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