Ajout retours PS
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10
annexe.tex
10
annexe.tex
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@ -76,6 +76,15 @@
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Thus $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$
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\end{frame}
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\begin{frame}
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\frametitle{On the BIC-L}
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\begin{align*}
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& \text{ICL}(\hat{\theta}) = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\log p(\mathbf{Y}|\mathbf{Z},\mathbf{W};\hat{\theta})] - \frac{1}{2} \text{pen}(\dots) \\
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& \text{BIC}(\hat{\theta}) = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\log p(\mathbf{Y}|\mathbf{Z},\mathbf{W};\hat{\theta})] + \mathcal{H}(p(\mathbf{Z},\mathbf{W}|\mathbf{Y})) - \frac{1}{2} \text{pen}(\dots) \\
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& \text{BIC-L}(\hat{\theta}, \hat{\tau}) = \Esp_{\mathcal{R}_{\mathbf{Y}, \hat{\tau}}}[\log \ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta}^{\text{var}})] + \mathcal{H}(\mathcal{R}_{\mathbf{Y}, \hat{\tau}}) - \frac{1}{2} \text{pen}(\dots) \\
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\end{align*}
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\end{frame}
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\section{Model selection}
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\begin{frame}
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@ -112,6 +121,7 @@
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}
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\frametitle{Choice of $(Q_1,Q_2)$ - Sliding window}
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\begin{columns}
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109
principal.tex
109
principal.tex
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@ -1,6 +1,5 @@
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\section{Model Context}
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\label{sec:context-of-the-model}
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\begin{frame}
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\frametitle{Why a network?}
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\begin{columns}
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@ -55,6 +54,7 @@
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\item node level: degree, centrality\dots
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\item network level: density, nestedness\dots
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\end{itemize}
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\cite{kolaczykStatisticalAnalysisNetwork2009}
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\item \textbf<2>{Node embedding and/or clustering with latent variable models}
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\\\cite{snijdersEstimationPredictionStochastic1997,hoffLatentSpaceApproaches2002}
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\item Node or network embedding with Graph Convolutional Networks
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@ -64,8 +64,7 @@
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\begin{frame}
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\addtocounter{footnote}{1}
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\frametitle{Latent Block Model (LBM\footnotemark[\thefootnote])}
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\cite{govaertEMAlgorithmBlock2005}.
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\frametitle{Bipartite Stochastic Block Model (BiSBM\footnotemark[\thefootnote])}\framesubtitle{\cite{govaertEMAlgorithmBlock2005}}
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\begin{columns}
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\begin{column}{0.40\linewidth}
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\begin{figure}[H]
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@ -108,7 +107,7 @@
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\end{column}}
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\end{columns}
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\footnotetext[\thefootnote]{Which I will henceforth call BiSBM}
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\footnotetext[\thefootnote]{Commonly Known as \emph{Latent Block Model} (LBM) in the literature.}
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\end{frame}
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\begin{frame}
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@ -160,11 +159,11 @@
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\label{sec:extension-of-colsbm-to-bipartite-networks}
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\begin{frame}
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\frametitle{Model 0: sep-BiSBM}
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\footnotesize
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$
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\forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1\alert<2>{^m}, Q_2\alert<2>{^m}, \pi\alert<2>{^m}, \rho\alert<2>{^m}, \alpha\alert<2>{^m})
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$
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\onslide<3>{
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\only<1-2>{
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\begin{equation*}
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\forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1\alert<2->{^m}, Q_2\alert<2->{^m}, \pi\alert<2->{^m}, \rho\alert<2->{^m}, \alpha\alert<2->{^m})
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\end{equation*}}
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\only<3>{
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[ht]{0.42\textwidth}
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@ -188,6 +187,7 @@
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Reading.pdf}
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\caption{Reading}
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\end{subfigure}
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\vspace{-\baselineskip}
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\caption{Reordered adjacency matrices, using BiSBM for each network}
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\label{fig:adj-reord}
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\end{figure}
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@ -249,32 +249,32 @@
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% \end{alertblock}}
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% \end{frame}
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\section{Inference and model selection}
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\label{sec:inference-and-model-selection}
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\begin{frame}{Parameter estimation}
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By \emph{Variational EM}, as proposed
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by~\cite{daudinMixtureModelRandom2008,
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chabert-liddellLearningCommonStructures2024}.
