From b7d83204210adf1b3a75855bdb1800a8de3d0c6b Mon Sep 17 00:00:00 2001 From: Louis Lacoste Date: Mon, 22 Jul 2024 15:58:47 +0200 Subject: [PATCH] rapport : updating rapport --- rapport/appendices.tex | 2 +- rapport/chapter3-structure-detection.tex | 73 +- rapport/chapter4-simulations/inference.tex | 22 +- .../information-transfer.tex | 4 +- .../chapter4-simulations/model-selection.tex | 26 +- .../chapter4-simulations/na-robustness.tex | 16 +- .../network-clustering.tex | 10 +- rapport/chapter5-applications.tex | 3 + rapport/conclusions.tex | 34 +- rapport/rapport.pdf | Bin 813023 -> 818826 bytes rapport/rapport.tex | 50 +- references.bib | 16 + tables/simulations/inference/iid.tex | 6 +- tables/simulations/inference/pi.tex | 6 +- tables/simulations/inference/pirho.tex | 6 +- tables/simulations/inference/rho.tex | 6 +- tables/simulations/inference/sep.tex | 4 +- .../model_selection/model-selection.tex | 160 +- tikz/clustering.tex | 4 +- tikz/simulations/inference/ari-plots.tex | 4 +- .../inference/model-proportions.tex | 4 +- .../model_selection/eps-pi-rho-preferred.tex | 3445 ++++++++--------- 22 files changed, 1981 insertions(+), 1920 deletions(-) create mode 100644 rapport/chapter5-applications.tex diff --git a/rapport/appendices.tex b/rapport/appendices.tex index 4f06e1b..96706f1 100644 --- a/rapport/appendices.tex +++ b/rapport/appendices.tex @@ -1,6 +1,6 @@ \clearpage \pagenumbering{arabic}% resets `page` counter to 1 -\renewcommand*{\thepage}{A\arabic{page}} +\renewcommand*{\thepage}{A-\arabic{page}} \appendix \chapter{Supplementary for~\nameref{chap:simulation-studies}} Below are the supplementary material for the~\nameref{chap:simulation-studies}. diff --git a/rapport/chapter3-structure-detection.tex b/rapport/chapter3-structure-detection.tex index d051296..15d1f0a 100644 --- a/rapport/chapter3-structure-detection.tex +++ b/rapport/chapter3-structure-detection.tex @@ -67,7 +67,7 @@ network $m$ is assumed to follow a $BiSBM$ with its own parameters ($\bm{\pi}^m, \bm{\rho}^m, \bm{\alpha}^m$). % DONE Finish explaining -\section{Definition of the colBiSBM models}\label{sec:definition-of-the-colbisbm-models} +\section{Definition of the \emph{colBiSBM} models}\label{sec:definition-of-the-colbisbm-models} % Here are some common notations and conventions that we will use in the following % sections. @@ -77,7 +77,7 @@ all the networks are the independent realizations of the same $Q_1$-$Q_2$-BiSBM with identical parameters. The \emph{iid-colBiSBM} is defined as follows: \begin{align} - \tag{\emph{iid-colBiSBM}} + \tag{\emph{iid}-colBiSBM} 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 \end{align} where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$, @@ -99,7 +99,7 @@ $\pi$-colBiSBM model still assumes that the networks share a common connectivity structure represented by $\bm{\alpha}$ but that each network has its own row block proportions. For $m \in \{1,\dots,M\}$, the $X^m$ are independent and \begin{align} - \tag{\emph{$\pi$-colBiSBM}} + \tag{\emph{$\pi$}-colBiSBM} 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 \end{align} where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$, @@ -120,7 +120,7 @@ there is no freedom on the column dimension. 