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Adding final version of the presentation

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Louis Lacoste 2023-06-26 21:50:48 +02:00
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\usepackage{url} % pour une gestion efficace des url \usepackage{url} % pour une gestion efficace des url
\usepackage{hyperref} % pour les hyperliens dans le document \usepackage{hyperref} % pour les hyperliens dans le document
\usepackage{tikz} % For graph plots \usepackage{tikz} % For graph plots
\usepackage{adjustbox} % To resize tikzpictures
\usepackage{fontawesome5}
\usepackage{makecell}
% Beamer % Beamer
\setbeamertemplate{headline}{% \setbeamertemplate{headline}{%
@ -64,10 +67,21 @@
\definecolor{gray}{RGB}{128,128,128} \definecolor{gray}{RGB}{128,128,128}
% Footnote
\makeatletter
\newcommand\blfootnote[1]{%
\begingroup
\renewcommand{\@makefntext}[1]{\noindent\makebox[1.8em][r]#1}
\renewcommand\thefootnote{}\footnote{#1}%
\addtocounter{footnote}{-1}%
\endgroup
}
\makeatother
\title{Séminaire des stagiaires}
\subtitle{Adaptation de colSBM aux réseaux bipartites} \subtitle{Séminaire des stagiaires}
\author[L. Lacoste]{Louis \textsc{Lacoste}} \title[Collections de réseaux bipartites]{Détection de structure dans des réseaux bipartites}
\author[L. Lacoste]{Louis \textsc{Lacoste}} % Sous la supervision de Pierre
\date{29 juin 2023} \date{29 juin 2023}
\begin{document} \begin{document}
@ -78,19 +92,20 @@
\end{frame} \end{frame}
\section{Contexte du modèle} \section{Contexte du modèle}
\label{sec:contexte-du-modele}
\begin{frame} \begin{frame}
\frametitle{Contexte écologique} \frametitle{Contexte écologique}
\begin{itemize} \begin{itemize}
\item Faire de la détection de structure sur un réseau (SBM, LBM) mais intérêt à le faire sur plusieurs
\item De nombreux réseaux disponibles \parencite{WebLifeEcological} et décrivant des interactions similaires \item De nombreux réseaux disponibles \parencite{WebLifeEcological} et décrivant des interactions similaires
\item Re-grouper les réseaux selon leur similarité (\emph{clustering} de réseaux) \item Re-grouper les réseaux selon leur similarité (\emph{clustering} de réseaux)
\item Compléter d'éventuelles informations manquantes grâce à la collection \item Transférer de l'information grâce à la collection (par exemple reconstitution de données manquantes)
\item Déterminer des structures d'interactions fines de manière agnostique \item Déterminer des structures d'interactions fines de manière agnostique % Pas d'idee preco
\item Vérifier si le regroupement est lié à des co-variables %\item Vérifier si le regroupement est lié à des co-variables
\end{itemize} \end{itemize}
\footnotetext[0]{Pour combler les lacunes de\\\cite{chabert-liddellLearningCommonStructures2023}}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Réseaux bipartites\footnote{Ou \emph{bipartis}. Voir \cite{larousseDefinitionsBipartiBipartite}.}} \frametitle{Réseaux bipartites\footnote{Ou \emph{bipartis}. Voir~\cite{larousseDefinitionsBipartiBipartite}.}}
\begin{columns}[c] \begin{columns}[c]
\begin{column}{0.48\textwidth} \begin{column}{0.48\textwidth}
\centering \centering
@ -126,7 +141,7 @@
\begin{column}{0.48\linewidth} \begin{column}{0.48\linewidth}
Matrice d'incidence Matrice d'incidence
\smallskip \smallskip
$B=\left( $X=\left(
\begin{array}{rrrrr} \begin{array}{rrrrr}
1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\
@ -142,12 +157,13 @@
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Latent Block Model (LBM\footnotemark[2])} \frametitle{Latent Block Model (LBM\footnotemark[2])}
Proposé par \cite{govaertEMAlgorithmBlock2005}. %DONE remplacer i \in bullet par Zi = \bullet
Proposé par~\cite{govaertEMAlgorithmBlock2005}.
