Finished main content

presentation.tex

@@ -7,14 +7,25 @@
 \usepackage[T1]{fontenc} % pour les font postscript
 \usepackage[cyr]{aeguill} % Police vectorielle TrueType, guillemets francais
 \usepackage{epsfig} % pour gérer les images
-\usepackage{amsmath,amsthm} % très bon mode mathématique
+\usepackage{amsmath,amsthm, stmaryrd} % très bon mode mathématique
 \usepackage{amsfonts,amssymb,bm, bbold}% permet la definition des ensembles
 \usepackage{algorithm2e} % pour les algorithmes
 \usepackage{algpseudocode} % pour les algorithmes
+\usepackage{graphicx}
 \usepackage{float} % pour le placement des figure
 \usepackage{url} % pour une gestion efficace des url
 \usepackage{hyperref}  % pour les hyperliens dans le document
 \usepackage{tikz} % For graph plots
+
+% Beamer
+\setbeamertemplate{headline}{%
+    \begin{beamercolorbox}[ht=2.25ex,dp=3.75ex]{section in head/foot}
+        \insertnavigation{\paperwidth}
+    \end{beamercolorbox}%
+}%
+\beamertemplatenavigationsymbolsempty % Pas de bar de navigation
+\setbeamerfont{caption}{size=\scriptsize} % Petit titre de figures
+
 % bibliographie
 \usepackage[style=apa,sorting=none]{biblatex}
 \addbibresource{references.bib}
@@ -49,7 +60,7 @@
 
 \definecolor{burntorange}{RGB}{204, 85, 0}
 \definecolor{goldenyellow}{RGB}{255, 192, 0}
-\definecolor{peach}{RGB}{255, 229, 180}
+\definecolor{peach}{RGB}{255,255,0}
 
 \definecolor{gray}{RGB}{128,128,128}
 
@@ -86,8 +97,8 @@
             Réseau bipartite\\
             \begin{tikzpicture}[scale=.6]
                 \tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt]
-                \tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
+                \tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
-                \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape]
+                \tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape]
                 \tikzstyle{every node}=[fill=blueind]
 
                 \node[state, draw=black!50] (A1) at (0,5) {\textbf{R1}};
@@ -95,7 +106,7 @@
                 \node[state, draw=black!50] (A3) at (5,5) {\textbf{R3}};
 
                 \tikzstyle{every node}=[fill=greenind, shape=rectangle]
-                \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
+                \tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle]
                 \node[state, draw=black!50] (B1) at (0,0) {\textbf{C1}};
                 \node[state, draw=black!50] (B2) at (1.25,0) {\textbf{C2}};
                 \node[state, draw=black!50] (B3) at (2.5,0) {\textbf{C3}};
@@ -130,15 +141,15 @@
     Par exemple : hôtes-parasites, plantes-pollinisateurs, graines-disperseurs \dots
 \end{frame}
 \begin{frame}
-    \frametitle{Latent Block Model (LBM)}
+    \frametitle{Latent Block Model (LBM\footnotemark[2])}
     Proposé par \cite{govaertEMAlgorithmBlock2005}.
     \begin{columns}
         \begin{column}{0.5\linewidth}
             \begin{figure}[H]
                 \center
                 \begin{tikzpicture}[scale=.45]
-                    \tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
+                    \tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
-                    \tikzstyle{every state}=[draw=none,text=white,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}}
 
                     \tikzstyle{every node}=[fill=blueind]
@@ -158,7 +169,7 @@
 
                     \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]
+                    \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]
@@ -197,15 +208,17 @@
                     \path (R31) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5);
 
                 \end{tikzpicture}
-                \caption{Exemple de LBM}
+                \caption{Exemple de LBM\footnotemark[2]}
                 \label{fig:LBMvisu}
             \end{figure}
         \end{column}
         \begin{column}{0.5\linewidth}
+            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 $\mathcal{K}_1 = \{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}$ 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}
@@ -213,17 +226,17 @@
         \end{column}
     \end{columns}
 
+    \footnotetext[2]{Que j'appellerai par la suite BiSBM}
 
 \end{frame}
 \begin{frame}
     \frametitle{\emph{colSBM}}
     Le modèle \emph{colSBM} \parencite{chabert-liddellLearningCommonStructures2023}.\\
     \smallskip
-
     \definecolor{yellow}{RGB}{255,190,60}
-    \begin{tikzpicture}[scale=.33]
+    \begin{tikzpicture}[scale=.28]
         \tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
-        \tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
+        \tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
         \tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
 
         \tikzstyle{every node}=[fill=yellow]
@@ -264,13 +277,14 @@
 
         \node[font=\small, text justified,draw=none, fill=none] at (4.5,-1.5) {SBM};
 
-        \node[font=\small, text justified, fill=none] at (11.5, 1.5) {$\Longrightarrow$};
+
 
         % Sampled network
-        \begin{scope}[xshift=14.5cm, yshift=4cm]
+        \begin{scope}[xshift=18.5cm, yshift=4cm]
+            \node[font=\small, text justified, fill=none] at (-4, -2.5) {$\backsim$};
             \tikzstyle{every node}=[fill=gray, scale=0.95]
             \tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=.5pt, bend left]
-            \tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
+            \tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
 
             \node[state, draw=black!50] (A1) at (0,0) {\textbf{10}};
             \node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
@@ -310,10 +324,10 @@
 
 
         \end{scope}
-        \begin{scope}[xshift=14.5cm, yshift=-4cm]
+        \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=white,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{9}};
             \node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
@@ -349,29 +363,21 @@
             \path (C2) edge[bend right] (B4);
         \end{scope}
     \end{tikzpicture}
-    \begin{columns}
+
-        \begin{column}{0.48\linewidth}
+    Pour $Q = |\{{\color{yellow}\bullet},{\color{blueind}\bullet},{\color{greenind}\bullet}\}|$ blocs fixés :
     \begin{block}{Paramètres}
         \begin{itemize}
-                    \item $\mathcal{K} = \{{\color{yellow}\bullet},{\color{blueind}\bullet},{\color{greenind}\bullet}\}$ blocs
             \item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$
             \item $\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{greenind}\bullet}, j \in {\color{blueind}\bullet})$
         \end{itemize}
     \end{block}
-        \end{column}
-        \begin{column}{0.52\linewidth}
-        \end{column}
-    \end{columns}
-
 \end{frame}
 \section{Extension de \emph{colSBM} aux réseaux bipartites}
 \begin{frame}
     \frametitle{Collections bipartites}
-    \begin{columns}
+    \begin{tikzpicture}[scale=.33]
-        \begin{column}{0.5\linewidth}
+        \begin{scope}[xshift=-3cm, yshift=2cm]
-            \begin{tikzpicture}[scale=.38]
+            \tikzstyle{every state}=[draw=none, text=black,scale=0.75, transform shape]
-                \tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
-                \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape]
             \tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
 