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\begin{block}{Variational approximation of $\bm{Z,W|Y},\theta^{(t-1)}$}
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$\mathcal{R}_{Y^m,\tau}(\mathbf{Z}^m, \mathbf{W}^m) =
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\mathcal{R}^1_{Y^m,\tau}(\mathbf{Z}^m)
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$\mathcal{R}_{Y^m,\tau}(Z^m, W^m) =
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\mathcal{R}^1_{Y^m,\tau}(Z^m)
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{\color{red}\times}
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\mathcal{R}^2_{Y^m,\tau}(\mathbf{W}^m) \Rightarrow$ independence rows, columns.
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\mathcal{R}^2_{Y^m,\tau}(W^m) \Rightarrow$ independence between rows and columns.
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\end{block}
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\begin{multline*}
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\ell (\bm{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg(
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\color{black} \mathcal{Q}^m(\theta\mid\theta^{(t)}) +
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\mathcal{H}(\mathcal{R}_{Y^m,\theta^{(t)}}
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(\mathbf{Z}^m, \mathbf{W}^m))
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(Z^m, W^m))
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\color{red}\bigg) \color{black}
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\eqcolon \mathcal{J}(\tau;\theta)
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\end{multline*}
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where $\mathcal{Q}^m(\theta\mid\theta^{(t)}) =
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\mathbb{E}_{\mathbf{Z}^m,\mathbf{W}^m
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\mathbb{E}_{Z^m,W^m
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\sim \mathcal{R}_{Y^m,\tau}(.)}
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\left[ \ell_c(Y^m,\mathbf{Z}^m,\mathbf{W}^m | \theta) \right] \,$
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\left[ \ell_c(Y^m,Z^m,W^m | \theta) \right] \,$
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\end{frame}
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\section{Model selection}
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\begin{frame}
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\frametitle{Problem of choosing $(Q_1, Q_2)$}
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Need to select $Q_1$ and $Q_2$. BIC-Like criterion\footnote{ICL + entropy - penalty}
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@ -289,7 +289,7 @@
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\item Exploration of a 2D grid is costly. \uncover<2->{$\rightarrow$ \textbf{Greedy
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approach} and \textbf{sliding window}}
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\item Sensitivity to initializations. \uncover<3->{$\rightarrow$ \textbf{Spectral
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clustering} and \textbf{reuse of previous inits}}
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clustering} and \textbf{split \& merge} approach}
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\end{itemize}
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\end{alertblock}
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\end{frame}
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@ -328,25 +328,26 @@
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\begin{frame}
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\frametitle{Results~\cite{baldockSystemsApproachReveals2019} focus on Leeds}
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\captionsetup{font=normalsize}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Leeds.pdf}
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\caption{Separate model}
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\caption{Leeds with sep-BiSBM}
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\end{subfigure}\hfill
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=1\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
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\caption{Joint model}
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\caption{Leeds with \emph{iid}-colBiSBM}
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\end{subfigure}
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\caption{Reordered adjacency matrix by sep-BiSBM (left) and by \emph{iid}-colBiSBM (right),~\cite{baldockSystemsApproachReveals2019}}
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\end{figure}
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\end{frame}
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\begin{frame}{\emph{Bombus}}
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\only<1>{
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\begin{figure}
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\captionsetup{font=normalsize}
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{img/baldock/bombus-hortorum.jpeg}
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@ -420,66 +421,21 @@
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}
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\end{frame}
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\begin{frame}
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\frametitle{Network clustering}
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\begin{figure}[ht]
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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2011_TB+Baldock2011_JN.pdf}
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\caption{Adjacency matrix,~\cite{baldockDailyTemporalStructure2011}}
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\end{figure}
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\end{frame}
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\section{Extension and conclusion}
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\begin{frame}[allowframebreaks]
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\frametitle{Application to~\cite{baldockDailyTemporalStructure2011,
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baldockSystemsApproachReveals2019}}
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TODO pivot or remove slide
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TODO Put $\alpha$ plots and tree structure of partition
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\begin{figure}[t]
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\centering
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\begin{subfigure}{0.5\textwidth}
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\centering
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\includegraphics[scale=0.1]{backup-app-iid-struct1.png}
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\includegraphics[scale=0.2]{backup-app-iid-struct2.png}
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\caption{Model $iid$,\\
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separate African (left) and English (right) networks}
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\end{subfigure}%
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~