\begin{columns} \begin{columns}
\begin{column}{0.5\linewidth} \begin{column}{0.40\linewidth}
\begin{figure}[H] \begin{figure}[H]
\center \center
\begin{tikzpicture}[scale=.45] \begin{tikzpicture}[scale=0.35]
\tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape] \tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
\tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape] \tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape]
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}} \tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
@ -208,334 +224,308 @@
\path (R31) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5); \path (R31) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5);
\end{tikzpicture} \end{tikzpicture}
\caption{Exemple de LBM\footnotemark[2]} \caption{Exemple de LBM\footnotemark}
\label{fig:LBMvisu} \label{fig:LBMvisu}
\end{figure} \end{figure}
\end{column} \end{column}
\begin{column}{0.5\linewidth} \begin{column}{0.51\linewidth}
Pour \begin{itemize} Pour \begin{itemize}
\item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne \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 \item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne
\end{itemize} \end{itemize}
\begin{block}{Paramètres} \begin{block}{Paramètres}
\begin{itemize} \begin{itemize}
\item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(j\in\bullet)$ en colonne \item $\pi_{\bullet} = \mathbb{P}(Z_i = \bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne
\item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{blueind}\bullet}, j \in {\color{burntorange}\bullet})$ \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$
\end{itemize} \end{itemize}
\end{block} \end{block}
\end{column} \end{column}
\end{columns} \end{columns}
\footnotetext[2]{Que j'appellerai par la suite BiSBM} \footnotetext{Que j'appellerai par la suite BiSBM}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{\emph{colSBM}} \frametitle{\emph{colSBM}}
Le modèle \emph{colSBM} \parencite{chabert-liddellLearningCommonStructures2023}.\\ Le modèle \emph{colSBM} \parencite{chabert-liddellLearningCommonStructures2023}.\\
% Difficulté estimer les parametres
% DONE Modifier les realisations pour variabilite, mettre iid au dessus du sim et inverser modele et realisations
\smallskip \smallskip
\definecolor{yellow}{RGB}{255,190,60} \definecolor{yellow}{RGB}{255,190,60}
\begin{tikzpicture}[scale=.28] \begin{center}
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left] \begin{adjustbox}{trim=0 0 0 1cm}
\tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape] \begin{tikzpicture}[scale=.32]
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}} \tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
\tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
\tikzstyle{every node}=[fill=yellow] \tikzstyle{every node}=[fill=yellow]
\node[state, draw=black!50] (A1) at (0,2) {\textbf{A1}}; \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] (A2) at (1.5, 2) {\textbf{A2}};
\node[state, draw=black!50] (A3) at (0.75,3.25) {\textbf{A3}}; \node[state, draw=black!50] (A3) at (0.75,3.25) {\textbf{A3}};
\tikzstyle{every node}=[fill=blueind] \tikzstyle{every node}=[fill=blueind]
\node[state, draw=black!50] (B1) at (4.5,3) {\textbf{B1}}; \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] (B2) at (4,4.75) {\textbf{B2}};
\node[state, draw=black!50] (B3) at (5.5,6) {\textbf{B3}}; \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] (B4) at (7,4.75) {\textbf{B4}};
\node[state, draw=black!50] (B5) at (6.5,3) {\textbf{B5}}; \node[state, draw=black!50] (B5) at (6.5,3) {\textbf{B5}};
\tikzstyle{every node}=[fill=greenind] \tikzstyle{every node}=[fill=greenind]
\node[state, draw=black!50] (C1) at (5,0) {\textbf{C1}}; \node[state, draw=black!50] (C1) at (5,0) {\textbf{C1}};
\node[state, draw=black!50] (C2) at (7,1) {\textbf{C2}}; \node[state, draw=black!50] (C2) at (7,1) {\textbf{C2}};
\path (A1) edge[bend right] (A2); \path (A1) edge[bend right] (A2);
\path (A1) edge node[midway, left, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{yellow}\bullet}}$} (A3); \path (A1) edge node[midway, left, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{yellow}\bullet}}$} (A3);
