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rapport : sub tables pour l'inférence

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14
rapport/chapter4-simulations/model-selection.tex
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@ -1,12 +1,12 @@
\section[Capacity to distinguish models]{Capacity to distinguish \section[Capacity to distinguish models]{Capacity to distinguish
$\pi\rho\text{-}colBiSBM$~from\newline $\pi\rho\text{-}colBiSBM$~from\newline
$iid\text{-}colBiSBM$ and other $iid\text{-}colBiSBM$ and other
models}\label{capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants} models}\label{sec:capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}
The idea of this model selection simulations is to assess how the model The idea of this model selection simulations is to assess how the model
select the correct \emph{colBiSBM} model among the possible ones: select the correct \emph{colBiSBM} model among the possible ones:
\textit{iid, pi, rho, pirho}. This difference being based on the row and \textit{$iid, \pi, \rho, \pi\rho$}. This difference being based on the row and
col block proportions.\\ col block proportions.\\
For this task we choose the same simulation settings as \paragraph{Simulation settings} For this task we choose the same simulation settings as
\cite{chabert-liddellLearningCommonStructures2024a}.\\ \cite{chabert-liddellLearningCommonStructures2024a}.\\
Namely, $n_{1}^{m} = 90, n_{2}^{m} = 90, Q_1 = Q_2 = 3$, 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:\\ $\bm{\alpha}, \bm{\pi}$ and $\bm{\rho}$ are set as follows:\\
@ -55,9 +55,8 @@ $\bm{\pi}^1 \neq \bm{\pi}^2$ and $\eps[\rho] > 0$ or
$\bm{\rho}^1 \neq \bm{\rho}^2$, the model is a $\bm{\rho}^1 \neq \bm{\rho}^2$, the model is a
$\pi\rho\text{-}colBiSBM$. $\pi\rho\text{-}colBiSBM$.
\input{../tables/simulations/model_selection/model-selection.tex}
\begin{figure}[H] \begin{figure}[!ht]
\centering \centering
\input{../tikz/simulations/model_selection/eps-pi-rho-preferred.tex} \input{../tikz/simulations/model_selection/eps-pi-rho-preferred.tex}
\caption{\label{fig:pref_model_func_eps}Plot of model selection proportions \caption{\label{fig:pref_model_func_eps}Plot of model selection proportions
@ -75,6 +74,9 @@ very often and after $0.2$ the $\pi\text{-}colBiSBM$ (resp.
$\rho\text{-}colBiSBM$) and $\pi\rho\text{-}colBiSBM$ gets more and $\rho\text{-}colBiSBM$) and $\pi\rho\text{-}colBiSBM$ gets more and
more selected. Moreover, the number of blocks are correctly detected in most more selected. Moreover, the number of blocks are correctly detected in most
of the case. of the case.
These two results highlight our capacity to recover the simulated These two results highlight our capacity to recover the simulated
structure. structure.
As $\eps[\pi]$ and $\eps[\rho]$ need to be above $0.2$ to see $\pi\rho$ model
being preferred this may indicate the need of a strong difference between blocks
to select this model.

22
rapport/chapter4-simulations/network-clustering.tex
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@ -1,13 +1,12 @@
\section{Network clustering of simulated networks}\label{sec:network-clustering-of-simulated-networks} \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 \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 $\forall m \in \{ 1 \dots M \} , n^m_1 = n^m_2 = 75$ with $Q_1 = Q_2 = 3$. For
the simulations the proportions are the following: the simulations the proportions are the following:
\begin{align*} \begin{align*}
\bm{\pi}^1 = \left( 0.2, 0.3, 0.5 \right) & & \bm{\rho}^1 = \left( 0.2, 0.3, 0.5 \right) \bm{\pi}^1 = \left( 0.2, 0.3, 0.5 \right) & & \bm{\rho}^1 = \left( 0.2, 0.3, 0.5 \right) \\
\end{align*} \end{align*} and for all $m = 2,\dots,9$
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\text{-}colBiSBM \\
@ -24,23 +23,22 @@ $\sigma^1 (\pi)= {(\pi_{\sigma^1 (i)})}_{i=\{1,\dots,3\}}$
and $\sigma^2 (\rho)= {(\rho_{\sigma^2 (i)})}_{i=\{1,\dots,3\}}$. and $\sigma^2 (\rho)= {(\rho_{\sigma^2 (i)})}_{i=\{1,\dots,3\}}$.
