\section{Model Context}
\label{sec:context-of-the-model}

\begin{frame}
    \frametitle{Why a network?}
    \begin{columns}
        \begin{column}{0.5\textwidth}
            \begin{columns}
                \begin{column}{0.5\textwidth}
                    \begin{figure}[ht]
                        \centering
                        \begin{tikzpicture}[scale=.6,rotate=270]
                            \input{tikz/plantpollinatornetwork.tex}
                        \end{tikzpicture}
                        \caption{Example of a network}
                        \label{fig:plants-pollin}
                    \end{figure}
                \end{column}
                \begin{column}{0.3\textwidth}
                    \centering
                    \begin{align*}
                        \begin{pmatrix}
                            1 & 0 & 1 \\
                            1 & 0 & 0 \\
                            1 & 0 & 0 \\
                            1 & 1 & 0
                        \end{pmatrix}
                    \end{align*}
                    \footnotesize
                    Associated adjacency matrix
                \end{column}
            \end{columns}
            \begin{figure}[ht]
                \centering
                \includegraphics[width=0.7\textwidth]{tikz/applications/baldock/graph-Baldock2019_Bristol.pdf}
                \caption{Plant-pollinator network of
                    Bristol\newline\cite{baldockSystemsApproachReveals2019}}
                \label{fig:label}
            \end{figure}
        \end{column}
        \begin{column}{0.5\textwidth}
            \begin{itemize}
                \item Modeling of various interactions, here ecosystems
                \item Structure necessary for: biodiversity monitoring, robustness, risk
                      of collapse
                \item Increasingly available
            \end{itemize}
        \end{column}
    \end{columns}
\end{frame}

\begin{frame}{Analysis methods for a network}
    Several methods~:
    \begin{itemize}
        \item Metrics~: degree, centrality, nesting \dots
        \item Network embedding with GNN
        \item \textbf<2>{\emph{Clustering} of nodes with latent variable models}
    \end{itemize}
\end{frame}

\begin{frame}
    \addtocounter{footnote}{1}
    \frametitle{Latent Block Model (LBM\footnotemark[\thefootnote])}
    %DONE remplacer i \in bullet par Zi = \bullet
    \cite{govaertEMAlgorithmBlock2005}.
    \begin{columns}
        \begin{column}{0.40\linewidth}
            \begin{figure}[H]
                \center
                \begin{tikzpicture}[scale=0.35]
                    \input{tikz/lbm.tex}
                \end{tikzpicture}
                \caption{Example of LBM\footnotemark[\thefootnote]}
                \label{fig:LBMvisu}
            \end{figure}
        \end{column}
        \only<1>{
            \begin{column}{0.51\linewidth}
                \begin{block}{Hierarchical model}
                    \vspace{-\baselineskip}
                    \begin{align*}
                        \forall q\in[\![ 1, Q_1]\!],~ & \mathbb{P}(Z_i = q) = \pi_q                          \\
                        \forall r\in[\![ 1, Q_2]\!],~ & \mathbb{P}(W_j = r) =  \rho_r                        \\
                                                      & Y_{ij} | Z_i, W_j \sim \mathcal{F}(\alpha_{Z_i,W_j})
                    \end{align*}
                    where $|\pi| = Q_1, |\rho| = Q_2, |\alpha| = Q_1 \times Q_2$
                \end{block}
                \begin{block}{Concise LBM formula}
                    $Y \sim \mathcal{F}\text{-BiSBM}_{n_1,n_2}(Q_1, Q_2, \pi, \rho, \alpha)$
                \end{block}
            \end{column}}
        \only<2>{
            \begin{column}{0.51\linewidth}
                With \begin{itemize}
                    \item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ fixed row blocks
                    \item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ fixed column blocks
                \end{itemize}
                \begin{block}{Parameters}
                    \begin{itemize}
                        \item $\pi_{{\color{blueind}\bullet}} = \mathbb{P}(Z_i = {\color{blueind}\bullet})$
                        \item $\rho_{{\color{burntorange}\bullet}} = \mathbb{P}(W_j = {\color{burntorange}\bullet})$
                        \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(Y_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$
                    \end{itemize}
                \end{block}
            \end{column}}
    \end{columns}