\path (A3) edge (A2); \path (A3) edge (A2);
\path (A3) edge node[midway, above, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{blueind}\bullet}}$} (B3); \path (A3) edge node[midway, above, fill=none] {$\alpha_{{\color{yellow}\bullet}{\color{blueind}\bullet}}$} (B3);
\path (B1) edge (B2); \path (B1) edge (B2);
\path (B2) edge (B3); \path (B2) edge (B3);
\path (B3) edge (B4); \path (B3) edge (B4);
\path (B4) edge (B5); \path (B4) edge (B5);
\path (B5) edge (B1); \path (B5) edge (B1);
\path (B1) edge[bend left=0] (B4); \path (B1) edge[bend left=0] (B4);
\path (B5) edge[bend left=0] (B2); \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 (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 (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); \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,draw=none, fill=none] at (4.5,-1.5) {SBM};
% Sampled network % Sampled network
\begin{scope}[xshift=18.5cm, yshift=4cm] \begin{scope}[xshift=-16cm,yshift=4cm]
\node[font=\small, text justified, fill=none] at (-4, -2.5) {$\backsim$}; \node[font=\small, text justified, fill=none] at (10, -2.5) {$\overset{iid}{\sim}$};
\tikzstyle{every node}=[fill=gray, scale=0.95] \tikzstyle{every node}=[fill=gray, scale=0.95]
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left] \tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
\tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape] \tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
\node[state, draw=black!50] (A1) at (0,0) {\textbf{10}}; \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] (A2) at (1, 0) {\textbf{2}};
\node[state, draw=black!50] (A3) at (0.5,1) {\textbf{5}}; \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] (B2) at (2,2.75) {\textbf{9}}; \node[state, draw=black!50] (B3) at (3.5,4) {\textbf{6}};
\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] (B4) at (5,2.75) {\textbf{3}}; \node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
\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] (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[bend right] (A2);
\path (A1) edge (A3); \path (A1) edge (A3);
\path (A3) edge (A2); \path (A3) edge (A2);
\path (A3) edge (B3); \path (A3) edge (B3);
\path (B1) edge (B2); \path (B2) edge (B3);
\path (B2) edge (B3); \path (B3) edge (B4);
\path (B3) edge (B4); \path (B4) edge (B5);
\path (B4) edge (B5);
\path (B5) edge (B1);
\path (B1) edge[bend left=0] (B4); \path (B5) edge[bend left=0] (B2);
\path (B5) edge[bend left=0] (B2);
\path (A2) edge[bend right] (C1); \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}; \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=-16cm,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=black,scale=0.95, transform shape]
\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] (B4) at (5,2.75) {\textbf{8}};
\node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
\node[state, draw=black!50] (C2) at (5,0) {\textbf{3}};
\end{scope}
\begin{scope}[xshift=18.5cm, 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=black,scale=0.95, transform shape]
\node[state, draw=black!50] (A1) at (0,0) {\textbf{9}}; \path (A3) edge (A2);
\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 (B1) edge (B2);
\path (A1) edge (A3); \path (B4) edge (B5);
\path (A3) edge (A2); \path (B5) edge (B1);
\path (A3) edge (B3); \path (B1) edge[bend left=0] (B4);
\path (B5) edge[bend left=0] (B2);
\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}
\path (C2) edge[bend right] (B4);
\end{scope}
\end{tikzpicture}
\end{adjustbox}
\end{center}
Pour $Q = |\{{\color{yellow}\bullet},{\color{blueind}\bullet},{\color{greenind}\bullet}\}|$ blocs fixés : Pour $Q = |\{{\color{yellow}\bullet},{\color{blueind}\bullet},{\color{greenind}\bullet}\}|$ blocs fixés :
\begin{block}{Paramètres} \begin{block}{Paramètres}
\begin{itemize} \begin{itemize}
\item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ \item $\pi_{\bullet} = \mathbb{P}(Z_i =\bullet)$
\item $\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{greenind}\bullet}, j \in {\color{blueind}\bullet})$ \item $\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{greenind}\bullet}, Z_j = {\color{blueind}\bullet})$
