rapport : updating inference, information transfer, model selection, na robustness, network clustering

rapport/chapter4-simulations/inference.tex

@@ -1,9 +1,10 @@
 \section{Efficiency of the inference}
+\label{sec:efficiency-of-the-inference}
 
 The goal here is to assess the quality of the inference procedure.
 
 \paragraph{Simulation settings} For this simulation the data is simulated with
-$M = 2, n_{1}^{m} = 120, n_{2}^{m} = 120, Q_1 = Q_2 = 4$, $\bm{\alpha}, \bm{\pi}$
+$M = 2, n_{1}^{m} = 120,~n_{2}^{m} = 120,~Q_1 = Q_2 = 4$, $\bm{\alpha}, \bm{\pi}$
 and $\bm{\rho}$ are set as follows:
 \begin{align*}
      & \bm{\alpha} = .25 +
@@ -76,7 +77,7 @@ use the following indicators:
           the real ones by computing the mean of the ARI per axis over the two
           networks
           \begin{equation*}
-              \overline{\text{ARI}}_d = \frac{1}{2} \text{ARI}\big( \text{ARI}(\widehat{\bm{Z}^1_d},\bm{Z}^1_d) + \text{ARI}(\widehat{\bm{Z}^2_d},\bm{Z}^2_d) \big),
+              \overline{\text{ARI}}_d = \frac{1}{2} \big( \text{ARI}(\widehat{\bm{Z}^1_d},\bm{Z}^1_d) + \text{ARI}(\widehat{\bm{Z}^2_d},\bm{Z}^2_d) \big),
           \end{equation*}
           where $d$ is the dimension or axis (i.e., rows, $d=1$, or columns, $d=2$) of
           the block memberships.
@@ -88,20 +89,26 @@ use the following indicators:
 \end{itemize}
 
 All these quality indicators are averaged over the 108 datasets. The results are
-provided in the tables \ref{tab:per_model_sep} to \ref{tab:per_model_pirho}. Each line corresponds to the
+provided in the tables \ref{tab:inference_results_iid} to \ref{tab:inference_results_pirho}. Each line corresponds to the
-108 datasets for a given value of $\eps[\alpha]$.
+108 datasets for a given value of $\eps[\alpha]$. Graphical representation of
+some results are shown on figures~\ref{fig:inference-prop-modele-pref}
+and~\ref{fig:inference-ari-plots}.
 
 \begin{figure}[ht]
     \centering
     \input{../tikz/simulations/inference/model-proportions.tex}
     \caption{Preferred model proportions over all datasets in function of
         $\eps[\alpha]$}
-    \label{fig:prop-modele-pref}
+    \label{fig:inference-prop-modele-pref}
 \end{figure}
 
-\foreach \modelname in {sep, iid, pi, rho, pirho}{
+\begin{figure}[H]
-        \input{../tables/simulations/inference/\modelname.tex}
+    \centering
-    }
+    \input{../tikz/simulations/inference/ari-plots}
+    \caption{Plot of the ARI quality indicators in function of
+        $\eps[\alpha]$}
+    \label{fig:inference-ari-plots}
+\end{figure}
 
 \paragraph{Results}
 For the model comparison, when $\eps[\alpha]$ is small
@@ -109,14 +116,12 @@ For the model comparison, when $\eps[\alpha]$ is small
 Erd\H{o}s-Reńyi network, and it is very hard to find any structure beyond the one
 of a single block on each dimension.
 
-On the figure \ref{fig:inference-proportion-preferred} and table
+On the figure \ref{fig:inference-prop-modele-pref} one can see that from
-\ref{tab:proportion-preferred-table} we can see that from
+$\eps[\alpha] = 0.06$ around $70\%$ of the time the
-$\eps[\alpha] = 0.06$ around $70\%$ of the time the $\pi\rho\text{-}colBiSBM$
+$\pi\rho\text{-}colBiSBM$ model (i.e., the correct one) is selected.
-model (i.e., the correct one) is selected.
 
 An interesting result we can read in the tables is that our models outperform
 the $sep\text{-}BiSBM$ when considering the ARI on the whole set of nodes
 ($\text{ARI}_d$). This means that our models are able to recover the block
 pairing \emph{between the networks} in addition to recovering the blocks and
-their parameters.
+their parameters.
-\clearpage

rapport/chapter4-simulations/information-transfer.tex

@@ -4,5 +4,6 @@ between the networks, allowing robustness to missing data and enabling the
 finding of finer structures in small networks with the help of bigger ones.
 
