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@ -16,7 +16,7 @@
\usepackage{hyperref} % pour les hyperliens dans le document
\usepackage{tikz} % For graph plots
% bibliographie
\usepackage[style=apa]{biblatex}
\usepackage[style=apa,sorting=none]{biblatex}
\addbibresource{references.bib}
@ -24,6 +24,7 @@
%% Tikz Related
\usetikzlibrary{calc,shapes,backgrounds,arrows,automata,shadows,positioning}
\usetikzlibrary{arrows,shapes,positioning,shadows,trees,calc,backgrounds,automata,positioning}
\usetikzlibrary{decorations.pathreplacing,calligraphy}
@ -61,8 +62,8 @@
\begin{document}
% titre
\begin{frame}
\maketitle
\begin{frame}[noframenumbering,plain]
\maketitle
\end{frame}
\section{Contexte du modèle}
@ -71,153 +72,484 @@
\begin{itemize}
\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 Compléter d'éventuelles informations manquantes grâce à la collection
\item Déterminer des structures d'interactions fines de manière agnostique
\item Vérifier si le regroupement est lié à des co-variables
\end{itemize}
\footnotetext[0]{Pour combler les lacunes de\\\cite{chabert-liddellLearningCommonStructures2023}}
\end{frame}
\begin{frame}
\frametitle{Réseaux bipartites\footnote{Ou \emph{bipartis}. Voir \cite{larousseDefinitionsBipartiBipartite}.}}
\begin{columns}[c]
\begin{column}{0.48\textwidth}
\centering
Réseau bipartite\\
\begin{tikzpicture}[scale=.6]
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt]
\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 node}=[fill=blueind]
\node[state, draw=black!50] (A1) at (0,5) {\textbf{R1}};
\node[state, draw=black!50] (A2) at (2.5,5) {\textbf{R2}};
\node[state, draw=black!50] (A3) at (5,5) {\textbf{R3}};
\tikzstyle{every node}=[fill=greenind, 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{C1}};
\node[state, draw=black!50] (B2) at (1.25,0) {\textbf{C2}};
\node[state, draw=black!50] (B3) at (2.5,0) {\textbf{C3}};
\node[state, draw=black!50] (B4) at (3.75,0) {\textbf{C4}};
\node[state, draw=black!50] (B5) at (5,0) {\textbf{C5}};
\path (A1) edge [] (B1);
\path (A1) edge (B2);
\path (A1) edge (B3);
\path (A1) edge (B4);
\path (A2) edge (B3);
\path (A2) edge (B4);
\path (A3) edge (B5);
\path (A2) edge (B5);
\end{tikzpicture}
\end{column}
\hfill
\begin{column}{0.48\linewidth}
Matrice d'incidence
\smallskip
$B=\left(
\begin{array}{rrrrr}
1 & 1 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 \\
\end{array}\right)
$\\
\end{column}
\begin{column}{0.48\textwidth}
\centering
Réseau bipartite\\
\begin{tikzpicture}[scale=.6]
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt]
\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 node}=[fill=blueind]
\node[state, draw=black!50] (A1) at (0,5) {\textbf{R1}};
\node[state, draw=black!50] (A2) at (2.5,5) {\textbf{R2}};
\node[state, draw=black!50] (A3) at (5,5) {\textbf{R3}};
\tikzstyle{every node}=[fill=greenind, 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{C1}};
\node[state, draw=black!50] (B2) at (1.25,0) {\textbf{C2}};
\node[state, draw=black!50] (B3) at (2.5,0) {\textbf{C3}};
\node[state, draw=black!50] (B4) at (3.75,0) {\textbf{C4}};
\node[state, draw=black!50] (B5) at (5,0) {\textbf{C5}};
\path (A1) edge [] (B1);
\path (A1) edge (B2);
\path (A1) edge (B3);
\path (A1) edge (B4);
\path (A2) edge (B3);
\path (A2) edge (B4);
\path (A3) edge (B5);
\path (A2) edge (B5);
\end{tikzpicture}
\end{column}
\hfill
\begin{column}{0.48\linewidth}
Matrice d'incidence
\smallskip
$B=\left(
\begin{array}{rrrrr}
1 & 1 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 \\
\end{array}\right)
$\\
\end{column}
\end{columns}
\smallskip
Permet de décrire toute interaction impliquant deux agents dont les rôles
Permet de décrire des interactions impliquant deux agents dont les rôles
sont de natures différentes.\\
Par exemple : hôtes-parasites, plantes-pollinisateurs, graines-disperseurs \dots
\end{frame}
\begin{frame}
\frametitle{Latent Block Model (LBM)}
Proposé par \cite{govaertEMAlgorithmBlock2005}.