\end{itemize} \end{itemize}
\end{block} \end{block}
\end{frame} \end{frame}
\section{Extension de \emph{colSBM} aux réseaux bipartites} \section{Extension de \emph{colSBM} aux réseaux bipartites}
\label{sec:extension-de-colsbm-aux-reseaux-bipartites}
\begin{frame} \begin{frame}
\frametitle{Collections bipartites} \frametitle{Collections bipartites}
\begin{tikzpicture}[scale=.33] \begin{center}
\begin{scope}[xshift=-3cm, yshift=2cm] \begin{adjustbox}{trim=0 0 1 1.5cm}
\tikzstyle{every state}=[draw=none, text=black,scale=0.75, transform shape] \begin{tikzpicture}[scale=.33]
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}} \begin{scope}[xshift=18cm, 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] \tikzstyle{every node}=[fill=blueind]
\node[edge_proba] (pi1) at (1,5.7) {\textbf{$\pi_{{\color{blueind}\bullet}}$}}; \node[edge_proba] (pi1) at (1,5.7) {\textbf{$\pi_{{\color{blueind}\bullet}}$}};
\node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}}; \node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}};
\node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}}; \node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}};
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}}; \node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
\tikzstyle{every node}=[fill=cyanind] \tikzstyle{every node}=[fill=cyanind]
\node[edge_proba] (pi2) at (6.75,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}}; \node[edge_proba] (pi2) at (6.75,5.7) {\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
\node[state, draw=black!50] (R21) at (6.25,5) {\textbf{R21}}; \node[state, draw=black!50] (R21) at (6.25,5) {\textbf{R21}};
\node[state, draw=black!50] (R22) at (7.25,5) {\textbf{R22}}; \node[state, draw=black!50] (R22) at (7.25,5) {\textbf{R22}};
\tikzstyle{every node}=[fill=electricblue] \tikzstyle{every node}=[fill=electricblue]
\node[edge_proba] (pi3) at (10,5.7) {\textbf{$\pi_{{\color{electricblue}\bullet}}$}}; \node[edge_proba] (pi3) at (10,5.7) {\textbf{$\pi_{{\color{electricblue}\bullet}}$}};
\node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}}; \node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
\tikzstyle{every node}=[fill=burntorange, shape=rectangle] \tikzstyle{every node}=[fill=burntorange, shape=rectangle]
\node[edge_proba] (pi3) at (0.5,-1) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}}; \node[edge_proba] (rho1) at (0.5,-1) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
\tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle] \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] (B1) at (0,0) {\textbf{C11}};
\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}}; \node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle] \tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
\node[edge_proba] (pi3) at (4,-1) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}}; \node[edge_proba] (rho2) 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] (B3) at (3.5,0) {\textbf{C21}};
\node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}}; \node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
\tikzstyle{every node}=[fill=peach, shape=rectangle] \tikzstyle{every node}=[fill=peach, shape=rectangle]
\node[edge_proba] (pi3) at (10,-1) {\textbf{$\rho_{{\color{peach}\bullet}}$}}; \node[edge_proba] (rho3) at (10,-1) {\textbf{$\rho_{{\color{peach}\bullet}}$}};
\node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}}; \node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}};
\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt,draw opacity=0.2] \node[font=\small, text justified,draw=none, fill=none, below = 0.05cm of rho2] {BiSBM};
\path (R11) edge (B2); \tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt,draw opacity=0.2]
\path (R11) edge (B3);
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\path (R12) edge (B2); \path (R11) edge (B3);
\path (R12) edge (B3); \path (R11) edge (B4);
\path (R12) edge (B4);
\path (R13) edge [] (B1); \path (R12) edge [] (B1);
\path (R13) edge (B2); \path (R12) edge (B2);
\path (R13) edge (B3); \path (R12) edge (B3);
\path (R12) edge (B4);
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\path (R21) edge (B5); \path (R13) edge (B2);
