structure detection : relecture
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@ -7,8 +7,8 @@
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We define a collection of bipartite networks as
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$\bm{X} = (X^1,\dots X^m,\dots, X^M)$
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the collection of incidence matrix. Moreover, all the networks in the
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collection have the same type of interaction (e.g., all interactions are
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binary).
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collection have the same valuation of the interactions (e.g., they are
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all binary).
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\section{Separate BiSBM (sep-BiSBM)}\label{sec:separate-bisbm-sepbisbm}
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@ -51,21 +51,21 @@ Equations~\eqref{eqn:lbm-block-membership-prob},
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\eqref{eqn:lbm-conditional-to-latent} and \eqref{eqn:lbm-emission} defines the
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BiSBM model and we will now use a short notation:
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\begin{equation}
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\begin{align}
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\tag{\emph{sep-BiSBM}}
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X^m \sim \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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\end{equation}
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where $\mathcal{F}$ encodes the emission distribution, $n_1^m,n_2^m$ are the row
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X^m \sim \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}) & & \forall m = 1, \dots M
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\end{align}
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where $\mathcal{F}$ encodes the emission distribution, $n_1^m,n_2^m$ are the number of row
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and column nodes, $Q_1^m, Q_2^m$ are the number of row and column blocks in
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network $m$, $\bm{\pi}^m~=~{(\pi^m_q)}_{q=1,\dots,Q_1^m}$ and
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$\bm{\rho}^m~=~{(\rho^m_r)}_{r=1,\dots,Q_2^m}$ are the vectors of their
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proportions. The $Q_1^m \times Q_2^m$ matrix
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$\bm{\alpha}^m = {(\alpha^m_{qr})}_{\substack{q = 1,\dots,Q_1^m \\ r = 1,\dots,Q_2^m}}$
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are the connectivity parameters, the parameters of the emission distribution.
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are the connectivity parameters, i.e.~the parameters of the emission distribution.
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$\alpha^m_{qr}\in\mathcal{A}_{\mathcal{F}}$ where, for the Bernoulli
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(resp. Poisson) emission distribution, $\mathcal{A}_{\mathcal{F}} = (0,1)$ (resp.
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$\mathcal{A}_{\mathcal{F}} = \mathbb{R}^{*+}$). In this $sep$-$BiSBM$ each
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network $m$ is assumed to follow a $BiSBM$ with its own parameters ($\bm{\pi}^m,
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$\mathcal{A}_{\mathcal{F}} = \mathbb{R}^{*+}$). In this $sep$-BiSBM model each
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network $m$ is assumed to follow a BiSBM with its own parameters ($\bm{\pi}^m,
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\bm{\rho}^m, \bm{\alpha}^m$).
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% DONE Finish explaining
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@ -76,7 +76,7 @@ network $m$ is assumed to follow a $BiSBM$ with its own parameters ($\bm{\pi}^m,
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\subsection{A collection of iid bipartite SBM}\label{ssec:a-collection-of-i-i-d-bipartite-sbm}
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As for \emph{colSBM} this first model is the most constrained. It assumes that
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all the networks are the independent realizations of the same $Q_1$-$Q_2$-BiSBM
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with identical parameters. The \emph{iid-colBiSBM} is defined as follows:
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with identical parameters. The \emph{iid}-colBiSBM is defined as follows:
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\begin{align}
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\tag{\emph{iid}-colBiSBM}
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@ -85,7 +85,7 @@ with identical parameters. The \emph{iid-colBiSBM} is defined as follows:
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where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$,
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$\pi_q \in \left( 0,1 \right], \sum_{q=1}^{Q_1} \pi_q = 1 $ and $\rho_r \in \left( 0,1 \right], \sum_{r=1}^{Q_2} \rho_r = 1 $.
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This model involves $(Q_1 - 1) + (Q_2 - 1) + Q_1\times Q_2$ parameters, the two
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first terms corresponding to block proportions on the row and column dimensions
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first terms corresponding to block proportions on the row and column
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and the third term to connectivity parameters.
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But the assumption that block proportions are the same among the networks is a
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@ -106,9 +106,9 @@ block proportions. For $m \in \{1,\dots,M\}$, the $X^m$ are independent and
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\end{align}
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where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$,
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$\pi^m_q \in \left[ 0,1 \right], \sum_{q=1}^{Q_1} \pi^m_q~=~1, \forall m \in \{1,\dots,M\}$ and $\rho_r \in \left( 0,1 \right], \sum_{r=1}^{Q_2} \rho_r = 1 $.