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 parameters is: \begin{equation*} - \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}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0} + \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}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0} \end{equation*} The first term corresponds to the non-null block proportions in each network. The third quantity accounts for the fact that some blocks may never be @@ -131,7 +131,7 @@ $\rho$-colBiSBM model still assumes that the networks share a common connectivit structure represented by $\bm{\alpha}$ but that each network has its own column block proportions. For $m \in \{1,\dots,M\}$, the $X^m$ are independent and \begin{align} - \tag{\emph{$\rho$-colBiSBM}} + \tag{\emph{$\rho$}-colBiSBM} 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 \end{align} where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$, @@ -142,13 +142,13 @@ proportions to be null in certain networks ($\rho^m_r\in\left[ 0,1 \right]$): if $\rho_r^m = 0$ then the column block $r$ is not represented in the network $m$. -\enquote{Mirroring} the formulas for the $\pi$-$colBiSBM$ we relax the constraints on +\enquote{Mirroring} the formulas for the $\pi$-colBiSBM we relax the constraints on the column dimension. 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 parameters is: \begin{equation*} - \text{NP}(\rho\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}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0} + \text{NP}(\rho\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}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0} \end{equation*} $\pi\rho$-colBiSBM model still assumes that the networks share a common connectivity @@ -156,7 +156,7 @@ structure represented by $\bm{\alpha}$ but that each network has its own row and column block proportions, it is the less constrained model. For $m \in \{1,\dots,M\}$, the $X^m$ are independent and \begin{align} - \tag{\emph{$\pi\rho$-colBiSBM}} + \tag{\emph{$\pi\rho$}-colBiSBM} 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 \end{align} where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$, @@ -166,17 +166,18 @@ $\rho^m_r \in \left[ 0,1 \right], \sum_{r=1}^{Q_2} \rho^m_r = 1 $. For a given number of blocks $Q_1$, $Q_2$ and matrices $S^1$, $S^2$, the number of parameters is: \begin{equation*} - \text{NP}(\pi\rho\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}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0} + \text{NP}(\pi\rho\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}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0} \end{equation*} \section{Variational estimation of the parameters}\label{sec:variational-estimation-of-the-parameters} In practice, the estimation of the likelihood is not tractable. Following the classical approach defined in~\cite{daudinMixtureModelRandom2008} we use a -variatonal version of the Expectation Maximization (VEM) algorithm. +variational version of the Expectation Maximization (VEM) algorithm. We maximize a variational lower bound of the log-likelihood of the observed -data by approximating $p(\bm{Z,W}|\bm{X};\bm{\theta})$ with a distribution on +data, the so-called Evidence Lower Bound (or ELBO), by approximating +$p(\bm{Z,W}|\bm{X};\bm{\theta})$ with a distribution on $\bm{Z}$ and $\bm{W}$ named $\mathcal{R}$ defined as $\mathcal{R} = \otimes_{m=1}^M \mathcal{R}_m$.\ @@ -185,7 +186,7 @@ The lower bound is defined as: \mathcal{J}(\mathcal{R};\bm{\theta}) \coloneqq \sum_{m=1}^{M} \bigg( \mathbb{E}_{\mathcal{R}_m}[\ell(X^m,Z^m,W^m;\bm{\theta})] + \mathcal{H}(\mathcal{R}_m) \bigg) \leq \ell(\bm{X};\bm{\theta}) \end{equation*} -$\bm{Z}$ and $\bm{W}$ are +$(Z^m_i)_{i=1\dots n_1^m}$ and $(W^m_j)_{j=1\dots n_2^m}$ are redefined using the \emph{one-hot encoded} conversion (i.e., $Z_i^m = q \rightarrow Z_{iq}^m = 1$ and $W_j^m = r \rightarrow W_{jr}^m = 1$).