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rapport/chapter2-context.tex
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@ -102,8 +102,8 @@ adapts the Stochastic Block Model (SBM)
to bipartite graphs. to bipartite graphs.
\textit{Note :}\begin{small} \textit{Note :}\begin{small}
Please note that we prefer the term ``BiSBM`` and will use both LBM and BiSBM to Please note that we prefer the term \enquote{BiSBM} and will use both LBM and BiSBM to
designate the Stochastic Block Model applied on bipartite networks. designate the Stochastic Block Model adapted to bipartite networks.
\end{small} \end{small}
This model supposes that: This model supposes that:
@ -130,10 +130,11 @@ This model supposes that:
Parameters Parameters
\begin{itemize} \begin{itemize}
\item $\pi_{\bullet} = \mathbb{P}(Z_i = \bullet)$ for rows and $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ for columns \item $\pi_{\bullet} = \mathbb{P}(Z_i = \bullet)$ for rows and $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ for columns
\item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$, probability of connectivity knowing node membership blocks. \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$, parameter influencing the probability and value of a link knowing node
membership blocks.
\end{itemize} \end{itemize}
On \ref{fig:LBMvisu}, $\bm{\pi}$ are the probabilities for a row node to belong On figure~\ref{fig:LBMvisu}, $\bm{\pi}$ are the probabilities for a row node to belong
to the row block of corresponding color, $\bm{\rho}$ are the probabilities for to the row block of corresponding color, $\bm{\rho}$ are the probabilities for
a column node to belong to the column block of corresponding color and a column node to belong to the column block of corresponding color and
$\bm{\alpha}$ is a matrix $Q_1 \times Q_2$ of the connectivity parameters $\bm{\alpha}$ is a matrix $Q_1 \times Q_2$ of the connectivity parameters
@ -143,20 +144,19 @@ of the network we are referring to this connectivity matrix.
This model can be used to easily generate bipartite graphs with complex and This model can be used to easily generate bipartite graphs with complex and
very varied structures. But when trying to determine the structure of a given very varied structures. But when trying to determine the structure of a given
network we need to find those parameters and as the row and column block network we need to find those parameters and as the row and column block
memberships are \emph{latent} i.e.,\ they are not known and must be inferred. memberships are \emph{latent} i.e.,\ they are not known, they must be inferred.
For this a common approach is to use a \emph{variational} EM algorithm (proposed For this a common approach is to use a \emph{variational} EM algorithm, proposed
for SBM in~\cite{daudinMixtureModelRandom2008} and for LBM in for SBM in~\cite{daudinMixtureModelRandom2008} and for LBM in
~\cite{govaertEMAlgorithmBlock2005}) those groups and the required parameters ~\cite{govaertEMAlgorithmBlock2005}. The groups and required parameters
can be inferred by maximizing a lower bound of the likelihood. can be inferred by maximizing a lower bound of the likelihood.
\section{colSBM model, a joint model for a collection of networks} \section{colSBM model, a joint model for a collection of networks}
\label{sec:colsbm-model-a-joint-model-for-a-collection-of-networks} \label{sec:colsbm-model-a-joint-model-for-a-collection-of-networks}
The \emph{colSBM} model introduced by ~\cite{chabert-liddellLearningCommonStructures2024a} The \emph{colSBM} model introduced by ~\cite{chabert-liddellLearningCommonStructures2024a}
propose an extension of the SBM model to collections of simple (or unipartite) propose an extension of the SBM model to collections of simple (or unipartite)
networks. A collection is a set of networks which nodes are not common or linked networks. A collection is a set of networks which nodes are not in common nor
between different networks, the interactions have the same valuations and linked between different networks and the interactions have the same valuations.
are of the same type.
The model can retrieve the shared structure in a collection, indicate if The model can retrieve the shared structure in a collection, indicate if
networks should be grouped in a collection and in a large pool of networks, networks should be grouped in a collection and in a large pool of networks,
@ -164,3 +164,5 @@ collections with common structures.
The next step after designing this collection model for unipartite networks was The next step after designing this collection model for unipartite networks was
to extend it to the bipartite case. to extend it to the bipartite case.
% DONE Relu