rapport : changing margins, explaining ARI negatives, adding NA robustness sims,

rapport/chapter4-simulations/inference.tex

@@ -70,8 +70,12 @@ use the following indicators:
           their true values. ($Q_1 = 4$ and $Q_2 = 4$)
     \item Finally, we assess the quality of the node grouping by computing the
           Adjusted Rand Index \parencite{hubertComparingPartitions1985}, ARI = 0
-          for a random grouping, ARI = 1 for a perfect recovery. For each
-          network, for the
+          for a random grouping, ARI = 1 for a
+          perfect match between groupings\footnote{Please note that even if Rand
+              Index can only yield values between 0 and 1, ARI can return
+              negative values if the RI is less than the expected value. This
+              indicates a structure in grouping discordance.}.
+          For each network, for the
           $\pi\text{-}colBiSBM$, $\rho\text{-}colBiSBM$,
           $\pi\rho\text{-}colBiSBM$ we compare the inferred block memberships to
           the real ones by computing the mean of the ARI per axis over the two

rapport/chapter4-simulations/na-robustness.tex

@@ -36,11 +36,15 @@ less with the other communities. And $\bm{\alpha}^{nested}$ represents a common
 structure detected in ecology with generalist and specialist species and a
 \enquote{nested} structure.
 
-The collections contain two networks of size $n^{m=1}_1 = n^{m=1}_2 = 40$ and
+The collections contain two networks ($M=2$) of size $n^{m=1}_1 =
+    n^{m=1}_2 = 40$ and
 $n^{m=2}_1 = n^{m=2}_2 = 120$. One collection is generated for each $colBiSBM$
 model. And the nodes block memberships (i.e., the row and column blocks they
 belong to) are saved.
 
+Per $colBiSBM$ model, 10 collections are generated and their results are
+averaged.
+
 In the network $m=1$ (i.e., the smaller one) a proportion of the edges
 $p_{\texttt{NA}}$ see their values replaced by \texttt{NA}s, the
 \enquote{forgotten} values are stored.
@@ -57,6 +61,39 @@ and we store the same predictions.
 \emph{Area Under the Curve} (AUC) for predicted versus real link values and the
 ARI for predicted versus real block memberships.
 
+For the comparison we subtract the metric given by the LBM to the one
+given by $colBiSBM$ and denote it $\Delta\mbox{metric}$.
+
+\begin{figure}[ht]
+    \centering
+    \input{../tikz/simulations/na_robustness/auc-model}
+    \caption{$\Delta\mbox{AUC}$ in function of $p_{\texttt{NA}}$. The dashed red
+        lines indicate the value 0 for which
+        $\mbox{AUC}_{LBM} = \mbox{AUC}_{colBiSBM}$}
+    \label{fig:auc-plot}
+\end{figure}
+
+\begin{figure}[ht]
+    \centering
+    \input{../tikz/simulations/na_robustness/ari-dim-model}
+    \caption{$\Delta\mbox{ARI}$ in function of $p_{\texttt{NA}}$. The dashed red
+        lines indicate the value 0 for which
+        $\mbox{ARI}_{LBM} = \mbox{ARI}_{colBiSBM}$}
+    \label{fig:ari-dim-plot-na}
+\end{figure}
+
+
 \paragraph{Results}
+On figure~\ref{fig:auc-plot} one can see that overall the nested structure seems
+to be the one benefitting most from the collection model having generally
+slightly higher $\Delta$AUC than the modular one.
+But in general it seems that for $\epsilon\in[0.1,0.7]$ there are no clear
+differences between LBM and colBiSBM regarding link prediction. For $\epsilon
+    \in[0.8,0.9]$ this is where the collection model seems to be most effective.
 
-
+For the ARI, figure~\ref{fig:ari-dim-plot-na} suggests that collection model
+does at least as well as LBM and improves nodes memberships recovery for modular
+structure starting from $\epsilon = 0.7$. Again, nested structure benefits
+of collection model for smaller $\epsilon$ values but those increase in
+$\Delta$ARI are also smaller than what can be observed for modular structure.
+\clearpage

rapport/rapport.tex

@@ -19,6 +19,7 @@
 \usepackage[citecolor=blueind,urlcolor=blueps,bookmarks=false,hypertexnames=true]{hyperref}  % pour les hyperliens dans le document
 \usepackage{tocbibind} % Pour avoir des index pour table des matières, biblio
 \usepackage{geometry}
+\geometry{bmargin=25mm}
 
 \usepackage{tikz} % For graph plots
 \usepackage[outline]{contour}