\path (R13) edge (B3);
\path (R22) edge (B3); \path (R21) edge (B4);
\path (R22) edge (B4); \path (R21) edge (B5);
\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 (R22) edge (B3);
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\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] \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);
\node[text justified, fill=none] at (-3, 3.5) {$\backsim$}; \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);
\node[text width=2.5cm, font=\small, text justified, fill=none] at (10,3.75) {$M$ réalisations indépendantes du BiSBM}; \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);
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\tikzstyle{every state}=[draw, text=black,scale=0.75, transform shape] \end{scope}
\tikzstyle{every node}=[fill=gray] \begin{scope}[xshift=3cm, yshift = 1cm]
\node[state, draw=black!50] (R11) at (0,1.25) {\textbf{1}}; \node[text justified, fill=none] at (10, 3.5) {$\overset{iid}{\sim}$};
\node[state, draw=black!50] (R12) at (1,1.25) {\textbf{2}}; \begin{scope}[yshift = 6cm]
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\node[state, draw=black!50] (B4) at (3.5,-1) {\textbf{4}}; \node[state, draw=black!50] (B1) at (0.5,-1) {\textbf{1}};
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\end{scope}
\node[text width=3cm,font=\small, text justified, rotate=90, fill=none] (dots) at (2.5, 7.5){\dots};
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\tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle] \tikzstyle{every node}=[fill=gray]
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\path (R12) edge (B2); \path (R11) edge (B1);
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\end{scope}
\end{tikzpicture}
\end{adjustbox}
\end{center}
\path (R31) edge (B5);
\end{scope}
\end{scope}
\end{tikzpicture}
Pour Pour
\begin{itemize} \begin{itemize}
\item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne \item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne
@ -543,53 +533,52 @@
\end{itemize} \end{itemize}
\begin{block}{Paramètres} \begin{block}{Paramètres}
\begin{itemize} \begin{itemize}
\item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(j\in\bullet)$ en colonne \item $\pi_{\bullet} = \mathbb{P}(Z_i =\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne
\item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{blueind}\bullet}, j \in {\color{burntorange}\bullet})$ \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$
\end{itemize} \end{itemize}
\end{block} \end{block}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Différents modèles I} \frametitle{Différents modèles}
\begin{block}{\emph{iid-colBiSBM}} \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 $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$} $\bm{\pi} = (\pi_1, \dots \pi_{Q_1})$ et $\bm{\rho} = (\rho_1, \dots \rho_{Q_2})$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
, tous les réseaux partagent les mêmes paramètres\footnotemark[3] , tous les réseaux partagent les mêmes paramètres\footnotemark
\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} \end{block}
\begin{block}{\emph{$\pi\rho$-colBiSBM}} \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 $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$} $\bm{\pi} = ((\pi_{\color{black}1}^{\color{red}m}, \dots \pi_{\color{black}Q_1}^{\color{red}m}))_{m=1,\dots M}$ et $\bm{\rho} = ((\rho_{\color{black}1}^{\color{red}m}, \dots \rho_{\color{black}Q_2}^{\color{red}m}))_{m=1,\dots M}$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
avec $\forall q,m \in \llbracket 1, Q_1-1 \rrbracket \times \llbracket 1, M \rrbracket, \pi_q^m \in \left[ 0,1 \right]$ \small \\
et $\forall r,m \in \llbracket 1, Q_2-1 \rrbracket \times \llbracket 1, M \rrbracket, \rho_r^m \in \left[ 0,1 \right]$ avec $\forall q,m \in \llbracket 1, Q_1 \rrbracket \times \llbracket 1, M \rrbracket, \pi_q^m \in \left[ 0,1 \right]$
et $\forall r,m \in \llbracket 1, Q_2 \rrbracket \times \llbracket 1, M \rrbracket, \rho_r^m \in \left[ 0,1 \right]$
\end{block} \end{block}
Et également deux autres modèles ($\pi$-colBiSBM et $\rho$-colBiSBM) où seulement une des deux dimensions est libre.
