165 lines
7.6 KiB
TeX
165 lines
7.6 KiB
TeX
\addtocounter{customchapter}{1}
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\chapter{Introduction}
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\section{Usage and importance of bipartite graphs}\label{sec:usage-and-importance-of-bipartite-graphs}
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Bipartite graphs, denoted as $G = (U,V,E)$ with $U$ and $V$ two disjoint and
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independent sets of vertices and $E$ the set of edges connecting $U$ vertices to
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$V$ vertices.
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\begin{minipage}{0.5\linewidth}
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\centering
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Bipartite network\\
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\begin{tikzpicture}[scale=.6]
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\tikzstyle{every edge}=[-,>=stealth',shorten >=1pt,auto,draw,line width=1.5pt]
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\tikzstyle{every state}=[draw, text=black,scale=0.95, transform shape]
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\tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape]
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\tikzstyle{every node}=[fill=blueind]
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\node[state, draw=black!50] (A1) at (0,5) {\textbf{R1}};
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\node[state, draw=black!50] (A2) at (2.5,5) {\textbf{R2}};
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\node[state, draw=black!50] (A3) at (5,5) {\textbf{R3}};
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\tikzstyle{every node}=[fill=greenind, shape=rectangle]
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\tikzstyle{every state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle]
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\node[state, draw=black!50] (B1) at (0,0) {\textbf{C1}};
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\node[state, draw=black!50] (B2) at (1.25,0) {\textbf{C2}};
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\node[state, draw=black!50] (B3) at (2.5,0) {\textbf{C3}};
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\node[state, draw=black!50] (B4) at (3.75,0) {\textbf{C4}};
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\node[state, draw=black!50] (B5) at (5,0) {\textbf{C5}};
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\path (A1) edge [] (B1);
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\path (A1) edge (B2);
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\path (A1) edge (B3);
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\path (A1) edge (B4);
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\path (A2) edge (B3);
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\path (A2) edge (B4);
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\path (A3) edge (B5);
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\path (A2) edge (B5);
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\end{tikzpicture}
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\end{minipage}
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\begin{minipage}{0.5\linewidth}
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\begin{center}
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$X=
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\begin{pmatrix}
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1 & 1 & 1 & 1 & 0 \\
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0 & 0 & 1 & 1 & 1 \\
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0 & 0 & 0 & 0 & 1 \\
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\end{pmatrix}
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$\\
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\vspace*{\baselineskip}
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Incidence matrix
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\end{center}
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\end{minipage}
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\vspace*{\baselineskip}
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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 observations (like presence-absence) 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
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and a numeric representation is the incidence matrix, in the above example
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$X$.\\
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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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nodes are the movies and the columns are the user, at the intersection the
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value is the review of the user $j$ for the movie $i$.\\
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Another use is the representation of ecological interactions like
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plant-pollinator \parencite{ramos-jilibertoTopologicalChangeAndean2010},
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birds-seed dispersion, prey-predator or host-parasite
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\parencite{kaszewska-gilasGlobalStudiesHostParasite2021}. For plant-pollinator
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interactions, the rows are pollinator species and the columns are plant species,
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and the intersection is a value, binary if it is a presence/absence or a value
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if it is an abundance count.
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Bipartite graphs are widely used in biology in general, in various fields, among
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which the previously cited ecological networks, but also in medicine with
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biomedical networks, biomolecular networks or epidemiological networks.
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\parencite{pavlopoulosBipartiteGraphsSystems2018}
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Some interesting results can arise when applying a tool widely used on a
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particular kind of interactions is used on another kind of interactions.
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Companies like Netflix or Amazon use recommender system, to recommend other
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products to consumers based on their previous interactions. In
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~\cite{desjardins-proulxEcologicalInteractionsNetflix2017} the authors use the
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\emph{K-nearest neighbour} (KNN) algorithm as a Recommender to predict missing
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preys for predators in a predator-prey network.
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\section{Latent Block Model}
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\label{sec:latent-block-model}
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The Latent Block Model (LBM) introduced by ~\cite{govaertLatentBlockModel2010}
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adapts the Stochastic Block Model (SBM)
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\parencite{hollandStochasticBlockmodelsFirst1983, snijdersEstimationPredictionStochastic1997}
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to bipartite graphs.
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\textit{Note :}\begin{small}
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Please note that we prefer the term ``BiSBM`` and will use both LBM and BiSBM to
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designate the Stochastic Block Model applied on bipartite networks.
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\end{small}
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This model supposes that:
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\begin{itemize}
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\item Row nodes are members of row blocks and column nodes are members of column
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blocks.
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\item The connectivity of two individuals is determined by their block memberships.
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\item An interaction can only occur between a row and a column node.
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\end{itemize}
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\begin{figure}[H]
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\center
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\begin{tikzpicture}[scale=.6]
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\input{../tikz/lbm.tex}
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\end{tikzpicture}
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\caption{An LBM model visualization}
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\label{fig:LBMvisu}
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\end{figure}
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\begin{itemize}
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\item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ \emph{given} blocks in rows
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\item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ \emph{given} blocks in columns
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\end{itemize}
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Parameters
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\begin{itemize}
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\item $\pi_{\bullet} = \mathbb{P}(Z_i = \bullet)$ for rows and $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ for columns
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\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.
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\end{itemize}
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On \ref{fig:LBMvisu}, $\bm{\pi}$ are the probabilities for a row node to belong
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to the row block of corresponding color, $\bm{\rho}$ are the probabilities for
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a column node to belong to the column block of corresponding color and
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$\bm{\alpha}$ is a matrix $Q_1 \times Q_2$ of the connectivity parameters
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between the row and column blocks.
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This model can be used to easily generate bipartite graphs with complex and
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very varied structures. But when trying to determine the structure of a given
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network we need to find those parameters and as the row and column block
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memberships are \emph{latent} i.e.,\ they are not known and must be inferred.
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For this a common approach is to use a \emph{variational} EM algorithm (proposed
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for SBM in~\cite{daudinMixtureModelRandom2008} and for LBM in
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~\cite{govaertEMAlgorithmBlock2005}) those groups and the required parameters
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can be inferred by maximizing a lower bound of the likelihood.
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\section{colSBM model, a joint model for a collection of networks}
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\label{sec:colsbm-model-a-joint-model-for-a-collection-of-networks}
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The \emph{colSBM} model introduced by ~\cite{chabert-liddellLearningCommonStructures2024a}
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propose an extension of the SBM model to collections of simple (or unipartite)
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networks. A collection is a set of networks which nodes are not common or linked
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between different networks, the interactions have the same valuations and
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are of the same type.
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The model can retrieve the shared structure in a collection, indicate if
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networks should be grouped in a collection and in a large pool of networks,
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collections with common structures.
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The next step after designing this collection model for unipartite networks was
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to extend it to the bipartite case.
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