             \tikzstyle{every node}=[fill=blueind]
@@ -390,21 +396,20 @@
             \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{$\rho_{{\color{burntorange}\bullet}}$}};
+            \node[edge_proba] (pi3) at (0.5,-1) {\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
-                \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
+            \tikzstyle{every state}=[draw=none,text=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,-0.7) {\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
+            \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,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}};
+            \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[-,>=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);
@@ -417,33 +422,26 @@
             \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 (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}
 
-            \end{tikzpicture}
+        \begin{scope}[xshift = 16cm, yshift = 1cm]
-            \begin{block}{Paramètres}
+            \node[text justified, fill=none] at (-3, 3.5) {$\backsim$};
-                \begin{itemize}
+            \node[text width=2.5cm, font=\small, text justified, fill=none] at (10,3.75) {$M$ réalisations indépendantes du BiSBM};
-                    \item $\mathcal{K}_1 = \{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}$ blocs en ligne
+            \draw [decorate, decoration = {brace}] (5.5, 7.6) --  (5.5,-0.4);
-                    \item $\mathcal{K}_2 = \{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}$ blocs en colonne
+            \begin{scope}[yshift = 6cm]
-                    \item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(j\in\bullet)$ en colonne
+                \tikzstyle{every state}=[draw, text=black,scale=0.75, transform shape]
-                    \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}};
@@ -453,14 +451,14 @@
                 \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]
+                \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{1}};
+                \node[state, draw=black!50] (B1) at (0.5,-1) {\textbf{1}};
-                    \node[state, draw=black!50] (B2) at (1.5,0) {\textbf{2}};
+                \node[state, draw=black!50] (B2) at (1.5,-1) {\textbf{2}};
 
-                    \node[state, draw=black!50] (B31) at (2.5,0) {\textbf{3}};
+                \node[state, draw=black!50] (B31) at (2.5,-1) {\textbf{3}};
-                    \node[state, draw=black!50] (B4) at (3.5,0) {\textbf{4}};
+                \node[state, draw=black!50] (B4) at (3.5,-1) {\textbf{4}};
 
-                    \node[state, draw=black!50] (B5) at (4.5,0) {\textbf{5}};
+                \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);
@@ -488,20 +486,20 @@
 
                 \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};
+            \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=white,scale=0.75, transform shape]
+                \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{4}};
+                \node[state, draw=black!50] (R11) at (0,2.25) {\textbf{4}};
-                    \node[state, draw=black!50] (R12) at (1,1.25) {\textbf{1}};
+                \node[state, draw=black!50] (R12) at (1,2.25) {\textbf{1}};
-                    \node[state, draw=black!50] (R13) at (2,1.25) {\textbf{6}};
+                \node[state, draw=black!50] (R13) at (2,2.25) {\textbf{6}};
-                    \node[state, draw=black!50] (R21) at (3,1.25) {\textbf{3}};
+                \node[state, draw=black!50] (R21) at (3,2.25) {\textbf{3}};
-                    \node[state, draw=black!50] (R22) at (4,1.25) {\textbf{5}};
+                \node[state, draw=black!50] (R22) at (4,2.25) {\textbf{5}};
-                    \node[state, draw=black!50] (R31) at (5,1.25) {\textbf{2}};
+                \node[state, draw=black!50] (R31) at (5,2.25) {\textbf{2}};
 
-                    \tikzstyle{every state}=[draw=none,text=white,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 (0.5,0) {\textbf{5}};
                 \node[state, draw=black!50] (B2) at (1.5,0) {\textbf{1}};
 
@@ -536,14 +534,278 @@
 
                 \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}
 
-\section*{Bibliographie}
 \begin{frame}[noframenumbering,plain,allowframebreaks]
     \frametitle{Bibliographie}
     \hfill

rapport.tex

@@ -13,7 +13,7 @@
 \usepackage{algpseudocode} % pour les algorithmes
 \usepackage{float} % pour le placement des figure
 \usepackage{url} % pour une gestion efficace des url
-\usepackage[colorlinks,citecolor=blueind,urlcolor=blue,bookmarks=false,hypertexnames=true]{hyperref}  % pour les hyperliens dans le document
+\usepackage[citecolor=blueind,urlcolor=blue,bookmarks=false,hypertexnames=true]{hyperref}  % pour les hyperliens dans le document
 \usepackage{tocbibind} % Pour avoir des index pour table des matières, biblio
 \usepackage{tikz} % For graph plots
 
@@ -50,7 +50,7 @@
 
 \definecolor{burntorange}{RGB}{204, 85, 0}
 \definecolor{goldenyellow}{RGB}{255, 192, 0}
-\definecolor{peach}{RGB}{255, 229, 180}
+\definecolor{yellow}{RGB}{255,255,0}
 
 \definecolor{gray}{RGB}{128,128,128}
 
@@ -80,8 +80,8 @@ $V$ vertices.
     Bipartite network\\
 \begin{tikzpicture}[scale=.6]
     \tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt]
-    \tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
+    \tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
-    \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape]
+    \tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape]
     \tikzstyle{every node}=[fill=blueind]
 
     \node[state, draw=black!50] (A1) at (0,5) {\textbf{R1}};
@@ -89,7 +89,7 @@ $V$ vertices.
     \node[state, draw=black!50] (A3) at (5,5) {\textbf{R3}};
 
     \tikzstyle{every node}=[fill=greenind, shape=rectangle]
-    \tikzstyle{every state}=[draw=none,text=white,scale=0.75, transform shape, shape=rectangle]
+    \tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle]
     \node[state, draw=black!50] (B1) at (0,0) {\textbf{C1}};
     \node[state, draw=black!50] (B2) at (1.25,0) {\textbf{C2}};
     \node[state, draw=black!50] (B3) at (2.5,0) {\textbf{C3}};
@@ -175,8 +175,8 @@ This model supposes that:
 \begin{figure}[H]
     \center
     \begin{tikzpicture}[scale=.6]
-        \tikzstyle{every state}=[draw, text=white,scale=0.95, transform shape]
+        \tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
-        \tikzstyle{every state}=[draw=none,text=white,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}}
         
         \tikzstyle{every node}=[fill=blueind]
@@ -196,20 +196,19 @@ This model supposes that:
     
