end{equation} + +\begin{itemize} +\item{$x$ - the number of already active peer neighbours} +\item{$y$ - the number of already active authorities} +\item{$\alpha$ - \emph{peer coefficient}, measure of internal influence} +\item{$\beta$ - \emph{authority coefficient}, measure of external influence} +\item{$\gamma$ - \emph{externality coefficient}, the impact of other factors} +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{Assumptions} +\begin{itemize} +\item{Propagation happens in discrete time steps, from timestamp $1$ to timestamp $T$.} +\item{For every time stamp and every currently inactive node, success of the node trying to become active is observed as a Bernoulli random variable. +} +\item{All observed events are independant.} +\item{From the time a node gets activated, its influence is constant and does not fade away with time. } +\item{Every external node has an equal probability of influencing any peer node.} +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{Model estimation} +\begin{equation} +\label{Pxy} + P(x, y; \alpha, \beta, \gamma) = \frac{1}{1 + e ^ {-\alpha \ln(x + 1) - \beta \ln(y + 1) - \gamma}}, +\end{equation} + +\begin{itemize} +\item{Estimate parameters $\alpha$, $\beta$ and $\gamma$ with observed events of activation tryouts at every timestep for every currently unactive node using the principle of maximal likelihood.} +\end{itemize} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.2]{figs/sig.png} +\caption{Function $\sigma(z) = \frac{1}{1 + e ^ {-z}}$.} +\label{sig} +\end{figure} +\end{frame} + +\begin{frame} +\frametitle{Principle of maximal likelihood} +Take $\alpha$, $\beta$ and $\gamma$ that maximize +\begin{equation} +\label{LF} + L(\alpha, \beta, \gamma) = \prod_{x, y} P(x, y) ^ {A(x, y)} (1 - P(x, y)) ^ { N(x, y)} +\end{equation} +or minimize +\begin{equation} +\centering + -\ln L(\alpha, \beta, \gamma) = -\sum_{x, y} A(x, y) \ln P(x, y) - \sum_{x, y} N(x, y) \ln (1 - P(x, y)). +\end{equation} + +\begin{itemize} +\item{$A(x,y)$ - number of observed succesful activations.} +\item{$N(x,y)$ - number of observed unsuccesful activations.} +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{Randomization Test} +\begin{itemize} +\item{Given $\alpha$, $\beta$, $\gamma$ can we just conclude \\ +$(\alpha > \beta) => $ internal influence is dominant?} +\item{\emph{time-shuffle} test is used to verify this conclusion.} +\end{itemize} +\textbf{\emph{time-shuffle} test} +\begin{itemize} +\item{randomize activation time of all eventually active nodes in the dataset} +\item{obtain $\alpha(D')$ and $\beta(D)'$ for the new dataset $D'$} +\item{update $S_\alpha$ and $S_\beta$} +\item{repeat} +\end{itemize} +\end{frame} + +\begin{frame} +\textbf{\emph{time-shuffle} test} +\begin{itemize} +\item{randomize activation time of all eventually active nodes in the dataset} +\item{obtain $\alpha(D')$ and $\beta(D)'$ for the new dataset $D'$} +\item{update $S_\alpha$ and $S_\beta$} +\item{repeat} +\end{itemize} +\vspace{1cm} +The \emph{strength of peer influence}, $S_\alpha$, is defined as +\begin{equation} +S_{\alpha} = P_{D' \in \mathcal{D}} (\alpha > \alpha(D')). +\end{equation} +and similarly for the \emph{strength of authority influence}, $S_\beta$: +\begin{equation} +S_{\beta} = P_{D' \in \mathcal{D}} (\beta > \beta(D')). +\end{equation} +\end{frame} + +\section{Results} +\subsection{Restricted Referendum Dataset} +\begin{frame} +\frametitle{Restricted Referendum Dataset} +$\alpha = 0.16432$, $\beta = 0.41826$, $\gamma = -8.84820$ \\ +$S_\alpha = 0.996$, $S_\beta = 0.633$ +\begin{figure}[htp] +\centering +\includegraphics[scale=0.3]{figs/abg1000.png} +\caption{Frequency histograms of estimated values of $\alpha$ (blue) and $\beta$ (green) obtained for the Restricted Referendum Dataset with time-shuffle test based on $1000$ randomized instances.} +\label{hist_full} +\end{figure} +\end{frame} + +\begin{frame} +\frametitle{Complete Referendum Dataset} +$\alpha = 1.44990$, $\beta = -1.02360$, $\gamma = -13.52627$ +\begin{table}[htp] +\resizebox{\columnwidth}{!}{ +\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 & $1697340$ & $1694519$ & $1693785$ & $1690362$ & $1689491$ & $1689169$ \\ + \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 + \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}} + \label{T2} + \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 Complete Referendum Datasets.