\\ % W_{jr\prime}^m pour r != r égal 0 @@ -201,7 +202,7 @@ we have: $\mathbb{P}_{\mathcal{R}_m} (Z_{iq}^m = 1, W_{jr}^m = 1|X^m) = The formula for the entropy per network is thus: \begin{equation*} - \mathcal{H}(\mathcal{R}_m) = - \sum_{i=1}^{n_1} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} + \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} \end{equation*} And the expectation of the completed log-likelihood under the $\mathcal{R}_m$ @@ -216,7 +217,7 @@ And thus the lower bound becomes: \begin{align*} \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}) \\ + \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} \\ - - \sum_{i=1}^{n_1} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} \bigg) \color{black} + - \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} \end{align*} where we identify the variational distribution $\mathcal{R}$ with its parameter @@ -240,11 +241,13 @@ $\bm{\tau}$: $$\widehat{\bm{\tau}}^{(t+1)} = \arg \max_{\bm{\tau}} \mathcal{J}(\mathcal{\bm{\tau}},\bm{\widehat{\theta}}^{(t)}).$$ And we obtain the following formulae for the $\bm{\tau^m}$: +\begin{equation*} + \begin{cases} + \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 \\ + \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 + \end{cases} +\end{equation*} -\begin{align*} - \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 \\ - \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 -\end{align*} which are used to update iteratively the values by a fixed point algorithm with only one step. @@ -269,13 +272,13 @@ The block proportions, in free mixture models, $(\pi_q^m)_{q\in\mathcal{Q}_1^m}, (\rho_r^m)_{r\in\mathcal{Q}_2^m}$ are estimated as \begin{align*} - \widehat{\pi}_q^{m}= \frac{n^{1,m}_{q}}{n_1^m} & & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM \\ - \widehat{\rho}_r^{m}= \frac{n^{2,m}_{r}}{n_2^m} & & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM + \widehat{\pi}_q^{m}= \frac{n^{1,m}_{q}}{n_1^m} & & \text{for } \pi\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM} \\ + \widehat{\rho}_r^{m}= \frac{n^{2,m}_{r}}{n_2^m} & & \text{for } \rho\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM} \end{align*} while on the other hand, \begin{align*} - \widehat{\pi}_q = \frac{\sum_{m=1}^{M} n^{1,m}_{q}}{\sum_{m=1}^{M} n_1^m} & & \text{for } iid\text{-}colBiSBM \text{ and } \rho\text{-}colBiSBM \\ - \widehat{\rho}_r = \frac{\sum_{m=1}^{M} n^{2,m}_{r}}{\sum_{m=1}^{M} n_2^m} & & \text{for } iid\text{-}colBiSBM \text{ and } \pi\text{-}colBiSBM + \widehat{\pi}_q = \frac{\sum_{m=1}^{M} n^{1,m}_{q}}{\sum_{m=1}^{M} n_1^m} & & \text{for } iid\text{-colBiSBM} \text{ and } \rho\text{-colBiSBM} \\ + \widehat{\rho}_r = \frac{\sum_{m=1}^{M} n^{2,m}_{r}}{\sum_{m=1}^{M} n_2^m} & & \text{for } iid\text{-colBiSBM} \text{ and } \pi\text{-colBiSBM} \end{align*} the parameters takes into account all the networks at the same time. The connectivity parameters $\alpha_{qr}$ for all models are estimated as the ratio @@ -344,7 +347,7 @@ This leads us to formulate a BIC-like criterion in the following manner: We provide below the expression for the penalties for the 4 models that we propose. \begin{description} - \item[\textit{iid-colBiSBM}] For the $\bm\pi$ and $\bm\rho$: + \item[\textit{iid}-colBiSBM] For the $\bm\pi$ and $\bm\rho$: \begin{align*} \text{pen}_{\pi}(Q_1) = (Q_1 - 1)\log(\sum_{m=1}^{M}n_{1}^{m}) & , & \text{pen}_{\rho}(Q_2) = (Q_2 - 1)\log(\sum_{m=1}^{M}n_{2}^{m}) @@ -358,15 +361,15 @@ propose. \mathcal{J} (\mathcal{\hat{R}}, \bm{\theta}) - \frac{1}{2} [\text{pen}_{\pi}(Q_1) + \text{pen}_{\rho}(Q_2) + \text{pen}_{\alpha}(Q_1, Q_2)]\] - \item[\textit{$\bm{\pi\rho}$-colBiSBM}] The support penalties are + \item[$\bm{\pi\rho}$-colBiSBM] The support penalties are \begin{align*} \text{pen}_{S_1}(Q_1) = -2 \log p_{Q_1} (S_1) & , & \text{pen}_{S_2}(Q_2) = -2 \log p_{Q_2} (S_2) \end{align*} with \begin{align*} - \log p_{Q_1}(S_1) = - M \log(Q_1) - \sum_{m=1}^{M} \log {Q_1 - \choose Q_1^{(m)}}, & - \log p_{Q_2}(S_2) = - M \log(Q_2) - \sum_{m=1}^{M} \log {Q_2 + \textstyle \log p_{Q_1}(S_1) = - M \log(Q_1) - \sum_{m=1}^{M} \log {Q_1 + \choose Q_1^{(m)}}, \\ + \textstyle \log p_{Q_2}(S_2) = - M \log(Q_2) - \sum_{m=1}^{M} \log {Q_2 \choose Q_2^{(m)}}. \end{align*} And penalties for the $\bm\rho$ and $\bm\pi$ are @@ -689,12 +692,12 @@ partition $\mathcal{G}$. \label{ssec:dissimilarity-between-two-networks} The parameters for the dissimilarity are defined as follow: \begin{align*} - \widetilde{n}_{qr}^m = \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \widehat{\tau}_{iq}^{1,m} \widehat{\tau}_{jr}^{2,m}, - & & \widetilde{\alpha}_{qr}^m = \frac{\sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \widehat{\tau}_{iq}^{1,m} \widehat{\tau}_{jr}^{2,m} X_{ij}^m}{\widetilde{n}_{qr}^m}, \\ - \widetilde{\pi}_q^m = \frac{\sum_{i=1}^{n_1^m} \widehat{\tau}_{iq}^{1,m}}{n_1^m}, - & & \widetilde{\rho}_r^m = \frac{\sum_{j=1}^{n_2^m} \widehat{\tau_{jr}}^{2,m}}{n_2^m} + \widetilde{n}_{qr}^m & = \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \widehat{\tau}_{iq}^{1,m} \widehat{\tau}_{jr}^{2,m}, + & & \widetilde{\alpha}_{qr}^m = \frac{\sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \widehat{\tau}_{iq}^{1,m} \widehat{\tau}_{jr}^{2,m} X_{ij}^m}{\widetilde{n}_{qr}^m}, \\ + \widetilde{\pi}_q^m & = \frac{\sum_{i=1}^{n_1^m} \widehat{\tau}_{iq}^{1,m}}{n_1^m}, + & & \widetilde{\rho}_r^m = \frac{\sum_{j=1}^{n_2^m} \widehat{\tau_{jr}}^{2,m}}{n_2^m}. \end{align*} -And the dissimilarity between any pair of networks $(m,m')\in\mathcal{M}^2$ is then: +And the pairwise dissimilarity for networks $(m,m')\in\mathcal{M}^2$ is then: \[ D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \widetilde{\alpha}_{qr}^{m} - \widetilde{\alpha}_{qr}^{m'}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'}) \] @@ -710,10 +713,10 @@ And the dissimilarity between any pair of networks $(m,m')\in\mathcal{M}^2$ is t \tikzstyle{es}=[font=\small, text justified, rectangle,draw,rounded corners=4pt,fill=cyanind!25] \node[es] (liste) at (0,4) {Supply a collection to partition}; - \node[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Fit \emph{colBiSBM}}; + \node[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Fit colBiSBM}; \node[first_col, right = 0.5cm of 1-collection] (1-col-obj) {}; \node[instruct, text width=5cm, below = 0.45cm of 1-collection] (dissimi) {Compute a dissimilarity matrix over the collection}; - \node[instruct, text width=5cm, below = 0.45cm of dissimi] (2-sous-collection) {Split the \emph{collection in 2 sub-collections} and fit the \emph{colBiSBM}}; + \node[instruct, text width=5cm, below = 0.45cm of dissimi] (2-sous-collection) {Split the \emph{collection in 2 sub-collections} and fit the colBiSBM}; \node[second_col, right = 0.25cm of 2-sous-collection] (1-sec-col-obj) {1}; \node[second_col, right = 0.25cm of 1-sec-col-obj] (1-sec-col-obj) {2}; \node[test,below = 0.45cm of 2-sous-collection, scale=0.7] (BICL-test) {$\sum_{i=1}^{2} (\text{BIC-L}(\tikz[baseline=-0.25cm]{\node[second_col] {i};} )) > \text{BIC-L}(\tikz[baseline=-0.25cm]{\node[first_col] {};})$?