817 lines
No EOL
40 KiB
TeX
817 lines
No EOL
40 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, stmaryrd} % 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{graphicx}
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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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% Beamer
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\setbeamertemplate{headline}{%
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\begin{beamercolorbox}[ht=2.25ex,dp=3.75ex]{section in head/foot}
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\insertnavigation{\paperwidth}
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\end{beamercolorbox}%
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}%
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\beamertemplatenavigationsymbolsempty % Pas de bar de navigation
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\setbeamerfont{caption}{size=\scriptsize} % Petit titre de figures
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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}{255,255,0}
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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=black,scale=0.95, transform shape]
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\tikzstyle{every state}=[draw=none,text=black,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=black,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\footnotemark[2])}
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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=black,scale=0.95, transform shape]
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\tikzstyle{every state}=[draw=none,text=black,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=black,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\footnotemark[2]}
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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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Pour \begin{itemize}
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\item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne
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\item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne
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\end{itemize}
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\begin{block}{Paramètres}
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\begin{itemize}
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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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\footnotetext[2]{Que j'appellerai par la suite BiSBM}
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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=.28]
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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=black,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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% Sampled network
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\begin{scope}[xshift=18.5cm, yshift=4cm]
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\node[font=\small, text justified, fill=none] at (-4, -2.5) {$\backsim$};
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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=black,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=18.5cm, 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=black,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);
|
|
\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}
|
|
|
|
Pour $Q = |\{{\color{yellow}\bullet},{\color{blueind}\bullet},{\color{greenind}\bullet}\}|$ blocs fixés :
|
|
\begin{block}{Paramètres}
|
|
\begin{itemize}
|
|
\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{frame}
|
|
\section{Extension de \emph{colSBM} aux réseaux bipartites}
|
|
\begin{frame}
|
|
\frametitle{Collections bipartites}
|
|
\begin{tikzpicture}[scale=.33]
|
|
\begin{scope}[xshift=-3cm, yshift=2cm]
|
|
\tikzstyle{every state}=[draw=none, text=black,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,-1) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
|
|
\tikzstyle{every state}=[draw=none,text=black,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,-1) {\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,-1) {\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 (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 (R21) edge (B4);
|
|
\path (R21) edge (B5);
|
|
|
|
\path (R22) edge (B3);
|
|
\path (R22) edge (B4);
|
|
|
|
