\section{Clustering
  of~\cite{baldockSystemsApproachReveals2019a,baldockDailyTemporalStructure2011}}
\label{sec:baldock-clustering}

\cite{baldockSystemsApproachReveals2019a} study the diversity, robustness and
impact of the type of environment on the ecological aspect of plant-pollinator
networks in four major english cities. The networks are presented in
figures~\ref{subfig:baldock-english-network-bristol} to~\ref{subfig:baldock-english-network-reading}

\cite{baldockDailyTemporalStructure2011} aim to study the daily temporal
structure in a savanna pollination network. The data was collected in 2004
and in two sites, \emph{Turkana Boma} (TB) and \emph{Junction} (JN) in Kenya.
We will not look for a temporal structure but only use the full networks merged
in one
due to their small sizes. The network obtained is presented in
figure~\ref{fig:baldock-african-networks}.

In the following results, the row nodes represent the plants and the column
nodes represent the pollinators (mostly insects in this case).

\emph{Note:} those networks were extracted from a bigger dataset
from~\cite{doreRelativeEffectsAnthropogenic2021}. The full dataset was also
clusterized but issues arose that are discussed in
sub-section~\ref{ssec:difficulties-encountered}.

\begin{figure}[ht]
  \centering
  \begin{subfigure}[htb]{0.40\textwidth}
    \includestandalone[width=0.86\textwidth]{tikz/applications/baldock/graph-Baldock2019_Bristol}
    \caption{Bristol}
    \label{subfig:baldock-english-network-bristol}
  \end{subfigure}
  \hfill
  \begin{subfigure}[htb]{0.40\textwidth}
    \includestandalone[width=0.86\textwidth]{tikz/applications/baldock/graph-Baldock2019_Edinburgh}
    \caption{Edinburgh}
  \end{subfigure}
  \newline
  \begin{subfigure}[htb]{0.40\textwidth}
    \includestandalone[width=0.86\textwidth]{tikz/applications/baldock/graph-Baldock2019_Leeds}
    \caption{Leeds}
  \end{subfigure}
  \hfill
  \begin{subfigure}[htb]{0.40\textwidth}
    \includestandalone[width=0.86\textwidth]{tikz/applications/baldock/graph-Baldock2019_Reading}
    \caption{Reading}
    \label{subfig:baldock-english-network-reading}
  \end{subfigure}
  \caption{English networks from~\cite{baldockSystemsApproachReveals2019a}}
  \label{fig:baldock-english-networks}
\end{figure}

\begin{figure}[ht]
  \centering
  \includestandalone[width=0.43\textwidth]{tikz/applications/baldock/graph-Baldock2011_TB+Baldock2011_JN}
  \caption{African network from~\cite{baldockDailyTemporalStructure2011}}
  \label{fig:baldock-african-networks}
\end{figure}

We applied our clustering method on those 6 networks, using the four models.
Interesting results arose from \emph{iid} and $\pi\rho$ models, which are
presented below in figures~\ref{fig:baldock-clust-iid}
and~\ref{fig:baldock-clust-pirho}.

\begin{figure}
  \begin{minipage}{.5\textwidth}
    \begin{subfigure}[htb]{\textwidth}
      \includestandalone{tikz/applications/baldock/iid-clust-struct}
      \caption{$\bm{\alpha}$ structures of the\newline
        collections identified}
      \label{subfig:baldock-clust-iid-struct}
    \end{subfigure}
    \newline
    \begin{subfigure}[htb]{\textwidth}
      \includestandalone[width=0.86\textwidth]{tikz/applications/baldock/iid-clust-tree}
      \caption{Tree of splits}
      \label{subfig:baldock-clust-iid-split}
    \end{subfigure}
    \captionof{figure}{Results for \emph{iid} clustering}
    \label{fig:baldock-clust-iid}
  \end{minipage}%
  \begin{minipage}{.5\textwidth}
    \begin{subfigure}[htb]{\textwidth}
      \includestandalone{tikz/applications/baldock/pirho-clust-struct}
      \caption{$\bm{\alpha}$ structure of the\newline
        collection identified}
      \label{subfig:baldock-clust-pirho-struct}
    \end{subfigure}
    \newline
    \begin{subfigure}[htb]{\textwidth}
      \includestandalone[width=0.86\textwidth]{tikz/applications/baldock/pirho-clust-tree}
      \caption{Tree of splits}
      \label{subfig:baldock-clust-pirho-split}
    \end{subfigure}
    \captionof{figure}{Results for $\pi\rho$ clustering}
    \label{fig:baldock-clust-pirho}
  \end{minipage}
\end{figure}

\paragraph{Results} The main thing one can see when comparing the two
clusterings is that while \emph{iid} do not find a common structure, $\pi\rho$
manage to find one.

When comparing figures~\ref{subfig:baldock-clust-iid-struct},~\ref{subfig:baldock-clust-pirho-struct} and with the
figures~\ref{fig:struct-mixture-iid},~\ref{fig:struct-mixture-pirho}
in appendix which show the block proportions we can deduce the following differences between the two models.

The TB+JN network doesn't have the \nth{3} column block and the first and second column blocks coincides with the English networks. But interestingly, the \nth{3} column block is the larger one in the English networks.

For the row blocks, where with the \emph{iid} model only two groups where detected in the African network, it is refined with the $\pi\rho$ detecting 4 blocks in its row nodes. The African network does not have the first row block, a few row nodes in row blocks 2 and 3, more in the \nth{4} block and the majority in the \nth{5}.
This contrasts, once again, with the English networks that have the majority of their row nodes in the \nth{4} block.

Two interesting points can be made with those results, firstly the $\pi\rho$
collection model allowed the detection of a finer structure in the African
network regarding the row block memberships. And secondly, whereas the English
networks are really similar, the African network while presenting the same
blocks fills them differently.
Those observations invite us to look at the species that are in those different
groups.