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This model is more flexible than the iid-colBiSBM as it allows some row block
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proportions to be null
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in certain networks ($\pi^m_q\in\left[ 0,1 \right]$): if $\pi_q^m = 0$ then the
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This model is more flexible than the iid-colBiSBM as it allows the row block
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proportions to vary between networks and even to be null
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($\pi^m_q\in\left[ 0,1 \right]$): if $\pi_q^m = 0$ then the
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block $q$ is not represented in the network $m$. The connectivity structure is
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thus a subset of a large connectivity structure common to all networks. We face
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the same problems as~\cite{chabert-liddellLearningCommonStructures2024a} and
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@ -139,9 +139,9 @@ block proportions. For $m \in \{1,\dots,M\}$, the $X^m$ are independent and
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where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$,
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$\pi_q \in \left( 0,1 \right], \sum_{q=1}^{Q_1} \pi_q = 1 $ and
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$\rho^m_r \in \left[ 0,1 \right], \sum_{r=1}^{Q_2} \rho^m_r = 1 $.
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This model is more flexible than the iid-colBiSBM as it allows some column block
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proportions to be
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null in certain networks ($\rho^m_r\in\left[ 0,1 \right]$): if $\rho_r^m = 0$
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This model is more flexible than the iid-colBiSBM as it allows
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proportions to vary between networks and even to be null
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($\rho^m_r\in\left[ 0,1 \right]$): if $\rho_r^m = 0$
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then the column block $r$ is not represented in the network $m$.
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\enquote{Mirroring} the formulas for the $\pi$-colBiSBM we relax the constraints on
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@ -155,7 +155,7 @@ case the matrix full of ones), the number of parameters is:
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$\pi\rho$-colBiSBM model still assumes that the networks share a common connectivity
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structure represented by $\bm{\alpha}$ but that each network has its own row and
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column block proportions, it is the less constrained model.
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column block proportions, it is the least constrained model.
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For $m \in \{1,\dots,M\}$, the $X^m$ are independent and
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\begin{align}
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\tag{\emph{$\pi\rho$}-colBiSBM}
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@ -204,22 +204,23 @@ we have: $\mathbb{P}_{\mathcal{R}_m} (Z_{iq}^m = 1, W_{jr}^m = 1|X^m) =
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The formula for the entropy per network is thus:
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\begin{equation*}
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\mathcal{H}(\mathcal{R}_m) = - \sum_{i=1}^{n_1^m} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2^m} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r}
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\mathcal{H}(\mathcal{R}_m) = - \sum_{i=1}^{n_1^m} \tau_{iq}^{1,m} \log \tau_{iq}^{1,m} - \sum_{j=1}^{n_2^m} \tau_{jr}^{2,m} \log \tau_{jr}^{2,m}
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\end{equation*}
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And the expectation of the completed log-likelihood under the $\mathcal{R}_m$
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variational distribution for network $m$ is:
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\begin{align*}
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\mathbb{E}_{\mathcal{R}_m}[\ell(X^m,Z^m,W^m;\bm{\theta})] = \sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in \mathcal{Q}_{1,m}} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{1,m}_{i,q} \tau^{2,m}_{j,r} \log f(X^{m}_{ij}; \alpha_{qr}) \\
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+ \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_{1,m}} \tau^{1,m}_{i,q} \log \pi_{\color{black}q}^{\color{gray}m} + \sum_{j=1}^{n_2^m} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{2,m}_{j,r} \log \rho_{\color{black}r}^{\color{gray}m}
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\mathbb{E}_{\mathcal{R}_m}[\ell(X^m,Z^m,W^m;\bm{\theta})] = \sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in \mathcal{Q}_1^m} \sum_{r \in \mathcal{Q}_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m} \log f(X^{m}_{ij}; \alpha_{qr}) \\
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+ \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_1^m} \tau_{iq}^{1,m} \log \pi_{\color{black}q}^{\color{gray}m} + \sum_{j=1}^{n_2^m} \sum_{r \in \mathcal{Q}_2^m} \tau_{jr}^{2,m} \log \rho_{\color{black}r}^{\color{gray}m}
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\end{align*}
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with $\mathcal{Q}_1^m = \{q\in \{1 \dots, Q_1\}|\pi_q^m > 0\}$ and
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$\mathcal{Q}_2^m = \{r\in \{1 \dots, Q_2\}|\rho_r^m > 0\}$
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And thus the lower bound becomes:
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\begin{align*}
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\mathcal{J}(\bm{\tau};\bm{\theta}) \coloneqq \sum_{m=1}^{M} \bigg(\sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in \mathcal{Q}_{1,m}} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{1,m}_{i,q} \tau^{2,m}_{j,r} \log f(X^{m}_{ij}; \alpha_{qr}) \\