         \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]
+        \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,-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]
+        \tikzstyle{every node}=[fill=yellow, shape=rectangle]
-        \node[edge_proba] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{peach}\bullet}}$}};
+        \node[edge_proba] (pi3) at (10,-0.7) {\textbf{$\rho_{{\color{yellow}\bullet}}$}};
         \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);
@@ -222,17 +221,19 @@ This model supposes that:
         \path (R13) edge [] (B1);
         \path (R13) edge  (B2);
         \path (R13) edge  (B3);
-        \path (R13) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[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);
+        \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, right, 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] {$\alpha_{{\color{cyanind}\bullet}{\color{yellow}\bullet}}$} (B5);
+        \path (R31) edge[-,>=stealth',shorten >=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, right, fill=none] {$\alpha_{{\color{electricblue}\bullet}{\color{yellow}\bullet}}$} (B5);
 
     \end{tikzpicture}
 \caption{An LBM model visualization}
@@ -241,9 +242,9 @@ This model supposes that:
 
 Parameters
 \begin{itemize}
-    \item $\mathcal{K}_1 = \{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}$ blocks in rows
+    \item $Q_1 = \{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}$ blocks in rows
-    \item $\mathcal{K}_2 = \{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}$ blocks in columns
+    \item $Q_2 = \{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{yellow}\bullet}\}$ blocks in columns
-    \item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ in row et $\rho_{\bullet} = \mathbb{P}(j\in\bullet)$ in column
+    \item $\pi_{\bullet} = \mathbb{P}(i\in\bullet)$ in row and $\rho_{\bullet} = \mathbb{P}(j\in\bullet)$ in column
     \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(i \leftrightarrow j | i \in {\color{blueind}\bullet}, j \in {\color{burntorange}\bullet})$ connectivity probability between two nodes, given their clustering
 \end{itemize}
 

