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rapport.tex
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rapport.tex
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@ -110,7 +110,7 @@ $V$ vertices.
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\begin{minipage}{0.5\linewidth}
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\begin{center}
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Incidence matrix
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$B=\left(
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$X=\left(
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\begin{array}{rrrrr}
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1 & 1 & 1 & 1 & 0 \\
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0 & 0 & 1 & 1 & 1 \\
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@ -120,11 +120,21 @@ $V$ vertices.
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\end{center}
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\end{minipage}
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$X$ is the \emph{incidence matrix} and is the mathematical object on which
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computations are performed. It is filled with the following rule:
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\begin{equation*}
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\begin{cases}
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X_{ij} = 0 & \text{if no interaction is observed between species }i\text{ and }j\\
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X_{ij} \neq 0 & \text{otherwise}
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\end{cases}
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\end{equation*}
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If the network represents binary observation (like presence-absence observation) then
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$X_{ij}\in\mathcal{K}=\{0,1\},\forall(i,j)$; if the interactions are weighted
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(like an abundance count), $X_{ij}\in\mathcal{K}=\mathbb{N},\forall(i,j)$.
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This representation can be used to represent various forms of interactions were
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two kinds of ``actors`` interact. Those interactions can be binary or valued and
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a numeric representation is the incidence matrix, in the above example $B$.\\
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a numeric representation is the incidence matrix, in the above example $X$.\\
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Among the use case of bipartite graphs one can find the Netflix Problem, which
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was a prize organized by Netflix to improve its Recommender system. The row
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@ -279,24 +289,73 @@ The next step after designing this collection model for unipartite was to adapt
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it to the bipartite case.
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\chapter{Structure detection in a collection of bipartite networks : Adjustment of colSBM to the bipartite case}
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\section{Definition of a collection}\label{sec:definition-of-a-collection}
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We define a collection of bipartite networks as $\bm{X} = (X^1, \dots, X^M)$
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the collection of incidence matrix. Moreover, all the networks in the collection
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have the same type of interaction (e.g., all interactions are binary).
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\section{Separate BiSBM (sepBiSBM)}\label{sec:separate-bisbm-sepbisbm}
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A first approach to deal with a collection of networks is to adjust separate
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BiSBM for each network of the collection.
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For network $m$, let $n_1^m$ (resp. $n_2^m$) be the number of nodes in row
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(resp. column) divided into $Q_{1,m}$ row clusters (resp. $Q_{2,m}$ column
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(resp. column) divided into $Q_1^m$ row clusters (resp. $Q_2^m$ column
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clusters).\\
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Let $Z^m~=~(Z^m_i, \dots, Z^m_{n_1^m})$ and $W^m~=~(W^m_j, \dots, W^m_{n_2^m})$
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be independent latent variables such that $Z^m_i = q$ if row node $i$ of network
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$m$ belongs to cluster $q$
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$m$ belongs to row cluster $q$ ($q\in\{1,\dots,Q_1^m\}$) and $W^m_j = r$ if column node $j$ of network $m$
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belong to column block $r$ ($r\in\{1,\dots,Q_2^m\}$). And we have
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\begin{align}\label{eqn:lbm-block-membership-prob}
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\mathbb{P}(Z_i^m=q)=\pi_q^m,&&\mathbb{P}(W_j^m=r)=\rho_r^m
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\end{align}
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where $\pi_q^m > 0$, $\rho_r^m > 0$, $\sum_{q=1}^{Q_1^m}\pi_q^m = 1$ and
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$\sum_{r=1}^{Q_2^m}\rho_r^m = 1$. Given the latent variables
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$Z^m, W^m$, the $X_{ij}^m$s are assumed to be independent and distributed
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as
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% TODO Finish explaining
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\begin{align}\label{eqn:lbm-conditional-to-latent}
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X_{ij}^m|Z_i^m = q,W_j^m = r \sim \mathcal{F}(.;\alpha_{qr}^m)
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\end{align}
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where $\mathcal{F}$ is referred to as the emission distribution. $\mathcal{F}$ is chosen to
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be the Bernoulli distribution for binary interactions, and the Poisson
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distribution for weighted interactions such as counts. Let $f$ be the density of
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the emission distribution, then:
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\section{Definition of the model}
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\label{sec:definition-of-the-model}
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Here are some common notations and conventions that we will use in the following
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sections.