} +\label{tab} +\end{table} +\end{frame} + +\begin{frame} +\frametitle{Redefining the model of external influence} +\begin{equation} +\label{Pxy} + P(x, y; \alpha, \beta, \gamma) = \frac{1}{1 + e ^ {-\alpha \ln(x + 1) - \beta \ln(y + 1) - \gamma}}, +\end{equation} + +\begin{itemize} +\item{$x$ - the number of already active peer neighbours} +\item{\textbf{$y$ - the number of activations of nodes with $0$ friends active at the time of the event}} +\vspace{1cm} +\item{now $y$ doesn't necessarily have positive correlation with time} +\end{itemize} +\end{frame} + +\begin{frame} +\begin{columns} +\column{.5\textwidth} +\textbf{Restricted Referendum Dataset}\\ +\begin{small} +$\alpha = 0.16697$, $\beta = 0.92495$, $\gamma = -8.60957$, $S_\alpha = 0.018$, $S_\beta = 0.959$ \\ +- based on $1000$ randomized instances +\end{small}\begin{figure}[htp] +\centering +\includegraphics[scale=0.3]{figs/abg1000_2.png} +\label{hist_full2} +\end{figure} +\column{.5\textwidth} +\textbf{Complete Referendum Dataset}\\ +\begin{small}$\alpha = 0.89215$, $\beta = 1.36409$, $\gamma = -15.7230$, $S_\alpha = 1.000$, $S_\beta = 0.998$\\ +- based on $500$ randomized instances \end{small} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.3]{figs/abg500v.png} +\label{hist_full3} +\end{figure} +\end{columns} +\end{frame} + +\begin{frame} +\begin{itemize} +\item{\bf{Q: Is the adoption of an information item for the given network mainly internal or external influence driven?}} +\item{\bf{A: The adoption is mainly external influence driven.}} +\end{itemize} +\end{frame} +\end{document} diff --git a/presentation/prezentacija.tex.bak b/presentation/prezentacija.tex.bak new file mode 100644 index 0000000..72505d7 --- /dev/null +++ b/presentation/prezentacija.tex.bak @@ -0,0 +1,329 @@ +\documentclass{beamer} +\usetheme{Antibes} +\usecolortheme{dolphin} + +\setbeamertemplate{footline}[frame number] +\definecolor{g2}{RGB}{25, 0, 51} +\setbeamercolor{title}{fg=g2} +\setbeamercolor{frametitle}{fg=g2} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{subcaption} + +\institute{Department of Electronic Systems and Information Processing + \\ Faculty of Electrical Engineering and Computing \\ University of Zagreb} +\author{Iva Miholić \\ Advisor: Asst. Prof. Mile Šikić} +\title{ Bachelor Thesis No. 3743 \\ Information propagation in online social networks} +\setbeamercovered{transparent} +%\setbeamertemplate{navigation symbols}{} +%\logo{} +\date{9 Jul 2014} +%\subject{} +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +%\begin{frame} +%\tableofcontents +%\end{frame} + +\section{Introduction} +\subsection{Complex networks} +\begin{frame} +\frametitle{Networks} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.2]{figs/alters2.png} +\caption{A network graph of Paul Erd\~os and his collaborators. The nodes represent mathematicians and the edges represent the relationship "wrote a paper with".} +\label{net} +\end{figure} +\end{frame} + +\begin{frame} +\begin{figure}[htp] +\centering +\includegraphics[width=0.7 \textheight]{figs/complex.jpg} +\caption{Comparison of a random and a scale-free network. $p(k)$ stands for the probabilty that a vertex chosen unformly at random has degree $k$.} +\label{net} +\end{figure} +\end{frame} + +\subsection{Information propagation} +\begin{frame} +\begin{itemize} +\item{propagation of the information item amongst the nodes of the social network} +\item{closed world $+$ social influence $=>$ information propagation similar to diffusion} +\item{In the real world, external influence is present. That is visible on the propagation dynamic.} +\item{Model of propagation that takes into account both internal and external influence.} +\end{itemize} +\end{frame} + +\section{Dataset} +\begin{frame} +\begin{figure}[htp] +\centering +\includegraphics[width=\textwidth]{figs/screenshot.png} +\caption{A screenshot of \emph{referendum2013.hr} site taken on 8 Jul 2014} +\label{screenshot} +\end{figure} +\end{frame} + +\begin{frame} +\frametitle{Registration dynamic} +\begin{itemize} +\item{$1.695$ million nodes, $4.461$ million edges} +\item{$11606$ registered users, $11$ recorded articles} +\end{itemize} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.3]{figs/glas_ponuto.png} +\caption{Dynamic of registrations for the first two days of application being active.