}; @@ -736,7 +739,7 @@ And the dissimilarity between any pair of networks $(m,m')\in\mathcal{M}^2$ is t The above figure (\ref{fig:netclustering-procedure}) shows a condensed explanation of the network clustering algorithm. -The idea is to adjust the \emph{colBiSBM} model over the full collection of $M$ +The idea is to adjust the colBiSBM model over the full collection of $M$ networks and then compute the dissimilarity matrix between all networks of the collection. We obtain the collection $\mathcal{G} = \{\mathcal{M}\}$ the trivial partition in a unique group. diff --git a/rapport/chapter4-simulations/inference.tex b/rapport/chapter4-simulations/inference.tex index 93a85f4..773d872 100644 --- a/rapport/chapter4-simulations/inference.tex +++ b/rapport/chapter4-simulations/inference.tex @@ -48,7 +48,7 @@ community and dis-assortative community structures, depending on which 3 of the blocks are selected for each network. $\eps[\alpha]$ represents the strength of these structures, the larger, the easier it is to tell apart one block from another. -The true model of all the simulation is a $\pi\rho\text{-}colBiSBM$. +The true model of all the simulation is a $\pi\rho$-colBiSBM. \paragraph{Inference} We want to measure the quality of the inference procedure, for this we use the inference described in the section @@ -57,15 +57,15 @@ inference procedure, for this we use the inference described in the section \paragraph{Quality indicators} To assess the quality of the inference, we will use the following indicators: \begin{itemize} - \item First, for each dataset, we put in competition $\pi\text{-}colBiSBM$ with - $sep\text{-}BiSBM$, $iid\text{-}colBiSBM$, $\rho\text{-}colBiSBM$, - $\pi\rho\text{-}colBiSBM$ + \item First, for each dataset, we put in competition $\pi$-colBiSBM with + $sep\text{-}BiSBM$, $iid$-colBiSBM, $\rho$-colBiSBM, + $\pi\rho$-colBiSBM respectively. To do so, for each dataset, we compute the - BIC-L of each model $\pi\text{-}colBiSBM$ is preferred to $sep\text{-}BiSBM$ - (resp. $iid\text{-}colBiSBM$, $\rho\text{-}colBiSBM$, - $\pi\rho\text{-}colBiSBM$) if + BIC-L of each model $\pi$-colBiSBM is preferred to $sep\text{-}BiSBM$ + (resp. $iid$-colBiSBM, $\rho$-colBiSBM, + $\pi\rho$-colBiSBM) if its BIC-L is greater. - \item When considering our \emph{colBiSBM} models we compare + \item When considering our colBiSBM models we compare $\widehat{Q_1}$, $\widehat{Q_2}$ to their true values. ($Q_1 = 4$ and $Q_2 = 4$) \item Finally, we assess the quality of the node grouping by computing the @@ -76,8 +76,8 @@ use the following indicators: negative values if the RI is less than the expected value. This indicates a structure in grouping discordance.}. For each network, for the - $\pi\text{-}colBiSBM$, $\rho\text{-}colBiSBM$, - $\pi\rho\text{-}colBiSBM$ we compare the inferred block memberships to + $\pi$-colBiSBM, $\rho$-colBiSBM, + $\pi\rho$-colBiSBM we compare the inferred block memberships to the real ones by computing the mean of the ARI per axis over the two networks \begin{equation*} @@ -122,7 +122,7 @@ of a single block on each dimension. On the figure \ref{fig:inference-prop-modele-pref} one can see that from $\eps[\alpha] = 0.06$ around $70\%$ of the time the -$\pi\rho\text{-}colBiSBM$ model (i.e., the correct one) is selected. +$\pi\rho$-colBiSBM model (i.e., the correct one) is selected. An interesting result we can read in the tables is that our models outperform the $sep\text{-}BiSBM$ when considering the ARI on the whole set of nodes diff --git a/rapport/chapter4-simulations/information-transfer.tex