The networks are divided into 3 sub-collections of 3 The networks are divided into 3 sub-collections of 3
networks with connectivity parameters as follows: networks with connectivity parameters as follows:
\begin{align*} \begin{align*}
\bm{\alpha}^{as} = .3 + \begin{pmatrix} \bm{\alpha}^{as} = .3 + \begin{pmatrix}
\epsilon & - \frac{\epsilon}{2} & - \frac{\epsilon}{2} \\ \epsilon & - \frac{\epsilon}{2} & - \frac{\epsilon}{2} \\
- \frac{\epsilon}{2} & \epsilon & - \frac{\epsilon}{2} \\ - \frac{\epsilon}{2} & \epsilon & - \frac{\epsilon}{2} \\
- \frac{\epsilon}{2} & - \frac{\epsilon}{2} & \epsilon - \frac{\epsilon}{2} & - \frac{\epsilon}{2} & \epsilon
\end{pmatrix}, & & \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}, & &
\bm{\alpha}^{dis} = .3 + \begin{pmatrix} \bm{\alpha}^{dis} = .3 + \begin{pmatrix}
- \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}
\frac{3 \epsilon}{2} & \epsilon & \frac{\epsilon}{2} \\
\epsilon & \frac{\epsilon}{2} & 0 \\
\frac{\epsilon}{2} & 0 & - \frac{\epsilon}{2}
\end{pmatrix} &
\end{align*} \end{align*}
with $\epsilon \in [.1, .4]$. $\bm{\alpha}^{as}$ represents a classical with $\epsilon \in [.1, .4]$. $\bm{\alpha}^{as}$ represents a classical
assortative community structure, assortative community structure,

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rapport/rapport.pdf
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55
tables/simulations/inference/iid.tex
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@ -1,22 +1,51 @@
\begin{table}[!h]
\begin{table}[!htb]
\centering \centering
\caption{\label{tab:per_model_iid}Quality metrics for $iid$$\text{-}colBiSBM$} \caption{\label{tab:inference_results_iid}Inference results for $iid$}
\begin{subtable}{\textwidth}
\centering
\caption{\label{subtab:ari_per_model_iid}Quality metrics for $iid$$\text{-}colBiSBM$}
\centering \centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{ \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{rllllllllll} \begin{tabular}[t]{rllll}
\toprule \toprule
$\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$ & $\mathbbb{1}_{\widehat{Q_1}<Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}=Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}>Q_1}$ & $\mathbbb{1}_{\widehat{Q_2}<Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}=Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}>Q_2}$\\ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$\\
\midrule \midrule
0.00 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 0.00 & 0 & 0 & 0 & 0\\