} +%TODO generiraj sliku +\label{glas_ponuto} +\end{figure} +\end{frame} + +\begin{frame} +\frametitle{Datasets} +\begin{columns} +\column{.5\textwidth} +Restricted Referendum Dataset +\begin{figure} +\includegraphics[scale=0.4]{figs/graphs1.png} +\end{figure} +\column{.5\textwidth} +Complete Referendum Dataset +\begin{figure} +\includegraphics[scale=0.3]{figs/graphs2.png} +\end{figure} +\end{columns} +\end{frame} + +\begin{frame} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.4]{figs/log_mali.png} +\caption{The degree distribution for the network of registered users. $p(k)$ stands for the probabilty that a vertex chosen unformly at random has degree $k$.} +\label{glas_ponuto} +\end{figure} +\end{frame} + +\begin{frame} +\begin{itemize} +\item{\bf{Q: Is the adoption of an information item for the given network mainly internal or external influence driven?}} +\item{information item = information on the existance of the application} +\item{network = Restricted and Complete Referendum Dataset} +\end{itemize} +\end{frame} + +\section{Model} +\subsection{The proces of information propagation} + +\subsection{The structure of the network} +\begin{frame} +\frametitle{The structure of the network} +\begin{itemize} +\item{nodes of social network + external nodes} +\item{every external node can influence every peer node (a node in social network)} +\item{set of external nodes form a complete graph} +\item{peer nodes do not influence the external nodes} +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{Probability of activation} +\begin{equation} +\label{Pxy} + P(x, y; \alpha, \beta, \gamma) = \frac{1}{1 + e ^ {-\alpha \ln(x + 1) - \beta \ln(y + 1) - \gamma}}, +\end{equation} + +\begin{itemize} +\item{$x$ - the number of already active peer neighbours} +\item{$y$ - the number of already active authorities} +\item{$\alpha$ - \emph{peer coefficient}, measure of internal influence} +\item{$\beta$ - \emph{authority coefficient}, measure of external influence} +\item{$\gamma$ - \emph{externality coefficient}, the impact of other factors} +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{Assumptions} +\begin{itemize} +\item{Propagation happens in discrete time steps, from timestamp $1$ to timestamp $T$.} +\item{For every time stamp and every currently inactive node, success of the node trying to become active is observed as a Bernoulli random variable. +} +\item{All observed events are independant.} +\item{From the time a node gets activated, its influence is constant and does not fade away with time. } +\item{Every external node has an equal probability of influencing any peer node.} +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{Model estimation} +\begin{equation} +\label{Pxy} + P(x, y; \alpha, \beta, \gamma) = \frac{1}{1 + e ^ {-\alpha \ln(x + 1) - \beta \ln(y + 1) - \gamma}}, +\end{equation} + +\begin{itemize} +\item{Estimate parameters $\alpha$, $\beta$ and $\gamma$ with observed events of activation tryouts at every timestep for every currently unactive node using the principle of maximal likelihood.} +\end{itemize} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.2]{figs/sig.png} +\caption{Function $\sigma(z) = \frac{1}{1 + e ^ {-z}}$.} +\label{sig} +\end{figure} +\end{frame} + +\begin{frame} +\frametitle{Principle of maximal likelihood} +Take $\alpha$, $\beta$ and $\gamma$ that maximize +\begin{equation} +\label{LF} + L(\alpha, \beta, \gamma) = \prod_{x, y} P(x, y) ^ {A(x, y)} (1 - P(x, y)) ^ { N(x, y)} +\end{equation} +or minimize +\begin{equation} +\centering + -\ln L(\alpha, \beta, \gamma) = -\sum_{x, y} A(x, y) \ln P(x, y) - \sum_{x, y} N(x, y) \ln (1 - P(x, y)). +\end{equation} + +\begin{itemize} +\item{$A(x,y)$ - number of observed succesful activations.} +\item{$N(x,y)$ - number of observed unsuccesful activations.} +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{Randomization Test} +\begin{itemize} +\item{Given $\alpha$, $\beta$, $\gamma$ can we just conclude \\ +$(\alpha > \beta) => $ internal influence is dominant?} +\item{\emph{time-shuffle} test is used to verify this conclusion.