\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}^{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

\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 $colBiSBM$.