\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 (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, anchor=center, fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3);
|
|
\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{scope}
|
|
|
|
\begin{scope}[xshift = 16cm, yshift = 1cm]
|
|
\node[text justified, fill=none] at (-3, 3.5) {$\backsim$};
|
|
\node[text width=2.5cm, font=\small, text justified, fill=none] at (10,3.75) {$M$ réalisations indépendantes du BiSBM};
|
|
\draw [decorate, decoration = {brace}] (5.5, 7.6) -- (5.5,-0.4);
|
|
\begin{scope}[yshift = 6cm]
|
|
\tikzstyle{every state}=[draw, text=black,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=black,scale=0.75, transform shape, shape=rectangle]
|
|
\node[state, draw=black!50] (B1) at (0.5,-1) {\textbf{1}};
|
|
\node[state, draw=black!50] (B2) at (1.5,-1) {\textbf{2}};
|
|
|
|
\node[state, draw=black!50] (B31) at (2.5,-1) {\textbf{3}};
|
|
\node[state, draw=black!50] (B4) at (3.5,-1) {\textbf{4}};
|
|
|
|
\node[state, draw=black!50] (B5) at (4.5,-1) {\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, 7.5){\dots};
|
|
|
|
\begin{scope}[yshift = 0cm]
|
|
\tikzstyle{every state}=[draw, text=black,scale=0.75, transform shape]
|
|
|
|
\tikzstyle{every node}=[fill=gray]
|
|
\node[state, draw=black!50] (R11) at (0,2.25) {\textbf{4}};
|
|
\node[state, draw=black!50] (R12) at (1,2.25) {\textbf{1}};
|
|
\node[state, draw=black!50] (R13) at (2,2.25) {\textbf{6}};
|
|
\node[state, draw=black!50] (R21) at (3,2.25) {\textbf{3}};
|
|
\node[state, draw=black!50] (R22) at (4,2.25) {\textbf{5}};
|
|
\node[state, draw=black!50] (R31) at (5,2.25) {\textbf{2}};
|
|
|
|
\tikzstyle{every state}=[draw=none,text=black,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{scope}
|
|
\end{tikzpicture}
|
|
Pour
|
|
\begin{itemize}
|
|
\item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne
|
|
\item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne
|
|
\end{itemize}
|
|
\begin{block}{Paramètres}
|
|
\begin{itemize}
|
|
\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{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Différents modèles I}
|
|
\begin{block}{\emph{iid-colBiSBM}}
|
|
$\bm{\pi} = (\pi_1, \dots \pi_{Q_1-1})$ et $\bm{\rho} = (\rho_1, \dots \rho_{Q_2-1})$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
|
|
, tous les réseaux partagent les mêmes paramètres\footnotemark[3]
|
|
\end{block}
|
|
\begin{block}{\emph{$\pi$-colBiSBM}}
|
|
$\bm{\pi} = ((\pi_1^m, \dots \pi_{Q_1-1}^m))_{m=1,\dots M}$ et $\bm{\rho} = (\rho_1, \dots \rho_{Q_2-1})$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
|
|
avec $\forall q,m \in \llbracket 1, Q_1-1 \rrbracket \times \llbracket 1, M \rrbracket, \pi_q^m \in \left[ 0,1 \right] $
|
|
\end{block}
|
|
\footnotetext[3]{Dans tous les modèles la structure de connectivité est supposée identique au sein de la collection.}
|
|
\end{frame}
|
|
\begin{frame}
|
|
\frametitle{Différents modèles II}
|
|
\begin{block}{\emph{$\rho$-colBiSBM}}
|
|
$\bm{\pi} = (\pi_1, \dots \pi_{Q_1-1})$ et $\bm{\rho} = ((\rho_1^m, \dots \rho_{Q_2-1}^m))_{m=1,\dots M}$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
|
|
avec $\forall r,m \in \llbracket 1, Q_2-1 \rrbracket \times \llbracket 1, M \rrbracket, \rho_r^m \in \left[ 0,1 \right] $
|
|
\end{block}
|
|
\begin{block}{\emph{$\pi\rho$-colBiSBM}}
|
|
$\bm{\pi} = ((\pi_1^m, \dots \pi_{Q_1-1}^m))_{m=1,\dots M}$ et $\bm{\rho} = ((\rho_1^m, \dots \rho_{Q_2-1}^m))_{m=1,\dots M}$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
|
|
avec $\forall q,m \in \llbracket 1, Q_1-1 \rrbracket \times \llbracket 1, M \rrbracket, \pi_q^m \in \left[ 0,1 \right]$
|
|
et $\forall r,m \in \llbracket 1, Q_2-1 \rrbracket \times \llbracket 1, M \rrbracket, \rho_r^m \in \left[ 0,1 \right]$
|
|
\end{block}
|
|
\end{frame}
|
|
\begin{frame}
|
|
\frametitle{Borne inférieure de la vraisemblance}
|
|
Maximisation de la borne inférieure de la log-vraisemblance des données observées.
|
|
\begin{multline*}
|
|
\ell (\bm{X};\bm{\theta}) \geq \sum_{m=1}^{M} (\sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in \mathcal{Q}_{1,m}} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{1,m}_{i,q} \tau^{2,m}_{j,r} \log f(X^{m}_{ij}; \alpha_{qr}) \\
|
|
+ \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_{1,m}} \tau^{1,m}_{i,q} \log \pi_{q}^m + \sum_{j=1}^{n_2^m} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{2,m}_{j,r} \log \rho_{r}^m \\
|
|
\overbrace{- \sum_{i=1}^{n_1} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} }^{\text{entropie de la distribution}}) =: J(\bm{\tau};\bm{\theta}) $$
|
|
\end{multline*}
|
|
|
|
Le premier terme correspond à la log-vraisemblance complétée et marginalisée sur la famille des distributions factorisables.