0.03 & 0.01 & 0.01 & 0.01 & 0.01 & 1 & 0 & 0 & 1 & 0 & 0\\ 0.03 & 0.01 & 0.01 & 0.01 & 0.01\\
0.06 & 0.71 $\pm$ 0.01 & 0.7 $\pm$ 0.01 & 0.56 $\pm$ 0.02 & 0.54 $\pm$ 0.02 & 0.29 $\pm$ 0.04 & 0.71 $\pm$ 0.04 & 0 & 0.42 $\pm$ 0.05 & 0.57 $\pm$ 0.05 & 0.01 $\pm$ 0.01\\ 0.06 & 0.71 $\pm$ 0.01 & 0.7 $\pm$ 0.01 & 0.56 $\pm$ 0.02 & 0.54 $\pm$ 0.02\\
0.09 & 0.96 & 0.96 & 0.9 $\pm$ 0.02 & 0.89 $\pm$ 0.02 & 0 & 0.99 $\pm$ 0.01 & 0.01 $\pm$ 0.01 & 0 & 0.96 $\pm$ 0.02 & 0.04 $\pm$ 0.02\\ 0.09 & 0.96 & 0.96 & 0.9 $\pm$ 0.02 & 0.89 $\pm$ 0.02\\
0.12 & 0.99 & 0.99 & 0.92 $\pm$ 0.02 & 0.91 $\pm$ 0.02 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.91 $\pm$ 0.03 & 0.09 $\pm$ 0.03\\ 0.12 & 0.99 & 0.99 & 0.92 $\pm$ 0.02 & 0.91 $\pm$ 0.02\\
\addlinespace \addlinespace
0.15 & 1 & 0.99 & 0.92 $\pm$ 0.02 & 0.92 $\pm$ 0.02 & 0 & 0.97 $\pm$ 0.02 & 0.03 $\pm$ 0.02 & 0 & 0.87 $\pm$ 0.03 & 0.13 $\pm$ 0.03\\ 0.15 & 1 & 0.99 & 0.92 $\pm$ 0.02 & 0.92 $\pm$ 0.02\\
0.18 & 1 & 1 & 0.94 $\pm$ 0.02 & 0.94 $\pm$ 0.02 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.93 $\pm$ 0.03 & 0.07 $\pm$ 0.03\\ 0.18 & 1 & 1 & 0.94 $\pm$ 0.02 & 0.94 $\pm$ 0.02\\
0.21 & 1 & 1 & 0.93 $\pm$ 0.02 & 0.93 $\pm$ 0.02 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.92 $\pm$ 0.03 & 0.08 $\pm$ 0.03\\ 0.21 & 1 & 1 & 0.93 $\pm$ 0.02 & 0.93 $\pm$ 0.02\\
0.24 & 1 & 1 & 0.93 $\pm$ 0.02 & 0.93 $\pm$ 0.02 & 0 & 0.93 $\pm$ 0.03 & 0.07 $\pm$ 0.03 & 0 & 0.93 $\pm$ 0.03 & 0.07 $\pm$ 0.03\\ 0.24 & 1 & 1 & 0.93 $\pm$ 0.02 & 0.93 $\pm$ 0.02\\
\bottomrule \bottomrule
\end{tabular}} \end{tabular}}
\end{subtable}
\begin{subtable}{\textwidth}
\centering
\caption{\label{subtab:blocrecov_per_model_iid}Bloc recovery for $iid$$\text{-}colBiSBM$}
\centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{rllllll}
\toprule
$\eps[\alpha]$ & $\mathbbb{1}_{\widehat{Q_1}<Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}=Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}>Q_1}$ & $\mathbbb{1}_{\widehat{Q_2}<Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}=Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}>Q_2}$\\
\midrule
0.00 & 1 & 0 & 0 & 1 & 0 & 0\\
0.03 & 1 & 0 & 0 & 1 & 0 & 0\\
0.06 & 0.29 $\pm$ 0.04 & 0.71 $\pm$ 0.04 & 0 & 0.42 $\pm$ 0.05 & 0.57 $\pm$ 0.05 & 0.01 $\pm$ 0.01\\
0.09 & 0 & 0.99 $\pm$ 0.01 & 0.01 $\pm$ 0.01 & 0 & 0.96 $\pm$ 0.02 & 0.04 $\pm$ 0.02\\
0.12 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.91 $\pm$ 0.03 & 0.09 $\pm$ 0.03\\
\addlinespace
0.15 & 0 & 0.97 $\pm$ 0.02 & 0.03 $\pm$ 0.02 & 0 & 0.87 $\pm$ 0.03 & 0.13 $\pm$ 0.03\\
