555 lines
No EOL
26 KiB
TeX
555 lines
No EOL
26 KiB
TeX
\documentclass{beamer}
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\usetheme{Boadilla}
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% importations
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\usepackage[french]{babel} % pour dire que le texte est en francais
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\usepackage{csquotes}
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\usepackage[T1]{fontenc} % pour les font postscript
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\usepackage[cyr]{aeguill} % Police vectorielle TrueType, guillemets francais
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\usepackage{epsfig} % pour gérer les images
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\usepackage{amsmath,amsthm} % très bon mode mathématique
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\usepackage{amsfonts,amssymb,bm, bbold}% permet la definition des ensembles
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\usepackage{algorithm2e} % pour les algorithmes
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\usepackage{algpseudocode} % pour les algorithmes
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\usepackage{float} % pour le placement des figure
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\usepackage{url} % pour une gestion efficace des url
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\usepackage{hyperref} % pour les hyperliens dans le document
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\usepackage{tikz} % For graph plots
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% bibliographie
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\usepackage[style=apa,sorting=none]{biblatex}
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\addbibresource{references.bib}
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% Tikz
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%% Tikz Related
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\usetikzlibrary{calc,shapes,backgrounds,arrows,automata,shadows,positioning}
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\usetikzlibrary{arrows,shapes,positioning,shadows,trees,calc,backgrounds,automata,positioning}
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\usetikzlibrary{decorations.pathreplacing,calligraphy}
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\tikzset{
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basic/.style = {draw, text width=3cm, font=\sffamily, rectangle},
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root/.style = {basic, rounded corners=2pt, thin, align=center,
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fill=green!30},
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level 2/.style = {basic, rounded corners=6pt, thin,align=center, fill=green!60,
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text width=8em},
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level 3/.style = {basic, thin, align=left, fill=pink!60, text width=3.5cm}
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}
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% Couleurs
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% pour tickz multilevel
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\definecolor{redorg}{RGB}{215,48,39}
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\definecolor{orangeorg}{RGB}{253,174,97}
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\definecolor{blueind}{RGB}{69,117,233}
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\definecolor{cyanind}{RGB}{116,173,209}
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\definecolor{electricblue}{RGB}{125, 249, 255}
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\definecolor{greenind}{RGB}{112,130,56}
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\definecolor{burntorange}{RGB}{204, 85, 0}
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\definecolor{goldenyellow}{RGB}{255, 192, 0}
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\definecolor{peach}{RGB}{255, 229, 180}
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\definecolor{gray}{RGB}{128,128,128}
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\title{Séminaire des stagiaires}
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\subtitle{Adaptation de colSBM aux réseaux bipartites}
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\author[L. Lacoste]{Louis \textsc{Lacoste}}
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\date{29 juin 2023}
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\begin{document}
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% titre
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\begin{frame}[noframenumbering,plain]
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\maketitle
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\end{frame}
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\section{Contexte du modèle}
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\begin{frame}
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\frametitle{Contexte écologique}
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\begin{itemize}
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\item De nombreux réseaux disponibles \parencite{WebLifeEcological} et décrivant des interactions similaires
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\item Re-grouper les réseaux selon leur similarité (\emph{clustering} de réseaux)
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\item Compléter d'éventuelles informations manquantes grâce à la collection
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\item Déterminer des structures d'interactions fines de manière agnostique
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\item Vérifier si le regroupement est lié à des co-variables
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\end{itemize}
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\footnotetext[0]{Pour combler les lacunes de\\\cite{chabert-liddellLearningCommonStructures2023}}
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\end{frame}
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\begin{frame}
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\frametitle{Réseaux bipartites\footnote{Ou \emph{bipartis}. Voir \cite{larousseDefinitionsBipartiBipartite}.}}
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\begin{columns}[c]
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\begin{column}{0.48\textwidth}
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\centering
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Réseau bipartite\\
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\begin{tikzpicture}[scale=.6]
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\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt]
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\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
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\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape]
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\tikzstyle{every node}=[fill=blueind]
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\node[state, draw=black!50] (A1) at (0,5) {\textbf{R1}};
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\node[state, draw=black!50] (A2) at (2.5,5) {\textbf{R2}};
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\node[state, draw=black!50] (A3) at (5,5) {\textbf{R3}};
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\tikzstyle{every node}=[fill=greenind, shape=rectangle]
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\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
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\node[state, draw=black!50] (B1) at (0,0) {\textbf{C1}};
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\node[state, draw=black!50] (B2) at (1.25,0) {\textbf{C2}};
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\node[state, draw=black!50] (B3) at (2.5,0) {\textbf{C3}};
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\node[state, draw=black!50] (B4) at (3.75,0) {\textbf{C4}};
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\node[state, draw=black!50] (B5) at (5,0) {\textbf{C5}};
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\path (A1) edge [] (B1);
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\path (A1) edge (B2);
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\path (A1) edge (B3);
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\path (A1) edge (B4);
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\path (A2) edge (B3);
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\path (A2) edge (B4);
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\path (A3) edge (B5);
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\path (A2) edge (B5);
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\end{tikzpicture}
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\end{column}
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\hfill
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\begin{column}{0.48\linewidth}
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Matrice d'incidence
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\smallskip
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$B=\left(
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\begin{array}{rrrrr}
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1 & 1 & 1 & 1 & 0 \\
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0 & 0 & 1 & 1 & 1 \\
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0 & 0 & 0 & 0 & 1 \\
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\end{array}\right)
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$\\
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\end{column}
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\end{columns}
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\smallskip
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Permet de décrire des interactions impliquant deux agents dont les rôles
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sont de natures différentes.\\
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Par exemple : hôtes-parasites, plantes-pollinisateurs, graines-disperseurs \dots
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\end{frame}
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\begin{frame}
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\frametitle{Latent Block Model (LBM)}
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Proposé par \cite{govaertEMAlgorithmBlock2005}.