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Parcours de la grille $(Q_1,Q_2)$ - Approche gloutonne}
|
|
Le VEM se fait à $Q_1, Q_2$ fixés, il faut donc déterminer les \enquote*{meilleurs} coordonnées.
|
|
Pour cela nous utilisons un BIC-L\footnote[4]{\emph{Bayesian Information Criterion - Like}} en adaptant les formules de \cite{chabert-liddellLearningCommonStructures2023}.
|
|
|
|
Détermination d'un premier mode par approche \emph{gloutonne} \smallskip
|
|
\begin{columns}
|
|
\begin{column}{0.5\linewidth}
|
|
\begin{tikzpicture}
|
|
\draw[step=1cm, help lines] (-2,-2) grid (2,2);
|
|
\draw[fill=gray, draw=gray] (0,0) circle [radius=0.225cm];
|
|
\draw[fill=red, draw=red] (1,0) circle [radius=0.225cm];
|
|
\draw[fill=red, draw=red] (0,1) circle [radius=0.225cm];
|
|
\draw[fill=blueind, draw=blueind] (-1,0) circle [radius=0.225cm];
|
|
\draw[fill=blueind, draw=blueind] (0,-1) circle [radius=0.225cm];
|
|
|
|
% Légende
|
|
\node[font=\tiny, text justified,fill=none, rotate=-45] (Splits) at (0.5,0.5){{\color{red} Splits}};
|
|
\node[font=\tiny, text justified,fill=none, rotate=-45] (Merges) at (-0.5,-0.5){{\color{blueind} Merges}};
|
|
|
|
% Splitting
|
|
\draw[>=stealth,->,thick, draw=red] (0.225,0) -- +(0.55,0);
|
|
\draw[>=stealth,->,thick, draw=red] (0,0.225) -- +(0,0.55);
|
|
|
|
% Merging
|
|
\draw[>=stealth,->,thick, draw=blueind] (-0.225,0) -- +(-0.55,0);
|
|
\draw[>=stealth,->,thick, draw=blueind] (0,-0.225) -- +(0,-0.55);
|
|
|
|
% Axes
|
|
\draw[>=to,->,thick] (-2,-2) -- +(1,0);
|
|
\node[font=\small, fill=none] (Q_1) at (-0.75,-2) {$Q_1$};
|
|
\draw[>=to,->,thick] (-2,-2) -- +(0,1);
|
|
\node[font=\small, fill=none] (Q_2) at (-2,-0.75) {$Q_2$};
|
|
|
|
\end{tikzpicture}
|
|
\end{column}
|
|
\begin{column}{0.5\linewidth}
|
|
\begin{block}{Exploration gloutonne}
|
|
Pendant cette phase, après l'initialisation, pour chaque position $Q_1,Q_2$ nous calculons tous les modèles possible depuis le point courant.
|
|
Le meilleur est alors celui avec le plus haut BIC-L et nous recommençons depuis ce point.
|
|
\end{block}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
\begin{frame}
|
|
\frametitle{Parcours de la grille $(Q_1,Q_2)$ - Fenêtre glissante}
|
|
\begin{columns}
|
|
\begin{column}{0.60\linewidth}
|
|
\begin{figure}
|
|
\includegraphics[scale=0.22]{img/moving_window.png}
|
|
\caption{Exemple de parcours de fenêtre glissante}
|
|
\end{figure}
|
|
\end{column}
|
|
\begin{column}{0.4\linewidth}
|
|
\definecolor{mypurple}{RGB}{128,0,128}
|
|
\begin{tikzpicture}
|
|
|
|
\tikzstyle{model}=[circle,draw=none,fill=gray]
|
|
\tikzstyle{split}=[>=stealth,->,thick, draw=blueind]
|
|
\tikzstyle{merge}=[>=stealth,->,thick, draw=red]
|
|
\draw[step=1cm, help lines] (-2,-2) grid (2,2);
|
|
\node[model] (mode) at (0,0) {{\color{red}X}};
|
|
|
|
\onslide<2->{
|
|
\draw[color=red, line width=1pt] (-1.5,-1.5) rectangle ++(3,3);
|
|
}
|
|
\onslide<3-3>{
|
|
\node[model] (bottom_left) at (-1,-1) {};
|
|
\node[model, opacity=0.6] (row_1) at (0,-1) {};
|
|
\node[model, opacity=0.6] (col_1) at (-1,0) {};