references-Stage MIA Paris-Saclay 2023.bib

@@ -0,0 +1,315 @@
+@misc{anakokDisentanglingStructureEcological2022,
+  title = {Disentangling the Structure of Ecological Bipartite Networks from Observation Processes},
+  author = {Anakok, Emre and Barbillon, Pierre and Fontaine, Colin and Thebault, Elisa},
+  year = {2022},
+  month = nov,
+  number = {arXiv:2211.16364},
+  eprint = {2211.16364},
+  primaryclass = {stat},
+  publisher = {{arXiv}},
+  urldate = {2023-06-14},
+  abstract = {The structure of a bipartite interaction network can be described by providing a clustering for each of the two types of nodes. Such clusterings are outputted by fitting a Latent Block Model (LBM) on an observed network that comes from a sampling of species interactions in the field. However, the sampling is limited and possibly uneven. This may jeopardize the fit of the LBM and then the description of the structure of the network by detecting structures which result from the sampling and not from actual underlying ecological phenomena. If the observed interaction network consists of a weighted bipartite network where the number of observed interactions between two species is available, the sampling efforts for all species can be estimated and used to correct the LBM fit. We propose to combine an observation model that accounts for sampling and an LBM for describing the structure of underlying possible ecological interactions. We develop an original inference procedure for this model, the efficiency of which is demonstrated in simulation studies. The practical interest in ecology of our model is highlighted on a large dataset of plant-pollinator network.},
+  archiveprefix = {arxiv},
+  langid = {english},
+  keywords = {Statistics - Methodology},
+  file = {/home/polarolouis/Zotero/storage/LQ3FINZG/Anakok et al. - 2022 - Disentangling the structure of ecological bipartit.pdf}
+}
+
+@article{celisseConsistencyMaximumlikelihoodVariational2012,
+  title = {Consistency of Maximum-Likelihood and Variational Estimators in the Stochastic Block Model},
+  author = {Celisse, Alain and Daudin, Jean-Jacques and Pierre, Laurent},
+  year = {2012},
+  month = jan,
+  journal = {Electronic Journal of Statistics},
+  volume = {6},
+  number = {none},
+  pages = {1847--1899},
+  publisher = {{Institute of Mathematical Statistics and Bernoulli Society}},
+  issn = {1935-7524, 1935-7524},
+  doi = {10.1214/12-EJS729},
+  urldate = {2023-06-06},
+  abstract = {The stochastic block model (SBM) is a probabilistic model designed to describe heterogeneous directed and undirected graphs. In this paper, we address the asymptotic inference in SBM by use of maximum-likelihood and variational approaches. The identifiability of SBM is proved while asymptotic properties of maximum-likelihood and variational estimators are derived. In particular, the consistency of these estimators is settled for the probability of an edge between two vertices (and for the group proportions at the price of an additional assumption), which is to the best of our knowledge the first result of this type for variational estimators in random graphs.},
+  keywords = {62E17,62G05,62G20,62H30,Concentration inequalities,consistency,maximum likelihood estimators,Random graphs,Stochastic block model,variational estimators},
+  file = {/home/polarolouis/Zotero/storage/JNWRIYKG/celisse2012.pdf.pdf;/home/polarolouis/Zotero/storage/XG463B5I/Celisse et al. - 2012 - Consistency of maximum-likelihood and variational .pdf}
+}
+
+@misc{chabert-liddellLearningCommonStructures2023,
+  type = {Article},
+  title = {Learning Common Structures in a Collection of Networks. {{An}} Application to Food Webs},
+  author = {{Chabert-Liddell}, Saint-Clair and Barbillon, Pierre and Donnet, Sophie},
+  year = {2023},
+  month = mar,
+  number = {arXiv:2206.00560},
+  eprint = {2206.00560},
+  primaryclass = {stat},
+  publisher = {{arXiv}},
+  doi = {10.48550/arXiv.2206.00560},
+  urldate = {2023-05-22},
+  abstract = {Let a collection of networks represent interactions within several (social or ecological) systems. We pursue two objectives: identifying similarities in the topological structures that are held in common between the networks and clustering the collection into sub-collections of structurally homogeneous networks. We tackle these two questions with a probabilistic model based approach. We propose an extension of the Stochastic Block Model (SBM) adapted to the joint modeling of a collection of networks. The networks in the collection are assumed to be independent realizations of SBMs. The common connectivity structure is imposed through the equality of some parameters. The model parameters are estimated with a variational Expectation-Maximization (EM) algorithm. We derive an ad-hoc penalized likelihood criterion to select the number of blocks and to assess the adequacy of the consensus found between the structures of the different networks. This same criterion can also be used to cluster networks on the basis of their connectivity structure. It thus provides a partition of the collection into subsets of structurally homogeneous networks. The relevance of our proposition is assessed on two collections of ecological networks. First, an application to three stream food webs reveals the homogeneity of their structures and the correspondence between groups of species in different ecosystems playing equivalent ecological roles. Moreover, the joint analysis allows a finer analysis of the structure of smaller networks. Second, we cluster 67 food webs according to their connectivity structures and demonstrate that five mesoscale structures are sufficient to describe this collection.},
+  archiveprefix = {arxiv},
+  keywords = {Statistics - Applications,Statistics - Methodology},
+  file = {/home/polarolouis/Zotero/storage/M74TXGCF/Chabert-Liddell et al. - 2023 - Learning common structures in a collection of netw.pdf;/home/polarolouis/Zotero/storage/A35M8KNP/2206.html}
+}
+
+@article{daudinMixtureModelRandom2008,
+  title = {A Mixture Model for Random Graphs},
+  author = {Daudin, J.-J. and Picard, F. and Robin, S.},
+  year = {2008},
+  month = jun,
+  journal = {Statistics and Computing},
+  volume = {18},
+  number = {2},
+  pages = {173--183},
+  issn = {1573-1375},
+  doi = {10.1007/s11222-007-9046-7},
+  urldate = {2023-06-16},
+  abstract = {The Erd\"os\textendash R\'enyi model of a network is simple and possesses many explicit expressions for average and asymptotic properties, but it does not fit well to real-world networks. The vertices of those networks are often structured in unknown classes (functionally related proteins or social communities) with different connectivity properties. The stochastic block structures model was proposed for this purpose in the context of social sciences, using a Bayesian approach. We consider the same model in a frequentest statistical framework. We give the degree distribution and the clustering coefficient associated with this model, a variational method to estimate its parameters and a model selection criterion to select the number of classes. This estimation procedure allows us to deal with large networks containing thousands of vertices. The method is used to uncover the modular structure of a network of enzymatic reactions.},
+  langid = {english},
+  keywords = {Mixture models,Random graphs,Variational~method},
+  file = {/home/polarolouis/Zotero/storage/439HK27B/Daudin et al. - 2008 - A mixture model for random graphs.pdf;/home/polarolouis/Zotero/storage/HVVF5MNY/daudin2007.pdf.pdf}
+}
+
+@article{desjardins-proulxEcologicalInteractionsNetflix2017,
+  title = {Ecological Interactions and the {{Netflix}} Problem},
+  author = {{Desjardins-Proulx}, Philippe and Laigle, Idaline and Poisot, Timoth{\'e}e and Gravel, Dominique},
+  year = {2017},
+  month = aug,
+  journal = {PeerJ},
+  volume = {5},
+  pages = {e3644},
+  publisher = {{PeerJ Inc.}},
