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Louis Lacoste 2023-06-19 17:34:26 +02:00
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@ -16,7 +16,7 @@
\usepackage{hyperref} % pour les hyperliens dans le document \usepackage{hyperref} % pour les hyperliens dans le document
\usepackage{tikz} % For graph plots \usepackage{tikz} % For graph plots
% bibliographie % bibliographie
\usepackage[style=apa]{biblatex} \usepackage[style=apa,sorting=none]{biblatex}
\addbibresource{references.bib} \addbibresource{references.bib}
@ -24,6 +24,7 @@
%% Tikz Related %% Tikz Related
\usetikzlibrary{calc,shapes,backgrounds,arrows,automata,shadows,positioning} \usetikzlibrary{calc,shapes,backgrounds,arrows,automata,shadows,positioning}
\usetikzlibrary{arrows,shapes,positioning,shadows,trees,calc,backgrounds,automata,positioning} \usetikzlibrary{arrows,shapes,positioning,shadows,trees,calc,backgrounds,automata,positioning}
\usetikzlibrary{decorations.pathreplacing,calligraphy}
@ -61,7 +62,7 @@
\begin{document} \begin{document}
% titre % titre
\begin{frame} \begin{frame}[noframenumbering,plain]
\maketitle \maketitle
\end{frame} \end{frame}
@ -71,9 +72,11 @@
\begin{itemize} \begin{itemize}
\item De nombreux réseaux disponibles \parencite{WebLifeEcological} et décrivant des interactions similaires \item De nombreux réseaux disponibles \parencite{WebLifeEcological} et décrivant des interactions similaires
\item Re-grouper les réseaux selon leur similarité (\emph{clustering} de réseaux) \item Re-grouper les réseaux selon leur similarité (\emph{clustering} de réseaux)
\item Compléter d'éventuelles informations manquantes grâce à la collection
\item Déterminer des structures d'interactions fines de manière agnostique \item Déterminer des structures d'interactions fines de manière agnostique
\item Vérifier si le regroupement est lié à des co-variables \item Vérifier si le regroupement est lié à des co-variables
\end{itemize} \end{itemize}
\footnotetext[0]{Pour combler les lacunes de\\\cite{chabert-liddellLearningCommonStructures2023}}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Réseaux bipartites\footnote{Ou \emph{bipartis}. Voir \cite{larousseDefinitionsBipartiBipartite}.}} \frametitle{Réseaux bipartites\footnote{Ou \emph{bipartis}. Voir \cite{larousseDefinitionsBipartiBipartite}.}}
@ -122,51 +125,53 @@
\end{column} \end{column}
\end{columns} \end{columns}
\smallskip \smallskip
Permet de décrire toute interaction impliquant deux agents dont les rôles Permet de décrire des interactions impliquant deux agents dont les rôles
sont de natures différentes.\\ sont de natures différentes.\\
Par exemple : hôtes-parasites, plantes-pollinisateurs, graines-disperseurs \dots Par exemple : hôtes-parasites, plantes-pollinisateurs, graines-disperseurs \dots
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Latent Block Model (LBM)} \frametitle{Latent Block Model (LBM)}
Proposé par \cite{govaertEMAlgorithmBlock2005}. Proposé par \cite{govaertEMAlgorithmBlock2005}.
\begin{columns}
\begin{column}{0.5\linewidth}
\begin{figure}[H] \begin{figure}[H]
\center \center
\begin{tikzpicture}[scale=.6] \begin{tikzpicture}[scale=.45]
\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape] \tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape] \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape]
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
\tikzstyle{every node}=[fill=blueind] \tikzstyle{every node}=[fill=blueind]
\node[state, draw=white, fill=none, text=black, scale=2] (pi1) at (1,5.7) {\textbf{$\pi_{{\color{blueind}\bullet}}$}}; \node[edge_proba] (pi1) at (1,5.7) {\textbf{$\pi_{{\color{blueind}\bullet}}$}};
\node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}}; \node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}};
\node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}}; \node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}};
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}}; \node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
\tikzstyle{every node}=[fill=cyanind] \tikzstyle{every node}=[fill=cyanind]
\node[state, draw=white, fill=none, text=black, scale=2] (pi2) at (6.5,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}}; \node[edge_proba] (pi2) at (6.75,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
\node[state, draw=black!50] (R21) at (6,5) {\textbf{R21}}; \node[state, draw=black!50] (R21) at (6.25,5) {\textbf{R21}};
\node[state, draw=black!50] (R22) at (7,5) {\textbf{R22}}; \node[state, draw=black!50] (R22) at (7.25,5) {\textbf{R22}};
\tikzstyle{every node}=[fill=electricblue] \tikzstyle{every node}=[fill=electricblue]
\node[state, draw=white, fill=none, text=black, scale=2] (pi3) at (10,5.7) {\textbf{$\pi_{{\color{electricblue}\bullet}}$}}; \node[edge_proba] (pi3) at (10,5.7) {\textbf{$\pi_{{\color{electricblue}\bullet}}$}};
\node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}}; \node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
\tikzstyle{every node}=[fill=burntorange, shape=rectangle] \tikzstyle{every node}=[fill=burntorange, shape=rectangle]
\node[state, draw=white, fill=none, text=black, scale=2] (pi3) at (0.5,-0.7) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}}; \node[edge_proba] (pi3) at (0.5,-0.7) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle] \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
\node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}}; \node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}};
\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}}; \node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle] \tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
\node[state, draw=white, fill=none, text=black, scale=2] (pi3) at (4.5,-0.7) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}}; \node[edge_proba] (pi3) at (4,-0.7) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
\node[state, draw=black!50] (B3) at (4,0) {\textbf{C21}}; \node[state, draw=black!50] (B3) at (3.5,0) {\textbf{C21}};
\node[state, draw=black!50] (B4) at (5,0) {\textbf{C22}}; \node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
\tikzstyle{every node}=[fill=peach, shape=rectangle] \tikzstyle{every node}=[fill=peach, shape=rectangle]
\node[state, draw=white, fill=none, text=black, scale=2] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}}; \node[edge_proba] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}};
\node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}}; \node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}};
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt,draw opacity=0.2] \tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt,draw opacity=0.2]
\path (R11) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}}$} (B1); \path (R11) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[left, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}}$} (B1);
\path (R11) edge (B2); \path (R11) edge (B2);
\path (R11) edge (B3); \path (R11) edge (B3);
\path (R11) edge (B4); \path (R11) edge (B4);
@ -179,45 +184,372 @@
\path (R13) edge [] (B1); \path (R13) edge [] (B1);
\path (R13) edge (B2); \path (R13) edge (B2);
\path (R13) edge (B3); \path (R13) edge (B3);
\path (R13) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{goldenyellow}\bullet}}$} (B4); \path (R13) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{goldenyellow}\bullet}}$} (B4);
\path (R21) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3); \path (R21) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3);
\path (R21) edge (B4); \path (R21) edge (B4);
\path (R21) edge (B5); \path (R21) edge (B5);
\path (R22) edge (B3); \path (R22) edge (B3);
\path (R22) edge (B4); \path (R22) edge (B4);
\path (R22) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{peach}\bullet}}$} (B5); \path (R22) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{peach}\bullet}}$} (B5);
\path (R31) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5); \path (R31) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5);
\end{tikzpicture} \end{tikzpicture}
\caption{Exemple de LBM} \caption{Exemple de LBM}
\label{fig:LBMvisu} \label{fig:LBMvisu}
\end{figure} \end{figure}
\end{column}
\end{frame} \begin{column}{0.5\linewidth}
\begin{frame} \begin{block}{Paramètres}
\frametitle{Extension de \emph{colSBM} aux réseaux bipartites} \begin{itemize}
Le modèle \emph{colSBM} \parencite{chabert-liddellLearningCommonStructures2023} \item $\mathcal{K}_1 = \{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}$ blocs en ligne
\item $\mathcal{K}_2 = \{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}$ blocs en colonne
\begin{equation*} \item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(j\in\bullet)$ en colonne
\overbrace{ \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{blueind}\bullet}, j \in {\color{burntorange}\bullet})$
}^\text{Collection} \end{itemize}
\end{equation*} \end{block}
\end{column}
\end{columns}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Collections en bipartites} \frametitle{\emph{colSBM}}
Le modèle \emph{colSBM} \parencite{chabert-liddellLearningCommonStructures2023}.\\
\smallskip
\definecolor{yellow}{RGB}{255,190,60}
\begin{tikzpicture}[scale=.33]
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
\tikzstyle{every node}=[fill=yellow]
\node[state, draw=black!50] (A1) at (0,2) {\textbf{A1}};
\node[state, draw=black!50] (A2) at (1.5, 2) {\textbf{A2}};
\node[state, draw=black!50] (A3) at (0.75,3.25) {\textbf{A3}};
\tikzstyle{every node}=[fill=blueind]
\node[state, draw=black!50] (B1) at (4.5,3) {\textbf{B1}};
\node[state, draw=black!50] (B2) at (4,4.75) {\textbf{B2}};
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\node[state, draw=black!50] (B4) at (7,4.75) {\textbf{B4}};
\node[state, draw=black!50] (B5) at (6.5,3) {\textbf{B5}};
\tikzstyle{every node}=[fill=greenind]
\node[state, draw=black!50] (C1) at (5,0) {\textbf{C1}};
\node[state, draw=black!50] (C2) at (7,1) {\textbf{C2}};
\path (A1) edge[bend right] (A2);
\path (A1) edge node[midway, left, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{yellow}\bullet}}$} (A3);
\path (A3) edge (A2);
\path (A3) edge node[midway, above, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{blueind}\bullet}}$} (B3);
\path (B1) edge (B2);
\path (B2) edge (B3);
\path (B3) edge (B4);
\path (B4) edge (B5);
\path (B5) edge (B1);
\path (B1) edge[bend left=0] (B4);
\path (B5) edge[bend left=0] (B2);
\path (A2) edge[bend right] node[midway, below, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{greenind}\bullet}}$} (C1);
\path (C1) edge[bend right] node[midway, below, fill=none] {$\alpha_{{\color{greenind}\bullet}{\color{greenind}\bullet}}$} (C2);
\path (C2) edge[bend right] node[midway, right, fill=none] {$\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}}$} (B4);
\node[font=\small, text justified,draw=none, fill=none] at (4.5,-1.5) {SBM};
\node[font=\small, text justified, fill=none] at (11.5, 1.5) {$\Longrightarrow$};
% Sampled network
\begin{scope}[xshift=14.5cm, yshift=4cm]
\tikzstyle{every node}=[fill=gray, scale=0.95]
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
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\node[state, draw=black!50] (B1) at (2.5,1) {\textbf{1}};
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\node[state, draw=black!50] (B3) at (3.5,4) {\textbf{6}};
\node[state, draw=black!50] (B4) at (5,2.75) {\textbf{3}};
\node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
\node[state, draw=black!50] (C1) at (3,-0.5) {\textbf{4}};
\node[state, draw=black!50] (C2) at (5,0) {\textbf{8}};
\path (A1) edge[bend right] (A2);
