\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.