\section[Capacity to distinguish models]{Capacity to distinguish
  $\pi\rho\text{-}colBiSBM$~from\newline$iid\text{-}colBiSBM$ and other
  models}\label{capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}

The idea of this model selection simulations is to assess how the model
select the correct \emph{colBiSBM} model among the possible ones:
\textit{iid, pi, rho, pirho}. This difference being based on the row and
col block proportions.

For this task we choose the same simulation settings as
\cite{chabert-liddellLearningCommonStructures2024a}.

Namely $n_{1}^{m} = 90, n_{2}^{m} = 90, Q_1 = Q_2 = 3$,
$\bm{\alpha}, \bm{\pi}$ and $\bm{\rho}$ are set as follows:
\begin{minipage}[l]{0.4\linewidth}
    \begin{align*}
        \bm{\alpha} =.25 +  \begin{pmatrix}
                                3 \eps[\alpha] & 2 \eps[\alpha] & \eps[\alpha]   \\
                                2 \eps[\alpha] & 2 \eps[\alpha] & - \eps[\alpha] \\
                                \eps[\alpha]   & - \eps[\alpha] & \eps[\alpha]
                            \end{pmatrix},
    \end{align*}
\end{minipage}
\hfill
\begin{minipage}[r]{0.4\linewidth}
    \begin{align*}
        \bm{\pi}^1 = \begin{pmatrix}
                         \frac{1}{3}, & \frac{1}{3}, & \frac{1}{3}
                     \end{pmatrix}, &  & \bm{\pi}^2 = \sigma\begin{pmatrix}
                                                                \frac{1}{3} - \eps[\pi], & \frac{1}{3}, & \frac{1}{3} + \eps[\pi]
                                                            \end{pmatrix},     \\
        \bm{\rho}^1 = \begin{pmatrix}
                          \frac{1}{3}, & \frac{1}{3}, & \frac{1}{3}
                      \end{pmatrix}, &  & \bm{\rho}^2 = \sigma\begin{pmatrix}
                                                                  \frac{1}{3} - \eps[\rho], & \frac{1}{3}, & \frac{1}{3} + \eps[\rho]
                                                              \end{pmatrix},
    \end{align*}
\end{minipage}
with $\eps[\alpha] = 0.16$, $\eps[\pi]$ and
$\eps[\rho]$ taking 9 values equally spaced in
$\left[ 0, .28\right]$. We simulate 324 different collections for each
value of $\eps[\pi]$ and $\eps[\rho]$.

$\pi\rho\text{-}colBiSBM$, $\pi\text{-}colBiSBM$,
$\rho\text{-}colBiSBM$, $iid\text{-}colBiSBM$ and
$sep\text{-}BiSBM$ are put in competition and the model with the
greater BIC-L is selected as the \emph{preferred model}.

When $\eps[\pi] = 0$, $\bm{\pi}^1 = \bm{\pi}^2$, $\eps[\rho] = 0$
and $\bm{\rho}^1 = \bm{\rho}^2$, the generated collection is an
$iid\text{-}colBiSBM$. When $\eps[\pi] > 0$ or
$\bm{\pi}^1 \neq \bm{\pi}^2$, the model is a $\pi\text{-}colBiSBM$.
When $\eps[\rho] > 0$ or $\bm{\rho}^1 \neq \bm{\rho}^2$, the model
is a $\rho\text{-}colBiSBM$. Finally, when $\eps[\pi] > 0$ or
$\bm{\pi}^1 \neq \bm{\pi}^2$ and $\eps[\rho] > 0$ or
$\bm{\rho}^1 \neq \bm{\rho}^2$, the model is a
$\pi\rho\text{-}colBiSBM$.

\begin{table}[!h]