b/rapport/chapter4-simulations/information-transfer.tex index 51cd89f..4d4b8e8 100644 --- a/rapport/chapter4-simulations/information-transfer.tex +++ b/rapport/chapter4-simulations/information-transfer.tex @@ -3,7 +3,7 @@ One of the motivation for collections of networks is \emph{information transfer} between the networks, allowing robustness to missing data and enabling the finding of finer structures in small networks with the help of bigger ones. -\subsection{Missing edges robustness} +% \subsection{Missing edges robustness} \input{chapter4-simulations/na-robustness} -\subsection{Finer structure detection on small networks} \ No newline at end of file +% \subsection{Finer structure detection on small networks} \ No newline at end of file diff --git a/rapport/chapter4-simulations/model-selection.tex b/rapport/chapter4-simulations/model-selection.tex index 6f89b94..acad147 100644 --- a/rapport/chapter4-simulations/model-selection.tex +++ b/rapport/chapter4-simulations/model-selection.tex @@ -1,9 +1,9 @@ \section[Capacity to distinguish models]{Capacity to distinguish - $\pi\rho\text{-}colBiSBM$~from\newline - $iid\text{-}colBiSBM$ and other + $\pi\rho$-colBiSBM~from\newline + $iid$-colBiSBM and other models}\label{sec:capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants} The idea of this model selection simulations is to assess how the model -select the correct \emph{colBiSBM} model among the possible ones: +select the correct colBiSBM model among the possible ones: \textit{$iid, \pi, \rho, \pi\rho$}. This difference being based on the row and col block proportions.\\ \paragraph{Simulation settings} For this task we choose the same simulation settings as @@ -40,20 +40,20 @@ $\left[ 0, .28\right]$.\newline We simulate 324 different collections for each value of $\eps[\pi]$ and $\eps[\rho]$. -$\pi\rho\text{-}colBiSBM$, $\pi\text{-}colBiSBM$, -$\rho\text{-}colBiSBM$, $iid\text{-}colBiSBM$ and +$\pi\rho$-colBiSBM, $\pi$-colBiSBM, +$\rho$-colBiSBM, $iid$-colBiSBM and $sep\text{-}BiSBM$ are put in competition and the model with the greater BIC-L is selected as the \emph{preferred model}. When $\eps[\pi] = 0$, $\bm{\pi}^1 = \bm{\pi}^2$, $\eps[\rho] = 0$ and $\bm{\rho}^1 = \bm{\rho}^2$, the generated collection is an -$iid\text{-}colBiSBM$. When $\eps[\pi] > 0$ or -$\bm{\pi}^1 \neq \bm{\pi}^2$, the model is a $\pi\text{-}colBiSBM$. +$iid$-colBiSBM. When $\eps[\pi] > 0$ or +$\bm{\pi}^1 \neq \bm{\pi}^2$, the model is a $\pi$-colBiSBM. When $\eps[\rho] > 0$ or $\bm{\rho}^1 \neq \bm{\rho}^2$, the model -is a $\rho\text{-}colBiSBM$. Finally, when $\eps[\pi] > 0$ or +is a $\rho$-colBiSBM. Finally, when $\eps[\pi] > 0$ or $\bm{\pi}^1 \neq \bm{\pi}^2$ and $\eps[\rho] > 0$ or $\bm{\rho}^1 \neq \bm{\rho}^2$, the model is a -$\pi\rho\text{-}colBiSBM$. +$\pi\rho$-colBiSBM. \begin{figure}[!ht] @@ -68,10 +68,10 @@ $\pi\rho\text{-}colBiSBM$. On the figure \ref{fig:pref_model_func_eps} and table \ref{tab:model-selection}, one can see that there is a turning point around $\eps[\pi] = 0.2$ (resp. -$\eps[\rho] = 0.2$), before which $iid\text{-}colBiSBM$ and -$\rho\text{-}colBiSBM$ (resp. $\pi\text{-}colBiSBM$) are selected -very often and after $0.2$ the $\pi\text{-}colBiSBM$ (resp. -$\rho\text{-}colBiSBM$) and $\pi\rho\text{-}colBiSBM$ gets more and +$\eps[\rho] = 0.2$), before which $iid$-colBiSBM and +$\rho$-colBiSBM (resp. $\pi$-colBiSBM) are selected +very often and after $0.2$ the $\pi$-colBiSBM (resp. +$\rho$-colBiSBM) and $\pi\rho$-colBiSBM gets more and more selected. Moreover, the