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90
annexe.tex
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@ -1,3 +1,16 @@
\section{Clustering}
\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}
\section{VEM} \section{VEM}
\begin{frame}{Developed formula of variational EM} \begin{frame}{Developed formula of variational EM}
@ -75,6 +88,7 @@
\end{align*} \end{align*}
Thus $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$ Thus $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$
\end{frame} \end{frame}
\section{Model selection}
\begin{frame} \begin{frame}
\frametitle{On the BIC-L} \frametitle{On the BIC-L}
@ -97,8 +111,6 @@
\] \]
\end{frame} \end{frame}
\section{Model selection}
\begin{frame} \begin{frame}
\frametitle{Choice of $(Q_1,Q_2)$ - Greedy approach} \frametitle{Choice of $(Q_1,Q_2)$ - Greedy approach}
\begin{columns} \begin{columns}
@ -155,78 +167,4 @@
\end{block}} \end{block}}
\end{column} \end{column}
\end{columns} \end{columns}
\end{frame}
\section{Clustering}
\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}
\section{Results~\cite{baldockSystemsApproachReveals2019,baldockDailyTemporalStructure2011}}
\begin{frame}[allowframebreaks]
\begin{figure}[ht]
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol.pdf}
\caption{Donnée}
\end{subfigure}\hfil
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol.pdf}
\caption{Reordered}
\end{subfigure}
\caption{Bristol}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh.pdf}
\caption{Donnée}
\end{subfigure}\hfil
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh.pdf}
\caption{Reordered}
\end{subfigure}
\caption{Edinburgh}
\end{figure}
\begin{figure}
\begin{subfigure}[ht]{0.5\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds.pdf}
\caption{Donnée}
\end{subfigure}\hfil
\begin{subfigure}[ht]{0.5\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
\caption{Réordonnée}
\end{subfigure}
\caption{Leeds}
\end{figure}
\begin{figure}
\begin{subfigure}[ht]{0.5\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Reading.pdf}
\caption{Donnée}
\end{subfigure}\hfil
\begin{subfigure}[ht]{0.5\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading.pdf}
\caption{Réordonnée}
\end{subfigure}
\caption{Reading}
\end{figure}
\end{frame} \end{frame}

6
presentation.tex
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@ -112,10 +112,10 @@
\subtitle{JdS 2025} \subtitle{JdS 2025}
\title[Bipartite networks collection]{Joint analysis of bipartite networks collection} \title[Bipartite networks collection]{Joint estimation of bipartite network collections. Application to plant-pollinator networks.}
\author[L. Lacoste]{\underline{Louis Lacoste}, Pierre Barbillon and \author[L. Lacoste]{\underline{Louis Lacoste}, Pierre Barbillon and
Sophie Donnet\newline Laboratoire MIA Paris-Saclay\newline\ccbysa} Sophie Donnet\newline UMR MIA Paris-Saclay, AgroParisTech, INRAE, Université Paris-Saclay\newline\ccbysa}
\date{\today} \date{03 Juin 2025}
\begin{document} \begin{document}

76
principal.tex
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@ -56,14 +56,14 @@
\begin{column}{0.4\textwidth} \begin{column}{0.4\textwidth}
\only<1>{ \only<1>{
\begin{itemize} \begin{itemize}
\item A bipartite graph $G = (U,V,E)$ \item Bipartite graph $G = (U,V,E)$
\item Can be encoded by a bi-adjacency matrix $Y \in \{0,1\}^{n_1 \times n_2}$ \item Encoded in bi-adjacency matrix $Y \in \{0,1\}^{n_1 \times n_2}$
\end{itemize}} \end{itemize}}
\only<2>{ \only<2>{
\begin{itemize} \begin{itemize}
\item Increasingly available \item Increasingly available
\item Modeling of various interactions, here ecosystems \item Ecosystems described by their interactions