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\begin{columns}
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\begin{column}{0.5\linewidth}
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\begin{figure}[H]
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\center
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\begin{tikzpicture}[scale=.45]
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\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
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\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape]
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\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
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\tikzstyle{every node}=[fill=blueind]
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\node[edge_proba] (pi1) at (1,5.7) {\textbf{$\pi_{{\color{blueind}\bullet}}$}};
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\node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}};
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\node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}};
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\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
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\tikzstyle{every node}=[fill=cyanind]
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\node[edge_proba] (pi2) at (6.75,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
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\node[state, draw=black!50] (R21) at (6.25,5) {\textbf{R21}};
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\node[state, draw=black!50] (R22) at (7.25,5) {\textbf{R22}};
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\tikzstyle{every node}=[fill=electricblue]
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\node[edge_proba] (pi3) at (10,5.7) {\textbf{$\pi_{{\color{electricblue}\bullet}}$}};
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\node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
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\tikzstyle{every node}=[fill=burntorange, shape=rectangle]
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\node[edge_proba] (pi3) at (0.5,-0.7) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
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\tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
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\node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}};
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\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
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\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
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\node[edge_proba] (pi3) at (4,-0.7) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
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\node[state, draw=black!50] (B3) at (3.5,0) {\textbf{C21}};
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\node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
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\tikzstyle{every node}=[fill=peach, shape=rectangle]
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\node[edge_proba] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}};
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\node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}};
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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 (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);
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\path (R11) edge (B2);
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\path (R11) edge (B3);
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\path (R11) edge (B4);
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\path (R12) edge [] (B1);
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\path (R12) edge (B2);
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\path (R12) edge (B3);
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\path (R12) edge (B4);
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\path (R13) edge [] (B1);
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\path (R13) edge (B2);
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\path (R13) edge (B3);
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\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);
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\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);
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\path (R21) edge (B4);
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\path (R21) edge (B5);
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\path (R22) edge (B3);
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\path (R22) edge (B4);
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\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);
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\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);
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\end{tikzpicture}
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\caption{Exemple de LBM}
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\label{fig:LBMvisu}
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\end{figure}
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\end{column}
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\begin{column}{0.5\linewidth}
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\begin{block}{Paramètres}
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\begin{itemize}
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\item $\mathcal{K}_1 = \{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}$ blocs en ligne
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\item $\mathcal{K}_2 = \{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}$ blocs en colonne
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\item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(j\in\bullet)$ en colonne
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\item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{blueind}\bullet}, j \in {\color{burntorange}\bullet})$
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\end{itemize}
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\end{block}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}
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\frametitle{\emph{colSBM}}
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Le modèle \emph{colSBM} \parencite{chabert-liddellLearningCommonStructures2023}.\\
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\smallskip
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\definecolor{yellow}{RGB}{255,190,60}
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\begin{tikzpicture}[scale=.33]
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\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
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\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
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\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
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\tikzstyle{every node}=[fill=yellow]
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\node[state, draw=black!50] (A1) at (0,2) {\textbf{A1}};
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\node[state, draw=black!50] (A2) at (1.5, 2) {\textbf{A2}};
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\node[state, draw=black!50] (A3) at (0.75,3.25) {\textbf{A3}};
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\tikzstyle{every node}=[fill=blueind]
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\node[state, draw=black!50] (B1) at (4.5,3) {\textbf{B1}};
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\node[state, draw=black!50] (B2) at (4,4.75) {\textbf{B2}};
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\node[state, draw=black!50] (B3) at (5.5,6) {\textbf{B3}};
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\node[state, draw=black!50] (B4) at (7,4.75) {\textbf{B4}};
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\node[state, draw=black!50] (B5) at (6.5,3) {\textbf{B5}};
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\tikzstyle{every node}=[fill=greenind]
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\node[state, draw=black!50] (C1) at (5,0) {\textbf{C1}};
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\node[state, draw=black!50] (C2) at (7,1) {\textbf{C2}};
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\path (A1) edge[bend right] (A2);
