appendix : adding proof of ident results
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4 changed files with 128 additions and 75 deletions
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\clearpage
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\pagenumbering{arabic}% resets `page` counter to 1
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\renewcommand*{\thepage}{A-\arabic{page}}
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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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@ -31,14 +81,18 @@ 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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\pagestyle{empty}
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\input{../tables/simulations/model_selection/model-selection.tex}
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\end{landscape}
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\pagestyle{fancy}
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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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@ -1,5 +1,6 @@
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\addtocounter{customchapter}{1}
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\chapter[Structure detection in bipartite collection]{Structure detection in a collection of bipartite networks}
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\label{chap:struct-detection}
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\section{Definition of a collection}
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\label{sec:definition-of-a-collection}
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@ -351,49 +352,49 @@ We provide below the expression for the penalties for the 4 models that we
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propose.
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\begin{description}
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\item[\textit{iid}-colBiSBM] For the $\bm\pi$ and $\bm\rho$:
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\begin{align*}
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\text{pen}_{\pi}(Q_1) = (Q_1 - 1)\log(\sum_{m=1}^{M}n_{1}^{m}) & , &
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\text{pen}_{\rho}(Q_2) = (Q_2 - 1)\log(\sum_{m=1}^{M}n_{2}^{m})
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\end{align*}
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For the $\bm\alpha$:
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\[\text{pen}_{\alpha}(Q_1, Q_2) = Q_1 \times Q_2 \log(N_M)\]
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with
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\[ N_M = \sum_{m = 1}^{M} n_{1}^{m} \times n_{2}^{m} \]
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And thus the $\text{BIC-L}$ formula is the following:
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\[ \text{BIC-L}(\bm{X},Q_1, Q_2) = \max_{\theta}
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\mathcal{J} (\mathcal{\hat{R}}, \bm{\theta})
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- \frac{1}{2} [\text{pen}_{\pi}(Q_1) + \text{pen}_{\rho}(Q_2) +
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\text{pen}_{\alpha}(Q_1, Q_2)]\]
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\begin{align*}
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\text{pen}_{\pi}(Q_1) = (Q_1 - 1)\log(\sum_{m=1}^{M}n_{1}^{m}) & , &
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\text{pen}_{\rho}(Q_2) = (Q_2 - 1)\log(\sum_{m=1}^{M}n_{2}^{m})
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\end{align*}
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For the $\bm\alpha$:
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\[\text{pen}_{\alpha}(Q_1, Q_2) = Q_1 \times Q_2 \log(N_M)\]
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with
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\[ N_M = \sum_{m = 1}^{M} n_{1}^{m} \times n_{2}^{m} \]
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And thus the $\text{BIC-L}$ formula is the following:
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\[ \text{BIC-L}(\bm{X},Q_1, Q_2) = \max_{\theta}
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\mathcal{J} (\mathcal{\hat{R}}, \bm{\theta})
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- \frac{1}{2} [\text{pen}_{\pi}(Q_1) + \text{pen}_{\rho}(Q_2) +
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\text{pen}_{\alpha}(Q_1, Q_2)]\]
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\item[$\bm{\pi\rho}$-colBiSBM] The support penalties are
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\begin{align*}
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\text{pen}_{S_1}(Q_1) = -2 \log p_{Q_1} (S_1) & , &
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\text{pen}_{S_2}(Q_2) = -2 \log p_{Q_2} (S_2)
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\end{align*}
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with \begin{align*}
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\textstyle \log p_{Q_1}(S_1) = - M \log(Q_1) - \sum_{m=1}^{M} \log {Q_1
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\choose Q_1^{(m)}}, \\
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\textstyle \log p_{Q_2}(S_2) = - M \log(Q_2) - \sum_{m=1}^{M} \log {Q_2
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\choose Q_2^{(m)}}.
