173 lines
9.4 KiB
TeX
173 lines
9.4 KiB
TeX
% --------------------------------------------------------------------------- %
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% Poster for the ECCS 2011 Conference about Elementary Dynamic Networks. %
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% --------------------------------------------------------------------------- %
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% Created with Brian Amberg's LaTeX Poster Template. Please refer for the %
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% attached README.md file for the details how to compile with `pdflatex`. %
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% --------------------------------------------------------------------------- %
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% $LastChangedDate:: 2011-09-11 10:57:12 +0200 (V, 11 szept. 2011) $ %
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% $LastChangedRevision:: 128 $ %
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% $LastChangedBy:: rlegendi $ %
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% $Id:: poster.tex 128 2011-09-11 08:57:12Z rlegendi $ %
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% --------------------------------------------------------------------------- %
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\documentclass[a0paper,portrait]{baposter}
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\usepackage{relsize} % For \smaller
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\usepackage{url} % For \url
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\usepackage{epstopdf} % Included EPS files automatically converted to PDF to include with pdflatex
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%%% Global Settings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\graphicspath{{pix/}} % Root directory of the pictures
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\tracingstats=2 % Enabled LaTeX logging with conditionals
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%%% Color Definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% Utility functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% Save space in lists. Use this after the opening of the list %%%%%%%%%%%%%%%%
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% Document Start %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\typeout{Poster rendering started}
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%%% Setting Background Image %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\background{
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\begin{tikzpicture}[remember picture,overlay]%
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{\includegraphics[height=1.1\textheight]{background}};
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\end{tikzpicture}
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%%% General Poster Settings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%% Eye Catcher, Title, Authors and University Images %%%%%%%%%%%%%%%%%%%%%%
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\begin{poster}{
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grid=false,
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% Option is left on true though the eyecatcher is not used. The reason is
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% that we have a bit nicer looking title and author formatting in the headercol
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% this way
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%eyecatcher=false,
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borderColor=bordercol,
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headerColorOne=headercol1,
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% Only simple background color used, no shading, so boxColorTwo isn't necessary
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boxColorOne=boxcolor,
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headerfont=\Large\sf\bf,
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textborder=rectangle,
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%%% Eye Cacther %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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{
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Eye Catcher, empty if option eyecatcher=false - unused
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}
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%%% Title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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{\sf\bf
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Properties of Elementary Random and Preferential Dynamic Networks
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}
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%%% Authors %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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{
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\vspace{1em} Richard O. Legendi, Laszlo Gulyas, George Kampis\\
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{\smaller legendi@inf.elte.hu, lgulyas@colbud.hu, gkampis@colbud.hu}
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}
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%%% Logo %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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{
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% The logos are compressed a bit into a simple box to make them smaller on the result
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% (Wasn't able to find any bigger of them.)
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\setlength\fboxsep{0pt}
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\setlength\fboxrule{0.5pt}
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\fbox{
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\begin{minipage}{14em}
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\includegraphics[width=10em,height=4em]{colbud_logo}
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\includegraphics[width=4em,height=4em]{elte_logo} \\
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\includegraphics[width=10em,height=4em]{dynanets_logo}
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\includegraphics[width=4em,height=4em]{aitia_logo}
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\end{minipage}
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}
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}
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\headerbox{Problem}{name=problem,column=0,row=0}{
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Sampling networks always involves the act of aggregation (e.g., when collecting longitudinal samples of networks). We sutdy how the cumulation window length effects the properties of the aggregated network.
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\includegraphics[width=\linewidth]{time_windows}
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}
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\headerbox{Basic Concepts}{name=definitions,column=0,below=problem}{
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In our work the dynamic network is a series of graphs, that is, $DN = G_t(V_t,E_t)$, where $E_t \subseteq V_t \times V_t$ ($\forall t \geq 0$). The initial network, $G_0$, is considered as a parameter of the process. The \textbf{node set fixed} and we worked with an about \textbf{constant number of edges}. We assume that the evolution of the network can be described as the result of an edge creation and an edge deletion process. We define $G_t$ as the \textbf{snapshot network} and
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{\smaller
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$$G_T = ( \bigcup^{T}_{t=0}V_t, \bigcup^{T}_{t=0}E_t) ~ \textnormal{for} ~ T \geq 0.$$
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}
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as the \textbf{cumulative network}.
