205 lines
10 KiB
TeX
205 lines
10 KiB
TeX
\hypertarget{efficiency-of-the-inference}{%
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\section{Efficiency of the
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inference}\label{efficiency-of-the-inference}}
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\paragraph{Simulation settings}
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For this simulation the data is simulated with
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\(M = 2, n_{1}^{m} = 120, n_{2}^{m} = 120, Q_1 = Q_2 = 4\),
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\(\bm{\alpha}, \bm{\pi}\) and \(\bm{\rho}\) are set as follows:
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\begin{align*}
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&&\bm{\alpha} = .25 +
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\begin{pmatrix}
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3 \eps[\alpha] & 2 \eps[\alpha] & \eps[\alpha] & - \eps[\alpha]\\
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2 \eps[\alpha] & 2 \eps[\alpha] & - \eps[\alpha] & \eps[\alpha]\\
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\eps[\alpha] & - \eps[\alpha] & \eps[\alpha] & 2 \eps[\alpha]\\
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- \eps[\alpha] & \eps[\alpha] & 2 \eps[\alpha] & 0
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\end{pmatrix}, \\ \bm{\pi}^1 = \sigma_1
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\begin{pmatrix}
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0.2 & 0.4 & 0.4 & 0
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\end{pmatrix},
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&& \bm{\pi}^2 =
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\begin{pmatrix}
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0.25 & 0.25 & 0.25 & 0.25
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\end{pmatrix}, \\
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\bm{\rho}^1 =
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\begin{pmatrix}
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0.25 & 0.25 & 0.25 & 0.25
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\end{pmatrix}, &&
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\bm{\rho}^2 = \sigma_2
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\begin{pmatrix}
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0 & 0.33 & 0.33 & 0.33
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\end{pmatrix}, &&
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\end{align*} with \(\eps[\alpha]\) taking nine equally spaced values
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ranging from 0 to 0.24. For each value of \(\eps[\alpha]\), 108 datasets
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(\(X_1, X_2\)) are simulated, resulting in \(9 \times 108 = 972\)
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datasets. More precisely, for each dataset, we pick uniformly at random
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two permutations of \(\{ 1, \dots , 4 \}\) (\(\sigma_1, \sigma_2\)) with
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the constraint that \(\sigma_1(4) \neq \sigma_2(1)\). This ensures that
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each of the two networks have a non-empty block that is empty in the
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other one. Then the networks are simulated with
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\(\mathcal{B}\)ern-\(BiSBM_{120}(4, \bm{\alpha}, \bm{\pi}^m, \bm{\rho}^m)\)
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with the previous parameters. Each network has 2 blocks in common and
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their connectivity structures encompass a mix of core-periphery,
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assortative community and disassortative community structures, depending
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on which 3 of the 4 blocks are selected for each network.
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\(\eps[\alpha]\) represents the strength of these structures, the
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larger, the easier it is to tell apart one block from another.
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\paragraph{Inference}
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We want to measure the quality of the inference procedure, for this we
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use the inference described in the section
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\ref{sec:variational-estimation-of-the-parameters}.
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\paragraph{Quality indicators}
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To assess the quality of the inference, we will use the following
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indicators:
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\begin{itemize}
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\item First, for each dataset, we put in competition $\pi\text{-}colBiSBM$ with
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$sep\text{-}BiSBM$, $iid\text{-}colBiSBM$, $\rho\text{-}colBiSBM$,
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$\pi\rho\text{-}colBiSBM$
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respectively. To do so, for each dataset, we compute the
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BIC-L of each model $\pi\text{-}colBiSBM$ is preferred to $sep\text{-}BiSBM$
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(resp. $iid\text{-}colBiSBM$, $\rho\text{-}colBiSBM$,
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$\pi\rho\text{-}colBiSBM$) if
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its BIC-L is greater.
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\item When considering $\pi\text{-}colBiSBM$, $\rho\text{-}colBiSBM$,
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$\pi\rho\text{-}colBiSBM$ we compare $\widehat{Q_1}$, $\widehat{Q_2}$ to
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their true values. ($Q_1 = 4$ and $Q_2 = 4$)
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\item Finally, we assess the quality of the node grouping by computing the
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Adjusted Rand Index \parencite[][, ARI = 0 for a random grouping, ARI = 1 for a perfect recovery]{hubertComparingPartitions1985}. For each network, for the
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$\pi\text{-}colBiSBM$, $\rho\text{-}colBiSBM$,
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$\pi\rho\text{-}colBiSBM$ we compare the inferred block memberships to the
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real ones by computing the mean of the ARI per axis over the two networks
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\begin{equation*}
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\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)
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\end{equation*}
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where $d$ is the dimension or axis (i.e., rows, $d=1$, or columns, $d=2$) of
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the block memberships.
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And we compute the ARI of the whole set of nodes to account for block
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pairing between networks
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\begin{equation*}
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\text{ARI}_d = \text{ARI}\big((\widehat{\bm{Z}^1_d},\widehat{\bm{Z}^2_d}),(\bm{Z}^1_d,\bm{Z}^2_d) \big)
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\end{equation*}
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\end{itemize}
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All these quality indicators are averaged over the 108 datasets. The
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results are provided in the tables \ref{tab:per_model_sep} to
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\ref{tab:per_model_pirho}. Each line corresponds to the 108 datasets for
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a given value of value of \(\eps[\alpha]\).