+  issn = {2167-8359},
+  doi = {10.7717/peerj.3644},
+  urldate = {2023-06-15},
+  abstract = {Species interactions are a key component of ecosystems but we generally have an incomplete picture of who-eats-who in a given community. Different techniques have been devised to predict species interactions using theoretical models or abundances. Here, we explore the K nearest neighbour approach, with a special emphasis on recommendation, along with a supervised machine learning technique. Recommenders are algorithms developed for companies like Netflix to predict whether a customer will like a product given the preferences of similar customers. These machine learning techniques are well-suited to study binary ecological interactions since they focus on positive-only data. By removing a prey from a predator, we find that recommenders can guess the missing prey around 50\% of the times on the first try, with up to 881 possibilities. Traits do not improve significantly the results for the K nearest neighbour, although a simple test with a supervised learning approach (random forests) show we can predict interactions with high accuracy using only three traits per species. This result shows that binary interactions can be predicted without regard to the ecological community given only three variables: body mass and two variables for the species' phylogeny. These techniques are complementary, as recommenders can predict interactions in the absence of traits, using only information about other species' interactions, while supervised learning algorithms such as random forests base their predictions on traits only but do not exploit other species' interactions. Further work should focus on developing custom similarity measures specialized for ecology to improve the KNN algorithms and using richer data to capture indirect relationships between species.},
+  langid = {english},
+  file = {/home/polarolouis/Zotero/storage/3L7JALP4/Desjardins-Proulx et al. - 2017 - Ecological interactions and the Netflix problem.pdf}
+}
+
+@article{doreRelativeEffectsAnthropogenic2021,
+  title = {Relative Effects of Anthropogenic Pressures, Climate, and Sampling Design on the Structure of Pollination Networks at the Global Scale},
+  author = {Dor{\'e}, Ma{\"e}l and Fontaine, Colin and Th{\'e}bault, Elisa},
+  year = {2021},
+  journal = {Global Change Biology},
+  volume = {27},
+  number = {6},
+  pages = {1266--1280},
+  issn = {1365-2486},
+  doi = {10.1111/gcb.15474},
+  urldate = {2023-06-21},
+  abstract = {Pollinators provide crucial ecosystem services that underpin to wild plant reproduction and yields of insect-pollinated crops. Understanding the relative impacts of anthropogenic pressures and climate on the structure of plant\textendash pollinator interaction networks is vital considering ongoing global change and pollinator decline. Our ability to predict the consequences of global change for pollinator assemblages worldwide requires global syntheses, but these analytical approaches may be hindered by variable methods among studies that either invalidate comparisons or mask biological phenomena. Here we conducted a synthetic analysis that assesses the relative impact of anthropogenic pressures and climatic variability, and accounts for heterogeneity in sampling methodology to reveal network responses at the global scale. We analyzed an extensive dataset, comprising 295 networks over 123 locations all over the world, and reporting over 50,000 interactions between flowering plant species and their insect visitors. Our study revealed that anthropogenic pressures correlate with an increase in generalism in pollination networks while pollinator richness and taxonomic composition are more related to climatic variables with an increase in dipteran pollinator richness associated with cooler temperatures. The contrasting response of species richness and generalism of the plant\textendash pollinator networks stresses the importance of considering interaction network structure alongside diversity in ecological monitoring. In addition, differences in sampling design explained more variation than anthropogenic pressures or climate on both pollination networks richness and generalism, highlighting the crucial need to report and incorporate sampling design in macroecological comparative studies of pollination networks. As a whole, our study reveals a potential human impact on pollination networks at a global scale. However, further research is needed to evaluate potential consequences of loss of specialist species and their unique ecological interactions and evolutionary pathways on the ecosystem pollination function at a global scale.},
+  copyright = {\textcopyright{} 2020 John Wiley \& Sons Ltd},
+  langid = {english},
+  keywords = {anthropogenic pressures,climate,connectance,data,generalism,human impacts,plant-pollinator,pollination networks,richness,sampling effects,specialization},
+  file = {/home/polarolouis/Zotero/storage/89ZXBJQP/10.1111@gcb.15474.pdf.pdf;/home/polarolouis/Zotero/storage/IVR6RGG7/Doré et al. - 2021 - Relative effects of anthropogenic pressures, clima.pdf;/home/polarolouis/Zotero/storage/WSJ4DV98/gcb.html}
+}
+
+@article{govaertEMAlgorithmBlock2005,
+  title = {An {{EM}} Algorithm for the Block Mixture Model},
+  author = {Govaert, G. and Nadif, M.},
+  year = {2005},
+  month = apr,
+  journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
+  volume = {27},
+  number = {4},
+  pages = {643--647},
+  issn = {1939-3539},
+  doi = {10.1109/TPAMI.2005.69},
+  abstract = {Although many clustering procedures aim to construct an optimal partition of objects or, sometimes, of variables, there are other methods, called block clustering methods, which consider simultaneously the two sets and organize the data into homogeneous blocks. Recently, we have proposed a new mixture model called block mixture model which takes into account this situation. This model allows one to embed simultaneous clustering of objects and variables in a mixture approach. We have studied this probabilistic model under the classification likelihood approach and developed a new algorithm for simultaneous partitioning based on the classification EM algorithm. In this paper, we consider the block clustering problem under the maximum likelihood approach and the goal of our contribution is to estimate the parameters of this model. Unfortunately, the application of the EM algorithm for the block mixture model cannot be made directly; difficulties arise due to the dependence structure in the model and approximations are required. Using a variational approximation, we propose a generalized EM algorithm to estimate the parameters of the block mixture model and, to illustrate our approach, we study the case of binary data by using a Bernoulli block mixture.},
+  keywords = {Approximation algorithms,Classification algorithms,Clustering algorithms,Clustering methods,Data mining,EM algorithm,Index Terms- Block mixture model,Maximum likelihood estimation,Parameter estimation,Partitioning algorithms,Self organizing feature maps,Sparse matrices,variational approximation.},
+  file = {/home/polarolouis/Zotero/storage/6IG45HH2/govaert2005.pdf.pdf;/home/polarolouis/Zotero/storage/TL8M3XRF/Govaert et Nadif - 2005 - An EM algorithm for the block mixture model.pdf;/home/polarolouis/Zotero/storage/2Y48IB26/1401917.html}
+}
+
+@article{govaertLatentBlockModel2010,
+  title = {Latent {{Block Model}} for {{Contingency Table}}},
+  author = {Govaert, G{\'e}rard and Nadif, Mohamed},
+  year = {2010},
+  month = jan,
+  journal = {Communications in Statistics - Theory and Methods},
+  volume = {39},
+  number = {3},
+  pages = {416--425},
+  publisher = {{Taylor \& Francis}},
+  issn = {0361-0926},
+  doi = {10.1080/03610920903140197},
+  urldate = {2023-06-15},
