From d48c55ce4cf05be8468bee8d5ab0a3707aae7137 Mon Sep 17 00:00:00 2001 From: Louis Lacoste Date: Mon, 15 Jul 2024 16:56:50 +0200 Subject: [PATCH] rapport : ajout des autres section du chapitre Simulations --- rapport/chapter4-simulation-studies.tex | 5 +- .../information-transfer.tex | 8 +++ .../network-clustering.tex | 60 ++++++++++++++++++ rapport/rapport.pdf | Bin 701834 -> 722042 bytes 4 files changed, 71 insertions(+), 2 deletions(-) create mode 100644 rapport/chapter4-simulations/information-transfer.tex create mode 100644 rapport/chapter4-simulations/network-clustering.tex diff --git a/rapport/chapter4-simulation-studies.tex b/rapport/chapter4-simulation-studies.tex index fcf5a4f..3dfab71 100644 --- a/rapport/chapter4-simulation-studies.tex +++ b/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} diff --git a/rapport/chapter4-simulations/information-transfer.tex b/rapport/chapter4-simulations/information-transfer.tex new file mode 100644 index 0000000..52b932c --- /dev/null +++ b/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} \ No newline at end of file diff --git a/rapport/chapter4-simulations/network-clustering.tex b/rapport/chapter4-simulations/network-clustering.tex new file mode 100644 index 0000000..42e309a --- /dev/null +++ b/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. 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