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+ \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_{1,m}} \tau^{1,m}_{i,q} \log \pi_{\color{black}q}^{\color{gray}m} + \sum_{j=1}^{n_2^m} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{2,m}_{j,r} \log \rho_{\color{black}r}^{\color{gray}m} \\
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- \sum_{i=1}^{n_1^m} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2^m} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} \bigg) \color{black}
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\mathcal{J}(\bm{\tau};\bm{\theta}) \coloneqq \sum_{m=1}^{M} \bigg(\sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in \mathcal{Q}_1^m} \sum_{r \in \mathcal{Q}_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m} \log f(X^{m}_{ij}; \alpha_{qr}) \\
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+ \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_1^m} \tau_{iq}^{1,m} \log \pi_{\color{black}q}^{\color{gray}m} + \sum_{j=1}^{n_2^m} \sum_{r \in \mathcal{Q}_2^m} \tau_{jr}^{2,m} \log \rho_{\color{black}r}^{\color{gray}m} \\
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- \sum_{i=1}^{n_1^m} \tau_{iq}^{1,m} \log \tau_{iq}^{1,m} - \sum_{j=1}^{n_2^m} \tau_{jr}^{2,m} \log \tau_{jr}^{2,m} \bigg) \color{black}
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\end{align*}
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where we identify the variational distribution $\mathcal{R}$ with its parameter
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@ -284,8 +285,8 @@ while on the other hand,
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\end{align*}
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the parameters take into account all the networks at the same time.
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The connectivity parameters $\alpha_{qr}$ for all models are estimated as the
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ratio of the number of interactions between row block $q$ and column block $r$
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among all networks over the number of number of possible interactions:
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ratio of the number of observed interactions between row block $q$ and column block $r$
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among all networks over the number of possible interactions:
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\begin{align*}
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\widehat{\alpha}_{qr} = \frac{\sum_{m=1}^{M} e^{m}_{qr}}{\sum_{m=1}^{M} n^{m}_{qr}}
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\end{align*}
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@ -303,7 +304,7 @@ $Q_1$ and $Q_2$. But as they are in general not known we need to explore the
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latent space to find the \emph{best} values.
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As discussed in~\cite{chabert-liddellLearningCommonStructures2024a}, the
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algorithmic aspect becomes complex when dealing with the bipartite case. Due to
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the size of the latent space being $\mathbb{N}^2$, conducting a complete
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the latent space being $\mathbb{N}^2$, conducting a complete
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exploration of the latent space is practically infeasible. Therefore, in
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addition to adapting the existing formulas, our contribution to addressing this
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challenge involved making significant choices, which are outlined below.
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@ -315,7 +316,7 @@ The below procedures are implemented in the \emph{colSBM} package, available on
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\label{ssec:the-bic-l-criterion-for-model-selection}
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To select the best number of blocks we need a criterion to
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measure adequacy between our model and data. The ELBO might seem a good
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criterion at first but as for the likelihood, the more complex a model the
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criterion at first but as for the likelihood, the more complex the model, the
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higher it gets. And thus a good criterion should make a \emph{trade-off} between
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fitting to data and model complexity.
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@ -340,7 +341,7 @@ well-separated blocks by imposing a penalty on the entropy of node grouping.
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However, the objective of our study extends beyond grouping nodes into coherent
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blocks. We also aim to assess the similarity of connectivity patterns across
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different networks. Consequently, we aim to permit models that offer more
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flexible node grouping without penalizing entropy.
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flexible node grouping by not penalizing on entropy.
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This leads us to formulate a BIC-like criterion in the following manner:
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@ -352,49 +353,49 @@ 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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@ -420,11 +421,11 @@ For the memberships on the rows: $row~order_m = order\left(\rho_m \times
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Using this order we relabel the memberships for the $M$ fitted collection of a
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single network.