\footnotetext{Dans tous les modèles la structure de connectivité est supposée identique au sein de la collection.}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Borne inférieure de la vraisemblance} \frametitle{Estimation des paramètres}
Maximisation de la borne inférieure de la log-vraisemblance des données observées. % DONE dire que tau i q m c' est la proba que Zim = q, approximation de la proba variationnelle. Parce qu on impose lindependance
Maximisation d'une borne inférieure de la log-vraisemblance des données observées.
\begin{multline*} \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}) \\ \ell (\bm{X};\bm{\theta}) \geq \color{red}\sum_{m=1}^{M} \bigg( \color{black} \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 \\ + \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_{1,m}} \tau^{1,m}_{i,q} \log \pi_{\color{black}q}^{\color{gray}m} + \sum_{j=1}^{n_2^m} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{2,m}_{j,r} \log \rho_{\color{black}r}^{\color{gray}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}) $$ - \sum_{i=1}^{n_1} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} \color{red}\bigg) \color{black} =: J(\bm{\tau};\bm{\theta}) $$
\end{multline*} \end{multline*}
Le premier terme correspond à la log-vraisemblance complétée et marginalisée sur la famille des distributions factorisables. \begin{block}{Approximation variationnelle}
$\tau_{i,q}^{1,m} = P(Z_i = q | X^m_{ij})$ et $\tau_{j,r}^{2,m} = P(W_j = r | X^m_{ij})$ tels que $P(Z_i = q, W_j = r | X^m_{ij}) = \tau_{i,q}^{1,m}\times\tau_{j,r}^{2,m}$
\end{block}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Parcours de la grille $(Q_1,Q_2)$ - Approche gloutonne} \frametitle{Sélection de modèle : choix de $(Q_1,Q_2)$ - Approche gloutonne}
Le VEM se fait à $Q_1, Q_2$ fixés, il faut donc déterminer les \enquote*{meilleurs} coordonnées. % DONE But maximiser un critere le BICL, deplacer voir St Clair dans la note
Pour cela nous utilisons un BIC-L\footnote[4]{\emph{Bayesian Information Criterion - Like}} en adaptant les formules de \cite{chabert-liddellLearningCommonStructures2023}. % VEM a Q1 Q2 fixer
% Choix de Q1 Q2 par maximisation du BICL
% Itemize dans la box : init, explo voisin, arrets
\underline{Le VEM se fait à $Q_1, Q_2$ fixés}, il faut donc déterminer les \enquote*{meilleures} coordonnées.
Nous maximisons un BIC-L\footnote{\emph{Bayesian Information Criterion - Like}, en adaptant les formules de~\cite{chabert-liddellLearningCommonStructures2023}}.
Détermination d'un premier mode par approche \emph{gloutonne} \smallskip Détermination d'un premier mode par approche \emph{gloutonne} \smallskip
\begin{columns} \begin{columns}
@ -624,14 +613,17 @@
\end{column} \end{column}
\begin{column}{0.5\linewidth} \begin{column}{0.5\linewidth}
\begin{block}{Exploration gloutonne} \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. \begin{itemize}
Le meilleur est alors celui avec le plus haut BIC-L et nous recommençons depuis ce point. \item Initialisation sur $(1,2)$ et $(2,1)$
\item Exploration des 4 voisins et déplacement sur le meilleur des 4
\item Arrêt après 2 étapes successives sans augmentation du BIC-L
\end{itemize}
\end{block} \end{block}
\end{column} \end{column}
\end{columns} \end{columns}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Parcours de la grille $(Q_1,Q_2)$ - Fenêtre glissante} \frametitle{Sélection de modèle : choix de $(Q_1,Q_2)$ - Fenêtre glissante}
\begin{columns} \begin{columns}
