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% Poster for the ECCS 2011 Conference about Elementary Dynamic Networks. %
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%%% Title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\sf\bf
Properties of Elementary Random and Preferential Dynamic Networks
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%%% Authors %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{
\vspace{1em} Richard O. Legendi, Laszlo Gulyas, George Kampis\\
{\smaller legendi@inf.elte.hu, lgulyas@colbud.hu, gkampis@colbud.hu}
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\headerbox{Problem}{name=problem,column=0,row=0}{
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.
\includegraphics[width=\linewidth]{time_windows}
}
\headerbox{Basic Concepts}{name=definitions,column=0,below=problem}{
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
{\smaller
$$G_T = ( \bigcup^{T}_{t=0}V_t, \bigcup^{T}_{t=0}E_t) ~ \textnormal{for} ~ T \geq 0.$$
}
as the \textbf{cumulative network}.
}
\headerbox{Models}{name=models,column=0,below=definitions}{
\textbf{ER1} $G_0$ is a random graph. Add each non-existing edge with $p_A$, delete each existing edge with $p_D$ probability. \\
\textbf{ER2} $G_0$ is a random graph. Add $k_A$ uniformly selected random new edges and delete $k_D$ existing edges. \\
\textbf{ER3} $G_0$ is a random graph. Rewire $k_{RW}$ edges. \\
\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. \\
\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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\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)
\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)
\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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\headerbox{Acknowledgements}{name=acknowledgements,column=0,below=references, above=bottom}{
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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.
}
\headerbox{Dynamic Networks are Sensitive to Aggregation}{name=density,span=2,column=1,row=0}{
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).
\includegraphics[angle=-90,width=0.98\linewidth]{PA_and_ER_Models_statisticalMeasures}
}
\headerbox{Degree Distribution Radically Changes}
{name=degreeDistribution,span=2,column=1,below=density,above=bottom}{
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.
\vspace{-0.2em}
\begin{center}
\includegraphics[angle=-90,width=0.49\linewidth]{CPA_3d_snapshot}
\includegraphics[angle=-90,width=0.49\linewidth]{CPA_3d_cumulative}
\end{center}
\vspace{-0.2em}
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.
\vspace{-0.2em}
\begin{center}
\includegraphics[angle=-90,width=0.49\linewidth]{ER1_cumulativeDegrees}
\includegraphics[angle=-90,width=0.49\linewidth]{CPA_cumulativeDegrees}
\end{center}
}
\end{poster}
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