} +\end{itemize} +\textbf{\emph{time-shuffle} test} +\begin{itemize} +\item{randomize activation time of all eventually active nodes in the dataset} +\item{obtain $\alpha(D')$ and $\beta(D)'$ for the new dataset $D'$} +\item{update $S_\alpha$ and $S_\beta$} +\item{repeat} +\end{itemize} +\end{frame} + +\begin{frame} +\textbf{\emph{time-shuffle} test} +\begin{itemize} +\item{randomize activation time of all eventually active nodes in the dataset} +\item{obtain $\alpha(D')$ and $\beta(D)'$ for the new dataset $D'$} +\item{update $S_\alpha$ and $S_\beta$} +\item{repeat} +\end{itemize} +\vspace{1cm} +The \emph{strength of peer influence}, $S_\alpha$, is defined as +\begin{equation} +S_{\alpha} = P_{D' \in \mathcal{D}} (\alpha > \alpha(D')). +\end{equation} +and similarly for the \emph{strength of authority influence}, $S_\beta$: +\begin{equation} +S_{\beta} = P_{D' \in \mathcal{D}} (\beta > \beta(D')). +\end{equation} +\end{frame} + +\section{Results} +\subsection{Restricted Referendum Dataset} +\begin{frame} +\frametitle{Restricted Referendum Dataset} +$\alpha = 0.16432$, $\beta = 0.41826$, $\gamma = -8.84820$ \\ +$S_\alpha = 0.996$, $S_\beta = 0.633$ +\begin{figure}[htp] +\centering +\includegraphics[scale=0.3]{figs/abg1000.png} +\caption{Frequency histograms of estimated values of $\alpha$ (blue) and $\beta$ (green) obtained for the Restricted Referendum Dataset with time-shuffle test based on $1000$ randomized instances.} +\label{hist_full} +\end{figure} +\end{frame} + +\begin{frame} +\frametitle{Complete Referendum Dataset} +$\alpha = 1.44990$, $\beta = -1.02360$, $\gamma = -13.52627$ +\begin{table}[htp] +\resizebox{\columnwidth}{!}{ +\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 & $1697340$ & $1694519$ & $1693785$ & $1690362$ & $1689491$ & $1689169$ \\ + \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 + \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}} + \label{T2} + \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 Complete Referendum Datasets.} +\label{tab} +\end{table} +\end{frame} + +\begin{frame} +\frametitle{Redefining the model of external influence} +\begin{equation} +\label{Pxy} + P(x, y; \alpha, \beta, \gamma) = \frac{1}{1 + e ^ {-\alpha \ln(x + 1) - \beta \ln(y + 1) - \gamma}}, +\end{equation} + +\begin{itemize} +\item{$x$ - the number of already active peer neighbours} +\item{\textbf{$y$ - the number of activations of nodes with $0$ friends active at the time of the event}} +\vspace{1cm} +\item{now $y$ doesn't necessarily have positive correlation with time} +\end{itemize} +\end{frame} + +\begin{frame} +\begin{columns} +\column{.5\textwidth} +\textbf{Restricted Referendum Dataset}\\ +\begin{small} +$\alpha = 0.16697$, $\beta = 0.92495$, $\gamma = -8.60957$, $S_\alpha = 0.018$, $S_\beta = 0.959$ \\ +- based on $1000$ randomized instances +\end{small}\begin{figure}[htp] +\centering +\includegraphics[scale=0.3]{figs/abg1000_2.png} +\label{hist_full2} +\end{figure} +\column{.5\textwidth} +\textbf{Complete Referendum Dataset}\\ +\begin{small}$\alpha = 0.89215$, $\beta = 1.36409$, $\gamma = -15.7230$, $S_\alpha = 1.000$, $S_\beta = 0.998$\\ +- based on $500$ randomized instances \end{small} +\begin{figure}[htp] +\centering +\includegraphics[scale=0.3]{figs/abg500v.png} +\label{hist_full3} +\end{figure} +\end{columns} +\end{frame} + +\begin{frame} +\begin{itemize} +\item{\bf{Q: Is the adoption of an information item for the given network mainly internal or external influence driven?}} +\item{\bf{A: The adoption is mainly external influence driven.}} +\end{itemize} +\end{frame} +\end{document} diff --git a/presentation/prezentacija.toc b/presentation/prezentacija.toc new file mode 100644 index 0000000..33d5b34 --- /dev/null +++ b/presentation/prezentacija.toc @@ -0,0 +1,10 @@ +\beamer@endinputifotherversion {3.10pt} +\beamer@sectionintoc {1}{Introduction}{2}{0}{1} +\beamer@subsectionintoc {1}{1}{Complex networks}{2}{0}{1} +\beamer@subsectionintoc {1}{2}{Information propagation}{4}{0}{1} +\beamer@sectionintoc {2}{Dataset}{5}{0}{2} +\beamer@sectionintoc {3}{Model}{10}{0}{3} +\beamer@subsectionintoc {3}{1}{The proces of information propagation}{10}{0}{3} +\beamer@subsectionintoc {3}{2}{The structure of the network}{10}{0}{3} +\beamer@sectionintoc {4}{Results}{17}{0}{4} +\beamer@subsectionintoc {4}{1}{Restricted Referendum Dataset}{17}{0}{4}