0.18 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.93 $\pm$ 0.03 & 0.07 $\pm$ 0.03\\
0.21 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.92 $\pm$ 0.03 & 0.08 $\pm$ 0.03\\
0.24 & 0 & 0.93 $\pm$ 0.03 & 0.07 $\pm$ 0.03 & 0 & 0.93 $\pm$ 0.03 & 0.07 $\pm$ 0.03\\
\bottomrule
\end{tabular}}
\end{subtable}
\end{table} \end{table}

55
tables/simulations/inference/pi.tex
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@ -1,22 +1,51 @@
\begin{table}[!h]
\begin{table}[!htb]
\centering \centering
\caption{\label{tab:per_model_pi}Quality metrics for $\pi$$\text{-}colBiSBM$} \caption{\label{tab:inference_results_pi}Inference results for $\pi$}
\begin{subtable}{\textwidth}
\centering
\caption{\label{subtab:ari_per_model_pi}Quality metrics for $\pi$$\text{-}colBiSBM$}
\centering \centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{ \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{rllllllllll} \begin{tabular}[t]{rllll}
\toprule \toprule
$\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$ & $\mathbbb{1}_{\widehat{Q_1}<Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}=Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}>Q_1}$ & $\mathbbb{1}_{\widehat{Q_2}<Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}=Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}>Q_2}$\\ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$\\
\midrule \midrule
0.00 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 0.00 & 0 & 0 & 0 & 0\\
0.03 & 0.01 & 0.01 & 0.01 & 0.01 & 1 & 0 & 0 & 1 & 0 & 0\\ 0.03 & 0.01 & 0.01 & 0.01 & 0.01\\
0.06 & 0.74 $\pm$ 0.01 & 0.71 $\pm$ 0.01 & 0.64 $\pm$ 0.02 & 0.59 $\pm$ 0.02 & 0.1 $\pm$ 0.03 & 0.65 $\pm$ 0.05 & 0.25 $\pm$ 0.04 & 0.38 $\pm$ 0.05 & 0.62 $\pm$ 0.05 & 0\\ 0.06 & 0.74 $\pm$ 0.01 & 0.71 $\pm$ 0.01 & 0.64 $\pm$ 0.02 & 0.59 $\pm$ 0.02\\
0.09 & 0.97 & 0.95 & 0.87 $\pm$ 0.02 & 0.84 $\pm$ 0.02 & 0 & 0.69 $\pm$ 0.04 & 0.31 $\pm$ 0.04 & 0 & 1 & 0\\ 0.09 & 0.97 & 0.95 & 0.87 $\pm$ 0.02 & 0.84 $\pm$ 0.02\\
0.12 & 1 & 0.98 & 0.92 $\pm$ 0.02 & 0.91 $\pm$ 0.02 & 0 & 0.8 $\pm$ 0.04 & 0.2 $\pm$ 0.04 & 0 & 1 & 0\\ 0.12 & 1 & 0.98 & 0.92 $\pm$ 0.02 & 0.91 $\pm$ 0.02\\
\addlinespace \addlinespace
0.15 & 1 & 0.99 & 0.93 $\pm$ 0.02 & 0.89 $\pm$ 0.02 & 0 & 0.81 $\pm$ 0.04 & 0.19 $\pm$ 0.04 & 0 & 1 & 0\\ 0.15 & 1 & 0.99 & 0.93 $\pm$ 0.02 & 0.89 $\pm$ 0.02\\
0.18 & 1 & 0.99 & 0.95 $\pm$ 0.01 & 0.93 $\pm$ 0.02 & 0 & 0.88 $\pm$ 0.03 & 0.12 $\pm$ 0.03 & 0 & 1 & 0\\ 0.18 & 1 & 0.99 & 0.95 $\pm$ 0.01 & 0.93 $\pm$ 0.02\\
