EOD 7/7

Rcodes/simulation/inference_analyze.tex

@@ -44,7 +44,8 @@ their connectivity structures encompass a mix of core-periphery,
 assortative community and disassortative community structures, depending
 on which 3 of the 4 blocks are selected for each network.
 \(\eps[\alpha]\) represents the strength of these structures, the
-larger, the easier it is to tell apart one block from another.
+larger, the easier it is to tell apart one block from another. The true
+model of all the simulation is a \(\pi\rho\text{-}colBiSBM\).
 
 \paragraph{Inference}
 
@@ -91,9 +92,10 @@ results are provided in the tables \ref{tab:per_model_sep} to
 \ref{tab:per_model_pirho}. Each line corresponds to the 108 datasets for
 a given value of value of \(\eps[\alpha]\).
 
+\tiny
 \begin{table}[!h]
 
-\caption{\label{tab:per_model_table}\label{tab:per_model_sep}Quality metrics for sep-colBiSBM}
+\caption{\label{tab:per_model_table}\label{tab:per_model_sep}Quality metrics for $sep\text{-}BiSBM$}
 \centering
 \begin{tabular}[t]{rllll}
 \toprule
@@ -114,7 +116,7 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
 \end{table}
 \begin{table}[!h]
 
-\caption{\label{tab:per_model_table}\label{tab:per_model_iid}Quality metrics for iid-colBiSBM}
+\caption{\label{tab:per_model_table}\label{tab:per_model_iid}Quality metrics for $iid$$\text{-}colBiSBM$}
 \centering
 \begin{tabular}[t]{rllllll}
 \toprule
@@ -135,7 +137,7 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
 \end{table}
 \begin{table}[!h]
 
-\caption{\label{tab:per_model_table}\label{tab:per_model_pi}Quality metrics for $\pi$-colBiSBM}
+\caption{\label{tab:per_model_table}\label{tab:per_model_pi}Quality metrics for $\pi$$\text{-}colBiSBM$}
 \centering
 \begin{tabular}[t]{rllllll}
 \toprule
@@ -156,7 +158,7 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
 \end{table}
 \begin{table}[!h]
 
-\caption{\label{tab:per_model_table}\label{tab:per_model_rho}Quality metrics for $\rho$-colBiSBM}
+\caption{\label{tab:per_model_table}\label{tab:per_model_rho}Quality metrics for $\rho$$\text{-}colBiSBM$}
 \centering
 \begin{tabular}[t]{rllllll}
 \toprule
@@ -177,7 +179,7 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
 \end{table}
 \begin{table}[!h]
 
-\caption{\label{tab:per_model_table}\label{tab:per_model_pirho}Quality metrics for $\pi\rho$-colBiSBM}
+\caption{\label{tab:per_model_table}\label{tab:per_model_pirho}Quality metrics for $\pi\rho$$\text{-}colBiSBM$}
 \centering
 \begin{tabular}[t]{rllllll}
 \toprule
@@ -196,6 +198,36 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
 \bottomrule
 \end{tabular}
 \end{table}
+\normalsize
+
+\begin{table}[!h]
+
+\caption{\label{tab:proportion-preferred_model}\label{tab:proportion-preferred-table}Proportions of models selected per \eps[\alpha] (data for Figure \ref{fig:inference-proportion-preferred})}
+\centering
+\begin{tabular}[t]{rccccc}
+\toprule
+\eps[\alpha] & $sep\text{-}BiSBM$ & $iid\text{-}colBiSBM$ & $\pi\text{-}colBiSBM$ & $\rho\text{-}colBiSBM$ & $\pi\rho\text{-}colBiSBM$\\
+\midrule
+0.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00\\
+0.03 & 0.95 & 0.04 & 0.01 & 0.00 & 0.00\\
+0.06 & 0.39 & 0.33 & 0.06 & 0.15 & 0.06\\
+0.09 & 0.07 & 0.01 & 0.12 & 0.25 & 0.55\\
+0.12 & 0.00 & 0.08 & 0.06 & 0.13 & 0.72\\
+\addlinespace
+0.15 & 0.00 & 0.12 & 0.08 & 0.08 & 0.71\\
+0.18 & 0.00 & 0.11 & 0.04 & 0.06 & 0.79\\
+0.21 & 0.00 & 0.19 & 0.04 & 0.07 & 0.69\\
+0.24 & 0.00 & 0.09 & 0.06 & 0.08 & 0.77\\
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{figure}
+\centering
+\includegraphics{./img/54eb0a21b143a53b6199a869d7a228ad7d158e57.png}
+\caption{\label{fig:inference-proportion-preferred}Plot of the
+proportions of different preferred models in function of \eps[\alpha]}
+\end{figure}
 
