82 lines
No EOL
3.8 KiB
TeX
82 lines
No EOL
3.8 KiB
TeX
\section[Capacity to distinguish models]{Capacity to distinguish
|
|
$\pi\rho$-colBiSBM~from\newline
|
|
$iid$-colBiSBM and other
|
|
models}\label{sec:capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}
|
|
The idea of this model selection simulations is to assess how the model
|
|
select the correct colBiSBM model among the possible ones:
|
|
\textit{$iid, \pi, \rho, \pi\rho$}. This difference being based on the row and
|
|
col block proportions.\\
|
|
\paragraph{Simulation settings} For this task we choose the same simulation settings as
|
|
\cite{chabert-liddellLearningCommonStructures2024a}.\\
|
|
Namely, $n_{1}^{m} = 90, n_{2}^{m} = 90, Q_1 = Q_2 = 3$,
|
|
$\bm{\alpha}, \bm{\pi}$ and $\bm{\rho}$ are set as follows:\\
|
|
\begin{minipage}[l]{0.4\linewidth}
|
|
\begin{align*}
|
|
\bm{\alpha} =.25 + \begin{pmatrix}
|
|
3 \eps[\alpha] & 2 \eps[\alpha] & \eps[\alpha] \\
|
|
2 \eps[\alpha] & 2 \eps[\alpha] & - \eps[\alpha] \\
|
|
\eps[\alpha] & - \eps[\alpha] & \eps[\alpha]
|
|
\end{pmatrix},
|
|
\end{align*}
|
|
\end{minipage}
|
|
\hfill
|
|
\begin{minipage}[r]{0.4\linewidth}
|
|
\begin{align*}
|
|
\bm{\pi}^1 = \begin{pmatrix}
|
|
\frac{1}{3}, & \frac{1}{3}, & \frac{1}{3}
|
|
\end{pmatrix}, & & \bm{\pi}^2 = \sigma\begin{pmatrix}
|
|
\frac{1}{3} - \eps[\pi], & \frac{1}{3}, & \frac{1}{3} + \eps[\pi]
|
|
\end{pmatrix}, \\
|
|
\bm{\rho}^1 = \begin{pmatrix}
|
|
\frac{1}{3}, & \frac{1}{3}, & \frac{1}{3}
|
|
\end{pmatrix}, & & \bm{\rho}^2 = \sigma\begin{pmatrix}
|
|
\frac{1}{3} - \eps[\rho], & \frac{1}{3}, & \frac{1}{3} + \eps[\rho]
|
|
\end{pmatrix},
|
|
\end{align*}
|
|
\end{minipage}
|
|
with $\eps[\alpha] = 0.16$, $\eps[\pi]$ and
|
|
$\eps[\rho]$ taking 9 values equally spaced in
|
|
$\left[ 0, .28\right]$.\newline
|
|
We simulate 324 different collections for each
|
|
value of $\eps[\pi]$ and $\eps[\rho]$.
|
|
|
|
$\pi\rho$-colBiSBM, $\pi$-colBiSBM,
|
|
$\rho$-colBiSBM, $iid$-colBiSBM and
|
|
$sep\text{-}BiSBM$ are put in competition and the model with the
|
|
greater BIC-L is selected as the \emph{preferred model}.
|
|
|
|
When $\eps[\pi] = 0$, $\bm{\pi}^1 = \bm{\pi}^2$, $\eps[\rho] = 0$
|
|
and $\bm{\rho}^1 = \bm{\rho}^2$, the generated collection is an
|
|
$iid$-colBiSBM. When $\eps[\pi] > 0$ or
|
|
$\bm{\pi}^1 \neq \bm{\pi}^2$, the model is a $\pi$-colBiSBM.
|
|
When $\eps[\rho] > 0$ or $\bm{\rho}^1 \neq \bm{\rho}^2$, the model
|
|
is a $\rho$-colBiSBM. Finally, when $\eps[\pi] > 0$ or
|
|
$\bm{\pi}^1 \neq \bm{\pi}^2$ and $\eps[\rho] > 0$ or
|
|
$\bm{\rho}^1 \neq \bm{\rho}^2$, the model is a
|
|
$\pi\rho$-colBiSBM.
|
|
|
|
|
|
\begin{figure}[!ht]
|
|
\centering
|
|
\input{../tikz/simulations/model_selection/eps-pi-rho-preferred.tex}
|
|
\caption{\label{fig:pref_model_func_eps}Plot of model selection proportions
|
|
over the different datasets in
|
|
function of $\eps[\pi]$ and $\eps[\rho]$}
|
|
\end{figure}
|
|
|
|
\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.
|
|
$\eps[\rho] = 0.2$), before which $iid$-colBiSBM and
|
|
$\rho$-colBiSBM (resp. $\pi$-colBiSBM) are selected
|
|
very often and after $0.2$ the $\pi$-colBiSBM (resp.
|
|
$\rho$-colBiSBM) and $\pi\rho$-colBiSBM gets more and
|
|
more selected. Moreover, the number of blocks are correctly detected in most
|
|
of the case.
|
|
These two results highlight our capacity to recover the simulated
|
|
structure.
|
|
|
|
As $\eps[\pi]$ and $\eps[\rho]$ need to be above $0.2$ to see $\pi\rho$ model
|
|
being preferred this may indicate the need of a strong difference between blocks
|
|
to select this model. |