\begin{columns}
\begin{column}{0.5\linewidth}
\begin{figure}[H]
\center
\begin{tikzpicture}[scale=.45]
\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape]
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
\begin{figure}[H]
\center
\begin{tikzpicture}[scale=.6]
\tikzstyle{every node}=[fill=blueind]
\node[edge_proba] (pi1) at (1,5.7) {\textbf{$\pi_{{\color{blueind}\bullet}}$}};
\node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}};
\node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}};
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
\tikzstyle{every node}=[fill=cyanind]
\node[edge_proba] (pi2) at (6.75,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
\node[state, draw=black!50] (R21) at (6.25,5) {\textbf{R21}};
\node[state, draw=black!50] (R22) at (7.25,5) {\textbf{R22}};
\tikzstyle{every node}=[fill=electricblue]
\node[edge_proba] (pi3) at (10,5.7) {\textbf{$\pi_{{\color{electricblue}\bullet}}$}};
\node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
\tikzstyle{every node}=[fill=burntorange, shape=rectangle]
\node[edge_proba] (pi3) at (0.5,-0.7) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
\node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}};
\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
\node[edge_proba] (pi3) at (4,-0.7) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
\node[state, draw=black!50] (B3) at (3.5,0) {\textbf{C21}};
\node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
\tikzstyle{every node}=[fill=peach, shape=rectangle]
\node[edge_proba] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}};
\node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}};
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt,draw opacity=0.2]
\path (R11) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[left, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}}$} (B1);
\path (R11) edge (B2);
\path (R11) edge (B3);
\path (R11) edge (B4);
\path (R12) edge [] (B1);
\path (R12) edge (B2);
\path (R12) edge (B3);
\path (R12) edge (B4);
\path (R13) edge [] (B1);
\path (R13) edge (B2);
\path (R13) edge (B3);
\path (R13) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left, fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{goldenyellow}\bullet}}$} (B4);
\path (R21) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3);
\path (R21) edge (B4);
\path (R21) edge (B5);
\path (R22) edge (B3);
\path (R22) edge (B4);
\path (R22) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left, fill=none] {$\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}
\caption{Exemple de LBM}
\label{fig:LBMvisu}
\end{figure}
\end{column}
\begin{column}{0.5\linewidth}
\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}
\end{columns}
\end{frame}
\begin{frame}
\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}};
\node[state, draw=black!50] (B3) at (5.5,6) {\textbf{B3}};
\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]
\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 node}=[fill=blueind]
\node[state, draw=white, fill=none, text=black, scale=2] (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] (R12) at (1,5) {\textbf{R12}};
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
\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[state, draw=black!50] (R21) at (6,5) {\textbf{R21}};
\node[state, draw=black!50] (R22) at (7,5) {\textbf{R22}};
\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[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
\tikzstyle{every node}=[fill=burntorange, shape=rectangle]
\node[state, draw=white, fill=none, text=black, scale=2] (pi3) at (0.5,-0.7) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
\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] (B2) at (1,0) {\textbf{C12}};
\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[state, draw=black!50] (B3) at (4,0) {\textbf{C21}};
\node[state, draw=black!50] (B4) at (5,0) {\textbf{C22}};
\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[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]
\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 (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[-,>=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 (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 (B4);
\path (R21) edge (B5);
\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[anchor=center, 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);
\end{tikzpicture}
\caption{Exemple de LBM}
\label{fig:LBMvisu}
\end{figure}
\node[state, draw=black!50] (A1) at (0,0) {\textbf{10}};
\node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
\node[state, draw=black!50] (A3) at (0.5,1) {\textbf{5}};
\node[state, draw=black!50] (B1) at (2.5,1) {\textbf{1}};
\node[state, draw=black!50] (B2) at (2,2.75) {\textbf{9}};
\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);
\path (A3) edge (A2);
\path (A3) edge (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] (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]
\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
\node[state, draw=black!50] (A1) at (0,0) {\textbf{9}};