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We then use the $M$ memberships to fit a collection containing
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the $M$ networks.
|
||||
We then use the $M$ memberships to compute first $\bm{\tau}$ to fit a collection
|
||||
containing the $M$ networks.
|
||||
\subsection{Greedy exploration to find an estimation of the mode}\label{ssec:greedy-exploration-to-find-an-estimation-of-the-mode}
|
||||
Using the previously fitted models for $Q = (1,2)$ and $Q = (2,1)$ we choose to
|
||||
perform a greedy exploration to find a first mode.
|
||||
perform a greedy exploration from each of those points to find a first mode.
|
||||
|
||||
Meaning that for a given $Q = (Q_1, Q_2)$ we will compute all the possible
|
||||
memberships for the points $Q \in \{(Q_1 + 1, Q_2),(Q_1, Q_2 + 1),(Q_1 - 1,
|
||||
|
|
@ -432,6 +433,10 @@ memberships for the points $Q \in \{(Q_1 + 1, Q_2),(Q_1, Q_2 + 1),(Q_1 - 1,
|
|||
maximizes the BIC-L as the next point from which to repeat the procedure. We
|
||||
repeat the procedure until the BIC-L stops increasing $2$ times in a row.
|
||||
|
||||
Let us denote the neighborhood in the latent space of a point $Q$ by
|
||||
$\mathcal{N}(Q) = Q + {(1,0), (0,1), (-1,0), (0,-1)}$, the four neighbors of $Q$
|
||||
in the grid.
|
||||
|
||||
\begin{algorithm}[H]
|
||||
\small
|
||||
\caption{Greedy Exploration for Mode Estimation}
|
||||
|
|
@ -443,28 +448,31 @@ repeat the procedure until the BIC-L stops increasing $2$ times in a row.
|
|||
\Output{Estimation of the mode using greedy exploration}
|
||||
|
||||
\BlankLine
|
||||
Initialize $Q = (1,2)$ as the starting point\\
|
||||
Initialize $\text{BIC-L}_{\text{max}}$ as the maximum achieved BIC-L value\\
|
||||
\For{$Q_{\text{start}} \in \{(1,2), (2,1)\}$}{ % and $Q = (2,1)$ as starting point
|
||||
\BlankLine
|
||||
Initialize $\text{BIC-L}_{\text{max}} \leftarrow \text{BIC-L}(Q_{\text{start}})$\\
|
||||
Initialize $consecutive\_count$ as 0
|
||||
|
||||
\BlankLine
|
||||
$Q_{\text{curr}} \leftarrow Q_{\text{start}}$
|
||||
|
||||
\While{$consecutive\_count < 2$}{
|
||||
Compute possible memberships for $Q \in \{(Q_1 + 1, Q_2), (Q_1, Q_2 + 1), (Q_1 - 1, Q_2), (Q_1, Q_2 - 1)\}$\;
|
||||
Fit models with the computed memberships
|
||||
Choose the model with the maximum BIC-L as the next point
|
||||
Fit models in $\mathcal{N}(Q_{\text{curr}})$\;
|
||||
|
||||
\BlankLine
|
||||
\If{$\text{BIC-L} > \text{BIC-L}_{\text{max}}$}{
|
||||
$\text{BIC-L}_{\text{max}} \leftarrow \text{BIC-L}$\\
|
||||
$Q \leftarrow \arg\max_{Q \in \mathcal{N}(Q_{\text{curr}})} \text{BIC-L}(Q)$
|
||||
|
||||
$\text{BIC-L}_{\text{curr}} \leftarrow \max_{Q \in \mathcal{N}(Q_{\text{curr}})} \text{BIC-L}(Q)$
|
||||
\BlankLine
|
||||
\If{$\text{BIC-L}_{\text{curr}} > \text{BIC-L}_{\text{max}}$}{
|
||||
$\text{BIC-L}_{\text{max}} \leftarrow \text{BIC-L}_{\text{curr}}$\\
|
||||
$consecutive\_count \leftarrow 0$
|
||||
}
|
||||
\Else{
|
||||
$consecutive\_count \leftarrow consecutive\_count + 1$
|
||||
}
|
||||
\BlankLine
|
||||
$Q \leftarrow$ Next selected point
|
||||
}
|
||||
|
||||
}
|
||||
\BlankLine
|
||||
\textbf{Output:} Estimation of the mode using greedy exploration
|
||||
\end{algorithm}
|
||||
|
|
@ -512,8 +520,7 @@ consists of two alternating steps:
|
|||
\For{$Q_1 \in \left[ Q_{1,\text{center}} - \text{depth} ; Q_{1,\text{center}} + \text{depth} \right]$}{
|
||||
\For{$Q_2 \in \left[ Q_{2,\text{center}} - \text{depth}; Q_{2,\text{center}} + \text{depth} \right] $}{
|
||||
Compute possible splits from predecessors $(Q_1 - 1, Q_2)$ and $(Q_1, Q_2 - 1)$\\
|
||||
Fit models with the block membership changes