    \footnotetext[\thefootnote]{Which I will henceforth call BiSBM}
\end{frame}

\begin{frame}
    \frametitle{Multiple networks}
    \begin{figure}[ht]
        \centering
        \begin{subfigure}[ht]{0.3\textwidth}
            \includegraphics[width=1.1\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol.pdf}
            \caption{Bristol}
        \end{subfigure}
        \begin{subfigure}[ht]{0.3\textwidth}
            \includegraphics[width=1.1\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh.pdf}
            \caption{Edinburgh}
        \end{subfigure}
        \begin{subfigure}[ht]{0.3\textwidth}
            \includegraphics[width=1.1\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds.pdf}
            \caption{Leeds}
        \end{subfigure}
        \caption{Adjacency matrices,~\cite{baldockSystemsApproachReveals2019}}
        \label{fig:adj}
    \end{figure}
\end{frame}

\section[Bipartite collection models]{Bipartite network collection models}
\label{sec:extension-of-colsbm-to-bipartite-networks}
\begin{frame}
    \frametitle{Bipartite collections}
    \[
        \forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1^m, Q_2^m, \pi^m, \rho^m, \alpha^m)
    \]
    \onslide<2>{
        \begin{figure}[ht]
            \centering
            \begin{subfigure}[ht]{0.3\textwidth}
                \includegraphics[width=1.1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Bristol.pdf}
                \caption{Bristol}
            \end{subfigure}
            \begin{subfigure}[ht]{0.3\textwidth}
                \includegraphics[width=1.1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Edinburgh.pdf}
                \caption{Edinburgh}
            \end{subfigure}
            \begin{subfigure}[ht]{0.3\textwidth}
                \includegraphics[width=1.1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Leeds.pdf}
                \caption{Leeds}
            \end{subfigure}
            \caption{Reordered adjacency matrices, thanks to LBM}
            \label{fig:adj-reord}
        \end{figure}
    }
\end{frame}

\begin{frame}
    \frametitle{Different models}
    \onslide<1->{    \begin{block}{\emph{iid}-colBiSBM}
            \[
                \forall m \in \{1\dots M\}, Y^m \overset{iid}{\sim}
                \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi, \rho, \alpha)
            \]

            with $\theta = (\pi, \rho, \alpha)$.
        \end{block}}
    \onslide<2>{    \begin{block}{$\pi\rho$-colBiSBM}
            \[
                \forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim}
                \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi^m, \rho^m, \alpha)
            \]

            with $\theta = ((\pi^m)_{m=1,\dots, M}, (\rho^m)_{m=1,\dots, M}, \alpha)$.
        \end{block}
    }
\end{frame}
\begin{frame}
    \frametitle{Parameter estimation}
    % DONE say that tau i q m c' is the probability that Zim = q, approximation of the variational probability. Because we impose independence
    % By maximizing a variational lower bound of the
    % log-likelihood of the observed data.
    Maximizing the log-likelihood?
    \begin{block}{log-likelihood and complete log-likelihood}
        \[
            \ell(\bm{Y};\theta) = \sum_{\bm{Z,W}\in \bm{\mathcal{Z}\times\mathcal{W}}} \ell_c(\bm{Y}, \bm{Z}, \bm{W};\theta)
        \]

        with $\bm{\mathcal{Z}} = \{1,\dots,\alert<2>{Q_1}\}^{\alert<2>{n}},
            \bm{\mathcal{W}} = \{1,\dots,\alert<2>{Q_2}\}^{\alert<2>{n}}$
    \end{block}
    \uncover<3>{So, classic algorithm $\Rightarrow$
        \emph{Expectation-Maximization} (EM).}
\end{frame}

\begin{frame}
    \frametitle{By classic EM}
    At iteration $(t)$:
    \begin{itemize}
        \item[$\bullet$]\textbf{E Step}: calculate
              $$ \mathcal{Q}(\theta | \theta^{(t-1)}) =   \mathbb E_{\alert<2>{\bm Z, \bm W | \bm Y, \theta^{(t-1)}} } \left[\ell_c(\bm Y, \bm W, \bm Z; \theta)  \right] $$
        \item[$\bullet$]\textbf{M Step}:
              $$ \theta^{(t)} = \arg \max_{\theta} \mathcal{Q}(\theta | \theta^{(t-1)})$$
    \end{itemize}
    \uncover<2>{
        \begin{alertblock}{Problem for classic EM}
            Law of $\bm{Z,W|Y},\theta^{(t-1)}$ inaccessible
        \end{alertblock}}