0.21 & 1 & 0.99 & 0.93 $\pm$ 0.02 & 0.9 $\pm$ 0.02 & 0 & 0.83 $\pm$ 0.04 & 0.17 $\pm$ 0.04 & 0 & 1 & 0\\ 0.21 & 1 & 0.99 & 0.93 $\pm$ 0.02 & 0.9 $\pm$ 0.02\\
0.24 & 1 & 0.99 & 0.95 $\pm$ 0.01 & 0.93 $\pm$ 0.02 & 0 & 0.88 $\pm$ 0.03 & 0.12 $\pm$ 0.03 & 0 & 1 & 0\\ 0.24 & 1 & 0.99 & 0.95 $\pm$ 0.01 & 0.93 $\pm$ 0.02\\
\bottomrule \bottomrule
\end{tabular}} \end{tabular}}
\end{subtable}
\begin{subtable}{\textwidth}
\centering
\caption{\label{subtab:blocrecov_per_model_pi}Bloc recovery for $\pi$$\text{-}colBiSBM$}
\centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{rllllll}
\toprule
$\eps[\alpha]$ & $\mathbbb{1}_{\widehat{Q_1}<Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}=Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}>Q_1}$ & $\mathbbb{1}_{\widehat{Q_2}<Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}=Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}>Q_2}$\\
\midrule
0.00 & 1 & 0 & 0 & 1 & 0 & 0\\
0.03 & 1 & 0 & 0 & 1 & 0 & 0\\
0.06 & 0.1 $\pm$ 0.03 & 0.65 $\pm$ 0.05 & 0.25 $\pm$ 0.04 & 0.38 $\pm$ 0.05 & 0.62 $\pm$ 0.05 & 0\\
0.09 & 0 & 0.69 $\pm$ 0.04 & 0.31 $\pm$ 0.04 & 0 & 1 & 0\\
0.12 & 0 & 0.8 $\pm$ 0.04 & 0.2 $\pm$ 0.04 & 0 & 1 & 0\\
\addlinespace
0.15 & 0 & 0.81 $\pm$ 0.04 & 0.19 $\pm$ 0.04 & 0 & 1 & 0\\
0.18 & 0 & 0.88 $\pm$ 0.03 & 0.12 $\pm$ 0.03 & 0 & 1 & 0\\
0.21 & 0 & 0.83 $\pm$ 0.04 & 0.17 $\pm$ 0.04 & 0 & 1 & 0\\
0.24 & 0 & 0.88 $\pm$ 0.03 & 0.12 $\pm$ 0.03 & 0 & 1 & 0\\
\bottomrule
\end{tabular}}
\end{subtable}
\end{table} \end{table}

55
tables/simulations/inference/pirho.tex
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@ -1,22 +1,51 @@
\begin{table}[!h]
\begin{table}[!htb]
\centering \centering
\caption{\label{tab:per_model_pirho}Quality metrics for $\pi\rho$$\text{-}colBiSBM$} \caption{\label{tab:inference_results_pirho}Inference results for $\pi\rho$}
\begin{subtable}{\textwidth}
\centering
\caption{\label{subtab:ari_per_model_pirho}Quality metrics for $\pi\rho$$\text{-}colBiSBM$}
\centering \centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{ \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{rllllllllll} \begin{tabular}[t]{rllll}
\toprule \toprule
$\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$ & $\mathbbb{1}_{\widehat{Q_1}<Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}=Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}>Q_1}$ & $\mathbbb{1}_{\widehat{Q_2}<Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}=Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}>Q_2}$\\ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$\\