    \caption{\label{tab:tables}\label{tab:pi-model-sel}Model selection for varying $\pi$ mixture parameters}
    \centering
    \begin{tabular}[t]{lccccl}
        \toprule
        \multicolumn{1}{c}{ } & \multicolumn{4}{c}{Models} & \multicolumn{1}{c}{Blocks}                                                                        \\
        \cmidrule(l{3pt}r{3pt}){2-5} \cmidrule(l{3pt}r{3pt}){6-6}
        $\eps[\pi]$           & $iid\text{-}colBiSBM$      & $\pi\text{-}colBiSBM$      & $\rho\text{-}colBiSBM$ & $\pi\rho\text{-}colBiSBM$ & Recovered $Q_1$ \\
        \midrule
        0.00                  & 0.65                       & 0.00                       & 0.35                   & 0.00                      & 3               \\
        0.04                  & 0.66                       & 0.00                       & 0.34                   & 0.00                      & 3               \\
        0.07                  & 0.64                       & 0.01                       & 0.34                   & 0.01                      & 3.01 $\pm$ 0.01 \\
        0.11                  & 0.63                       & 0.03                       & 0.31                   & 0.03                      & 3.01 $\pm$ 0.01 \\
        0.14                  & 0.55                       & 0.12                       & 0.28                   & 0.05                      & 3               \\
        \addlinespace
        0.18                  & 0.39                       & 0.26                       & 0.21                   & 0.13                      & 3.01            \\
        0.21                  & 0.23                       & 0.42                       & 0.13                   & 0.23                      & 3.01            \\
        0.25                  & 0.10                       & 0.56                       & 0.05                   & 0.29                      & 3.02 $\pm$ 0.01 \\
        0.28                  & 0.01                       & 0.65                       & 0.01                   & 0.33                      & 3.01 $\pm$ 0.01 \\
        \bottomrule
    \end{tabular}
\end{table}

\begin{table}[!h]

    \caption{\label{tab:tables}\label{tab:rho-model-sel}Model selection for varying $\rho$ mixture parameters}
    \centering
    \begin{tabular}[t]{lccccl}
        \toprule
        \multicolumn{1}{c}{ } & \multicolumn{4}{c}{Models} & \multicolumn{1}{c}{Blocks}                                                                        \\
        \cmidrule(l{3pt}r{3pt}){2-5} \cmidrule(l{3pt}r{3pt}){6-6}
        $\eps[\rho]$          & $iid\text{-}colBiSBM$      & $\pi\text{-}colBiSBM$      & $\rho\text{-}colBiSBM$ & $\pi\rho\text{-}colBiSBM$ & Recovered $Q_2$ \\
        \midrule
        0.00                  & 0.63                       & 0.37                       & 0.00                   & 0.00                      & 3               \\
        0.04                  & 0.65                       & 0.34                       & 0.00                   & 0.01                      & 3               \\
        0.07                  & 0.64                       & 0.33                       & 0.01                   & 0.01                      & 3               \\
        0.11                  & 0.64                       & 0.31                       & 0.03                   & 0.02                      & 3               \\
        0.14                  & 0.53                       & 0.29                       & 0.11                   & 0.06                      & 3               \\
        \addlinespace
        0.18                  & 0.42                       & 0.20                       & 0.24                   & 0.14                      & 3.01            \\
        0.21                  & 0.25                       & 0.12                       & 0.40                   & 0.22                      & 3.01            \\
        0.25                  & 0.08                       & 0.06                       & 0.58                   & 0.29                      & 3.01            \\
        0.28                  & 0.01                       & 0.01                       & 0.65                   & 0.32                      & 3               \\
        \bottomrule
    \end{tabular}
\end{table}

\begin{figure}[H]
    % \includegraphics{./Rcodes/simulation/img/plot_model_function_eps.png}
    \caption{Plot of preferred model in function of $\eps[\pi]$ and $\eps[\rho]$}
    \label{fig:pref_model_func_eps}
\end{figure}

\paragraph{Results:}

On the figure \ref{fig:pref_model_func_eps} and tables
\ref{tab:pi-model-sel} and \ref{tab:rho-model-sel}, one can see that
there is a turning point around $\eps[\pi] = 0.2$ (resp.
$\eps[\rho] = 0.2$), before which $iid\text{-}colBiSBM$ and
$\rho\text{-}colBiSBM$ (resp. $\pi\text{-}colBiSBM$) are selected
most of the times and after $0.2$ the $\pi\text{-}colBiSBM$ (resp.
$\rho\text{-}colBiSBM$) and $\pi\rho\text{-}colBiSBM$ gets more and
more selected, highlighting our capacity to recover the simulated
structure.

\paragraph*{Remark:}

Please note that when ``Recovered $Q_1$(or $Q_2$)'' is not an
integer it's because some procedures returned a value other than 3.