 \subsection{Missing edges robustness}
+\input{chapter4-simulations/na-robustness}
 
 \subsection{Finer structure detection on small networks}

rapport/chapter4-simulations/model-selection.tex

@@ -64,7 +64,7 @@ $\pi\rho\text{-}colBiSBM$.
         function of $\eps[\pi]$ and $\eps[\rho]$}
 \end{figure}
 
-\paragraph{Results:}
+\paragraph{Results}
 
 On the figure \ref{fig:pref_model_func_eps} and table \ref{tab:model-selection}, one can see that
 there is a turning point around $\eps[\pi] = 0.2$ (resp.

rapport/chapter4-simulations/na-robustness.tex

@@ -0,0 +1,62 @@
+\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 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.
+
+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.
+
+\paragraph{Results}
+
+

rapport/chapter4-simulations/network-clustering.tex

@@ -1,28 +1,30 @@
+\clearpage
 \section{Network clustering of simulated networks}
 \label{sec:network-clustering-of-simulated-networks}
 
-\paragraph{Simulation settings} For all models we simulate $M = 9$ networks with
+\paragraph{Simulation settings} For all models we simulate $M = 9$ networks
-$\forall m \in \{ 1 \dots M \} , n^m_1 = n^m_2 = 75$ with $Q_1 = Q_2 = 3$. For
+with~$\forall m \in \{ 1 \dots M \} , n^m_1 = n^m_2 = 75$ with $Q_1 = Q_2 = 3$.\newline
-the simulations the proportions are the following:
+For the simulations the proportions are the following:
 \begin{align*}
     \bm{\pi}^1 = \left( 0.2, 0.3, 0.5 \right) &  & \bm{\rho}^1 = \left( 0.2, 0.3, 0.5 \right) \\
 \end{align*} and for all $m = 2,\dots,9$
 \begin{align*}
     \bm{\pi}^m = \begin{cases}
                      \bm{\pi}^1             & \text{for } iid\text{-}colBiSBM                                      \\
-                     \sigma^1_m(\bm{\pi}^1) & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
+                     \sigma_1^m(\bm{\pi}^1) & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
                  \end{cases} \\
     \bm{\rho}^m =
     \begin{cases}
         \bm{\rho}^1             & \text{for } iid\text{-}colBiSBM                                       \\
-        \sigma^2_m(\bm{\rho}^1) & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
+        \sigma_2^m(\bm{\rho}^1) & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
     \end{cases}
 \end{align*}
-where $\sigma^1_m$ and $\sigma^2_m$ are permutations of {1, 2, 3} proper to network $m$ and
+where $\sigma_1^m$ and $\sigma_2^m$ are permutations of \{1, 2, 3\} proper to network $m$ and
-$\sigma^1 (\pi)= {(\pi_{\sigma^1 (i)})}_{i=\{1,\dots,3\}}$
+$\sigma_1 (\pi)= {(\pi_{\sigma_1 (i)})}_{i=\{1,\dots,3\}}$
-and $\sigma^2 (\rho)= {(\rho_{\sigma^2 (i)})}_{i=\{1,\dots,3\}}$.
+and $\sigma_2 (\rho)= {(\rho_{\sigma_2 (i)})}_{i=\{1,\dots,3\}}$.
-The networks are divided into 3 sub-collections of 3
+
-networks with connectivity parameters as follows:
+The networks are divided into 3 sub-collections of 3 networks with connectivity
+parameters as follows:
 \begin{align*}
     \bm{\alpha}^{as} = .3 + \begin{pmatrix}
                                 \epsilon             & - \frac{\epsilon}{2} & - \frac{\epsilon}{2} \\
@@ -50,6 +52,13 @@ matrices are equal and the 9 networks have the same connection structure.
 Increasing $\epsilon$ differentiates the 3 sub-collections of networks.
 
 % ARI boxplot
+\begin{figure}[H]
+    \centering
+    \input{../tikz/simulations/clustering/ari-clustering.tex}
+    \caption{ARI obtained for the clustering with the different models in
+        function of $\epsilon$}
+    \label{fig:ari-clustering-boxplot}
+\end{figure}
 
 \paragraph{Results} The evaluation of our method involves a comparison between
 the resulting partition of the network collection and the simulated partition