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\begin{equation}\label{eqn:lbm-emission}
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\log f(X^m_{ij};\alpha_{qr}^m) =
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\begin{cases}
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X_{ij}^m \log(\alpha_{qr}^m) + (1-X_{ij}^m) \log(1-\alpha_{qr}^m) & \text{for Bernoulli emission} \\
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-\alpha_{qr}^m + X_{ij}^m \log(\alpha_{qr}^m) - \log(X_{ij}^m!) & \text{for Poisson emission}
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\end{cases}
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\end{equation}
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Equations \eqref{eqn:lbm-block-membership-prob}, \eqref{eqn:lbm-conditional-to-latent}
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and \eqref{eqn:lbm-emission} defines the BiSBM model and we will now use a short
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notation:
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\begin{equation}
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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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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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$\alpha^m_{qr}\in\mathcal{A}_{\mathcal{F}}$ 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 ($\bm{\pi}^m,
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\bm{\rho}^m, \bm{\alpha}^m$).
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% DONE Finish explaining
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\section{Definition of the colBiSBM models}\label{sec:definition-of-the-colbisbm-models}
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% Here are some common notations and conventions that we will use in the following
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% sections.
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\subsection{A collection of i.i.d 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
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@ -305,13 +364,96 @@ 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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X^m \sim \mathcal{F}-BiSBM_{n_1,n_2} (Q_1, Q_2, \bm{\pi}, \bm{\rho}, \bm{\alpha}), \forall m = 1, \dots M,
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X^m \sim \mathcal{F}-BiSBM_{n_1^m,n_2^m} (Q_1, Q_2, \bm{\pi}, \bm{\rho}, \bm{\alpha}), && \forall m = 1, \dots M
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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_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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and the third term to connectivity parameters.
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% TODO Finish explaining
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But the assumption that block proportions are the same among the networks is a
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strong assumption. In plant-pollinator networks, the proportion of specialist
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species can differ between networks and thus the model may benefit from not
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having the same block proportions but sharing a common connectivity structure.
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The following models relaxes this assumption on either row, column or both.
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\subsection{A collection of bipartite SBM with varying block size on either rows or columns}\label{ssec:a-collection-of-bipartite-sbm-with-varying-block-size-on-either-rows-or-columns}
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% TODO Finish explaining
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% DONE Finish explaining
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$\pi$-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
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block proportions. For $m \in \{1,\dots,M\}$, the $X^m$ are independent and
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\begin{align}
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\tag{\emph{$\pi$-colBiSBM}}
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X^m \sim \mathcal{F}-BiSBM_{n_1^m,n_2^m} (Q_1, Q_2, \bm{\pi^m}, \bm{\rho}, \bm{\alpha}), && \forall m = 1, \dots, M
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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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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-liddellLearningCommonStructures2023} and
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adapt the support $S$ they define for the $\pi$-colSBM to the bipartite case by
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having $S^1$ of size $M\times Q_1$ the support for the rows and $S^2$ of size
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$M\times Q_2$ the support for the columns. Thus
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$S^1_{mq} = \mathbb{1}_{\pi^m_q > 0}$ and
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$S^2_{mr} = \mathbb{1}_{\rho^m_r > 0}$. In this case, $S^2 = \bm{1}$, because
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there is no freedom on the column dimension.
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For a given number of blocks $Q_1$, $Q_2$ and matrix $S^1$ ($S^2$ being in this case the matrix full of ones), the number of
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parameters is:
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\begin{equation*}
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\text{NP}(\pi\text{-}colBiSBM) = \sum_{m=1}^{M}\Bigg( \sum_{q=1}^{Q_1} S^1_{mq} - 1 \Bigg) + (Q_2 - 1) + \sum_{\substack{q=1,\dots,Q_1 \\ r=1,\dots,Q_2}} \mathbb{1}_{{(S^{1\prime}S^2)}_{qr}>0}
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\end{equation*}
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The first term corresponds to the non-null block proportions in each network.
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The third quantity accounts for the fact that some blocks may never be
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represented simultaneously in any network, so the corresponding connection
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parameters $\alpha_{qr}$ are not useful for defining the model.
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$\rho$-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 column
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block proportions. For $m \in \{1,\dots,M\}$, the $X^m$ are independent and
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\begin{align}
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\tag{\emph{$\rho$-colBiSBM}}
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X^m \sim \mathcal{F}-BiSBM_{n_1^m,n_2^m} (Q_1, Q_2, \bm{\pi}, \bm{\rho^m}, \bm{\alpha}), && \forall m = 1, \dots, M
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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_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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then the column block $r$ is not represented in the network $m$.
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"Mirroring" the formulas for the $\pi$-$colBiSBM$ we relax the constraints on
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the column dimension.