\midrule \midrule
0.00 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 0.00 & 0 & 0 & 0 & 0\\
0.03 & 0.01 & 0.01 & 0.01 & 0.01 & 1 & 0 & 0 & 1 & 0 & 0\\ 0.03 & 0.01 & 0.01 & 0.01 & 0.01\\
0.06 & 0.76 $\pm$ 0.01 & 0.75 $\pm$ 0.01 & 0.66 $\pm$ 0.02 & 0.66 $\pm$ 0.02 & 0.1 $\pm$ 0.03 & 0.88 $\pm$ 0.03 & 0.02 $\pm$ 0.01 & 0.08 $\pm$ 0.03 & 0.86 $\pm$ 0.03 & 0.06 $\pm$ 0.02\\ 0.06 & 0.76 $\pm$ 0.01 & 0.75 $\pm$ 0.01 & 0.66 $\pm$ 0.02 & 0.66 $\pm$ 0.02\\
0.09 & 0.97 & 0.97 & 0.9 $\pm$ 0.02 & 0.89 $\pm$ 0.02 & 0 & 0.9 $\pm$ 0.03 & 0.1 $\pm$ 0.03 & 0 & 0.87 $\pm$ 0.03 & 0.13 $\pm$ 0.03\\ 0.09 & 0.97 & 0.97 & 0.9 $\pm$ 0.02 & 0.89 $\pm$ 0.02\\
0.12 & 1 & 0.99 & 0.93 $\pm$ 0.02 & 0.93 $\pm$ 0.02 & 0 & 0.9 $\pm$ 0.03 & 0.1 $\pm$ 0.03 & 0 & 0.9 $\pm$ 0.03 & 0.1 $\pm$ 0.03\\ 0.12 & 1 & 0.99 & 0.93 $\pm$ 0.02 & 0.93 $\pm$ 0.02\\
\addlinespace \addlinespace
0.15 & 1 & 1 & 0.95 $\pm$ 0.01 & 0.94 $\pm$ 0.02 & 0 & 0.95 $\pm$ 0.02 & 0.05 $\pm$ 0.02 & 0 & 0.89 $\pm$ 0.03 & 0.11 $\pm$ 0.03\\ 0.15 & 1 & 1 & 0.95 $\pm$ 0.01 & 0.94 $\pm$ 0.02\\
0.18 & 1 & 1 & 0.95 $\pm$ 0.01 & 0.95 $\pm$ 0.02 & 0 & 0.93 $\pm$ 0.03 & 0.07 $\pm$ 0.03 & 0 & 0.91 $\pm$ 0.03 & 0.09 $\pm$ 0.03\\ 0.18 & 1 & 1 & 0.95 $\pm$ 0.01 & 0.95 $\pm$ 0.02\\
0.21 & 1 & 1 & 0.95 $\pm$ 0.01 & 0.94 $\pm$ 0.02 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.89 $\pm$ 0.03 & 0.11 $\pm$ 0.03\\ 0.21 & 1 & 1 & 0.95 $\pm$ 0.01 & 0.94 $\pm$ 0.02\\
0.24 & 1 & 1 & 0.94 $\pm$ 0.02 & 0.93 $\pm$ 0.02 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.88 $\pm$ 0.03 & 0.12 $\pm$ 0.03\\ 0.24 & 1 & 1 & 0.94 $\pm$ 0.02 & 0.93 $\pm$ 0.02\\
\bottomrule \bottomrule
\end{tabular}} \end{tabular}}
\end{subtable}
\begin{subtable}{\textwidth}
\centering
\caption{\label{subtab:blocrecov_per_model_pirho}Bloc recovery for $\pi\rho$$\text{-}colBiSBM$}
\centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{rllllll}
\toprule
$\eps[\alpha]$ & $\mathbbb{1}_{\widehat{Q_1}<Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}=Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}>Q_1}$ & $\mathbbb{1}_{\widehat{Q_2}<Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}=Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}>Q_2}$\\
\midrule
0.00 & 1 & 0 & 0 & 1 & 0 & 0\\
0.03 & 1 & 0 & 0 & 1 & 0 & 0\\
0.06 & 0.1 $\pm$ 0.03 & 0.88 $\pm$ 0.03 & 0.02 $\pm$ 0.01 & 0.08 $\pm$ 0.03 & 0.86 $\pm$ 0.03 & 0.06 $\pm$ 0.02\\
0.09 & 0 & 0.9 $\pm$ 0.03 & 0.1 $\pm$ 0.03 & 0 & 0.87 $\pm$ 0.03 & 0.13 $\pm$ 0.03\\