 \paragraph{Results}
 
@@ -203,3 +235,14 @@ For the model comparison, when \(\eps[\alpha]\) is small
 (\(\eps[\alpha]\in[0, .04]\)), the simulation model is close to the
 Erd\H{o}s-Reńyi network and it is very hard to find any structure beyond
 the one of a single block on each dimension.
+
+On the figure \ref{fig:inference-proportion-preferred} and table
+\ref{tab:proportion-preferred-table} we can see that from
+\(\eps[\alpha] = 0.12\) around \(70\%\) of the time the
+\(\pi\rho\text{-}colBiSBM\) model (i.e., the correct one) is selected.
+
+An interesting result we can read in the tables is that our models
+outperform the \(sep\text{-}BiSBM\) when considering the ARI on the
+whole set of nodes (\(\text{ARI}_d\)). This means that our models are
+able to recover the block pairing \emph{between the networks} in
+addition to recovering the blocks and their parameters.

Rcodes/simulation/model_selection_analyze.tex

@@ -36,7 +36,7 @@ value of \(\eps[\pi]\) and \(\eps[\rho]\).
 
 \(\pi\rho\text{-}colBiSBM\), \(\pi\text{-}colBiSBM\),
 \(\rho\text{-}colBiSBM\), \(iid\text{-}colBiSBM\) and
-\(sep\text{-}colBiSBM\) are put in competition and the model with the
+\(sep\text{-}BiSBM\) are put in competition and the model with the
 greater BIC-L is selected as the \emph{preferred model}.
 
 When \(\eps[\pi] = 0\), \(\bm{\pi}^1 = \bm{\pi}^2\), \(\eps[\rho] = 0\)
@@ -55,20 +55,20 @@ is a \(\rho\text{-}colBiSBM\). Finally, when \(\eps[\pi] > 0\) or
 \centering
 \begin{tabular}[t]{lccccl}
 \toprule
-\multicolumn{1}{c}{ } & \multicolumn{4}{c}{Proportions of model selection} & \multicolumn{1}{c}{Blocks} \\
+\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 & 3.00\\
-0.04 & 0.66 & 0.00 & 0.34 & 0.00 & 3.00\\
-0.07 & 0.64 & 0.01 & 0.34 & 0.01 & 3.01\\
-0.11 & 0.63 & 0.03 & 0.31 & 0.03 & 3.01\\
-0.14 & 0.55 & 0.12 & 0.28 & 0.05 & 3.00\\
+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\\
-0.28 & 0.01 & 0.65 & 0.01 & 0.33 & 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}
@@ -79,20 +79,20 @@ $\eps[\pi]$ & $iid\text{-}colBiSBM$     & $\pi\text{-}colBiSBM$ & $\rho\text{-}c
 \centering
 \begin{tabular}[t]{lccccl}
 \toprule
-\multicolumn{1}{c}{ } & \multicolumn{4}{c}{Proportions of model selection} & \multicolumn{1}{c}{Blocks} \\
+\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 & 3.00\\
-0.04 & 0.65 & 0.34 & 0.00 & 0.01 & 3.00\\
-0.07 & 0.64 & 0.33 & 0.01 & 0.01 & 3.00\\
-0.11 & 0.64 & 0.31 & 0.03 & 0.02 & 3.00\\
-0.14 & 0.53 & 0.29 & 0.11 & 0.06 & 3.00\\
+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 & 3.00\\
+0.28 & 0.01 & 0.01 & 0.65 & 0.32 & 3\\
 \bottomrule
 \end{tabular}
 \end{table}

Rcodes/simulation/netclustering_analyze.tex

@@ -1 +1,13 @@
 \section{Network clustering of simulated networks}\label{sec:network-clustering-of-simulated-networks}
+
+\begin{verbatim}
+## character(0)
+\end{verbatim}
+
+\begin{verbatim}
+## list()
+\end{verbatim}
+
+\begin{verbatim}
+## Error in eval(expr, envir, enclos): objet 'list_best_partition' introuvable
+\end{verbatim}