\end{frame}

\begin{frame}
    By \emph{Variational EM}, as proposed
    by~\cite{daudinMixtureModelRandom2008,
        chabert-liddellLearningCommonStructures2024}.
    \begin{block}{Variational approximation of $\bm{Z,W|Y},\theta^{(t-1)}$}
        $\mathcal{R}_{Y^m,\tau}(\mathbf{Z}^m, \mathbf{W}^m) =
            \mathcal{R}^1_{Y^m,\tau}(\mathbf{Z}^m)
            {\color{red}\times}
            \mathcal{R}^2_{Y^m,\tau}(\mathbf{W}^m) \Rightarrow$ independence rows, columns.
    \end{block}
    \begin{multline*}
        \ell (\bm{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg(
        \color{black} \mathcal{Q}^m(\theta\mid\theta^{(t)}) +
        \mathcal{H}(\mathcal{R}_{Y^m,\theta^{(t)}}
        (\mathbf{Z}^m, \mathbf{W}^m))
        \color{red}\bigg) \color{black}
        \eqcolon \mathcal{J}(\tau;\theta)
    \end{multline*}
    where $\mathcal{Q}^m(\theta\mid\theta^{(t)}) =
        \mathbb{E}_{\mathbf{Z}^m,\mathbf{W}^m
        \sim \mathcal{R}_{Y^m,\tau}(.)}
        \left[ \ell_c(Y^m,\mathbf{Z}^m,\mathbf{W}^m | \theta) \right] \,$
\end{frame}

\begin{frame}{Developed formula of variational EM}
    \begin{multline*}
        \ell (\bm{Y};\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(Y^{m}_{ij}; \alpha_{qr}) \\
        + \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} \\
        - \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} \eqcolon
        \mathcal{J}(\tau;\theta),
    \end{multline*}

    \begin{block}{Variational approximation}
        $\tau_{iq}^{1,m} = \mathcal{R}^1_{Y^m,\tau}(Z_{iq}^m = 1)$
        and $\tau_{jr}^{2,m} = \mathcal{R}^2_{Y^m,\tau}(W_{jr}^m = 1)$
    \end{block}
\end{frame}

\begin{frame}{\emph{Variational Expectation} Step}
    \[
        \widehat{\tau}^{(t+1)} = \arg \max_{\tau}
        \mathcal{J}(\mathcal{\tau},\bm{\widehat{\theta}}^{(t)})
        \Leftrightarrow \arg\min_{\tau\in\mathcal{T}} \mathbf{KL}[\mathcal{R}_{\mathbf{Y},\tau}, \mathbb{P}(.|\mathbf{Y})]
    \]

    \begin{equation*}
        \begin{cases}
            \widehat{\tau}_{iq}^{1,m} \propto \widehat{\pi}_{q}^{m(t)} \prod_{j=1}^{n_2^m}\prod_{r\in\mathcal{Q}_2^m} f(Y_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{jr}^{2,m(t+1)}}  & \forall i = 1, \dots , n_1^m, q \in \mathcal{Q}_1^m \\
            \widehat{\tau}_{jr}^{2,m} \propto \widehat{\rho}_{r}^{m(t)} \prod_{i=1}^{n_1^m}\prod_{q\in\mathcal{Q}_1^m} f(Y_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{iq}^{1,m(t+1)}} & \forall j = 1, \dots , n_2^m, r \in \mathcal{Q}_2^m
        \end{cases}
    \end{equation*}
    \footnotetext[2]{Initialization of $\widehat{\tau}$ with a
        \emph{spectral clustering} on the networks.}
\end{frame}

\begin{frame}{\emph{Maximization} Step}
    \[
        \widehat{\theta}^{(t+1)} = \arg \max_{\theta} \mathcal{J}(\mathcal{\bm{\widehat{\tau}}}^{(t+1)},\theta)
    \]

    \begin{block}{Connectivity parameters}
        \begin{align*}
            \widehat{\alpha}_{qr} = \frac{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m} \alert<2>{Y_{ij}^m}}{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m}}
        \end{align*}
    \end{block}
    \only<1>{
        \begin{block}{Proportions for \emph{iid}}
            \begin{align*}
                \widehat{\pi}_q = \frac{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \tau_{iq}^{1,m}}{\sum_{m=1}^{M} n_1^m} &  &
                \widehat{\rho}_r = \frac{\sum_{m=1}^{M} \sum_{j=1}^{n_2^m} \tau_{jr}^{2,m}}{\sum_{m=1}^{M} n_2^m}
            \end{align*}
        \end{block}
    }
    \only<2>{
        \begin{block}{Proportions for $\pi\rho$}
            \begin{align*}
                \widehat{\pi}^{\color{red}m}_q = \frac{\sum_{i=1}^{n_1^m} \tau_{iq}^{1,m}}{n_1^m} &  &
                \widehat{\rho}^{\color{red}m}_r = \frac{\sum_{j=1}^{n_2^m} \tau_{jr}^{2,m}}{n_2^m}
            \end{align*}
        \end{block}
    }