|
||||
Compare and keep the best model based on BIC-L
|
||||
Among the model generated from the splits choose the best in regard of the BIC-L
|
||||
}
|
||||
}
|
||||
|
||||
|
|
@ -523,13 +530,12 @@ consists of two alternating steps:
|
|||
\For{$Q_1 \in \left[ Q_{1,\text{center}} + \text{depth} ; Q_{1,\text{center}} - \text{depth} \right]$}{
|
||||
\For{$Q_2 \in \left[ Q_{2,\text{center}} + \text{depth}; Q_{2,\text{center}} - \text{depth} \right] $}{
|
||||
Compute possible merges from predecessors $(Q_1 + 1, Q_2)$ and $(Q_1, Q_2 + 1)$\\
|
||||
Fit models with the block membership changes
|
||||
Compare and keep the best model based on BIC-L
|
||||
Among the model generated from the merges choose the best in regard of the BIC-L
|
||||
}
|
||||
}
|
||||
|
||||
\BlankLine
|
||||
Update the best model based on the maximum BIC-L
|
||||
Choose the mode as the one that maximizes the BIC-L
|
||||
}
|
||||
|
||||
\BlankLine
|
||||
|
|
@ -637,6 +643,7 @@ The procedure then repeats for the point at $(Q_1 + 1, Q_2)$ until it reaches
|
|||
$(Q_{1,center} + depth, Q_2)$ from which it repeats from
|
||||
$(Q_{1,center} - depth, Q_2 + 1)$. This repeats until computing the best model
|
||||
for $(Q_{1,center} + depth, Q_{2,center} + depth)$.
|
||||
|
||||
\textit{Note on the initialization:} The forward pass starts from the point
|
||||
$(Q_{1,center} + depth, Q_{2,center} + depth)$, so this points needs to have at
|
||||
least a model fitted. In the best case, the greedy exploration will have visited
|
||||
|
|
@ -663,7 +670,7 @@ $(Q_{1,center} + depth, Q_{2,center} + depth)$, we know it was initialized at
|
|||
least by the forward pass, no special case here.\\
|
||||
|
||||
At the end of the moving window pass, the model of max BIC-L is the new best
|
||||
fit and the procedure can repeat until convergence.
|
||||
fit and the procedure repeats until convergence.
|
||||
|
||||
\section{Networks clustering}
|
||||
\label{sec:networks-clustering}
|
||||
|
|
@ -752,7 +759,7 @@ trivial partition in a unique group.
|
|||
Then using the \emph{Kmeans} we split the collection in two sub-collections
|
||||
with the dissimilarity matrix. The two sub-collections are fitted and we
|
||||
compute the score of this new partition $\mathcal{G}^{*} = \{G_1, G_2\}$.
|
||||
If $Sc(\mathcal{G}^{*}) > Sc(\mathcal{G})$ then we repeat the same procedure on
|
||||
If $Sc(\mathcal{G}^{*}) > Sc(\mathcal{G})$, we repeat the same procedure on
|
||||
$G_1$ and $G_2$. Else we return $\mathcal{G}$.
|
||||
We illustrate our capacity to perform a partition of a collection for all
|
||||
colBiSBM models in~\ref{sec:network-clustering-of-simulated-networks}.
|
||||
|
|
@ -772,11 +779,11 @@ we obtain the following result of identifiability\footnote{The proof is in appen
|
|||
\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 $X^{m^*}$).
|
||||
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.
|
||||
and the coordinates of vector $\bm{\pi}
|
||||
X^{m^*}$ are distinct.
|
||||
\end{itemize}
|
||||
\end{theorem}
|
||||
|
||||
|
|
|
|||
Binary file not shown.
|
|
@ -22,8 +22,8 @@ Maxime.
|
|||
Merci à tous les permanents du 3\ieme étage, parmi lesquels: Christophe,
|
||||
Stéphane et Vincent.
|
||||
|
||||
Merci à Hugo, Théodore, Éric, Jean-Benoist, Nicolas, Tristan, Sarah, Jade et
|
||||
Pierre Gloaguen.
|
||||
Merci à Liliane, Isabelle, Hugo, Théodore, Éric, Jean-Benoist, Nicolas, Lucia,
|
||||
Tristan, Sarah, Jade et Pierre Gloaguen.
|
||||
|
||||
Un grand merci à tous ceux qui ont participé de près ou de loin au bon
|
||||
déroulement de ce stage.
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue