\addtocounter{customchapter}{1}
\chapter{Conclusions and future work}
\label{chap:conclusions-and-future-work}
\section{Conclusion}
\label{sec:conclusion}

\subsection{State of work at the end of the internship}

At the end of this internship we now have:
\begin{itemize}
      \item A model capable to find structure in collections of bipartite network.
            Enabling the possibility to bring together networks that may have not
            seem evident to put together.
      \item A clustering method that is able to partition a collection into
            collections that are similar in their structures.
      \item All the described methods implemented into an \texttt{R} package.
\end{itemize}

\subsection{Difficulties encountered}
\label{ssec:difficulties-encountered}
\paragraph{Local optima} While using our clustering on data
from~\cite{doreRelativeEffectsAnthropogenic2021} we obtained quite interesting
results but investigating further, we noticed that the clustering on such big
collections ($M=123$) was not fully reproducible. It depends a lot on the random
generator seed and as there is no possibility to merge back
collections
the clustering dendrograms do not stabilize on large collections.
This, currently, prevents us to clusterize large collections.
We suspect that our model selection method gets stuck on local optima
of the BIC-L.

\paragraph{Large penalties with free mixture models}
We observed while testing clustering with the different models that
the $\pi$, $\rho$ and $\pi\rho$ model, with their increased number of parameters
for block memberships parameters tends to give smaller BIC-L criterion values
while having a higher Evidence Lower Bound than the \emph{iid}.
This arises because of the penalties on the block memberships and supports that
increase significantly and exceeds the gain on the ELBO and the diminution of
the connectivity parameters.

\section{Future work}
\label{sec:future-work}

\paragraph{Fixing local optima}
We are currently investigating the procedure and code to see if reducing or
escaping seed dependance is possible and would allow escaping the local optima.

\paragraph{Identifiability}
As stated in section~\ref{sec:model-identifiability}, we only have
identifiability for the \emph{iid}-colBiSBM and we will work on establishing
identifiability for $\pi$, $\rho$ and $\pi\rho$ models which are the most
challenging with regard to identifiability.

\paragraph{Finding a trade-off between \emph{iid} and $\pi\rho$}
An idea to tackle the problem of large penalties with $\pi$, $\rho$ and
$\pi\rho$ could be to suppose that the block memberships
for network $m$ are themselves the realizations of random variables and
thus introduce sort of a mixed effect model. This may allow a self-penalization
that could keep the flexibility intended in these models.

\paragraph{More ecological applications} We have leads to apply the models on
interesting cases, among them are the following:
\begin{itemize}
      \item Networks that are spaced along an altitude gradient, which could be
            accounted for in the dissimilarity measure for instance.
      \item Collection of different sorts of interactions (plants-seed dispersors,
            host-parasites, \dots)
\end{itemize}

\paragraph{Turning this work into an article} We will work in the upcoming
months to write an article about the method.

\paragraph{Comparison to other graphs clustering methods}
Recent work have been comparing
\texttt{colSBM}~\parencite{chabert-liddellLearningCommonStructures2024a} and
\texttt{graphclust}~\parencite{rebafkaModelbasedClusteringMultiple2023} assessing various
capabilities of the models and particularly focusing on networks clustering.
We will reproduce and adapt the analysis to test other simulation settings that
were not considered in this work.