383 lines
No EOL
16 KiB
TeX
383 lines
No EOL
16 KiB
TeX
\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 from
|
|
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}
|
|
TODO (Supprimable) 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 different BiSBM}
|
|
\[
|
|
\forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1^m, Q_2^m, \pi^m, \rho^m, \alpha^m)
|
|
\]
|
|
\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{Several joint 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}
|
|
And intermediate models freeing $\pi$ or $\rho$.
|
|
}
|
|
\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}{Parameter estimation}
|
|
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}
|
|
|
|
\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. \uncover<2->{$\rightarrow$ \textbf{Greedy
|
|
approach} and \textbf{sliding window}}
|
|
\item Sensitivity to initializations. \uncover<3->{$\rightarrow$ \textbf{Spectral
|
|
clustering} and \textbf{reuse of previous inits}}
|
|
\end{itemize}
|
|
\end{alertblock}
|
|
\end{frame}
|
|
|
|
\section{Application}
|
|
\label{sec:application}
|
|
|
|
\begin{frame}
|
|
\frametitle{Results~\cite{baldockSystemsApproachReveals2019}}
|
|
\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}
|
|
TODO Interesting structures detected, functional roles in the british networks.
|
|
\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=0.1]{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}{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}.
|
|
\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 submission
|
|
\item Integrate the possibility of an additional criterion for clustering (e.g.
|
|
urbanization gradient~\cite{fisogniSeasonalTrajectoriesPlantpollinator2022})
|
|
\item Apply clustering to data from
|
|
\cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021}
|
|
\end{itemize}
|
|
|
|
\end{block}
|
|
\bigskip
|
|
\centering
|
|
Thank you for your attention~!
|
|
\end{frame} |