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\end{align*}
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And penalties for the $\bm\rho$ and $\bm\pi$ are
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\[ \text{pen}_{\pi}(Q_1, S_1) = \sum_{m=1}^{M} (Q_{1}^{(m)} - 1)
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\log n_{1}^{m},
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~\text{pen}_{\rho}(Q_2, S_2) = \sum_{m=1}^{M} (Q_{2}^{(m)} - 1)
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\log n_{2}^{m}. \]
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Penalties for the $\bm\alpha$
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\[ \text{pen}_{\alpha}(Q_1, Q_2, S_1, S_2) = (\sum_{q=1}^{Q_1}
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\sum_{r=1}^{Q_2} \mathbbb{1}_{(S_1)'S_2 > 0}) \log (N_M). \]
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And the corresponding BIC-L formula,
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\[
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\begin{aligned}
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\text{BIC-L}(\bm{X},Q_1, Q_2) =
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\max_{S_1,S_2} [
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& \max_{\theta_{S_1,S_2} \in \Theta_{S_1,S_2}} \mathcal{J}(\mathcal{\hat{R}},\theta_{S_1,S_2}) \\
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- \frac{1}{2} & (\text{pen}_{\pi}(Q_1, S_1) + \text{pen}_{\rho}(Q_2, S_2) \\
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& + \text{pen}_{\alpha}(Q_1, Q_2, S_1, S_2) \\
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& + \text{pen}_{S_1}(Q_1) + \text{pen}_{S_2}(Q_2))] \\
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\end{aligned}
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\]
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\begin{align*}
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\text{pen}_{S_1}(Q_1) = -2 \log p_{Q_1} (S_1) & , &
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\text{pen}_{S_2}(Q_2) = -2 \log p_{Q_2} (S_2)
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\end{align*}
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with \begin{align*}
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\textstyle \log p_{Q_1}(S_1) = - M \log(Q_1) - \sum_{m=1}^{M} \log {Q_1
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\choose Q_1^{(m)}}, \\
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\textstyle \log p_{Q_2}(S_2) = - M \log(Q_2) - \sum_{m=1}^{M} \log {Q_2
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\choose Q_2^{(m)}}.
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\end{align*}
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And penalties for the $\bm\rho$ and $\bm\pi$ are
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\[ \text{pen}_{\pi}(Q_1, S_1) = \sum_{m=1}^{M} (Q_{1}^{(m)} - 1)
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\log n_{1}^{m},
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~\text{pen}_{\rho}(Q_2, S_2) = \sum_{m=1}^{M} (Q_{2}^{(m)} - 1)
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\log n_{2}^{m}. \]
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Penalties for the $\bm\alpha$
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\[ \text{pen}_{\alpha}(Q_1, Q_2, S_1, S_2) = (\sum_{q=1}^{Q_1}
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\sum_{r=1}^{Q_2} \mathbbb{1}_{(S_1)'S_2 > 0}) \log (N_M). \]
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And the corresponding BIC-L formula,
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\[
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\begin{aligned}
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\text{BIC-L}(\bm{X},Q_1, Q_2) =
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\max_{S_1,S_2} [
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& \max_{\theta_{S_1,S_2} \in \Theta_{S_1,S_2}} \mathcal{J}(\mathcal{\hat{R}},\theta_{S_1,S_2}) \\
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- \frac{1}{2} & (\text{pen}_{\pi}(Q_1, S_1) + \text{pen}_{\rho}(Q_2, S_2) \\
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& + \text{pen}_{\alpha}(Q_1, Q_2, S_1, S_2) \\
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& + \text{pen}_{S_1}(Q_1) + \text{pen}_{S_2}(Q_2))] \\
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\end{aligned}
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\]
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\end{description}
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\subsection{Initialization and pairing of the models}
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@ -708,7 +709,7 @@ And the pairwise dissimilarity for networks $(m,m')\in\mathcal{M}^2$ is then:
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\begin{figure}[t]
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\centering
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\begin{tikzpicture}
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\begin{tikzpicture}[scale=0.7]
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\tikzstyle{instruct}=[font=\small, text justified, rectangle,draw,fill=yellow!50]
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\tikzstyle{first_col}=[rectangle, text justified, draw,fill=gray!50]
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\tikzstyle{second_col}=[scale=0.55, circle, draw,fill=red!50]
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@ -751,12 +752,10 @@ trivial partition in a unique group.