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For a given number of blocks $Q_1$, $Q_2$ and matrix $S^2$ ($S^1$ being in this case the matrix full of ones), the number of
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parameters is:
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\begin{equation*}
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\text{NP}(\pi\text{-}colBiSBM) = (Q_1 - 1) + \sum_{m=1}^{M}\Bigg( \sum_{r=1}^{Q_2} S^2_{mr} - 1 \Bigg) + \sum_{\substack{q=1,\dots,Q_1 \\ r=1,\dots,Q_2}} \mathbb{1}_{{(S^{1\prime}S^2)}_{qr}>0}
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\end{equation*}
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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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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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X^m \sim \mathcal{F}-BiSBM_{n_1^m,n_2^m} (Q_1, Q_2, \bm{\pi^m}, \bm{\rho^m}, \bm{\alpha}), && \forall m = 1, \dots, M
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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
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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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For a given number of blocks $Q_1$, $Q_2$ and matrices $S^1$, $S^2$, the number of
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parameters is:
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\begin{equation*}
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\text{NP}(\pi\text{-}colBiSBM) = \sum_{m=1}^{M}\Bigg( \sum_{q=1}^{Q_1} S^1_{mq} - 1 \Bigg) + \sum_{m=1}^{M}\Bigg( \sum_{r=1}^{Q_2} S^2_{mr} - 1 \Bigg) + \sum_{\substack{q=1,\dots,Q_1 \\ r=1,\dots,Q_2}} \mathbb{1}_{{(S^{1\prime}S^2)}_{qr}>0}
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\end{equation*}
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\section{Variational estimation of the parameters}\label{sec:variational-estimation-of-the-parameters}
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@ -327,8 +469,14 @@ and $\bm{W}$ named $\mathcal{R}$ issued from a family of factorizable distributi
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\[
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\mathcal{J}(\mathcal{R};\bm{\theta}) \coloneqq \mathbb{E}_{\mathcal{R}}[\ell(\bm{X},\bm{Z},\bm{W};\bm{\theta})] + \mathcal{H}(\bm{Z,W}) \leq \ell(\bm{X};\bm{\theta})
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\]
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$\mathcal{H}$ is the entropy of the distribution. We define $\tau_{iq}^{1,m} = \mathbb{P}_{\mathcal{R}}(Z_{iq}^m = 1)$
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and $\tau_{jr}^{2,m} = \mathbb{P}_{\mathcal{R}}(W_{jr}^m = 1)$.
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$\mathcal{H}$ is the entropy of the distribution. $\bm{Z}$ and $\bm{W}$ are
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redefined using the \emph{one-hot encoded} conversion (i.e., $Z_i^m = q
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\rightarrow Z_{iq}^m = 1$ and $W_j^m = r \rightarrow W_{jr}^m = 1$)
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We define $\tau_{iq}^{1,m} = \mathbb{P}_{\mathcal{R}}(Z_{iq}^m = 1|X_{ij}^m)$
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and $\tau_{jr}^{2,m} = \mathbb{P}_{\mathcal{R}}(W_{jr}^m = 1|X_{ij}^m)$ and the
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variational approximation is
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$\mathbb{P}_{\mathcal{R}} (Z_{iq}^m = 1, W_{jr}^m = 1|X_{ij}^m) =
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\mathbb{P}_{\mathcal{R}}(Z_{iq}^m = 1|X_{ij}^m) {\color{red}\times} \mathbb{P}_{\mathcal{R}}(W_{jr}^m = 1|X_{ij}^m) = \tau_{iq}^{1,m} {\color{red}\times} \tau_{jr}^{2,m}$.