\node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
\node[state, draw=black!50] (A3) at (0.5,1) {\textbf{1}};
\node[state, draw=black!50] (B1) at (2.5,1) {\textbf{5}};
\node[state, draw=black!50] (B2) at (2,2.75) {\textbf{10}};
\node[state, draw=black!50] (B3) at (3.5,4) {\textbf{4}};
\node[state, draw=black!50] (B4) at (5,2.75) {\textbf{8}};
\node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
\node[state, draw=black!50] (C1) at (3,-0.5) {\textbf{6}};
\node[state, draw=black!50] (C2) at (5,0) {\textbf{3}};
\path (A1) edge[bend right] (A2);
\path (A1) edge (A3);
\path (A3) edge (A2);
\path (A3) edge (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] (C1);
\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{Extension de \emph{colSBM} aux réseaux bipartites}
Le modèle \emph{colSBM} \parencite{chabert-liddellLearningCommonStructures2023}
\frametitle{Collections bipartites}
\begin{columns}
\begin{column}{0.5\linewidth}
\begin{tikzpicture}[scale=.38]
\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape]
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
\begin{equation*}
\overbrace{
}^\text{Collection}
\end{equation*}
\tikzstyle{every node}=[fill=blueind]
\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] (R12) at (1,5) {\textbf{R12}};
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
\end{frame}
\begin{frame}
\frametitle{Collections en bipartites}
\tikzstyle{every node}=[fill=cyanind]
\node[edge_proba] (pi2) at (6.75,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
\node[state, draw=black!50] (R21) at (6.25,5) {\textbf{R21}};
\node[state, draw=black!50] (R22) at (7.25,5) {\textbf{R22}};
\tikzstyle{every node}=[fill=electricblue]
\node[edge_proba] (pi3) at (10,5.7) {\textbf{$\pi_{{\color{electricblue}\bullet}}$}};
\node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
\tikzstyle{every node}=[fill=burntorange, shape=rectangle]
\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]
\node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}};
\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
\node[edge_proba] (pi3) at (4,-0.7) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
\node[state, draw=black!50] (B3) at (3.5,0) {\textbf{C21}};
\node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
\tikzstyle{every node}=[fill=peach, shape=rectangle]
\node[edge_proba] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}};
\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]
\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 (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[-,>=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[midway, right, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3);
\path (R21) edge (B4);
\path (R21) edge (B5);
\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}
\section*{Bibliographie}
\begin{frame}
\begin{frame}[noframenumbering,plain,allowframebreaks]
\frametitle{Bibliographie}
\printbibliography
\hfill
\begin{minipage}{0.9\textwidth}
\printbibliography
\end{minipage}
\end{frame}
\end{document}

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rapport.tex
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@ -177,38 +177,39 @@ This model supposes that:
\begin{tikzpicture}[scale=.6]
\tikzstyle{every state}=[draw, text=white,scale=0.95, 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]
\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] (R12) at (1,5) {\textbf{R12}};
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
\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[state, draw=black!50] (R21) at (6,5) {\textbf{R21}};
\node[state, draw=black!50] (R22) at (7,5) {\textbf{R22}};
\node[edge_proba] (pi2) at (6.75,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
\node[state, draw=black!50] (R21) at (6.25,5) {\textbf{R21}};
\node[state, draw=black!50] (R22) at (7.25,5) {\textbf{R22}};
\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}};
\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]
\node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}};
\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
\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[state, draw=black!50] (B3) at (4,0) {\textbf{C21}};
\node[state, draw=black!50] (B4) at (5,0) {\textbf{C22}};
\node[edge_proba] (pi3) at (4,-0.7) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
\node[state, draw=black!50] (B3) at (3.5,0) {\textbf{C21}};
\node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
\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}};
\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 (B3);
\path (R11) edge (B4);
@ -221,23 +222,31 @@ This model supposes that:
\path (R13) edge [] (B1);
\path (R13) edge (B2);
\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 (B5);
\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[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}
\caption{An LBM model visualization}
\label{fig:LBMvisu}
\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
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