\item Structure necessary for: biodiversity monitoring, robustness, risk of collapse \item Functional structure for: biodiversity monitoring, robustness, risk of collapse
\end{itemize}} \end{itemize}}
\end{column} \end{column}
\end{columns} \end{columns}
@ -139,7 +139,7 @@
\begin{tikzpicture}[scale=0.35] \begin{tikzpicture}[scale=0.35]
\input{tikz/lbm.tex} \input{tikz/lbm.tex}
\end{tikzpicture} \end{tikzpicture}
\caption{Example of LBM\footnotemark[\thefootnote]} \caption{Example of BiSBM}
\label{fig:LBMvisu} \label{fig:LBMvisu}
\end{figure} \end{figure}
\end{column} \end{column}
@ -192,8 +192,7 @@
\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Reading.pdf} \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Reading.pdf}
\caption{Reading, $Q_1 = 3, Q_2 = 3$} \caption{Reading, $Q_1 = 3, Q_2 = 3$}
\end{subfigure} \end{subfigure}
\vspace{-\baselineskip} \caption{Separate BiSBM fit for each network}
\caption{Reordered adjacency matrices, using BiSBM for each network}
\label{fig:adj-reord} \label{fig:adj-reord}
\end{figure} \end{figure}
} }
@ -215,7 +214,7 @@
\onslide<2>{ \begin{block}{$\pi\rho$-colBiSBM} \onslide<2>{ \begin{block}{$\pi\rho$-colBiSBM}
\[ \[
\forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim}
\mathcal{B}ern\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi^m, \rho^m, \alpha) \mathcal{B}ern\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi\alert{^m}, \rho\alert{^m}, \alpha)
\] \]
with $\theta = ((\pi\alert{^m})_{m=1,\dots, M}, (\rho\alert{^m})_{m=1,\dots, with $\theta = ((\pi\alert{^m})_{m=1,\dots, M}, (\rho\alert{^m})_{m=1,\dots,
@ -264,21 +263,20 @@
\label{sec:inference-and-model-selection} \label{sec:inference-and-model-selection}
\begin{frame}{Parameter estimation}{How ?} \begin{frame}{Parameter estimation}{How ?}
\begin{align*} \begin{align*}
\ell(\mathbf{Y};\theta) = & \sum_{m=1}^{M} \ell(Y^m;\theta) \\ \ell(\mathbf{Y};\theta) = & \sum_{m=1}^{M} \ell(Y^m;\theta) \\
= & \sum_{m=1}^{M} \log \int_{\alert<2->{\mathcal{Z}^m\times\mathcal{W}^m}} \exp\{\ell_c(Y^m,Z^m,W^m;\theta)\} dZ^m dW^m \\ = & \sum_{m=1}^{M} \log \sum_{\alert<2->{Z^m \in \mathcal{Z}^m,W^m\in\mathcal{W}^m}} \exp\{\ell_c(Y^m,Z^m,W^m;\theta)\} \\
= & \sum_{m=1}^{M} \log\int_{\alert<2->{\mathcal{Z}^m\times\mathcal{W}^m}}\exp\{\ell(Y^m | Z^m,W^m;\alpha) + \\ = & \sum_{m=1}^{M} \log\sum_{\alert<2->{Z^m \in \mathcal{Z}^m,W^m\in\mathcal{W}^m}}\exp\{\ell(Y^m | Z^m,W^m;\alpha) + \\
& \ell(Z^m;\pi) + \ell(W^m;\rho)\} dZ^m dW^m & \ell(Z^m;\pi) + \ell(W^m;\rho)\}
% & = \sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{q=1}^{Q_1} Z_{iq} \log(\pi_q) + \sum_{j=1}^{n_2^m}\sum_{r=1}^{Q_2} W_{jr} \log(\rho_r) \\ % & = \sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{q=1}^{Q_1} Z_{iq} \log(\pi_q) + \sum_{j=1}^{n_2^m}\sum_{r=1}^{Q_2} W_{jr} \log(\rho_r) \\
% & + \sum_{i,j}\sum_{q,r} Z_{iq}W_{jr} \log \mathcal{B}ern(Y_{ij};\alpha_{qr}) % & + \sum_{i,j}\sum_{q,r} Z_{iq}W_{jr} \log \mathcal{B}ern(Y_{ij};\alpha_{qr})
\end{align*} \end{align*}
\onslide<3>{ \onslide<3>{
We would like to use Expectation-Maximization (EM) algorithm~\parencite{dempsterMaximumLikelihoodIncomplete1977} but the law of $\mathbf{Z,W|Y},\theta^{(t-1)}$ is untractable due to dependence between row and column groups.} EM impracticable since $\mathbf{Z,W|Y}$ intractable due to
conditional dependency.}
\end{frame} \end{frame}
\begin{frame}{Parameter estimation}{Solution} \begin{frame}{Parameter estimation}{Solution}
By \emph{Variational EM}, as proposed \emph{Variational EM}~\cite{daudinMixtureModelRandom2008,chabert-liddellLearningCommonStructures2024}.
by~\cite{daudinMixtureModelRandom2008} and adapted for joint simple networks
by~\cite{chabert-liddellLearningCommonStructures2024}.
