diff --git a/rapport/chapter4-simulations/inference.tex b/rapport/chapter4-simulations/inference.tex index 8582a52..93a85f4 100644 --- a/rapport/chapter4-simulations/inference.tex +++ b/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 diff --git a/rapport/chapter4-simulations/na-robustness.tex b/rapport/chapter4-simulations/na-robustness.tex index f9ffe98..ec9d949 100644 --- a/rapport/chapter4-simulations/na-robustness.tex +++ b/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 \ No newline at end of file diff --git a/rapport/rapport.pdf b/rapport/rapport.pdf index 1d8ac74..c754f8a 100644 Binary files a/rapport/rapport.pdf and b/rapport/rapport.pdf differ diff --git a/rapport/rapport.tex b/rapport/rapport.tex index 6c9e2f5..60652ff 100644 --- a/rapport/rapport.tex +++ b/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} diff --git a/tikz/simulations/na_robustness/ari-dim-model.tex b/tikz/simulations/na_robustness/ari-dim-model.tex new file mode 100644 index 0000000..f5c21a7 --- /dev/null +++ b/tikz/simulations/na_robustness/ari-dim-model.tex @@ -0,0 +1,3922 @@ +% Created by tikzDevice version 0.12.6 on 2024-07-18 16:44:40 +% !TEX encoding = UTF-8 Unicode +\begin{tikzpicture}[x=1pt,y=1pt] +\definecolor{fillColor}{RGB}{255,255,255} +\path[use as bounding box,fill=fillColor,fill opacity=0.00] (0,0) rectangle (397.48,180.67); 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+\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (361.35,180.67); +\definecolor{drawColor}{gray}{0.30} + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 51.77, 19.68) {0}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 76.24, 19.68) {0.1}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at (100.71, 19.68) {0.2}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at (125.18, 19.68) {0.3}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at (149.65, 19.68) {0.4}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at (174.12, 19.68) {0.5}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at (198.59, 19.68) {0.6}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at (223.06, 19.68) {0.7}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at (247.53, 19.68) {0.8}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 0.88] at (272.00, 19.68) {0.9}; +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (361.35,180.67); +\definecolor{drawColor}{gray}{0.30} + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14,104.75) {-0.1}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14,117.31) {0.0}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14,129.87) {0.1}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14,142.44) {0.2}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14,155.00) {0.3}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14,167.57) {0.4}; +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (361.35,180.67); +\definecolor{drawColor}{gray}{0.30} + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14, 29.75) {-0.1}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14, 42.32) {0.0}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14, 54.88) {0.1}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14, 67.44) {0.2}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14, 80.01) {0.3}; + +\node[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 32.14, 92.57) {0.4}; +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (361.35,180.67); +\definecolor{drawColor}{RGB}{0,0,0} + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.10] at (161.88, 7.64) {$p_{\texttt{NA}}$}; +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (361.35,180.67); +\definecolor{drawColor}{RGB}{0,0,0} + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.10] at ( 13.08,102.93) {$\Delta\mbox{AUC}$}; 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