\clearpage
\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 $Q_1 =
    Q_2 = 3$.\newline
For the simulations the proportions are the following:\newline
$\bm{\pi}^1 = \left( 0.2, 0.3, 0.5 \right),
    ~\bm{\rho}^1 = \left( 0.2, 0.3, 0.5 \right)$ and for all $m = 2,\dots,9$
\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}^{dis} = .3 + \begin{pmatrix}
                                 - \frac{\epsilon}{2} & \epsilon             & \epsilon             \\
                                 \epsilon             & - \frac{\epsilon}{2} & \epsilon             \\
                                 \epsilon             & \epsilon             & - \frac{\epsilon}{2}
                             \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} &
\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
\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
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.