\begin{block}{Variational approximation of $\bm{Z,W|Y},\theta^{(t-1)}$} \begin{block}{Variational approximation of $\bm{Z,W|Y},\theta^{(t-1)}$}
$\mathcal{R}_{Y^m,\tau}(Z^m, W^m) = $\mathcal{R}_{Y^m,\tau}(Z^m, W^m) =
\mathcal{R}^1_{Y^m,\tau}(Z^m) \mathcal{R}^1_{Y^m,\tau}(Z^m)
@ -298,26 +296,25 @@
\end{frame} \end{frame}
\begin{frame}{Selection criterion for $Q_1, Q_2$} \begin{frame}{Selection criterion for $Q_1, Q_2$}
\cite{biernackiAssessingMixtureModel2000} introduced the Integrated Classification Likelihood (ICL): Integrated Classification Likelihood (ICL)~\cite{biernackiAssessingMixtureModel2000}
\begin{align*} \begin{align*}
\text{ICL}(\bm{Y}, Q_1, Q_2) & = \mathbb{E} [\ell_c(\bm{Y,Z,W};\hat{\theta})] -\frac{1}{2}\text{pen}(Q_1, Q_2) \\ \text{ICL}(\bm{Y}, Q_1, Q_2) & = \mathbb{E} [\ell_c(\bm{Y,Z,W};\hat{\theta})] -\frac{1}{2}\text{pen}(Q_1, Q_2) \\
& = \ell(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z,W}|\mathbf{Y},\hat{\theta})) - \frac{1}{2}\text{pen}(Q_1, Q_2) & = \ell(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z,W}|\mathbf{Y},\hat{\theta})) - \frac{1}{2}\text{pen}(Q_1, Q_2)
\end{align*} leads to low entropy clustering. Common in literature for SBM. \end{align*} For SBM~\cite{daudinMixtureModelRandom2008}.
\onslide<2->{ \onslide<2->{
\begin{align*} \begin{align*}
\text{BIC-L}(\bm{Y}, & Q_1, Q_2) = \mathbb{E}_{\mathcal{R}_{\mathbf{Y},\hat{\tau}}} [\ell_c(\bm{Y,Z,W};\hat{\theta}^{\text{var}})] + \mathcal{H(\mathcal{R}_{\mathbf{Y},\hat{\tau}})} - \frac{1}{2}\text{pen}(Q_1, Q_2) \\ \text{BIC-L}(\bm{Y}, Q_1, Q_2) & = \mathbb{E}_{\mathcal{R}_{\mathbf{Y},\hat{\tau}}} [\ell_c(\bm{Y,Z,W};\hat{\theta}^{\text{var}})] + \mathcal{H(\mathcal{R}_{\mathbf{Y},\hat{\tau}})} - \frac{1}{2}\text{pen}(Q_1, Q_2) \\
& = \mathcal{J(\mathcal{R}_{\mathbf{Y},\hat{\tau}}, \hat{\theta}^{\text{var}})} - \frac{1}{2}\text{pen}(Q_1, Q_2) \textcolor{red}{\leq \log p(\mathbf{Y};\hat{\theta}^{\text{MV}})- \frac{1}{2}\text{pen}(Q_1, Q_2)} \\ & = \mathcal{J(\mathcal{R}_{\mathbf{Y},\hat{\tau}}, \hat{\theta}^{\text{var}})} - \frac{1}{2}\text{pen}(Q_1, Q_2)
\end{align*} \end{align*}
because we want fuzzier clustering.
} }
\end{frame} \end{frame}
\begin{frame}{Practical problems of choosing $Q_1, Q_2$} \begin{frame}{Practical problems of choosing $Q_1, Q_2$}
\begin{alertblock}{Exploration problems} \begin{alertblock}{Exploration problems}
\begin{itemize} \begin{itemize}
\item Exploration of a 2D grid is costly. \uncover<2->{$\rightarrow$ \textbf{Greedy \item Sensitivity to initializations. \uncover<2->{$\rightarrow$ \textbf{Spectral
approach} and \textbf{sliding window}}
\item Sensitivity to initializations. \uncover<3->{$\rightarrow$ \textbf{Spectral
clustering} and \textbf{split \& merge} approach} clustering} and \textbf{split \& merge} approach}
\item Exploration of a 2D grid is costly. \uncover<3->{$\rightarrow$ \textbf{Greedy
approach} and \textbf{sliding window}}
\end{itemize} \end{itemize}
\end{alertblock} \end{alertblock}
\end{frame} \end{frame}
@ -351,25 +348,25 @@
\begin{subfigure}[t]{0.5\textwidth} \begin{subfigure}[t]{0.5\textwidth}
\centering \centering
\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol.pdf} \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol.pdf}
\caption{Bristol, $Q_1 = 3, Q_2 = 5$} \caption{Bristol}
\end{subfigure}\hfil \end{subfigure}\hfil
\begin{subfigure}[t]{0.5\textwidth} \begin{subfigure}[t]{0.5\textwidth}
\centering \centering
\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh.pdf} \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh.pdf}
\caption{Edinburgh, $Q_1 = 3, Q_2 = 5$} \caption{Edinburgh}
\end{subfigure} \end{subfigure}
\newline \newline
\begin{subfigure}[ht]{0.5\textwidth} \begin{subfigure}[ht]{0.5\textwidth}
\centering \centering
\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf} \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
\caption{Leeds, $Q_1 = 3, Q_2 = 5$} \caption{Leeds}
\end{subfigure}\hfil \end{subfigure}\hfil
\begin{subfigure}[ht]{0.5\textwidth} \begin{subfigure}[ht]{0.5\textwidth}
