\clearpage
\pagenumbering{arabic}% resets `page` counter to 1
\renewcommand*{\thepage}{S-\arabic{page}}
\appendix

\chapter{Supplementary for~\nameref{chap:struct-detection}}
\section{Proof of the idenfiability result}
\label{sec:proof-identifiability}

We recall the following
\def\thetheorem{\ref{thm:identifiability-iid}}
\begin{theorem}[Identifiability of $iid$-colBiSBM]
    The parameters $(\bm{\pi}, \bm{\rho}, \bm{\alpha})$ are
    identifiable up to a label switching of the blocks if those
    conditions are achieved:
    \begin{itemize}
        \item[(1.1)] $\exists m^*\in\{1,\dots,M\} : n^1_{m^*} \geq 2 Q_2 - 1~\text{and}~n^2_{m^*} \geq 2 Q_1 - 1$.
        \item[(1.2)] $\forall 1\leq q \leq Q_1, \pi_q > 0$
            and the coordinates of vector $\bm{\rho}
                {X^{m^*}}^T$ are distinct (where ${X^{m^*}}^T$ is the transpose of $X^{m^*}$).
        \item[(1.3)] $\forall 1\leq r \leq Q_2, \rho_r > 0$
            and the coordinates of vector $\bm{\pi}
                X^{m^*}$ are distinct.
    \end{itemize}
\end{theorem}
\begin{proof}
    Following the tracks of~\cite{chabert-liddellLearningCommonStructures2024a}
    we derive the result in Properties~\ref{thm:identifiability-iid}.

    \cite{keribinEstimationSelectionLatent2015} building
    on~\cite{celisseConsistencyMaximumlikelihoodVariational2012}, proved that the
    parameters $(\bm{\pi}, \bm{\rho}, \bm{\alpha})$ of the
    $\mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1^m, Q_2^m, \bm{\pi^m}, \bm{\rho^m}, \bm{\alpha^m})$
    are identifiable from the observation of network $X^m$ when $\mathcal{F}$
    is the Bernoulli distribution and the following conditions are met:
    \begin{enumerate}
        \item $ n_1^m \geq 2 Q_2^m - 1~\text{and}~n_2^m \geq 2 Q_1^m - 1$.
        \item $\forall 1\leq q \leq Q_1^m, \pi_q^m > 0$
              and the coordinates of vector $\bm{\rho^m}
                  {X^{m^*}}^T$ are distinct (where ${X^{m^*}}^T$ is the transpose of $X^{m^*}$).
        \item $\forall 1\leq r \leq Q_2^m, \rho_r^m > 0$
              and the coordinates of vector $\bm{\pi^m}
                  X^{m^*}$ are distinct.
    \end{enumerate}

    Under the \emph{iid}-colBiSBM model, for all $m=1\dots M$,
    $X^m \sim \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2,
        \bm{\pi}, \bm{\rho}, \bm{\alpha})$. This means that
    following~\cite{keribinEstimationSelectionLatent2015}, the
    identifiability of $\bm{\alpha}$, $\bm{\pi}$ and $\bm{\rho}$ is obtained
    from the distribution of $X^{m^*}$ under assumptions (1.1), (1.2) and
    (1.3).
\end{proof}

\chapter{Supplementary for~\nameref{chap:simulation-studies}}
Below are the supplementary material for the~\nameref{chap:simulation-studies}.

\section{Supplementary for~\nameref{sec:efficiency-of-the-inference}}

The tables~\ref{tab:inference_results_iid} to \ref{tab:inference_results_pirho}
show detailed results for the inference of the model detailed in this section.

\foreach \modelname in {sep, iid, pi, rho, pirho}{
        \input{../tables/simulations/inference/\modelname.tex}
    }


\section{Supplementary for~\nameref{sec:capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}}

The table~\ref{tab:model-selection} present the results discussed in
section~\ref{sec:capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}
For the block number recovery part, the \emph{minimum} values are in
\textbf{bold} as they indicate conditions for which all the different models did
not recovered the correct structure.

For the model proportion part of the table, the \emph{maximum }values are in
\textbf{bold} and highlight the model that was selected the most among the
conditions.

Please note that blank space indicates that among all conditions
the corresponding model was not selected at all.

\begin{landscape}
    \input{../tables/simulations/model_selection/model-selection.tex}
\end{landscape}



\chapter{Supplementary for~\nameref{chap:applications-ecological-networks}}
\section{Additional information on~\nameref{sec:baldock-clustering}}

\fancypagestyle{fancy}

\renewcommand*{\thepage}{S-\arabic{page}}

Due to report size limitations we included these plots here as they are not crucial to understand what is going on in
the section~\ref{sec:baldock-clustering}.
Yet they are useful to confirm the explanation given.

\begin{figure}[!ht]
    \centering
    \begin{subfigure}[htb]{\textwidth}
        \centering
        \includestandalone[width=0.6\textwidth]{tikz/applications/baldock/app-iid-clust-struct-1}
        \caption{Small collection structure}
    \end{subfigure}
    \newline
    \begin{subfigure}[htb]{\textwidth}
        \includestandalone[width=0.9\textwidth]{tikz/applications/baldock/app-iid-clust-struct-2}
        \caption{English networks collection structure}
    \end{subfigure}
    \caption{Structure and mixture proportions for \emph{iid} clustering}
    \label{fig:struct-mixture-iid}
\end{figure}

\begin{figure}[!ht]
    \centering
    \includestandalone{tikz/applications/baldock/app-pirho-clust-struct}
    \caption{Structure and mixture proportions for $\pi\rho$ clustering}
    \label{fig:struct-mixture-pirho}
\end{figure}