0.12 & 0 & 0.9 $\pm$ 0.03 & 0.1 $\pm$ 0.03 & 0 & 0.9 $\pm$ 0.03 & 0.1 $\pm$ 0.03\\
\addlinespace
0.15 & 0 & 0.95 $\pm$ 0.02 & 0.05 $\pm$ 0.02 & 0 & 0.89 $\pm$ 0.03 & 0.11 $\pm$ 0.03\\
0.18 & 0 & 0.93 $\pm$ 0.03 & 0.07 $\pm$ 0.03 & 0 & 0.91 $\pm$ 0.03 & 0.09 $\pm$ 0.03\\
0.21 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.89 $\pm$ 0.03 & 0.11 $\pm$ 0.03\\
0.24 & 0 & 0.94 $\pm$ 0.02 & 0.06 $\pm$ 0.02 & 0 & 0.88 $\pm$ 0.03 & 0.12 $\pm$ 0.03\\
\bottomrule
\end{tabular}}
\end{subtable}
\end{table} \end{table}

55
tables/simulations/inference/rho.tex
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@ -1,22 +1,51 @@
\begin{table}[!h]
\begin{table}[!htb]
\centering \centering
\caption{\label{tab:per_model_rho}Quality metrics for $\rho$$\text{-}colBiSBM$} \caption{\label{tab:inference_results_rho}Inference results for $\rho$}
\begin{subtable}{\textwidth}
\centering
\caption{\label{subtab:ari_per_model_rho}Quality metrics for $\rho$$\text{-}colBiSBM$}
\centering \centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{ \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{rllllllllll} \begin{tabular}[t]{rllll}
\toprule \toprule
$\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$ & $\mathbbb{1}_{\widehat{Q_1}<Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}=Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}>Q_1}$ & $\mathbbb{1}_{\widehat{Q_2}<Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}=Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}>Q_2}$\\ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$\\
\midrule \midrule
0.00 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 0.00 & 0 & 0 & 0 & 0\\
0.03 & 0.01 & 0.01 & 0.01 & 0.01 & 1 & 0 & 0 & 1 & 0 & 0\\ 0.03 & 0.01 & 0.01 & 0.01 & 0.01\\
0.06 & 0.73 $\pm$ 0.01 & 0.76 $\pm$ 0.01 & 0.55 $\pm$ 0.02 & 0.58 $\pm$ 0.02 & 0.26 $\pm$ 0.04 & 0.74 $\pm$ 0.04 & 0 & 0.04 $\pm$ 0.02 & 0.58 $\pm$ 0.05 & 0.38 $\pm$ 0.05\\ 0.06 & 0.73 $\pm$ 0.01 & 0.76 $\pm$ 0.01 & 0.55 $\pm$ 0.02 & 0.58 $\pm$ 0.02\\
0.09 & 0.94 & 0.97 & 0.82 $\pm$ 0.02 & 0.85 $\pm$ 0.02 & 0 & 1 & 0 & 0 & 0.65 $\pm$ 0.05 & 0.35 $\pm$ 0.05\\ 0.09 & 0.94 & 0.97 & 0.82 $\pm$ 0.02 & 0.85 $\pm$ 0.02\\
0.12 & 0.97 & 0.99 & 0.84 $\pm$ 0.03 & 0.87 $\pm$ 0.02 & 0 & 1 & 0 & 0 & 0.68 $\pm$ 0.05 & 0.32 $\pm$ 0.05\\ 0.12 & 0.97 & 0.99 & 0.84 $\pm$ 0.03 & 0.87 $\pm$ 0.02\\
\addlinespace \addlinespace
0.15 & 0.98 & 1 & 0.88 $\pm$ 0.02 & 0.9 $\pm$ 0.02 & 0 & 1 & 0 & 0 & 0.78 $\pm$ 0.04 & 0.22 $\pm$ 0.04\\ 0.15 & 0.98 & 1 & 0.88 $\pm$ 0.02 & 0.9 $\pm$ 0.02\\