\path (A1) edge (A3);
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\path (B2) edge (B3);
\path (B3) edge (B4);
\path (B4) edge (B5);
\path (B5) edge (B1);
\path (B1) edge[bend left=0] (B4);
\path (B5) edge[bend left=0] (B2);
\path (A2) edge[bend right] (C1);
\path (C1) edge[bend right] (C2);
\path (C2) edge[bend right] (B4);
\node[text width=3cm,font=\small, text justified, rotate=90, fill=none, below = -0.8cm of C1] (dots) {\dots};
\draw [decorate, decoration = {brace}] (6, 4) -- (6,-8.5);
\node[text width=3cm, font=\small, text justified, fill=none] at (11.5,-2.25) {$M$ réalisations indépendantes du SBM};
\end{scope}
\begin{scope}[xshift=14.5cm, yshift=-4cm]
\tikzstyle{every node}=[fill=gray, scale=0.95]
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
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\path (A1) edge[bend right] (A2);
\path (A1) edge (A3);
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\path (B3) edge (B4);
\path (B4) edge (B5);
\path (B5) edge (B1);
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\path (C1) edge[bend right] (C2);
\path (C2) edge[bend right] (B4);
\end{scope}
\end{tikzpicture}
\begin{columns}
\begin{column}{0.48\linewidth}
\begin{block}{Paramètres}
\begin{itemize}
\item $\mathcal{K} = \{{\color{yellow}\bullet},{\color{blueind}\bullet},{\color{greenind}\bullet}\}$ blocs
\item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$
\item $\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{greenind}\bullet}, j \in {\color{blueind}\bullet})$
\end{itemize}
\end{block}
\end{column}
\begin{column}{0.52\linewidth}
\end{column}
\end{columns}
\end{frame}
\section{Extension de \emph{colSBM} aux réseaux bipartites}
\begin{frame}
\frametitle{Collections bipartites}
\begin{columns}
\begin{column}{0.5\linewidth}
\begin{tikzpicture}[scale=.38]
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\tikzstyle{every node}=[fill=burntorange, shape=rectangle]
\node[edge_proba] (pi3) at (0.5,-0.7) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
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\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
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\tikzstyle{every node}=[fill=peach, shape=rectangle]
\node[edge_proba] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}};
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\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt,draw opacity=0.2]
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\path (R22) edge (B3);
\path (R22) edge (B4);
\path (R22) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{peach}\bullet}}$} (B5);
\path (R31) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5);
\end{tikzpicture}
\begin{block}{Paramètres}
\begin{itemize}
\item $\mathcal{K}_1 = \{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}$ blocs en ligne
\item $\mathcal{K}_2 = \{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}$ blocs en colonne
\item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(j\in\bullet)$ en colonne
\item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{blueind}\bullet}, j \in {\color{burntorange}\bullet})$
\end{itemize}
\end{block}
\end{column}
\begin{column}{0.5\linewidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{scope}[yshift = 4cm]
\tikzstyle{every state}=[draw, text=white,scale=0.75, transform shape]
\tikzstyle{every node}=[fill=gray]
\node[state, draw=black!50] (R11) at (0,1.25) {\textbf{1}};
\node[state, draw=black!50] (R12) at (1,1.25) {\textbf{2}};
\node[state, draw=black!50] (R13) at (2,1.25) {\textbf{3}};
\node[state, draw=black!50] (R21) at (3,1.25) {\textbf{4}};
\node[state, draw=black!50] (R22) at (4,1.25) {\textbf{5}};
\node[state, draw=black!50] (R31) at (5,1.25) {\textbf{6}};
\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
\node[state, draw=black!50] (B1) at (0.5,0) {\textbf{1}};
\node[state, draw=black!50] (B2) at (1.5,0) {\textbf{2}};
\node[state, draw=black!50] (B31) at (2.5,0) {\textbf{3}};
\node[state, draw=black!50] (B4) at (3.5,0) {\textbf{4}};
\node[state, draw=black!50] (B5) at (4.5,0) {\textbf{5}};
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1pt, draw=gray, fill=gray]
\path (R11) edge (B1);
\path (R11) edge (B2);
\path (R11) edge (B31);
\path (R11) edge (B4);
\path (R12) edge [] (B1);