\begin{column}{0.60\linewidth} \begin{column}{0.60\linewidth}
\begin{figure} \begin{figure}
@ -705,36 +697,36 @@
\node[model, draw=blue] (top_right) at (1,1) {}; \node[model, draw=blue] (top_right) at (1,1) {};
} }
\onslide<8->{ \onslide<8->{
\node[model, draw=mypurple] (top_right) at (1,1) {}; \node[model, draw=mypurple] (top_right) at (1,1) {};
\node[model, draw=mypurple] (row_3) at (1,0) {}; \node[model, draw=mypurple] (row_3) at (1,0) {};
\node[model, draw=mypurple] (col_3) at (0,1) {}; \node[model, draw=mypurple] (col_3) at (0,1) {};
\node[model, draw=mypurple] (row_2) at (1,-1) {}; \node[model, draw=mypurple] (row_2) at (1,-1) {};
\node[model, draw=mypurple] (col_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=mypurple] (mode) at (0,0) {{\color{red}X}};
\node[model, draw=red] (bottom_left) at (-1,-1) {}; \node[model, draw=red] (bottom_left) at (-1,-1) {};
\node[model, draw=mypurple] (row_1) at (0,-1) {}; \node[model, draw=mypurple] (row_1) at (0,-1) {};
\node[model, draw=mypurple] (col_1) at (-1,0) {}; \node[model, draw=mypurple] (col_1) at (-1,0) {};
\draw[merge] (1,1.75) -- (top_right); \draw[merge] (1,1.75) -- (top_right);
\draw[merge] (1.75,1) -- (top_right); \draw[merge] (1.75,1) -- (top_right);
\draw[merge] (0,1.75) -- (col_3); \draw[merge] (0,1.75) -- (col_3);
\draw[merge] (1.75,0) -- (row_3); \draw[merge] (1.75,0) -- (row_3);
\draw[merge] (1.75,-1) -- (row_2); \draw[merge] (1.75,-1) -- (row_2);
\draw[merge] (-1,1.75) -- (col_2); \draw[merge] (-1,1.75) -- (col_2);
\draw[merge] (top_right) -- (col_3); \draw[merge] (top_right) -- (col_3);
\draw[merge] (top_right) -- (row_3); \draw[merge] (top_right) -- (row_3);
\draw[merge] (col_3) -- (col_2); \draw[merge] (col_3) -- (col_2);
\draw[merge] (row_3) -- (row_2) ; \draw[merge] (row_3) -- (row_2) ;
\draw[merge] (row_3) -- (mode); \draw[merge] (row_3) -- (mode);
\draw[merge] (col_3) -- (mode); \draw[merge] (col_3) -- (mode);
\draw[merge] (col_2) --(col_1); \draw[merge] (col_2) --(col_1);
\draw[merge] (row_2) -- (row_1); \draw[merge] (row_2) -- (row_1);
\draw[merge] (mode) -- (row_1); \draw[merge] (mode) -- (row_1);
\draw[merge] (mode) -- (col_1); \draw[merge] (mode) -- (col_1);
\draw[merge] (col_1) -- (bottom_left); \draw[merge] (col_1) -- (bottom_left);
\draw[merge] (row_1) -- (bottom_left); \draw[merge] (row_1) -- (bottom_left);
} }
\end{tikzpicture} \end{tikzpicture}
\end{column} \end{column}
\end{columns} \end{columns}
@ -757,11 +749,11 @@
draw=blue,fill=yellow!50,text=blue] draw=blue,fill=yellow!50,text=blue]
\tikzstyle{es}=[font=\small, text justified, rectangle,draw,rounded corners=4pt,fill=cyanind!25] \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[es] (liste) at (0,4) {Donner une collection à partitionner};
\node[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Calculer les paramètres de la collection}; \node[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Ajuster \emph{colBiSBM}};
\node[first_col, right = 0.5cm of 1-collection] (1-col-obj) {}; \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 1-collection] (dissimi) {Calculer une 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[instruct, text width=5cm, below = 0.45cm of dissimi] (2-sous-collection) {Séparer la \emph{collection en 2 sous-collections} et ajuster les \emph{colBiSBM}};
\node[second_col, right = 0.25cm of 2-sous-collection] (1-sec-col-obj) {1}; \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[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[test,below = 0.45cm of 2-sous-collection, scale=0.5] (BICL-test) {$\sum_{i=1}^{2} (\text{BIC-L}(\tikz[baseline=-0.25cm]{\node[second_col] {i};} )) > \text{BIC-L}(\tikz[baseline=-0.25cm]{\node[first_col] {};})$?};