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Then using the \emph{Kmeans} we split the collection in two sub-collections
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with the dissimilarity matrix. The two sub-collections are fitted and we
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compute the score of this new partition $\mathcal{G}^{*} = \{G_1, G_2\}$.
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If $Sc(\mathcal{G}^{*}) > Sc(\mathcal{G})$ then we repeat the same procedure on
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$G_1$ and $G_2$. Else we return $\mathcal{G}$.
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We illustrate our capacity to perform a partition of a collection for all
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colBiSBM models in %\ref{sec:network-clustering-of-simulated-networks}.
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colBiSBM models in~\ref{sec:network-clustering-of-simulated-networks}.
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\section{Model identifiability}
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\label{sec:model-identifiability}
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@ -764,7 +763,7 @@ colBiSBM models in %\ref{sec:network-clustering-of-simulated-networks}.
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The goal here is to prove that if $\ell(\bm{X};\bm{\theta}) = \ell(\bm{X};\bm{\theta}')$ for any collection $\bm{X}$ then $\bm{\theta} = \bm{\theta}'$.
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Following the proof proposed by~\cite{chabert-liddellLearningCommonStructures2024a}, that adapted it to the collection case and~\cite{keribinEstimationSelectionLatent2015} that extended the result of~\cite{celisseConsistencyMaximumlikelihoodVariational2012} to the LBM Bernoulli model,
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we obtain the following proof of identifiability for the $iid$-colBiSBM:
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we obtain the following result of identifiability\footnote{The proof is in appendix. \ref{sec:proof-identifiability}} for the $iid$-colBiSBM:
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\begin{theorem}[Identifiability of $iid$-colBiSBM]
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\label{thm:identifiability-iid}
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The parameters $(\bm{\pi}, \bm{\rho}, \bm{\alpha})$ are
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@ -773,11 +772,11 @@ we obtain the following proof of identifiability for the $iid$-colBiSBM:
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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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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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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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Binary file not shown.
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@ -26,7 +26,7 @@
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hypertexnames=true
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}
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\newtheorem{theorem}{Theorem}
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\newtheorem{theorem}{Properties}
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\usepackage{tocbibind} % Pour avoir des index pour table des matières, biblio
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\usepackage{geometry}
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\geometry{bmargin=25mm}
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@ -223,31 +223,31 @@ automata,positioning}
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% Pour activer les onglets
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\ActivateBG
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\begin{selectlanguage}{french}
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% \maketitle
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\pagenumbering{roman}
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\tableofcontents
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\include{remerciements}
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% \include{chapter1-presentation_UMR}
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% \maketitle
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\pagenumbering{roman}
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\tableofcontents
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\include{remerciements}
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% \include{chapter1-presentation_UMR}
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\end{selectlanguage}
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\begin{selectlanguage}{english}
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\pagenumbering{arabic}
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\include{chapter2-context}
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\include{chapter3-structure-detection}
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\include{chapter4-simulation-studies}
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\pagenumbering{arabic}
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\include{chapter2-context}
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\include{chapter3-structure-detection}
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\include{chapter4-simulation-studies}
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% \chapter{Applications}
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% \include{Rcodes/real_data/application_dore}
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% \include{Rcodes/real_data/CoOPLBM_completion_analyze}
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\include{chapter5-applications}
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\include{conclusions}
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% \chapter{Applications}
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% \include{Rcodes/real_data/application_dore}
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% \include{Rcodes/real_data/CoOPLBM_completion_analyze}
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\include{chapter5-applications}
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\include{conclusions}
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\addtocounter{maincontentend}{1}
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\addtocounter{customchapter}{1}
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\printbibliography
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\input{appendices.tex}
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% \listoffigures
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% \listoftables
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\addtocounter{maincontentend}{1}
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\addtocounter{customchapter}{1}
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\printbibliography
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\input{appendices.tex}
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% \listoffigures
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% \listoftables
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\end{selectlanguage}
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\end{document}
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