rapport : ajout des autres section du chapitre Simulations

rapport/chapter4-simulation-studies.tex

@@ -10,6 +10,7 @@ the report repository at
 \url{https://gitea.polarolouis.fr/polarolouis/rapport-CEI-MIA-2023}.
 
 
-\input{chapter4-simulations/inference.tex}
+\input{chapter4-simulations/inference}
 \input{chapter4-simulations/model-selection}
-% \include{Rcodes/simulation/netclustering_analyze}
+\input{chapter4-simulations/network-clustering}
+\input{chapter4-simulations/information-transfer}

rapport/chapter4-simulations/information-transfer.tex

@@ -0,0 +1,8 @@
+\section{Information transfer between networks}
+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{Finer structure detection on small networks}

rapport/chapter4-simulations/network-clustering.tex

@@ -0,0 +1,60 @@
+\section{Network clustering of simulated networks}\label{sec:network-clustering-of-simulated-networks}
+
+\paragraph{Simulation settings} For all models we simulate $M = 9$ networks with
+$\forall m \in \{ 1 \dots M \} , n^m_1 = n^m_2 = 75$ with $Q_1 = Q_2 = 3$. For
+the simulations the proportions are the following:
+
+\begin{align*}
+    \bm{\pi}^1 = \left( 0.2, 0.3, 0.5 \right) &  & \bm{\rho}^1 = \left( 0.2, 0.3, 0.5 \right)
+\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
+                 \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
+    \end{cases}
+\end{align*}
+where $\sigma^1_m$ and $\sigma^2_m$ are permutations of {1, 2, 3} proper to network $m$ and
+$\sigma^1 (\pi)= {(\pi_{\sigma^1 (i)})}_{i=\{1,\dots,3\}}$
+and $\sigma^2 (\rho)= {(\rho_{\sigma^2 (i)})}_{i=\{1,\dots,3\}}$.
+The networks are divided into 3 sub-collections of 3
+networks with connectivity parameters as follows:
+
+\begin{align*}
+    \bm{\alpha}^{as} = .3 + \begin{pmatrix}
+                                \epsilon             & - \frac{\epsilon}{2} & - \frac{\epsilon}{2} \\
+                                - \frac{\epsilon}{2} & \epsilon             & - \frac{\epsilon}{2} \\
+                                - \frac{\epsilon}{2} & - \frac{\epsilon}{2} & \epsilon
+                            \end{pmatrix}, &  &
+    \bm{\alpha}^{cp} = .3 + \begin{pmatrix}
+                                \frac{3 \epsilon}{2} & \epsilon           & \frac{\epsilon}{2}   \\
+                                \epsilon             & \frac{\epsilon}{2} & 0                    \\
+                                \frac{\epsilon}{2}   & 0                  & - \frac{\epsilon}{2}
+                            \end{pmatrix},   &  &
+    \bm{\alpha}^{dis} = .3 + \begin{pmatrix}
+                                 - \frac{\epsilon}{2} & \epsilon             & \epsilon             \\
+                                 \epsilon             & - \frac{\epsilon}{2} & \epsilon             \\
+                                 \epsilon             & \epsilon             & - \frac{\epsilon}{2}
+                             \end{pmatrix},
+\end{align*}
+with $\epsilon \in [.1, .4]$. $\bm{\alpha}^{as}$ represents a classical
+assortative community structure,
+while $\bm{\alpha}^{cp}$ is a layered core-periphery structure with block 2
+acting as a semi-core. Finally, $\bm{\alpha}^{dis}$ is a dis-assortative
+community structure with stronger
+connections between blocks than within blocks. If $\epsilon = 0$, the three
+matrices are equal and the 9 networks have the same connection structure.
+Increasing $\epsilon$ differentiates the 3 sub-collections of networks.
+
+% ARI boxplot
+
+\paragraph{Results} The evaluation of our method involves a comparison between
+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 $colBiSBM$.