\paragraph{Simulation settings} We want to compare the performance of retrieving
the nodes blocks with missing edges (that are labeled as \texttt{NA} in the
incidence matrix).

For this purpose we generate collections of networks with the following
parameters:
\begin{align*}
    \bm{\pi}^m = \begin{cases}
                     \bm{\pi} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-}colBiSBM                                      \\
                     \sigma_1^m(\bm{\pi})                    & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
                 \end{cases} \\
    \bm{\rho}^m =
    \begin{cases}
        \bm{\rho}  = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-}colBiSBM                                        \\
        \sigma_2^m(\bm{\rho})                     & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM,
    \end{cases}
\end{align*}
for the block proportions, and two different structures with the corresponding
$\bm{\alpha}$,
\begin{align*}
    \bm{\alpha}^{modular} = \begin{pmatrix}
                                0.9  & 0.05 & 0.05 \\
                                0.05 & 0.2  & 0.05 \\
                                0.05 & 0.05 & 0.8
                            \end{pmatrix}, &
    \bm{\alpha}^{nested} = \begin{pmatrix}
                               0.9 & 0.25 & 0.1  \\
                               0.3 & 0.15 & 0.05 \\
                               0.1 & 0.05 & 0.05
                           \end{pmatrix},
\end{align*}

where $\bm{\alpha}^{modular}$ represents networks where there are look-a-like
communities, which tends to interact preferentially within the community and
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 ($M=2$) 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.

\paragraph{Test procedure} A LBM is fitted on the first network, and the
predicted block memberships are saved, along with the predicted links using the
inferred parameters. This will serve as a baseline to see if the use of the
collection benefits the predictions.

A $colBiSBM$ model is then fitted (with a model matching the dataset considered)
and we store the same predictions.

\paragraph{Quality metrics} To benchmark the performance we use the
\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