@ -780,38 +772,71 @@
\end{tikzpicture} \end{tikzpicture}
\end{column} \end{column}
\end{columns} \end{columns}
\let\thefootnote\relax\footnote{{Même approche que \cite{chabert-liddellLearningCommonStructures2023}}} \blfootnote{Même approche que~\cite{chabert-liddellLearningCommonStructures2023}}
\end{frame} \end{frame}
\section{Application} \section{Application}
\label{sec:application}
\begin{frame} \begin{frame}
\frametitle{Application, données plantes pollinisateurs} \frametitle{Application, données plantes pollinisateurs}
Voici des résultats du modèles \emph{iid-colBiSBM} sur des données \small
plantes-pollinisateurs (\cite{doreRelativeEffectsAnthropogenic2021} Voici des résultats du modèle \emph{iid-colBiSBM} sur des données
et \cite{thebaultDatabasePlantpollinatorNetworks2020}) plantes-pollinisateurs (~\cite{doreRelativeEffectsAnthropogenic2021}
et~\cite{thebaultDatabasePlantpollinatorNetworks2020})
% DONE Ajouter un tableau avec le nombre de réseaux dans chaque sous-collection
\begin{columns} \begin{columns}
\begin{column}{0.48\linewidth} \begin{column}{0.48\linewidth}
\includegraphics[scale=0.32]{img/annual_time_span_vs_iid.png} \includegraphics[scale=0.32]{img/annual_time_span_vs_iid.png}
\begin{center}
\begin{tabular}{ |c|c|c|c|c|c| }
\hline
\thead{N° de \\collection} & 1 & 2 & 3 & 4 & 5 \\
\hline
\thead{Nombre de \\réseaux} & 38 & 45 & 1 & 20 & 19 \\
\hline
\end{tabular}
\end{center}
\end{column} \end{column}
\begin{column}{0.48\linewidth} \begin{column}{0.48\linewidth}
\begin{figure}[H] \begin{figure}[H]
\includegraphics[width=0.45\textwidth]{img/iid-meso-1.png} \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-2.png}
\includegraphics[width=0.45\textwidth]{img/iid-meso-3.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-4.png}
\includegraphics[width=0.45\textwidth]{img/iid-meso-5.png} \includegraphics[width=0.45\textwidth]{img/iid-meso-5.png}
\caption{Connectivités de la partition} \caption{Connectivités de la partition}
\end{figure} \end{figure}
\end{column} \end{column}
\end{columns} \end{columns}
\end{frame} \end{frame}
\section{Conclusion}
\label{sec:conclusion}
\begin{frame}
\frametitle{Conclusion et perspectives}
% DONE Ajouter une slide conclusion perspective
% Rappeler les modeles avec clustering
% Evoquer l'analyse de reseaux corrigés pour l'échantillonnage
% Lien vers le package
\begin{itemize}
\item 4 modèles dont 3 qui ont une flexibilité sur au moins une des dimensions (adaptabilité aux données)
\item Partitionner un ensemble de réseaux selon leurs structures
\item Comparer les \emph{clusterings} de réseaux obtenus entre données brutes et données corrigées (par exemple par la méthode \emph{CoOPLBM}\footnote{~\cite{anakokDisentanglingStructureEcological2022}})
\end{itemize}
\bigskip
\centering
Le package est disponible sur GitHub : \faGithub \url{https://github.com/Chabert-Liddell/colSBM}
\bigskip
\huge
Merci pour votre attention !
\end{frame}
\renewcommand{\pgfuseimage}[1]{\scalebox{.75}{\includegraphics{#1}}}
\begin{frame}[noframenumbering,plain,allowframebreaks] \begin{frame}[noframenumbering,plain,allowframebreaks]
\frametitle{Bibliographie} \frametitle{Bibliographie}
\hfill \printbibliography
\begin{minipage}{0.9\textwidth}
\printbibliography
\end{minipage}
\end{frame} \end{frame}
\end{document} \end{document}