|
|
|
|
\draw[split] (bottom_left) -- (col_1);
|
|
\draw[split] (-1.75,0) -- (col_1);
|
|
\draw[split] (bottom_left) -- (row_1);
|
|
\draw[split] (0,-1.75) -- (row_1);
|
|
}
|
|
\onslide<4->{
|
|
\node[model] (bottom_left) at (-1,-1) {};
|
|
\node[model, draw=blue] (row_1) at (0,-1) {};
|
|
\node[model, draw=blue] (col_1) at (-1,0) {};
|
|
}
|
|
\onslide<4-4>{
|
|
\node[model, opacity=0.6] (row_2) at (1,-1) {};
|
|
\node[model, opacity=0.6] (col_2) at (-1,1) {};
|
|
|
|
\draw[split] (col_1) -- (col_2);
|
|
\draw[split] (-1.75,1) -- (col_2);
|
|
\draw[split] (row_1) -- (row_2);
|
|
\draw[split] (1,-1.75) -- (row_2);
|
|
\draw[split] (row_1) -- (mode);
|
|
\draw[split] (col_1) -- (mode);
|
|
}
|
|
\onslide<5->{
|
|
\node[model, draw=blue] (row_2) at (1,-1) {};
|
|
\node[model, draw=blue] (col_2) at (-1,1) {};
|
|
\node[model, draw=blue] (mode) at (0,0) {{\color{red}X}};
|
|
}
|
|
\onslide<5-5>{
|
|
\node[model, opacity=0.6] (row_3) at (1,0) {};
|
|
\node[model, opacity=0.6] (col_3) at (0,1) {};
|
|
|
|
\draw[split] (col_2) -- (col_3);
|
|
\draw[split] (row_2) -- (row_3);
|
|
\draw[split] (mode) -- (row_3);
|
|
\draw[split] (mode) -- (col_3);
|
|
}
|
|
\onslide<6->{
|
|
\node[model, draw=blue] (row_3) at (1,0) {};
|
|
\node[model, draw=blue] (col_3) at (0,1) {};
|
|
}
|
|
\onslide<6-6>{
|
|
\node[model, opacity=0.6] (top_right) at (1,1) {};
|
|
\draw[split] (col_3) -- (top_right);
|
|
\draw[split] (row_3) -- (top_right);
|
|
}
|
|
\onslide<7->{
|
|
\node[model, draw=blue] (top_right) at (1,1) {};
|
|
}
|
|
\onslide<8->{
|
|
\node[model, draw=mypurple] (top_right) at (1,1) {};
|
|
\node[model, draw=mypurple] (row_3) at (1,0) {};
|
|
\node[model, draw=mypurple] (col_3) at (0,1) {};
|
|
\node[model, draw=mypurple] (row_2) at (1,-1) {};
|
|
\node[model, draw=mypurple] (col_2) at (-1,1) {};
|
|
\node[model, draw=mypurple] (mode) at (0,0) {{\color{red}X}};
|
|
\node[model, draw=red] (bottom_left) at (-1,-1) {};
|
|
\node[model, draw=mypurple] (row_1) at (0,-1) {};
|
|
\node[model, draw=mypurple] (col_1) at (-1,0) {};
|
|
|
|
\draw[merge] (1,1.75) -- (top_right);
|
|
\draw[merge] (1.75,1) -- (top_right);
|
|
\draw[merge] (0,1.75) -- (col_3);
|
|
\draw[merge] (1.75,0) -- (row_3);
|
|
\draw[merge] (1.75,-1) -- (row_2);
|
|
\draw[merge] (-1,1.75) -- (col_2);
|
|
|
|
\draw[merge] (top_right) -- (col_3);
|
|
\draw[merge] (top_right) -- (row_3);
|
|
\draw[merge] (col_3) -- (col_2);
|
|
\draw[merge] (row_3) -- (row_2) ;
|
|
\draw[merge] (row_3) -- (mode);
|
|
\draw[merge] (col_3) -- (mode);
|
|
\draw[merge] (col_2) --(col_1);
|
|
\draw[merge] (row_2) -- (row_1);
|
|
\draw[merge] (mode) -- (row_1);
|
|
\draw[merge] (mode) -- (col_1);
|
|
\draw[merge] (col_1) -- (bottom_left);
|
|
\draw[merge] (row_1) -- (bottom_left);
|
|
}
|
|
\end{tikzpicture}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Clustering de réseaux}
|
|
\begin{columns}
|
|
\begin{column}{0.2\linewidth}
|
|
\begin{block}{Objectif}
|
|
Déterminer une partition qui maximise la somme du BICL de ses sous-collections.