\path (R12) edge (B2);
\path (R12) edge (B31);
\path (R12) edge (B4);
\path (R13) edge [] (B1);
\path (R13) edge (B2);
\path (R13) edge (B31);
\path (R13) edge (B4);
\path (R21) edge (B31);
\path (R21) edge (B4);
\path (R21) edge (B5);
\path (R22) edge (B31);
\path (R22) edge (B4);
\path (R22) edge (B5);
\path (R31) edge (B5);
\end{scope}
\node[text width=3cm,font=\small, text justified, rotate=90, fill=none] (dots) at (2.5, 4.75){\dots};
\begin{scope}[yshift = 0cm]
\tikzstyle{every state}=[draw, text=white,scale=0.75, transform shape]
\tikzstyle{every node}=[fill=gray]
\node[state, draw=black!50] (R11) at (0,1.25) {\textbf{4}};
\node[state, draw=black!50] (R12) at (1,1.25) {\textbf{1}};
\node[state, draw=black!50] (R13) at (2,1.25) {\textbf{6}};
\node[state, draw=black!50] (R21) at (3,1.25) {\textbf{3}};
\node[state, draw=black!50] (R22) at (4,1.25) {\textbf{5}};
\node[state, draw=black!50] (R31) at (5,1.25) {\textbf{2}};
\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
\node[state, draw=black!50] (B1) at (0.5,0) {\textbf{5}};
\node[state, draw=black!50] (B2) at (1.5,0) {\textbf{1}};
\node[state, draw=black!50] (B3) at (2.5,0) {\textbf{3}};
\node[state, draw=black!50] (B4) at (3.5,0) {\textbf{2}};
\node[state, draw=black!50] (B5) at (4.5,0) {\textbf{4}};
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1pt, draw=gray, fill=gray]
\path (R11) edge (B1);
\path (R11) edge (B2);
\path (R11) edge (B3);
\path (R11) edge (B4);
\path (R12) edge [] (B1);
\path (R12) edge (B2);
\path (R12) edge (B3);
\path (R12) edge (B4);
\path (R13) edge [] (B1);
\path (R13) edge (B2);
\path (R13) edge (B3);
\path (R13) edge (B4);
\path (R21) edge (B3);
\path (R21) edge (B4);
\path (R21) edge (B5);
\path (R22) edge (B3);
\path (R22) edge (B4);
\path (R22) edge (B5);
\path (R31) edge (B5);
\end{scope}
\end{tikzpicture}
\end{column}
\end{columns}
\end{frame} \end{frame}
\section*{Bibliographie} \section*{Bibliographie}
\begin{frame} \begin{frame}[noframenumbering,plain,allowframebreaks]
\frametitle{Bibliographie} \frametitle{Bibliographie}
\hfill
\begin{minipage}{0.9\textwidth}
\printbibliography \printbibliography
\end{minipage}
\end{frame} \end{frame}
\end{document} \end{document}

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rapport.tex
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@ -177,38 +177,39 @@ This model supposes that:
\begin{tikzpicture}[scale=.6] \begin{tikzpicture}[scale=.6]
\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape] \tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape] \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape]
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
\tikzstyle{every node}=[fill=blueind] \tikzstyle{every node}=[fill=blueind]
\node[state, draw=white, fill=none, text=black, scale=2] (pi1) at (1,5.7) {\textbf{$\pi_{{\color{blueind}\bullet}}$}}; \node[edge_proba] (pi1) at (1,5.7) {\textbf{$\pi_{{\color{blueind}\bullet}}$}};
\node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}}; \node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}};
\node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}}; \node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}};
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}}; \node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
\tikzstyle{every node}=[fill=cyanind] \tikzstyle{every node}=[fill=cyanind]
\node[state, draw=white, fill=none, text=black, scale=2] (pi2) at (6.5,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}}; \node[edge_proba] (pi2) at (6.75,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
\node[state, draw=black!50] (R21) at (6,5) {\textbf{R21}}; \node[state, draw=black!50] (R21) at (6.25,5) {\textbf{R21}};
\node[state, draw=black!50] (R22) at (7,5) {\textbf{R22}}; \node[state, draw=black!50] (R22) at (7.25,5) {\textbf{R22}};
\tikzstyle{every node}=[fill=electricblue] \tikzstyle{every node}=[fill=electricblue]
\node[state, draw=white, fill=none, text=black, scale=2] (pi3) at (10,5.7) {\textbf{$\pi_{{\color{electricblue}\bullet}}$}}; \node[edge_proba] (pi3) at (10,5.7) {\textbf{$\pi_{{\color{electricblue}\bullet}}$}};
\node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}}; \node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