0.18 & 0.98 & 1 & 0.86 $\pm$ 0.03 & 0.89 $\pm$ 0.02 & 0 & 1 & 0 & 0 & 0.74 $\pm$ 0.04 & 0.26 $\pm$ 0.04\\ 0.18 & 0.98 & 1 & 0.86 $\pm$ 0.03 & 0.89 $\pm$ 0.02\\
0.21 & 0.97 & 1 & 0.81 $\pm$ 0.03 & 0.86 $\pm$ 0.02 & 0 & 1 & 0 & 0 & 0.69 $\pm$ 0.04 & 0.31 $\pm$ 0.04\\ 0.21 & 0.97 & 1 & 0.81 $\pm$ 0.03 & 0.86 $\pm$ 0.02\\
0.24 & 0.98 & 1 & 0.84 $\pm$ 0.03 & 0.88 $\pm$ 0.02 & 0 & 0.99 $\pm$ 0.01 & 0.01 $\pm$ 0.01 & 0 & 0.73 $\pm$ 0.04 & 0.27 $\pm$ 0.04\\ 0.24 & 0.98 & 1 & 0.84 $\pm$ 0.03 & 0.88 $\pm$ 0.02\\
\bottomrule \bottomrule
\end{tabular}} \end{tabular}}
\end{subtable}
\begin{subtable}{\textwidth}
\centering
\caption{\label{subtab:blocrecov_per_model_rho}Bloc recovery for $\rho$$\text{-}colBiSBM$}
\centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{rllllll}
\toprule
$\eps[\alpha]$ & $\mathbbb{1}_{\widehat{Q_1}<Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}=Q_1}$ & $\mathbbb{1}_{\widehat{Q_1}>Q_1}$ & $\mathbbb{1}_{\widehat{Q_2}<Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}=Q_2}$ & $\mathbbb{1}_{\widehat{Q_2}>Q_2}$\\
\midrule
0.00 & 1 & 0 & 0 & 1 & 0 & 0\\
0.03 & 1 & 0 & 0 & 1 & 0 & 0\\
0.06 & 0.26 $\pm$ 0.04 & 0.74 $\pm$ 0.04 & 0 & 0.04 $\pm$ 0.02 & 0.58 $\pm$ 0.05 & 0.38 $\pm$ 0.05\\
0.09 & 0 & 1 & 0 & 0 & 0.65 $\pm$ 0.05 & 0.35 $\pm$ 0.05\\
0.12 & 0 & 1 & 0 & 0 & 0.68 $\pm$ 0.05 & 0.32 $\pm$ 0.05\\
\addlinespace
0.15 & 0 & 1 & 0 & 0 & 0.78 $\pm$ 0.04 & 0.22 $\pm$ 0.04\\
0.18 & 0 & 1 & 0 & 0 & 0.74 $\pm$ 0.04 & 0.26 $\pm$ 0.04\\
0.21 & 0 & 1 & 0 & 0 & 0.69 $\pm$ 0.04 & 0.31 $\pm$ 0.04\\
0.24 & 0 & 0.99 $\pm$ 0.01 & 0.01 $\pm$ 0.01 & 0 & 0.73 $\pm$ 0.04 & 0.27 $\pm$ 0.04\\
\bottomrule
\end{tabular}}
\end{subtable}
\end{table} \end{table}

10
tables/simulations/inference/sep.tex
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@ -1,6 +1,10 @@
\begin{table}[!h]
\begin{table}[!htb]
\centering \centering
\caption{\label{tab:per_model_sep}Quality metrics for $sep\text{-}BiSBM$} \caption{\label{tab:inference_results_sep}Inference results for $sep$}
\begin{subtable}{\textwidth}
\centering
\caption{\label{subtab:ari_per_model_sep}Quality metrics for $sep\text{-}BiSBM$}
\centering \centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{ \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{rllll} \begin{tabular}[t]{rllll}
@ -19,4 +23,6 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
0.24 & 1 & 1 & 0.57 $\pm$ 0.02 & 0.54 $\pm$ 0.02\\ 0.24 & 1 & 1 & 0.57 $\pm$ 0.02 & 0.54 $\pm$ 0.02\\
\bottomrule \bottomrule
\end{tabular}} \end{tabular}}
\end{subtable}
\end{table} \end{table}