\centering \centering
\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading.pdf} \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading.pdf}
\caption{Reading, $Q_1 = 3, Q_2 = 5$} \caption{Reading}
\end{subfigure} \end{subfigure}
\caption{Reordered adjacency matrices by \emph{iid}-colBiSBM,~\cite{baldockSystemsApproachReveals2019}} \caption{\emph{iid}-colBiSBM fit, $Q_1 = 3, Q_2 = 5$}
\end{figure}} \end{figure}}
\end{frame} \end{frame}
@ -412,14 +409,14 @@
\begin{column}{0.2\textwidth} \begin{column}{0.2\textwidth}
\begin{figure} \begin{figure}
\onslide<3>{ \onslide<2>{
\begin{subfigure}[t]{0.7\textwidth} \begin{subfigure}[t]{0.7\textwidth}
\centering \centering
\includegraphics[width=1\textwidth]{img/baldock/bombus-hortorum.jpeg} \includegraphics[width=1\textwidth]{img/baldock/bombus-hortorum.jpeg}
\caption{\emph{Bombus Hortorum} or garden bumblebee} \caption{\emph{Bombus Hortorum} or garden bumblebee}
\end{subfigure} \end{subfigure}
} }
\onslide<4>{ \onslide<3>{
\begin{subfigure}[t]{0.7\textwidth} \begin{subfigure}[t]{0.7\textwidth}
\centering \centering
\includegraphics[width=1\textwidth]{img/baldock/bombus-lapidarius.jpeg} \includegraphics[width=1\textwidth]{img/baldock/bombus-lapidarius.jpeg}
@ -448,13 +445,13 @@
bottom color=blue!1!white, bottom color=blue!1!white,
anchor=right corner, minimum height=42mm, label={[label distance = 2mm]207:Generalists}, label={[label distance = 12mm]357:Specialists}] (T) at ($(struct.north east)+(-1,-2.5)$) {}; anchor=right corner, minimum height=42mm, label={[label distance = 2mm]207:Generalists}, label={[label distance = 12mm]357:Specialists}] (T) at ($(struct.north east)+(-1,-2.5)$) {};
\only<3>{ \only<2>{
\node[left = 3mm of gen] (towns_gen_garden) {B, L}; \node[left = 3mm of gen] (towns_gen_garden) {B, L};
\node[left = 3mm of spe] (towns_spe_garden) {\phantom{B, }E, R}; \node[left = 3mm of spe] (towns_spe_garden) {\phantom{B, }E, R};
\path (towns_gen_garden) edge[->,thick] (gen); \path (towns_gen_garden) edge[->,thick] (gen);
\path (towns_spe_garden) edge[->,thick] (spe); \path (towns_spe_garden) edge[->,thick] (spe);
} }
\only<4>{ \only<3>{
\node[left = 3mm of interm] (towns_interm_red) {L}; \node[left = 3mm of interm] (towns_interm_red) {L};
\node[left = 3mm of spe] (towns_spe_red) {B, E, R}; \node[left = 3mm of spe] (towns_spe_red) {B, E, R};
\path (towns_interm_red) edge[->,thick] (interm); \path (towns_interm_red) edge[->,thick] (interm);
@ -471,22 +468,21 @@
\section{Conclusion} \section{Conclusion}
\begin{frame} \begin{frame}
\frametitle{Conclusion and perspectives} \frametitle{Conclusion and perspectives}
\begin{block}{Capabilities} \begin{block}{Summary}
\begin{itemize} \begin{itemize}
\item 4 models including 3 with flexibility on at least one of \item 4 models including 3 with flexibility on at least one of
the dimensions (adaptability to data). the dimensions (adaptability to data).
\item Detect classic and less classic structures in an agnostic way. \item Jointly detect classic and less classic structures agnostically.
\item Partition a set of networks according to their structures. \item Partition a collection in sub-collections with homogeneous structures.
\item \texttt{R} package \texttt{colSBM} at \url{https://github.com/GrossSBM/colSBM}
\end{itemize} \end{itemize}
\end{block} \end{block}
\begin{block}{Package and applications} \begin{block}{Future work}
\begin{itemize} \begin{itemize}
\item Article in redaction \item Article in redaction
\item \texttt{R} package \texttt{colSBM} on
Github\footnote{\url{https://github.com/GrossSBM/colSBM}}
\item Apply clustering to data from \item Apply clustering to data from
\cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021} \cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021}
to tell if interaction drives the structure of the network. to tell if interaction types drives the structure of the network.
\end{itemize} \end{itemize}
\end{block} \end{block}
\end{frame} \end{frame}