121 lines
No EOL
5.1 KiB
TeX
121 lines
No EOL
5.1 KiB
TeX
\clearpage
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\pagenumbering{arabic}% resets `page` counter to 1
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\renewcommand*{\thepage}{S-\arabic{page}}
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\appendix
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\chapter{Supplementary for~\nameref{chap:struct-detection}}
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\section{Proof of the idenfiability result}
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\label{sec:proof-identifiability}
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We recall the following
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\def\thetheorem{\ref{thm:identifiability-iid}}
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\begin{theorem}[Identifiability of $iid$-colBiSBM]
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The parameters $(\bm{\pi}, \bm{\rho}, \bm{\alpha})$ are
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identifiable up to a label switching of the blocks if those
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conditions are achieved:
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\begin{itemize}
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\item[(1.1)] $\exists m^*\in\{1,\dots,M\} : n^1_{m^*} \geq 2 Q_2 - 1~\text{and}~n^2_{m^*} \geq 2 Q_1 - 1$.
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\item[(1.2)] $\forall 1\leq q \leq Q_1, \pi_q > 0$
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and the coordinates of vector $\bm{\rho}
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{X^{m^*}}^T$ are distinct (where ${X^{m^*}}^T$ is the transpose of $X^{m^*}$).
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\item[(1.3)] $\forall 1\leq r \leq Q_2, \rho_r > 0$
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and the coordinates of vector $\bm{\pi}
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X^{m^*}$ are distinct.
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\end{itemize}
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\end{theorem}
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\begin{proof}
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Following the tracks of~\cite{chabert-liddellLearningCommonStructures2024a}
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we derive the result in Properties~\ref{thm:identifiability-iid}.
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\cite{keribinEstimationSelectionLatent2015} building
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on~\cite{celisseConsistencyMaximumlikelihoodVariational2012}, proved that the
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parameters $(\bm{\pi}, \bm{\rho}, \bm{\alpha})$ of the
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$\mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1^m, Q_2^m, \bm{\pi^m}, \bm{\rho^m}, \bm{\alpha^m})$
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are identifiable from the observation of network $X^m$ when $\mathcal{F}$
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is the Bernoulli distribution and the following conditions are met:
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\begin{enumerate}
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\item $ n_1^m \geq 2 Q_2^m - 1~\text{and}~n_2^m \geq 2 Q_1^m - 1$.
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\item $\forall 1\leq q \leq Q_1^m, \pi_q^m > 0$
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and the coordinates of vector $\bm{\rho^m}
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{X^{m^*}}^T$ are distinct (where ${X^{m^*}}^T$ is the transpose of $X^{m^*}$).
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\item $\forall 1\leq r \leq Q_2^m, \rho_r^m > 0$
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and the coordinates of vector $\bm{\pi^m}
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X^{m^*}$ are distinct.
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\end{enumerate}
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Under the \emph{iid}-colBiSBM model, for all $m=1\dots M$,
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$X^m \sim \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2,
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\bm{\pi}, \bm{\rho}, \bm{\alpha})$. This means that
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following~\cite{keribinEstimationSelectionLatent2015}, the
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identifiability of $\bm{\alpha}$, $\bm{\pi}$ and $\bm{\rho}$ is obtained
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from the distribution of $X^{m^*}$ under assumptions (1.1), (1.2) and
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(1.3).
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\end{proof}
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\chapter{Supplementary for~\nameref{chap:simulation-studies}}
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Below are the supplementary material for the~\nameref{chap:simulation-studies}.
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\section{Supplementary for~\nameref{sec:efficiency-of-the-inference}}
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The tables~\ref{tab:inference_results_iid} to~\ref{tab:inference_results_pirho}
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show detailed results for the inference of the model detailed in this section.
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\foreach \modelname in {sep, iid, pi, rho, pirho}{
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\input{../tables/simulations/inference/\modelname.tex}
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}
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\section{Supplementary for~\nameref{sec:capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}}
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The table~\ref{tab:model-selection} present the results discussed in
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section~\ref{sec:capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}
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For the block number recovery part, the \emph{minimum} values are in
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\textbf{bold} as they indicate conditions for which all the different models did
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not recovered the correct structure.
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For the model proportion part of the table, the \emph{maximum }values are in
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\textbf{bold} and highlight the model that was selected the most among the
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conditions.
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Please note that blank space indicates that among all conditions
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the corresponding model was not selected at all.
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\begin{landscape}
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\input{../tables/simulations/model_selection/model-selection.tex}
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\end{landscape}
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\chapter{Supplementary for~\nameref{chap:applications-ecological-networks}}
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\section{Additional information on~\nameref{sec:baldock-clustering}}
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\fancypagestyle{fancy}
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\renewcommand*{\thepage}{S-\arabic{page}}
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Due to report size limitations we included these plots here as they are not crucial to understand what is going on in
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the section~\ref{sec:baldock-clustering}.
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Yet they are useful to confirm the explanation given.
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\begin{figure}[!ht]
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\centering
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\begin{subfigure}[htb]{\textwidth}
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\centering
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\includestandalone[width=0.6\textwidth]{tikz/applications/baldock/app-iid-clust-struct-1}
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\caption{Small collection structure}
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\end{subfigure}
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\newline
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\begin{subfigure}[htb]{\textwidth}
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\includestandalone[width=0.9\textwidth]{tikz/applications/baldock/app-iid-clust-struct-2}
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\caption{English networks collection structure}
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\end{subfigure}
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\caption{Structure and mixture proportions for \emph{iid} clustering}
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\label{fig:struct-mixture-iid}
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\end{figure}
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\begin{figure}[!ht]
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\centering
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\includestandalone{tikz/applications/baldock/app-pirho-clust-struct}
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\caption{Structure and mixture proportions for $\pi\rho$ clustering}
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\label{fig:struct-mixture-pirho}
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\end{figure} |