+  abstract = {Although many clustering procedures aim to construct an optimal partition of objects or, sometimes, variables, there are other methods, called block clustering methods, which simultaneously consider the two sets and organize the data into homogeneous blocks. This kind of method has practical importance in a wide variety of applications such as text and market basket data analysis. Typically, the data that arise in these applications are arranged as a two-way contingency table. Using Poisson distributions, a latent block model for these data is proposed and, setting it under the maximum likelihood approach and the classification maximum likelihood approach, various algorithms are provided. Their performances are evaluated and compared to a simple use of EM or CEM applied separately on the rows and columns of the contingency table.},
+  keywords = {62H17,62H30,Block clustering,Block Poisson mixture model,CEM algorithm,Contingency table,EM algorithm},
+  file = {/home/polarolouis/Zotero/storage/PPHP33Z9/Govaert et Nadif - 2010 - Latent Block Model for Contingency Table.pdf;/home/polarolouis/Zotero/storage/UT8TARCX/govaert2010.pdf.pdf}
+}
+
+@article{hollandStochasticBlockmodelsFirst1983,
+  title = {Stochastic Blockmodels: {{First}} Steps},
+  shorttitle = {Stochastic Blockmodels},
+  author = {Holland, Paul W. and Laskey, Kathryn Blackmond and Leinhardt, Samuel},
+  year = {1983},
+  month = jun,
+  journal = {Social Networks},
+  volume = {5},
+  number = {2},
+  pages = {109--137},
+  issn = {0378-8733},
+  doi = {10.1016/0378-8733(83)90021-7},
+  urldate = {2023-06-15},
+  abstract = {A stochastic model is proposed for social networks in which the actors in a network are partitioned into subgroups called blocks. The model provides a stochastic generalization of the blockmodel. Estimation techniques are developed for the special case of a single relation social network, with blocks specified a priori. An extension of the model allows for tendencies toward reciprocation of ties beyond those explained by the partition. The extended model provides a one degree-of-freedom test of the model. A numerical example from the social network literature is used to illustrate the methods.},
+  langid = {english},
+  file = {/home/polarolouis/Zotero/storage/6F8YT8AD/holland1983.pdf.pdf;/home/polarolouis/Zotero/storage/7DSZ3KD9/Holland et al. - 1983 - Stochastic blockmodels First steps.pdf;/home/polarolouis/Zotero/storage/DUL2RV8Q/holland1983.pdf.pdf;/home/polarolouis/Zotero/storage/G9KZBG9W/0378873383900217.html}
+}
+
+@article{kaszewska-gilasGlobalStudiesHostParasite2021,
+  title = {Global {{Studies}} of the {{Host-Parasite Relationships}} between {{Ectoparasitic Mites}} of the {{Family Syringophilidae}} and {{Birds}} of the {{Order Columbiformes}}},
+  author = {{Kaszewska-Gilas}, Katarzyna and Kosicki, Jakub Ziemowit and Hromada, Martin and Skoracki, Maciej},
+  year = {2021},
+  month = dec,
+  journal = {Animals},
+  volume = {11},
+  number = {12},
+  pages = {3392},
+  publisher = {{Multidisciplinary Digital Publishing Institute}},
+  issn = {2076-2615},
+  doi = {10.3390/ani11123392},
+  urldate = {2023-06-15},
+  abstract = {The quill mites belonging to the family Syringophilidae (Acari: Prostigmata: Cheyletoidea) are obligate ectoparasites of birds. They inhabit different types of the quills, where they spend their whole life cycle. In this paper, we conducted a global study of syringophilid mites associated with columbiform birds. We examined 772 pigeon and dove individuals belonging to 112 species (35\% world fauna) from all zoogeographical regions (except Madagascan) where Columbiformes occur. We measured the prevalence (IP) and the confidence interval (CI) for all infested host species. IP ranges between 4.2 and 66.7 (CI 0.2\textendash 100). We applied a bipartite analysis to determine host\textendash parasite interaction, network indices, and host specificity on species and whole network levels. The Syringophilidae\textendash Columbiformes network was composed of 25 mite species and 65 host species. The bipartite network was characterized by a high network level specialization H2{${'}$} = 0.93, high nestedness N = 0.908, connectance C = 0.90, and high modularity Q = 0.83, with 20 modules. Moreover, we reconstructed the phylogeny of the quill mites associated with columbiform birds on the generic level. Analysis shows two distinct clades: Meitingsunes + Psittaciphilus, and Peristerophila + Terratosyringophilus.},
+  copyright = {http://creativecommons.org/licenses/by/3.0/},
+  langid = {english},
+  keywords = {Acari,biodiversity,bipartite-example,network,pigeons and doves,quill mites},
+  file = {/home/polarolouis/Zotero/storage/VXVQ5CPH/Kaszewska-Gilas et al. - 2021 - Global Studies of the Host-Parasite Relationships .pdf}
+}
+
+@misc{larousseDefinitionsBipartiBipartite,
+  title = {{D\'efinitions : biparti, bipartite - Dictionnaire de fran\c{c}ais Larousse}},
+  shorttitle = {{D\'efinitions}},
+  author = {Larousse, {\'E}ditions},
+  urldate = {2023-06-17},
+  abstract = {biparti, bipartite - D\'efinitions Fran\c{c}ais : Retrouvez la d\'efinition de biparti, bipartite, ainsi que les difficult\'es... - synonymes, homonymes, difficult\'es, citations.},
+  howpublished = {https://www.larousse.fr/dictionnaires/francais/biparti/9503},
+  langid = {french},
+  file = {/home/polarolouis/Zotero/storage/MA2VH6NX/9503.html}
+}
+
+@article{maeldoreMaelDorePollinationNetworks2020,
+  title = {{{MaelDore}}/{{Pollination}}\_networks: {{R}} Scripts for {{Dor\'e}} et al., 2020 - {{Relative}} Effects of Anthropogenic Pressures, Climate, and Sampling Design on the Structure of Pollination Networks at the Global Scale},
+  shorttitle = {{{MaelDore}}/{{Pollination}}\_networks},
+  author = {MaelDore},
+  year = {2020},
+  month = nov,
+  publisher = {{Zenodo}},
+  doi = {10.5281/ZENODO.4290503},
+  urldate = {2023-06-21},
+  abstract = {R scripts for Dor\'e et al., 2020 - Relative effects of anthropogenic pressures, climate, and sampling design on the structure of pollination networks at the global scale},
+  copyright = {Open Access}
+}
+
+@article{pavlopoulosBipartiteGraphsSystems2018,
+  title = {Bipartite Graphs in Systems Biology and Medicine: A Survey of Methods and Applications},
+  shorttitle = {Bipartite Graphs in Systems Biology and Medicine},
+  author = {Pavlopoulos, Georgios A and Kontou, Panagiota I and Pavlopoulou, Athanasia and Bouyioukos, Costas and Markou, Evripides and Bagos, Pantelis G},
+  year = {2018},
+  month = apr,
+  journal = {GigaScience},
+  volume = {7},
+  number = {4},
+  pages = {giy014},
+  issn = {2047-217X},
+  doi = {10.1093/gigascience/giy014},
+  urldate = {2023-06-15},
+  abstract = {The latest advances in high-throughput techniques during the past decade allowed the systems biology field to expand significantly. Today, the focus of biologists has shifted from the study of individual biological components to the study of complex biological systems and their dynamics at a larger scale. Through the discovery of novel bioentity relationships, researchers reveal new information about biological functions and processes. Graphs are widely used to represent bioentities such as proteins, genes, small molecules, ligands, and others such as nodes and their connections as edges within a network. In this review, special focus is given to the usability of bipartite graphs and their impact on the field of network biology and medicine. Furthermore, their topological properties and how these can be applied to certain biological case studies are discussed. Finally, available methodologies and software are presented, and useful insights on how bipartite graphs can shape the path toward the solution of challenging biological problems are provided.},
+  file = {/home/polarolouis/Zotero/storage/2KJFL3SB/Pavlopoulos et al. - 2018 - Bipartite graphs in systems biology and medicine .pdf;/home/polarolouis/Zotero/storage/A2Y2EGPA/pavlopoulos2018.pdf.pdf;/home/polarolouis/Zotero/storage/UK2MK5FW/pavlopoulos2018.pdf.pdf;/home/polarolouis/Zotero/storage/XP7G4PZF/4875933.html}