\tikzstyle{every node}=[fill=burntorange, shape=rectangle] \tikzstyle{every node}=[fill=burntorange, shape=rectangle]
\node[state, draw=white, fill=none, text=black, scale=2] (pi3) at (0.5,-0.7) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}}; \node[edge_proba] (pi3) at (0.5,-0.7) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle] \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
\node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}}; \node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}};
\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}}; \node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle] \tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
\node[state, draw=white, fill=none, text=black, scale=2] (pi3) at (4.5,-0.7) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}}; \node[edge_proba] (pi3) at (4,-0.7) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
\node[state, draw=black!50] (B3) at (4,0) {\textbf{C21}}; \node[state, draw=black!50] (B3) at (3.5,0) {\textbf{C21}};
\node[state, draw=black!50] (B4) at (5,0) {\textbf{C22}}; \node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
\tikzstyle{every node}=[fill=peach, shape=rectangle] \tikzstyle{every node}=[fill=peach, shape=rectangle]
\node[state, draw=white, fill=none, text=black, scale=2] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}}; \node[edge_proba] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}};
\node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}}; \node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}};
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt,draw opacity=0.2] \tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt,draw opacity=0.2]
\path (R11) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}}$} (B1); \path (R11) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[left, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}}$} (B1);
\path (R11) edge (B2); \path (R11) edge (B2);
\path (R11) edge (B3); \path (R11) edge (B3);
\path (R11) edge (B4); \path (R11) edge (B4);
@ -221,23 +222,31 @@ This model supposes that:
\path (R13) edge [] (B1); \path (R13) edge [] (B1);
\path (R13) edge (B2); \path (R13) edge (B2);
\path (R13) edge (B3); \path (R13) edge (B3);
\path (R13) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{goldenyellow}\bullet}}$} (B4); \path (R13) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{goldenyellow}\bullet}}$} (B4);
\path (R21) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3); \path (R21) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3);
\path (R21) edge (B4); \path (R21) edge (B4);
\path (R21) edge (B5); \path (R21) edge (B5);
\path (R22) edge (B3); \path (R22) edge (B3);
\path (R22) edge (B4); \path (R22) edge (B4);
\path (R22) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{peach}\bullet}}$} (B5); \path (R22) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{peach}\bullet}}$} (B5);
\path (R31) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[anchor=center, fill=none] {$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5); \path (R31) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5);
\end{tikzpicture} \end{tikzpicture}
\caption{An LBM model visualization} \caption{An LBM model visualization}
\label{fig:LBMvisu} \label{fig:LBMvisu}
\end{figure} \end{figure}
Parameters
\begin{itemize}
\item $\mathcal{K}_1 = \{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}$ blocks in rows
\item $\mathcal{K}_2 = \{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}$ blocks in columns
\item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ in row et $\rho_{\bullet} = \mathbb{P}(j\in\bullet)$ in column
\item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{blueind}\bullet}, j \in {\color{burntorange}\bullet})$ connectivity probability between two nodes, given their clustering
\end{itemize}
On \ref{fig:LBMvisu}, $\pi$ are the probabilities for a row node to belong to On \ref{fig:LBMvisu}, $\pi$ are the probabilities for a row node to belong to
the row block of corresponding color, $\rho$ are the probabilities for a column the row block of corresponding color, $\rho$ are the probabilities for a column
node to belong to the column block of corresponding color and $\alpha$ are the node to belong to the column block of corresponding color and $\alpha$ are the