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% TODO Develop the formula
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@ -347,31 +495,32 @@ $$\widehat{\bm{\tau}}^{(t+1)} = \arg \max_{\bm{\tau}} \mathcal{J}(\mathcal{\bm{\
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And we obtain the following formulae for the $\bm{\tau^m}$:
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\begin{align*}
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\widehat{\tau}_{iq}^{1,m} \propto \widehat{\pi}_{q}^{m(t)} \prod_{j=1}^{n_2^m}\prod_{r\in\mathcal{Q}_{2,m}} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{jr}^{2,m(t+1)}} & \forall i = 1, \dots , n_1^m, q \in \mathcal{Q}_{1,m} \\
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\widehat{\tau}_{jr}^{2,m} \propto \widehat{\rho}_{r}^{m(t)} \prod_{i=1}^{n_1^m}\prod_{q\in\mathcal{Q}_{1,m}} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{iq}^{1,m(t+1)}} & \forall j = 1, \dots , n_2^m, r \in \mathcal{Q}_{2,m}
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\widehat{\tau}_{iq}^{1,m} \propto \widehat{\pi}_{q}^{m(t)} \prod_{j=1}^{n_2^m}\prod_{r\in\mathcal{Q}_2^m} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{jr}^{2,m(t+1)}} & \forall i = 1, \dots , n_1^m, q \in \mathcal{Q}_1^m \\
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\widehat{\tau}_{jr}^{2,m} \propto \widehat{\rho}_{r}^{m(t)} \prod_{i=1}^{n_1^m}\prod_{q\in\mathcal{Q}_1^m} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{iq}^{1,m(t+1)}} & \forall j = 1, \dots , n_2^m, r \in \mathcal{Q}_2^m
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\end{align*}
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From the above formulae we obtain for the Bernoulli distribution:
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\begin{itemize}
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\item[-] \textit{iid} :
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\[ \bm{\tau}^{m,1} = ~^{t}\pi + \exp((\text{Mask}^{m} \odot A^{m})
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\bm{\tau}^{m,2} ~^{t}(\text{logit}(\alpha)) + \text{Mask}^{m}
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\bm{\tau}^{m,2} ~^{t}\log(\bm{1} - \alpha)) \]
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\[ \bm{\tau}^{m,2} = ~^{t}\rho + \exp(~^{t}(\text{Mask}^{m} \odot A^{m})
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\bm{\tau}^{m,1} \text{logit}(\alpha) + ~^{t}\text{Mask}^{m}
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\bm{\tau}^{m,1} \log(\bm{1} - \alpha)) \]
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\item[-] $\rho\pi$ :
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\[ \bm{\tau}^{m,1} = ~^{t}\pi^{m} + \exp((\text{Mask}^{m} \odot A^{m})
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\bm{\tau}^{m,2} ~^{t}(\text{logit}(\alpha)) + \text{Mask}^{m}
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\bm{\tau}^{m,2} ~^{t}\log(\bm{1} - \alpha)) \]
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\[ \bm{\tau}^{m,2} = ~^{t}\rho^{m} + \exp(~^{t}(\text{Mask}^{m} \odot A^{m})
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\bm{\tau}^{m,1} \text{logit}(\alpha) + ~^{t}\text{Mask}^{m}
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\bm{\tau}^{m,1} \log(\bm{1} - \alpha)) \]
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\end{itemize}
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% TODO move to technical.tex
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% \begin{itemize}
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% \item[-] \textit{iid} :
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% \[ \bm{\tau}^{m,1} = ~^{t}\pi + \exp((\text{Mask}^{m} \odot A^{m})
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% \bm{\tau}^{m,2} ~^{t}(\text{logit}(\alpha)) + \text{Mask}^{m}
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% \bm{\tau}^{m,2} ~^{t}\log(\bm{1} - \alpha)) \]
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% \[ \bm{\tau}^{m,2} = ~^{t}\rho + \exp(~^{t}(\text{Mask}^{m} \odot A^{m})
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% \bm{\tau}^{m,1} \text{logit}(\alpha) + ~^{t}\text{Mask}^{m}
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% \bm{\tau}^{m,1} \log(\bm{1} - \alpha)) \]
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% \item[-] $\rho\pi$ :
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% \[ \bm{\tau}^{m,1} = ~^{t}\pi^{m} + \exp((\text{Mask}^{m} \odot A^{m})
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% \bm{\tau}^{m,2} ~^{t}(\text{logit}(\alpha)) + \text{Mask}^{m}
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% \bm{\tau}^{m,2} ~^{t}\log(\bm{1} - \alpha)) \]
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% \[ \bm{\tau}^{m,2} = ~^{t}\rho^{m} + \exp(~^{t}(\text{Mask}^{m} \odot A^{m})
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% \bm{\tau}^{m,1} \text{logit}(\alpha) + ~^{t}\text{Mask}^{m}
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% \bm{\tau}^{m,1} \log(\bm{1} - \alpha)) \]
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% \end{itemize}
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with $\text{Mask}^{m}$ the matrix containing $0$ if the value is a NA and a 1
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otherwise.
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% with $\text{Mask}^{m}$ the matrix containing $0$ if the value is a NA and a 1
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% otherwise.
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\subsection{M step of the algorithm}
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\label{ssec:m-step-of-the-algorithm}
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@ -391,7 +540,7 @@ The following quantities are involved in the obtained formulae:
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\end{align*}
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The block proportions, in free mixture models,
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$(\pi_q^m)_{q\in\mathcal{Q}_{1,m}}, (\rho_r^m)_{r\in\mathcal{Q}_{2,m}}$ are estimated as
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$(\pi_q^m)_{q\in\mathcal{Q}_1^m}, (\rho_r^m)_{r\in\mathcal{Q}_2^m}$ are estimated as
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\begin{align*}
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\widehat{\pi}_q^{m}= \frac{n^{1,m}_{q}}{n_1^m} & & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM \\
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\widehat{\rho}_r^{m}= \frac{n^{2,m}_{r}}{n_2^m} & & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
|
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|
|
|
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Reference in a new issue