+}
+
+@article{ramos-jilibertoTopologicalChangeAndean2010,
+  title = {Topological Change of {{Andean}} Plant\textendash Pollinator Networks along an Altitudinal Gradient},
+  author = {{Ramos-Jiliberto}, Rodrigo and Dom{\'i}nguez, Daniela and Espinoza, Claudia and L{\'o}pez, Gioconda and Valdovinos, Fernanda S. and Bustamante, Ramiro O. and Medel, Rodrigo},
+  year = {2010},
+  month = mar,
+  journal = {Ecological Complexity},
+  volume = {7},
+  number = {1},
+  pages = {86--90},
+  issn = {1476-945X},
+  doi = {10.1016/j.ecocom.2009.06.001},
+  urldate = {2023-06-15},
+  abstract = {Pollination interaction networks exhibit structural regularities across a wide range of natural environments. Long-tailed degree distribution, nestedness, and modularity are the most prevalent topological patterns found in most bipartite networks analyzed up to day. In this work we evaluate the variation of these topological properties along an altitudinal gradient. To this end, we examined four plant\textendash pollinator networks from the Chilean Andes at 33\textdegree S, in range from 1800 to 3600m elevation. Our results indicate that network topology is strongly and systematically affected by elevation. At increasing altitude, the number of potential visitors per plant decreased, and species' degree distributions are closer to random expectations. On the other hand, the nested structure of mutualistic interactions systematically decreased with elevation, and network modularity was significantly higher than random expectations over the entire altitudinal range. In addition, at increasing elevations the pollination networks were organized in fewer and more strongly connected modules. Our results suggest that the severe abiotic conditions found at increased elevations translate into less organized pollination networks.},
+  langid = {english},
+  keywords = {bipartite-example,Chile,Complexity,Degree distribution,Modularity,Mutualistic networks,Nestedness,Power law},
+  file = {/home/polarolouis/Zotero/storage/ATY3ZP2X/Ramos-Jiliberto et al. - 2010 - Topological change of Andean plant–pollinator netw.pdf;/home/polarolouis/Zotero/storage/HPBGUP65/ramos-jiliberto2010.pdf.pdf;/home/polarolouis/Zotero/storage/I33MZQQ7/ramos-jiliberto2010.pdf.pdf;/home/polarolouis/Zotero/storage/YJX8XBNW/S1476945X09000622.html}
+}
+
+@article{snijdersEstimationPredictionStochastic1997,
+  title = {Estimation and {{Prediction}} for {{Stochastic Blockmodels}} for {{Graphs}} with {{Latent Block Structure}}},
+  author = {Snijders, Tom A.B. and Nowicki, Krzysztof},
+  year = {1997},
+  month = jan,
+  journal = {Journal of Classification},
+  volume = {14},
+  number = {1},
+  pages = {75--100},
+  issn = {1432-1343},
+  doi = {10.1007/s003579900004},
+  urldate = {2023-06-15},
+  abstract = {blockmodeling for graphs is proposed. The model assumes that the vertices of the graph are partitioned into two unknown blocks and that the probability of an edge between two vertices depends only on the blocks to which they belong. Statistical procedures are derived for estimating the probabilities of edges and for predicting the block structure from observations of the edge pattern only. ML estimators can be computed using the EM algorithm, but this strategy is practical only for small graphs. A Bayesian estimator, based on the Gibbs sampling, is proposed. This estimator is practical also for large graphs. When ML estimators are used, the block structure can be predicted based on predictive likelihood. When Gibbs sampling is used, the block structure can be predicted from posterior predictive probabilities. A side result is that when the number of vertices tends to infinity while the probabilities remain constant, the block structure can be recovered correctly with probability tending to 1.},
+  langid = {english},
+  keywords = {Bayesian Estimator,Block Structure,Gibbs Sampling,Large Graph,Statistical Procedure},
+  file = {/home/polarolouis/Zotero/storage/2GYRASW5/snijders1997.pdf.pdf;/home/polarolouis/Zotero/storage/JJNQV32Y/Snijders et Nowicki - 1997 - Estimation and Prediction for Stochastic Blockmode.pdf;/home/polarolouis/Zotero/storage/LXGG9SRP/snijders1997.pdf.pdf}
+}
+
+@misc{thebaultDatabasePlantpollinatorNetworks2020,
+  title = {A Database of Plant-Pollinator Networks},
+  author = {Th{\'e}bault, Elisa and Fontaine, Colin},
+  year = {2020},
+  month = dec,
+  publisher = {{Zenodo}},
+  doi = {10.5281/zenodo.4300427},
+  urldate = {2023-06-21},
+  abstract = {This database assembles different published datasets of observed interaction networks between plants and pollinators, which were extracted from articles, theses and existing online databases. Each row in the data table corresponds to an interaction between a plant and a pollinator species reported at a given site by a given publication.},
+  keywords = {diversity,flower visitors,mutualistic network,plant-pollinator interaction}
+}
+
+@misc{thebaultelisaDatabasePlantpollinatorNetworks2020,
+  title = {A Database of Plant-Pollinator Networks},
+  author = {Th{\'e}bault, Elisa and Fontaine, Colin},
+  year = {2020},
+  month = dec,
+  publisher = {{Zenodo}},
+  doi = {10.5281/ZENODO.4300427},
+  urldate = {2023-06-21},
+  abstract = {This database assembles different published datasets of observed interaction networks between plants and pollinators, which were extracted from articles, theses and existing online databases. Each row in the data table corresponds to an interaction between a plant and a pollinator species reported at a given site by a given publication.},
+  copyright = {Creative Commons Attribution 4.0 International, Open Access},
+  keywords = {data,diversity,flower visitors,mutualistic network,plant-pollinator,plant-pollinator interaction}
+}
+
+@misc{thebaultelisaDatabasePlantpollinatorNetworks2022,
+  title = {A Database of Plant-Pollinator Networks},
+  author = {Th{\'e}bault, Elisa and Fontaine, Colin},
+  year = {2022},
+  month = jun,
+  publisher = {{Zenodo}},
+  doi = {10.5281/ZENODO.6630184},
+  urldate = {2023-06-21},
+  abstract = {This database assembles different published datasets of observed interaction networks between plants and pollinators, which were extracted from articles, theses and existing online databases. Each row in the data table corresponds to an interaction between a plant and a pollinator species reported at a given site by a given publication.},
+  collaborator = {Dor{\'e}, Ma{\"e}l and Parra, Santiago},
+  copyright = {Creative Commons Attribution 4.0 International, Open Access},
+  keywords = {data,diversity,flower visitors,mutualistic network,plant-pollinator,plant-pollinator interaction}
+}
+
+@misc{thebaultelisaDatabasePlantpollinatorNetworks2022a,
+  title = {A Database of Plant-Pollinator Networks},
+  author = {Th{\'e}bault, Elisa and Fontaine, Colin},
+  year = {2022},
+  month = jun,
+  publisher = {{Zenodo}},
+  doi = {10.5281/ZENODO.4300426},
+  urldate = {2023-06-21},
+  abstract = {This database assembles different published datasets of observed interaction networks between plants and pollinators, which were extracted from articles, theses and existing online databases. Each row in the data table corresponds to an interaction between a plant and a pollinator species reported at a given site by a given publication.},
+  collaborator = {Dor{\'e}, Ma{\"e}l and Parra, Santiago},
+  copyright = {Creative Commons Attribution 4.0 International, Open Access},
+  keywords = {data,diversity,flower visitors,mutualistic network,plant-pollinator,plant-pollinator interaction}
+}
+
+@misc{WebLifeEcological,
+  title = {Web of {{Life}}: Ecological Networks Database},
+  urldate = {2023-06-17},
+  howpublished = {https://www.web-of-life.es/map.php},
+  keywords = {networks,site},
+  file = {/home/polarolouis/Zotero/storage/9WZE8QLQ/map.html}
+}