\end{frame}

\section{Model selection}
\begin{frame}
    \frametitle{Problem of choosing $(Q_1, Q_2)$}
    Need to select $Q_1$ and $Q_2$. BIC-Like criterion\footnote{ICL + Entropy + penalty}

    \begin{align*}
        \text{BIC-L}(\bm{Y}, Q_1, Q_2) & = \max_{\theta} \mathbb{E}_{\mathcal{R}_{\mathbf{Y},\hat{\tau}}} [\ell_c(\bm{Y,Z,W};\theta)] + \mathcal{H(\mathcal{R}_{\mathbf{Y},\hat{\tau}})} - \frac{1}{2}\text{pen}(\theta, Q_1, Q_2) \\
                                       & = \max_{\theta} \mathcal{J(\mathcal{R}_{\mathbf{Y},\hat{\tau}}, \theta)} - \frac{1}{2}\text{pen}(\theta, Q_1, Q_2)
    \end{align*}

    \begin{alertblock}{Exploration problems}
        \begin{itemize}
            \item Exploration of $\mathbb{N}^2$ costly.
            \item Sensitivity to initializations.
        \end{itemize}
    \end{alertblock}
\end{frame}

\begin{frame}
    \frametitle{Choice of $(Q_1,Q_2)$ - Greedy approach}
    \begin{columns}
        \begin{column}{0.5\linewidth}
            \begin{tikzpicture}
                \input{tikz/greedy-exploration.tex}
            \end{tikzpicture}
        \end{column}
        \begin{column}{0.35\linewidth}
            \begin{itemize}
                \item Initial model~:\\
                      \begin{tikzpicture}
                          \draw[fill=gray, draw=gray] circle [radius=0.225cm];
                      \end{tikzpicture}
                      \onslide<2->{
                \item Model after \emph{split}~:
                      \begin{tikzpicture}
                          \draw[fill=blueind, draw=blueind] circle [radius=0.225cm];
                      \end{tikzpicture}
                \item Model maximizing the criterion~:\\
                      \begin{tikzpicture}
                          \draw[fill=white, draw=green, very thick] circle [radius=0.225cm];
                      \end{tikzpicture}
                      }
                      \onslide<3->{
                \item Model after \emph{merge}~:
                      \begin{tikzpicture}
                          \draw[fill=red, draw=red] circle [radius=0.225cm];
                      \end{tikzpicture}
                      }
            \end{itemize}
        \end{column}
    \end{columns}
\end{frame}
\begin{frame}
    \frametitle{Choice of $(Q_1,Q_2)$ - Sliding window}
    \begin{columns}
        \begin{column}{0.6\textwidth}
            \begin{figure}
                \input{tikz/moving-window}
                \caption{Sliding window}
            \end{figure}
        \end{column}
        \begin{column}{0.4\textwidth}
            \only<3>{\begin{block}{}
                    Initialization of the model if necessary
                \end{block}}
            \only<9>{\begin{block}{}
                    Localization of the new mode
                \end{block}}
            \only<10>{\begin{block}{}
                    Move to the new mode then iterate
                \end{block}}
        \end{column}
    \end{columns}
\end{frame}

\section{Application}
\label{sec:application}

\begin{frame}
    \frametitle{Results~\cite{baldockSystemsApproachReveals2019}}

    \only<1>{
        \begin{figure}[ht]
            \centering
            \begin{subfigure}[t]{0.5\textwidth}
                \centering
                \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol.pdf}
                \caption{Bristol}
            \end{subfigure}\hfil
            \begin{subfigure}[t]{0.5\textwidth}
                \centering
                \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh.pdf}
                \caption{Edinburgh}
            \end{subfigure}
            \newline
            \begin{subfigure}[ht]{0.5\textwidth}
                \centering
                \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds.pdf}
                \caption{Leeds}
            \end{subfigure}\hfil
            \begin{subfigure}[ht]{0.5\textwidth}
                \centering
                \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Reading.pdf}
                \caption{Reading}
            \end{subfigure}
            \caption{Adjacency matrices,~\cite{baldockSystemsApproachReveals2019}}
        \end{figure}
    }
    \only<2>{
        \begin{figure}[ht]
            \centering
            \begin{subfigure}[t]{0.5\textwidth}
                \centering
                \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol.pdf}
                \caption{Bristol}
            \end{subfigure}\hfil
            \begin{subfigure}[t]{0.5\textwidth}
                \centering
                \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh.pdf}
                \caption{Edinburgh}
            \end{subfigure}
            \newline
            \begin{subfigure}[ht]{0.5\textwidth}
                \centering
                \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
                \caption{Leeds}
            \end{subfigure}\hfil
            \begin{subfigure}[ht]{0.5\textwidth}
                \centering
                \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading.pdf}
                \caption{Reading}
            \end{subfigure}
            \caption{Reordered adjacency matrices by \emph{iid}-colBiSBM,~\cite{baldockSystemsApproachReveals2019}}
        \end{figure}
    }
\end{frame}

\begin{frame}
    \frametitle{Network clustering}
    \begin{figure}[ht]
        \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2011_TB+Baldock2011_JN.pdf}
        \caption{Adjacency matrix,~\cite{baldockDailyTemporalStructure2011}}
    \end{figure}
\end{frame}

\begin{frame}[allowframebreaks]
    \frametitle{Application to~\cite{baldockDailyTemporalStructure2011,
            baldockSystemsApproachReveals2019}}
    \begin{figure}[t]
        \centering
        \begin{subfigure}{0.5\textwidth}
            \centering
            \includegraphics[scale=0.2,angle=-90]{backup-app-iid.png}
            \caption{Model $iid$}
        \end{subfigure}%
        ~
        \begin{subfigure}{0.5\textwidth}
            \centering
            \includegraphics[scale=0.2,angle=-90]{backup-app-pirho.png}
            \caption{Model $\pi\rho$}
        \end{subfigure}%
        \caption{Partitioning of networks
            of~\cite{baldockDailyTemporalStructure2011,
                baldockSystemsApproachReveals2019}}
    \end{figure}

    \begin{figure}[t]
        \centering
        \begin{subfigure}{0.5\textwidth}
            \centering
            \includegraphics[scale=0.1]{backup-app-iid-struct1.png}
            \includegraphics[scale=0.2]{backup-app-iid-struct2.png}
            \caption{Model $iid$,\\
                separate African network and English networks}
        \end{subfigure}%
        ~
        \begin{subfigure}{0.5\textwidth}
            \centering
            \includegraphics[scale=0.2]{backup-app-pirho-struct.png}
            \caption{Model $\pi\rho$,\\
                merge African and English networks}
        \end{subfigure}%
        \caption{Structures detected for networks
            of~\cite{baldockDailyTemporalStructure2011,
                baldockSystemsApproachReveals2019}}
    \end{figure}
\end{frame}

\begin{frame}{Clustering algorithm}
    \centering
    \vspace{0.25\baselineskip}
    \begin{tikzpicture}[scale=0.85]
        \input{tikz/clustering.tex}
    \end{tikzpicture}
    \[
        D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \widetilde{\alpha}_{qr}^{m} - \widetilde{\alpha}_{qr}^{m'}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'})
    \]
\end{frame}

\begin{frame}{Results}
    \begin{figure}[ht]
        \centering
        \begin{subfigure}{0.5\textwidth}
            \centering
            \includegraphics[width=1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2011_TB+Baldock2011_JN.pdf}
            \caption{Reordered by LBM}
        \end{subfigure}\hfil
        \begin{subfigure}{0.5\textwidth}
            \centering
            \includegraphics[width=1\textwidth]{tikz/applications/baldock/pirho-colbisbm-mat-Baldock2011_TB+Baldock2011_JN.pdf}
            \caption{Reordered by $\pi\rho$-colBiSBM}
        \end{subfigure}

        \caption{Reordered adjacency matrix by $\pi\rho$-colBiSBM,~\cite{baldockDailyTemporalStructure2011}}
    \end{figure}
\end{frame}

\section{Conclusion}
\label{sec:conclusion}
\begin{frame}
    \frametitle{Conclusion and perspectives}
    % DONE Add a conclusion perspective slide
    % Recall models with clustering
    % Mention analysis of corrected networks for sampling
    % Link to the package
    \begin{block}{Capabilities}
        \begin{itemize}
            \item 4 models including 3 with flexibility on at least one of
                  the dimensions (adaptability to data).
            \item Detect classic and less classic structures in an agnostic way.
            \item Partition a set of networks according to their structures.
        \end{itemize}
    \end{block}

\end{frame}

\begin{frame}{Perspectives}
    \begin{itemize}
        \item Investigate stability against randomness and local \emph{optima}.
        \item Proof of identifiability of the $\pi\rho$ model.
    \end{itemize}

    \begin{block}{Package and applications}
        \begin{itemize}
            \item Integration into the \texttt{colSBM} package, improvement of user interface and
                  addition of ecologists' feedback
            \item CRAN publication
            \item Integrate the possibility of an additional criterion for clustering
            \item Apply clustering to data from
                  \cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021}
        \end{itemize}

    \end{block}
    \bigskip
    \centering
    Thank you for your attention~!
\end{frame}