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Louis Lacoste 2024-08-16 18:03:59 +02:00
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3 changed files with 23 additions and 21 deletions
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12
rapport/chapter4-simulations/na-robustness.tex
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@ -6,13 +6,13 @@ For this purpose we generate collections of networks with the following
parameters: parameters:
\begin{align*} \begin{align*}
\bm{\pi}^m = \begin{cases} \bm{\pi}^m = \begin{cases}
\bm{\pi} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-colBiSBM} \\ \bm{\pi} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid \\
\sigma_1^m(\bm{\pi}) & \text{for } \pi\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM} \sigma_1^m(\bm{\pi}) & \text{for } \pi \text{ and } \pi\rho
\end{cases} \\ \end{cases}, &
\bm{\rho}^m = \bm{\rho}^m =
\begin{cases} \begin{cases}
\bm{\rho} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-colBiSBM} \\ \bm{\rho} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid \\
\sigma_2^m(\bm{\rho}) & \text{for } \rho\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM}, \sigma_2^m(\bm{\rho}) & \text{for } \rho \text{ and } \pi\rho,
\end{cases} \end{cases}
\end{align*} \end{align*}
for the block proportions, and two different structures with the corresponding for the block proportions, and two different structures with the corresponding
@ -105,5 +105,5 @@ $\pi\rho$ present smaller values and larger variances.
An explanation for the cases in which our models return lower values than An explanation for the cases in which our models return lower values than
expected could be to look for in our simulation parameters. They may, combined expected could be to look for in our simulation parameters. They may, combined
with the $\rho$ model be a difficult case for the estimation. with the $\rho$ model be a difficult case for the estimation.
As we currently do not have identifiability results this is just and As we currently do not have identifiability results this is just an
hypothesis. hypothesis.

32
rapport/chapter4-simulations/network-clustering.tex
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@ -10,13 +10,13 @@ $\bm{\pi}^1 = \left( 0.2, 0.3, 0.5 \right),
~\bm{\rho}^1 = \left( 0.2, 0.3, 0.5 \right)$ and for all $m = 2,\dots,9$ ~\bm{\rho}^1 = \left( 0.2, 0.3, 0.5 \right)$ and for all $m = 2,\dots,9$
\begin{align*} \begin{align*}
\bm{\pi}^m = \begin{cases} \bm{\pi}^m = \begin{cases}
\bm{\pi}^1 & \text{for } iid\text{-colBiSBM} \\ \bm{\pi}^1 & \text{for } iid \\
\sigma_1^m(\bm{\pi}^1) & \text{for } \pi\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM} \sigma_1^m(\bm{\pi}^1) & \text{for } \pi \text{ and } \pi\rho
\end{cases} \\ \end{cases},~ &
\bm{\rho}^m = \bm{\rho}^m =
\begin{cases} \begin{cases}
\bm{\rho}^1 & \text{for } iid\text{-colBiSBM} \\ \bm{\rho}^1 & \text{for } iid \\
\sigma_2^m(\bm{\rho}^1) & \text{for } \rho\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM} \sigma_2^m(\bm{\rho}^1) & \text{for } \rho \text{ and } \pi\rho
\end{cases} \end{cases}
\end{align*} \end{align*}
where $\sigma_1^m$ and $\sigma_2^m$ are permutations of \{1, 2, 3\} proper to network $m$ and where $\sigma_1^m$ and $\sigma_2^m$ are permutations of \{1, 2, 3\} proper to network $m$ and
@ -35,7 +35,7 @@ parameters as follows:
- \frac{\epsilon}{2} & \epsilon & \epsilon \\ - \frac{\epsilon}{2} & \epsilon & \epsilon \\
\epsilon & - \frac{\epsilon}{2} & \epsilon \\ \epsilon & - \frac{\epsilon}{2} & \epsilon \\
\epsilon & \epsilon & - \frac{\epsilon}{2} \epsilon & \epsilon & - \frac{\epsilon}{2}
\end{pmatrix}, \\ \end{pmatrix},
& \bm{\alpha}^{cp} = .3 + \begin{pmatrix} & \bm{\alpha}^{cp} = .3 + \begin{pmatrix}
\frac{3 \epsilon}{2} & \epsilon & \frac{\epsilon}{2} \\ \frac{3 \epsilon}{2} & \epsilon & \frac{\epsilon}{2} \\
\epsilon & \frac{\epsilon}{2} & 0 \\ \epsilon & \frac{\epsilon}{2} & 0 \\
@ -51,17 +51,19 @@ connections between blocks than within blocks. If $\epsilon = 0$, the three
matrices are equal and the 9 networks have the same connection structure. matrices are equal and the 9 networks have the same connection structure.
Increasing $\epsilon$ differentiates the 3 sub-collections of networks. Increasing $\epsilon$ differentiates the 3 sub-collections of networks.
% ARI boxplot
\begin{figure}[!ht]
\centering
\includestandalone{tikz/simulations/clustering/ari-clustering.tex}
\caption{ARI obtained for the clustering with the different models in
function of $\epsilon$}
\label{fig:ari-clustering-boxplot}
\end{figure}
\paragraph{Results} The evaluation of our method involves a comparison between \paragraph{Results} The evaluation of our method involves a comparison between
the resulting partition of the network collection and the simulated partition the resulting partition of the network collection and the simulated partition
using the ARI index. As the value of $\epsilon$ increases, our ability to using the ARI index. As the value of $\epsilon$ increases, our ability to
distinguish between the networks improves, and this distinction becomes nearly distinguish between the networks improves, and this distinction becomes nearly
perfect in all setups of the colBiSBM. perfect in all setups of the colBiSBM.
% ARI boxplot
\begin{figure}[!hb]
\centering
\includestandalone[height=0.25\textheight]{tikz/simulations/clustering/ari-clustering.tex}
\caption{ARI obtained for the clustering with the different models in
function of $\epsilon$}
\label{fig:ari-clustering-boxplot}
\end{figure}

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rapport/rapport.pdf
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