|
|
\end{block}
|
|
\end{column}
|
|
\begin{column}{0.78\linewidth}
|
|
\begin{tikzpicture}
|
|
\tikzstyle{instruct}=[font=\small, text justified, rectangle,draw,fill=yellow!50]
|
|
\tikzstyle{first_col}=[rectangle, text justified, draw,fill=gray!50]
|
|
\tikzstyle{second_col}=[scale=0.55, circle, draw,fill=red!50]
|
|
\tikzstyle{test}=[font=\small, text justified, diamond, aspect=2.5,thick,
|
|
draw=blue,fill=yellow!50,text=blue]
|
|
\tikzstyle{es}=[font=\small, text justified, rectangle,draw,rounded corners=4pt,fill=cyanind!25]
|
|
|
|
\node[es] (liste) at (0,4) {Entrer la liste de tous les réseaux à partitionner};
|
|
\node[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Calculer les paramètres de la collection};
|
|
\node[first_col, right = 0.5cm of 1-collection] (1-col-obj) {};
|
|
\node[instruct, text width=5cm, below = 0.45cm of 1-collection] (dissimi) {Calculer la matrice de dissimilarité de la collection};
|
|
\node[instruct, text width=5cm, below = 0.45cm of dissimi] (2-sous-collection) {Séparer la \emph{collection en 2 sous-collections} et calculer leurs paramètres};
|
|
\node[second_col, right = 0.25cm of 2-sous-collection] (1-sec-col-obj) {1};
|
|
\node[second_col, right = 0.25cm of 1-sec-col-obj] (1-sec-col-obj) {2};
|
|
\node[test,below = 0.45cm of 2-sous-collection, scale=0.5] (BICL-test) {$\sum_{i=1}^{2} (\text{BIC-L}(\tikz[baseline=-0.25cm]{\node[second_col] {i};} )) > \text{BIC-L}(\tikz[baseline=-0.25cm]{\node[first_col] {};})$?};
|
|
\node[es, right = 0.55cm of BICL-test] (sortie) {Renvoyer \tikz{\node[rectangle, draw, fill=gray!50, rounded corners=0pt] {};}};
|
|
\node[es, left = 0.45cm of dissimi, text width = 2cm] (recursion) {Recommencer sur \tikz{\node[second_col] {1};} et \tikz{\node[second_col] {2};} };
|
|
|
|
\tikzstyle{suite}=[->,>=stealth,thick,rounded corners=4pt]
|
|
\draw[suite] (liste) -- (1-collection);
|
|
\draw[suite] (1-collection) -- (dissimi);
|
|
\draw[suite] (dissimi) -- (2-sous-collection);
|
|
\draw[suite] (2-sous-collection) -- (BICL-test);
|
|
\draw[suite] (BICL-test) -| node[near start, above, fill=none] {Oui} (recursion);
|
|
\draw[suite] (recursion.north) |- (1-collection.west);
|
|
\draw[suite] (BICL-test) -- node[near start, above, fill=none] {Non} (sortie);
|
|
|
|
\end{tikzpicture}
|
|
\end{column}
|
|
\end{columns}
|
|
\let\thefootnote\relax\footnote{{Même approche que \cite{chabert-liddellLearningCommonStructures2023}}}
|
|
\end{frame}
|
|
|
|
\section{Application}
|
|
\begin{frame}
|
|
\frametitle{Application, données plantes pollinisateurs}
|
|
Voici des résultats du modèles \emph{iid-colBiSBM} sur des données
|
|
plantes-pollinisateurs (\cite{doreRelativeEffectsAnthropogenic2021}
|
|
et \cite{thebaultDatabasePlantpollinatorNetworks2020})
|
|
\begin{columns}
|
|
\begin{column}{0.48\linewidth}
|
|
\includegraphics[scale=0.32]{img/annual_time_span_vs_iid.png}
|
|
\end{column}
|
|
\begin{column}{0.48\linewidth}
|
|
\begin{figure}[H]
|
|
\includegraphics[width=0.45\textwidth]{img/iid-meso-1.png}
|
|
\includegraphics[width=0.45\textwidth]{img/iid-meso-2.png}
|
|
\includegraphics[width=0.45\textwidth]{img/iid-meso-3.png}
|
|
\includegraphics[width=0.45\textwidth]{img/iid-meso-4.png}
|
|
\includegraphics[width=0.45\textwidth]{img/iid-meso-5.png}
|
|
\caption{Connectivités de la partition}
|
|
\end{figure}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}[noframenumbering,plain,allowframebreaks]
|
|
\frametitle{Bibliographie}
|
|
\hfill
|
|
\begin{minipage}{0.9\textwidth}
|
|
\printbibliography
|
|
\end{minipage}
|
|
\end{frame}
|
|
|
|
\end{document} |