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}
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\headerbox{Models}{name=models,column=0,below=definitions}{
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\textbf{ER1} $G_0$ is a random graph. Add each non-existing edge with $p_A$, delete each existing edge with $p_D$ probability. \\
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\textbf{ER2} $G_0$ is a random graph. Add $k_A$ uniformly selected random new edges and delete $k_D$ existing edges. \\
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\textbf{ER3} $G_0$ is a random graph. Rewire $k_{RW}$ edges. \\
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\textbf{SPA} (\emph{Snapshot preferential}) $G_0$ is a scale free network. Add $k_A$ edges from a random node with preferential attachment based on the snapshot network. Delete $k_D$ existing edges. \\
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\textbf{CPA} (\emph{Cumulative preferential}) $G_0$ is a scale free network. Add $k_A$ edges from a random node with preferential attachment based on the cumulative network. Delete $k_D$ existing edges.
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}
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\headerbox{References}{name=references,column=0,below=models}{
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\smaller % Make the whole text smaller
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\vspace{-0.4em} % Save some space at the beginning
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\bibliographystyle{plain} % Use plain style
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\renewcommand{\section}[2]{\vskip 0.05em} % Omit "References" title
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\begin{thebibliography}{1} % Simple bibliography with widest label of 1
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\itemsep=-0.01em % Save space between the separation
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\setlength{\baselineskip}{0.4em} % Save space with longer lines
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\bibitem{prevWork1} Laszlo Gulyas, Richard Legendi: \emph{Effects of Sample Duration on Network Statistics in Elementary Models of Dynamic Networks}, International Conference on Computational Science, Singapore (2011)
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\bibitem{prevWork2} Laszlo Gulyas, Susan Khor, Richard Legendi and George Kampis \emph{Cumulative Properties of Elementary Dynamic Networks}, The International Sunbelt Social Network Conference XXXI (2011)
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\bibitem{gulya-kampis1} Gulyas, Laszlo et al.: \emph{Betweenness Centrality Dynamics in Networks of Changing Density}. Presented at the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010)
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\end{thebibliography}
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}
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\headerbox{Acknowledgements}{name=acknowledgements,column=0,below=references, above=bottom}{
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\smaller % Make the whole text smaller
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\vspace{-0.4em} % Save some space at the beginning
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This research was partially supported by the Hungarian Government (KMOP-1.1.2-08/1-2008-0002 ) and the European Union's Seventh Framework Programme: DynaNets, FET-Open project no. FET-233847 (\url{http://www.dynanets.org}). The supports are gratefully acknowledged.
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}
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\headerbox{Dynamic Networks are Sensitive to Aggregation}{name=density,span=2,column=1,row=0}{
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Network characteristics are extremely sensitive to minor changes in aggregation length. In our previous work \cite{prevWork1} \cite{prevWork2}, we studied the cumulative properties of Elementary Dynamic Network models over the complete time period (i.e., until they reach the stable point of a full network). Here we focus on the more realistc domain of sparse (cumulative) networks. We find that even when snapshot networks are stationary, \textbf{important network characteristics} (average path lenght, clustering, betwenness centrality) \textbf{are extremely sensitive to aggregation} (window length).
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\includegraphics[angle=-90,width=0.98\linewidth]{PA_and_ER_Models_statisticalMeasures}
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}
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\headerbox{Degree Distribution Radically Changes}
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{name=degreeDistribution,span=2,column=1,below=density,above=bottom}{
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Degree distributions are exceptionally sensitive to the length of the aggregation window. \textbf{The same dynamic network may produce a normal, lognormal or even power law distribution for different aggregation lenghts.} The digree distribution of the snapshot and cumulative network is inherently different. The following surfaces show the CPA model until it approaches the complete network.
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\vspace{-0.2em}
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\begin{center}
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\includegraphics[angle=-90,width=0.49\linewidth]{CPA_3d_snapshot}
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\includegraphics[angle=-90,width=0.49\linewidth]{CPA_3d_cumulative}
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\end{center}
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\vspace{-0.2em}
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Taking slices of the cumulative 3D charts shows us how the degree distribution changes. The log-log charts below show the progression of these changes as the aggregation window gets larger.
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\vspace{-0.2em}
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\begin{center}
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\includegraphics[angle=-90,width=0.49\linewidth]{ER1_cumulativeDegrees}
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\includegraphics[angle=-90,width=0.49\linewidth]{CPA_cumulativeDegrees}
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\end{center}
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}
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\end{poster}
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\end{document}
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