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\begin{subfigure}{0.5\textwidth}
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\centering
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\includegraphics[scale=0.2]{backup-app-pirho-struct.png}
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\caption{Model $\pi\rho$,\\
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merge African and English networks}
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\end{subfigure}%
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\caption{Structures detected for networks
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of~\cite{baldockDailyTemporalStructure2011,
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baldockSystemsApproachReveals2019}}
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\caption{Model $iid$, separate African (left) and English (right) networks}
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\end{figure}
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\end{frame}
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\begin{frame}{Results}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}{0.5\textwidth}
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\centering
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\includegraphics[width=1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2011_TB+Baldock2011_JN.pdf}
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\caption{Reordered by LBM}
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\end{subfigure}\hfil
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\begin{subfigure}{0.5\textwidth}
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\centering
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\includegraphics[width=1\textwidth]{tikz/applications/baldock/pirho-colbisbm-mat-Baldock2011_TB+Baldock2011_JN.pdf}
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\caption{Reordered by $\pi\rho$-colBiSBM}
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\end{subfigure}
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\caption{Reordered adjacency matrix by $\pi\rho$-colBiSBM,~\cite{baldockDailyTemporalStructure2011}}
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\end{figure}
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\end{frame}
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\section{Conclusion}
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\label{sec:conclusion}
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\begin{frame}
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\frametitle{Conclusion and perspectives}
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% DONE Add a conclusion perspective slide
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% Recall models with clustering
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% Mention analysis of corrected networks for sampling
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% Link to the package
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\begin{block}{Capabilities}
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\begin{itemize}
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\item 4 models including 3 with flexibility on at least one of
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@ -488,21 +444,10 @@
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\item Partition a set of networks according to their structures.
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}{Perspectives}
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\begin{block}{Future work}
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\begin{itemize}
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\item Multi-layer networks (account for sampling bias, presence/absence)
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\item Graph Convolutional Network to allow for scalability
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\end{itemize}
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\end{block}
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\begin{block}{Package and applications}
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\begin{itemize}
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\item CRAN submission
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\item \texttt{ArXiv} preprint in redaction
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\item \texttt{CRAN} submission
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\item Integrate the possibility of an additional criterion for clustering (e.g.
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urbanization gradient~\cite{fisogniSeasonalTrajectoriesPlantpollinator2022})
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\item Apply clustering to data from
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@ -708,6 +708,25 @@ Read\_Status\_Date: 2025-05-09T11:54:37.094Z},
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file = {/home/louis/snap/zotero-snap/common/Zotero/storage/5THEWLW6/Kipf et Welling - 2016 - Variational Graph Auto-Encoders.pdf;/home/louis/snap/zotero-snap/common/Zotero/storage/BBTHQNRZ/1611.html}
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}
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@book{kolaczykStatisticalAnalysisNetwork2009,
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title = {Statistical {{Analysis}} of {{Network Data}}: {{Methods}} and {{Models}}},
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shorttitle = {Statistical {{Analysis}} of {{Network Data}}},
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author = {Kolaczyk, Eric D.},
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date = {2009},
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series = {Springer {{Series}} in {{Statistics}}},
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publisher = {Springer New York},
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location = {New York, NY},
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doi = {10.1007/978-0-387-88146-1},
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url = {https://link.springer.com/10.1007/978-0-387-88146-1},
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urldate = {2025-05-26},
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isbn = {978-0-387-88145-4 978-0-387-88146-1},
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langid = {english},
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keywords = {/unread},
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annotation = {Read\_Status: New\\
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Read\_Status\_Date: 2025-05-26T11:42:27.939Z},
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file = {/home/louis/snap/zotero-snap/common/Zotero/storage/RQPMHFGB/Kolaczyk - 2009 - Statistical Analysis of Network Data Methods and Models.pdf}
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}
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@online{kumpulainenYourBlockOur2024,
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title = {From Your {{Block}} to Our {{Block}}: {{How}} to {{Find Shared Structure}} between {{Stochastic Block Models}} over {{Multiple Graphs}}},
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shorttitle = {From Your {{Block}} to Our {{Block}}},
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