references.bib

@@ -86,6 +86,24 @@
   file = {/home/polarolouis/Zotero/storage/3L7JALP4/Desjardins-Proulx et al. - 2017 - Ecological interactions and the Netflix problem.pdf}
 }
 
+@article{doreRelativeEffectsAnthropogenic2021,
+  title = {Relative Effects of Anthropogenic Pressures, Climate, and Sampling Design on the Structure of Pollination Networks at the Global Scale},
+  author = {Dor{\'e}, Ma{\"e}l and Fontaine, Colin and Th{\'e}bault, Elisa},
+  year = {2021},
+  journal = {Global Change Biology},
+  volume = {27},
+  number = {6},
+  pages = {1266--1280},
+  issn = {1365-2486},
+  doi = {10.1111/gcb.15474},
+  urldate = {2023-06-21},
+  abstract = {Pollinators provide crucial ecosystem services that underpin to wild plant reproduction and yields of insect-pollinated crops. Understanding the relative impacts of anthropogenic pressures and climate on the structure of plant\textendash pollinator interaction networks is vital considering ongoing global change and pollinator decline. Our ability to predict the consequences of global change for pollinator assemblages worldwide requires global syntheses, but these analytical approaches may be hindered by variable methods among studies that either invalidate comparisons or mask biological phenomena. Here we conducted a synthetic analysis that assesses the relative impact of anthropogenic pressures and climatic variability, and accounts for heterogeneity in sampling methodology to reveal network responses at the global scale. We analyzed an extensive dataset, comprising 295 networks over 123 locations all over the world, and reporting over 50,000 interactions between flowering plant species and their insect visitors. Our study revealed that anthropogenic pressures correlate with an increase in generalism in pollination networks while pollinator richness and taxonomic composition are more related to climatic variables with an increase in dipteran pollinator richness associated with cooler temperatures. The contrasting response of species richness and generalism of the plant\textendash pollinator networks stresses the importance of considering interaction network structure alongside diversity in ecological monitoring. In addition, differences in sampling design explained more variation than anthropogenic pressures or climate on both pollination networks richness and generalism, highlighting the crucial need to report and incorporate sampling design in macroecological comparative studies of pollination networks. As a whole, our study reveals a potential human impact on pollination networks at a global scale. However, further research is needed to evaluate potential consequences of loss of specialist species and their unique ecological interactions and evolutionary pathways on the ecosystem pollination function at a global scale.},
+  copyright = {\textcopyright{} 2020 John Wiley \& Sons Ltd},
+  langid = {english},
+  keywords = {anthropogenic pressures,climate,connectance,data,generalism,human impacts,plant-pollinator,pollination networks,richness,sampling effects,specialization},
+  file = {/home/polarolouis/Zotero/storage/89ZXBJQP/10.1111@gcb.15474.pdf.pdf;/home/polarolouis/Zotero/storage/IVR6RGG7/Doré et al. - 2021 - Relative effects of anthropogenic pressures, clima.pdf;/home/polarolouis/Zotero/storage/WSJ4DV98/gcb.html}
+}
+
 @article{govaertEMAlgorithmBlock2005,
   title = {An {{EM}} Algorithm for the Block Mixture Model},
   author = {Govaert, G. and Nadif, M.},
@@ -169,6 +187,19 @@
   file = {/home/polarolouis/Zotero/storage/MA2VH6NX/9503.html}
 }
 
+@article{maeldoreMaelDorePollinationNetworks2020,
+  title = {{{MaelDore}}/{{Pollination}}\_networks: {{R}} Scripts for {{Dor\'e}} et al., 2020 - {{Relative}} Effects of Anthropogenic Pressures, Climate, and Sampling Design on the Structure of Pollination Networks at the Global Scale},
+  shorttitle = {{{MaelDore}}/{{Pollination}}\_networks},
+  author = {MaelDore},
+  year = {2020},
+  month = nov,
+  publisher = {{Zenodo}},
+  doi = {10.5281/ZENODO.4290503},
+  urldate = {2023-06-21},
+  abstract = {R scripts for Dor\'e et al., 2020 - Relative effects of anthropogenic pressures, climate, and sampling design on the structure of pollination networks at the global scale},
+  copyright = {Open Access}
+}
+
 @article{pavlopoulosBipartiteGraphsSystems2018,
   title = {Bipartite Graphs in Systems Biology and Medicine: A Survey of Methods and Applications},
   shorttitle = {Bipartite Graphs in Systems Biology and Medicine},
@@ -222,6 +253,59 @@
   file = {/home/polarolouis/Zotero/storage/2GYRASW5/snijders1997.pdf.pdf;/home/polarolouis/Zotero/storage/JJNQV32Y/Snijders et Nowicki - 1997 - Estimation and Prediction for Stochastic Blockmode.pdf;/home/polarolouis/Zotero/storage/LXGG9SRP/snijders1997.pdf.pdf}
 }
 
+@misc{thebaultDatabasePlantpollinatorNetworks2020,
+  title = {A Database of Plant-Pollinator Networks},
+  author = {Th{\'e}bault, Elisa and Fontaine, Colin},
+  year = {2020},
+  month = dec,
+  publisher = {{Zenodo}},
+  doi = {10.5281/zenodo.4300427},
+  urldate = {2023-06-21},
+  abstract = {This database assembles different published datasets of observed interaction networks between plants and pollinators, which were extracted from articles, theses and existing online databases. Each row in the data table corresponds to an interaction between a plant and a pollinator species reported at a given site by a given publication.},
+  keywords = {diversity,flower visitors,mutualistic network,plant-pollinator interaction}
+}
+
+@misc{thebaultelisaDatabasePlantpollinatorNetworks2020,
+  title = {A Database of Plant-Pollinator Networks},
+  author = {Th{\'e}bault, Elisa and Fontaine, Colin},
+  year = {2020},
+  month = dec,
+  publisher = {{Zenodo}},
+  doi = {10.5281/ZENODO.4300427},
+  urldate = {2023-06-21},
+  abstract = {This database assembles different published datasets of observed interaction networks between plants and pollinators, which were extracted from articles, theses and existing online databases. Each row in the data table corresponds to an interaction between a plant and a pollinator species reported at a given site by a given publication.},
+  copyright = {Creative Commons Attribution 4.0 International, Open Access},
+  keywords = {data,diversity,flower visitors,mutualistic network,plant-pollinator,plant-pollinator interaction}
+}
+
+@misc{thebaultelisaDatabasePlantpollinatorNetworks2022,
+  title = {A Database of Plant-Pollinator Networks},
+  author = {Th{\'e}bault, Elisa and Fontaine, Colin},
+  year = {2022},
+  month = jun,
+  publisher = {{Zenodo}},
+  doi = {10.5281/ZENODO.6630184},
+  urldate = {2023-06-21},
+  abstract = {This database assembles different published datasets of observed interaction networks between plants and pollinators, which were extracted from articles, theses and existing online databases. Each row in the data table corresponds to an interaction between a plant and a pollinator species reported at a given site by a given publication.},
+  collaborator = {Dor{\'e}, Ma{\"e}l and Parra, Santiago},
+  copyright = {Creative Commons Attribution 4.0 International, Open Access},
+  keywords = {data,diversity,flower visitors,mutualistic network,plant-pollinator,plant-pollinator interaction}
+}
+
+@misc{thebaultelisaDatabasePlantpollinatorNetworks2022a,
+  title = {A Database of Plant-Pollinator Networks},
+  author = {Th{\'e}bault, Elisa and Fontaine, Colin},
+  year = {2022},
+  month = jun,
+  publisher = {{Zenodo}},
+  doi = {10.5281/ZENODO.4300426},
+  urldate = {2023-06-21},
+  abstract = {This database assembles different published datasets of observed interaction networks between plants and pollinators, which were extracted from articles, theses and existing online databases. Each row in the data table corresponds to an interaction between a plant and a pollinator species reported at a given site by a given publication.},
+  collaborator = {Dor{\'e}, Ma{\"e}l and Parra, Santiago},
+  copyright = {Creative Commons Attribution 4.0 International, Open Access},
+  keywords = {data,diversity,flower visitors,mutualistic network,plant-pollinator,plant-pollinator interaction}
+}
+
 @misc{WebLifeEcological,
   title = {Web of {{Life}}: Ecological Networks Database},
   urldate = {2023-06-17},