number of blocks are correctly detected in most of the case. These two results highlight our capacity to recover the simulated diff --git a/rapport/chapter4-simulations/na-robustness.tex b/rapport/chapter4-simulations/na-robustness.tex index ec9d949..839181f 100644 --- a/rapport/chapter4-simulations/na-robustness.tex +++ b/rapport/chapter4-simulations/na-robustness.tex @@ -6,13 +6,13 @@ For this purpose we generate collections of networks with the following parameters: \begin{align*} \bm{\pi}^m = \begin{cases} - \bm{\pi} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-}colBiSBM \\ - \sigma_1^m(\bm{\pi}) & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM + \bm{\pi} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-colBiSBM} \\ + \sigma_1^m(\bm{\pi}) & \text{for } \pi\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM} \end{cases} \\ \bm{\rho}^m = \begin{cases} - \bm{\rho} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-}colBiSBM \\ - \sigma_2^m(\bm{\rho}) & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM, + \bm{\rho} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-colBiSBM} \\ + \sigma_2^m(\bm{\rho}) & \text{for } \rho\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM}, \end{cases} \end{align*} for the block proportions, and two different structures with the corresponding @@ -38,11 +38,11 @@ structure detected in ecology with generalist and specialist species and a The collections contain two networks ($M=2$) of size $n^{m=1}_1 = n^{m=1}_2 = 40$ and -$n^{m=2}_1 = n^{m=2}_2 = 120$. One collection is generated for each $colBiSBM$ +$n^{m=2}_1 = n^{m=2}_2 = 120$. One collection is generated for each colBiSBM model. And the nodes block memberships (i.e., the row and column blocks they belong to) are saved. -Per $colBiSBM$ model, 10 collections are generated and their results are +Per colBiSBM model, 10 collections are generated and their results are averaged. In the network $m=1$ (i.e., the smaller one) a proportion of the edges @@ -54,7 +54,7 @@ predicted block memberships are saved, along with the predicted links using the inferred parameters. This will serve as a baseline to see if the use of the collection benefits the predictions. -A $colBiSBM$ model is then fitted (with a model matching the dataset considered) +A colBiSBM model is then fitted (with a model matching the dataset considered) and we store the same predictions. \paragraph{Quality metrics} To benchmark the performance we use the @@ -62,7 +62,7 @@ and we store the same predictions. ARI for predicted versus real block memberships. For the comparison we subtract the metric given by the LBM to the one -given by $colBiSBM$ and denote it $\Delta\mbox{metric}$. +given by colBiSBM and denote it $\Delta\mbox{metric}$. \begin{figure}[ht] \centering diff --git a/rapport/chapter4-simulations/network-clustering.tex b/rapport/chapter4-simulations/network-clustering.tex index 5d8b353..f04eff5 100644 --- a/rapport/chapter4-simulations/network-clustering.tex +++ b/rapport/chapter4-simulations/network-clustering.tex @@ -10,13 +10,13 @@ For the simulations the proportions are the following: \end{align*} and for all $m = 2,\dots,9$ \begin{align*} \bm{\pi}^m = \begin{cases} - \bm{\pi}^1 & \text{for } iid\text{-}colBiSBM \\ - \sigma_1^m(\bm{\pi}^1) & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM + \bm{\pi}^1 & \text{for } iid\text{-colBiSBM} \\ + \sigma_1^m(\bm{\pi}^1) & \text{for } \pi\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM} \end{cases} \\ \bm{\rho}^m = \begin{cases} - \bm{\rho}^1 & \text{for } iid\text{-}colBiSBM \\ - \sigma_2^m(\bm{\rho}^1) & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM + \bm{\rho}^1 & \text{for } iid\text{-colBiSBM} \\ + \sigma_2^m(\bm{\rho}^1) & \text{for } \rho\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM} \end{cases} \end{align*} where $\sigma_1^m$ and $\sigma_2^m$ are permutations of \{1, 2, 3\} proper to network $m$ and @@ -64,4 +64,4 @@ Increasing $\epsilon$ differentiates the 3 sub-collections of networks. the resulting partition of the network collection and the simulated partition using the ARI index. As the value of $\epsilon$ increases, our ability to distinguish between the networks improves, and this distinction becomes nearly -perfect in all setups of the $colBiSBM$. \ No newline at end of file +perfect in all setups of the colBiSBM. \ No newline at end of file diff --git a/rapport/chapter5-applications.tex b/rapport/chapter5-applications.tex new file mode 100644 index 0000000..19f8dff --- /dev/null +++ b/rapport/chapter5-applications.tex @@ -0,0 +1,3 @@ +\addtocounter{customchapter}{1} +\chapter{Applications on ecological networks} +\label{chap:applications-ecological-networks} \ No newline at end of file diff --git a/rapport/conclusions.tex b/rapport/conclusions.tex index c53af71..61460b8 100644 --- a/rapport/conclusions.tex +++ b/rapport/conclusions.tex @@ -1,3 +1,35 @@ \addtocounter{customchapter}{1} \chapter{Conclusions and future work} -\label{chap:conclusions-and-future-work} \ No newline at end of file +\label{chap:conclusions-and-future-work} +\section{Conclusion} +\label{sec:conclusion} + +\section{Future work} +\label{sec:future-work} + +\paragraph{Identifiability} +As stated in section~\ref{sec:model-identifiability}, we only have +identifiability for the \emph{iid}-colBiSBM and we will work on establishing +identifiability for $\pi$, $\rho$ and $\pi\rho$ models. + +\paragraph{Finding a trade-off between \emph{iid} and $\pi\rho$} +We observed while testing clustering with the different models that +the $\pi$, $\rho$ and $\pi\rho$ model, with their increased number of parameters +for block memberships parameters tends to give smaller BIC-L criterion values +while having a higher Evidence Lower Bound than the \emph{iid}. +This arises because of the penalties on the block memberships and support that +increase significantly and exceeds the gain on the ELBO and the diminution of +the connectivity parameters. +An idea to tackle this problem could be to suppose that the block memberships +for network $m$ are themselves the realizations of random variables and +thus introduce sort of a mixed effect model. + +\paragraph{Comparison to other graphs clustering methods} +Recent work have been comparing +colSBM~\parencite{chabert-liddellLearningCommonStructures2024a} and +graphclust~\parencite{rebafkaModelbasedClusteringMultiple2023} assessing various +capabilities of the models and particularly focusing on networks clustering. +We will reproduce and adapt the analysis to test other simulation settings that +were not considered in this work. + +\section*{Thank you for reading this work} \ No newline at end of file diff --git a/rapport/rapport.pdf b/rapport/rapport.pdf index c754f8abc32c60fa83c8356bf92c74af657a45e5..fa59792d6c1df84b97b41a2e46f3ef0c7a16bebb 100644 GIT binary patch delta 256715 zcmZs?Q*bU!6D=G(v2EM7ZQHi(Jh5%>*tTsOJ9e_OV<$VwdB6HE{&R6IrdQRfo|>Az z=$T&Av%kO+^TpE03JJ!-m1)cbh0!7nX$Syk;bKWfrpbgO2ct@TgH-}!;ZBu?lm8D| z(&2RB|A%*2u+;y#u_hyvr4k9N{Eu=G)_cmtNdh1y<6-~DNp^5?fL7gPzJ&u2Gq1_O 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