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\path (A1) edge node[midway, left, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{yellow}\bullet}}$} (A3);
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\path (A3) edge (A2);
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\path (A3) edge node[midway, above, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{blueind}\bullet}}$} (B3);
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\path (B1) edge (B2);
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\path (B2) edge (B3);
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\path (B3) edge (B4);
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\path (B4) edge (B5);
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\path (B5) edge (B1);
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\path (B1) edge[bend left=0] (B4);
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\path (B5) edge[bend left=0] (B2);
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\path (A2) edge[bend right] node[midway, below, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{greenind}\bullet}}$} (C1);
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\path (C1) edge[bend right] node[midway, below, fill=none] {$\alpha_{{\color{greenind}\bullet}{\color{greenind}\bullet}}$} (C2);
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\path (C2) edge[bend right] node[midway, right, fill=none] {$\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}}$} (B4);
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\node[font=\small, text justified,draw=none, fill=none] at (4.5,-1.5) {SBM};
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\node[font=\small, text justified, fill=none] at (11.5, 1.5) {$\Longrightarrow$};
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% Sampled network
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\begin{scope}[xshift=14.5cm, yshift=4cm]
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\tikzstyle{every node}=[fill=gray, scale=0.95]
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\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
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\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
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\node[state, draw=black!50] (A1) at (0,0) {\textbf{10}};
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\node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
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\node[state, draw=black!50] (A3) at (0.5,1) {\textbf{5}};
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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] (B2) at (2,2.75) {\textbf{9}};
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\node[state, draw=black!50] (B3) at (3.5,4) {\textbf{6}};
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\node[state, draw=black!50] (B4) at (5,2.75) {\textbf{3}};
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\node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
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\node[state, draw=black!50] (C1) at (3,-0.5) {\textbf{4}};
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\node[state, draw=black!50] (C2) at (5,0) {\textbf{8}};
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\path (A1) edge[bend right] (A2);
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\path (A1) edge (A3);
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\path (A3) edge (A2);
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\path (A3) edge (B3);
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\path (B1) edge (B2);
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\path (B2) edge (B3);
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\path (B3) edge (B4);
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\path (B4) edge (B5);
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\path (B5) edge (B1);
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\path (B1) edge[bend left=0] (B4);
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\path (B5) edge[bend left=0] (B2);
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\path (A2) edge[bend right] (C1);
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\path (C1) edge[bend right] (C2);
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\path (C2) edge[bend right] (B4);
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\node[text width=3cm,font=\small, text justified, rotate=90, fill=none, below = -0.8cm of C1] (dots) {\dots};
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\draw [decorate, decoration = {brace}] (6, 4) -- (6,-8.5);
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\node[text width=3cm, font=\small, text justified, fill=none] at (11.5,-2.25) {$M$ réalisations indépendantes du SBM};
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\end{scope}
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\begin{scope}[xshift=14.5cm, yshift=-4cm]
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\tikzstyle{every node}=[fill=gray, scale=0.95]
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\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
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\tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
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\node[state, draw=black!50] (A1) at (0,0) {\textbf{9}};
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\node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
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\node[state, draw=black!50] (A3) at (0.5,1) {\textbf{1}};
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\node[state, draw=black!50] (B1) at (2.5,1) {\textbf{5}};
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\node[state, draw=black!50] (B2) at (2,2.75) {\textbf{10}};
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\node[state, draw=black!50] (B3) at (3.5,4) {\textbf{4}};
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\node[state, draw=black!50] (B4) at (5,2.75) {\textbf{8}};
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\node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
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\node[state, draw=black!50] (C1) at (3,-0.5) {\textbf{6}};
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\node[state, draw=black!50] (C2) at (5,0) {\textbf{3}};
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\path (A1) edge[bend right] (A2);
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\path (A1) edge (A3);
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\path (A3) edge (A2);
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\path (A3) edge (B3);
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\path (B1) edge (B2);
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\path (B2) edge (B3);
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\path (B3) edge (B4);
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\path (B4) edge (B5);
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\path (B5) edge (B1);
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\path (B1) edge[bend left=0] (B4);
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\path (B5) edge[bend left=0] (B2);
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\path (A2) edge[bend right] (C1);
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\path (C1) edge[bend right] (C2);
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\path (C2) edge[bend right] (B4);
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\end{scope}
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\end{tikzpicture}
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\begin{columns}
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\begin{column}{0.48\linewidth}
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\begin{block}{Paramètres}
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\begin{itemize}
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\item $\mathcal{K} = \{{\color{yellow}\bullet},{\color{blueind}\bullet},{\color{greenind}\bullet}\}$ blocs
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\item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$
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\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]
|
|
\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[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}
|
|
\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}[noframenumbering,plain,allowframebreaks]
|
|
\frametitle{Bibliographie}
|
|
\hfill
|
|
\begin{minipage}{0.9\textwidth}
|
|
\printbibliography
|
|
\end{minipage}
|
|
\end{frame}
|
|
|
|
\end{document} |