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\begin{table}[!h]
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\caption{\label{tab:per_model_table}\label{tab:per_model_sep}Quality metrics for sep-colBiSBM}
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\centering
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\begin{tabular}[t]{rllll}
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\toprule
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$\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$\\
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\midrule
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0.00 & 0 & 0 & 0 & 0\\
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0.03 & 0 & 0 & 0 & 0\\
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0.06 & 0.1 $\pm$ 0.01 & 0.08 $\pm$ 0.01 & 0.06 $\pm$ 0.01 & 0.05 $\pm$ 0.01\\
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0.09 & 0.71 $\pm$ 0.02 & 0.7 $\pm$ 0.01 & 0.37 $\pm$ 0.02 & 0.37 $\pm$ 0.02\\
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0.12 & 0.94 $\pm$ 0.01 & 0.93 $\pm$ 0.01 & 0.5 $\pm$ 0.02 & 0.49 $\pm$ 0.02\\
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\addlinespace
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0.15 & 0.99 & 0.99 & 0.54 $\pm$ 0.02 & 0.49 $\pm$ 0.01\\
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0.18 & 0.99 & 0.99 & 0.52 $\pm$ 0.02 & 0.52 $\pm$ 0.02\\
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0.21 & 0.99 & 0.99 & 0.54 $\pm$ 0.02 & 0.52 $\pm$ 0.02\\
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0.24 & 1 & 1 & 0.55 $\pm$ 0.02 & 0.52 $\pm$ 0.02\\
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\bottomrule
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\end{tabular}
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\end{table}
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\begin{table}[!h]
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\caption{\label{tab:per_model_table}\label{tab:per_model_iid}Quality metrics for iid-colBiSBM}
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\centering
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\begin{tabular}[t]{rllllll}
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\toprule
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$\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$ & Recovered $Q_1$ & Recovered $Q_2$\\
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\midrule
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0.00 & 0 & 0 & 0 & 0 & 1 & 1\\
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0.03 & 0 & 0 & 0 & 0 & 1 & 1\\
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0.06 & 0.08 $\pm$ 0.01 & 0.08 $\pm$ 0.01 & 0.08 $\pm$ 0.01 & 0.07 $\pm$ 0.01 & 1.4 $\pm$ 0.05 & 1.49 $\pm$ 0.05\\
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0.09 & 0.72 $\pm$ 0.01 & 0.71 $\pm$ 0.01 & 0.53 $\pm$ 0.02 & 0.52 $\pm$ 0.02 & 3.4 $\pm$ 0.06 & 3.41 $\pm$ 0.06\\
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0.12 & 0.94 & 0.93 & 0.75 $\pm$ 0.03 & 0.72 $\pm$ 0.03 & 4.06 $\pm$ 0.02 & 3.97 $\pm$ 0.02\\
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\addlinespace
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0.15 & 0.98 & 0.98 & 0.77 $\pm$ 0.03 & 0.76 $\pm$ 0.03 & 4.11 $\pm$ 0.03 & 4.11 $\pm$ 0.03\\
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0.18 & 0.99 & 0.99 & 0.82 $\pm$ 0.03 & 0.82 $\pm$ 0.03 & 4.15 $\pm$ 0.04 & 4.13 $\pm$ 0.03\\
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0.21 & 0.99 & 0.99 & 0.8 $\pm$ 0.02 & 0.79 $\pm$ 0.03 & 4.35 $\pm$ 0.06 & 4.19 $\pm$ 0.04\\
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0.24 & 0.99 & 0.99 & 0.77 $\pm$ 0.03 & 0.77 $\pm$ 0.03 & 4.3 $\pm$ 0.06 & 4.43 $\pm$ 0.07\\
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\bottomrule
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\end{tabular}
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\end{table}
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\begin{table}[!h]
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\caption{\label{tab:per_model_table}\label{tab:per_model_pi}Quality metrics for $\pi$-colBiSBM}
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\centering
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\begin{tabular}[t]{rllllll}
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\toprule
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$\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$ & Recovered $Q_1$ & Recovered $Q_2$\\
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\midrule
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0.00 & 0 & 0 & 0 & 0 & 1 & 1\\
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0.03 & 0 & 0 & 0 & 0 & 1.01 $\pm$ 0.01 & 1\\
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0.06 & 0.07 $\pm$ 0.01 & 0.08 $\pm$ 0.01 & 0.07 $\pm$ 0.01 & 0.06 $\pm$ 0.01 & 1.49 $\pm$ 0.05 & 1.5 $\pm$ 0.05\\
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0.09 & 0.73 $\pm$ 0.02 & 0.72 $\pm$ 0.01 & 0.56 $\pm$ 0.02 & 0.53 $\pm$ 0.02 & 3.78 $\pm$ 0.07 & 3.37 $\pm$ 0.07\\
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0.12 & 0.96 & 0.93 & 0.79 $\pm$ 0.02 & 0.74 $\pm$ 0.03 & 4.46 $\pm$ 0.07 & 3.95 $\pm$ 0.02\\
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\addlinespace
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0.15 & 0.99 & 0.97 & 0.82 $\pm$ 0.02 & 0.76 $\pm$ 0.03 & 4.62 $\pm$ 0.08 & 4\\
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0.18 & 1 & 0.98 & 0.83 $\pm$ 0.02 & 0.79 $\pm$ 0.03 & 4.65 $\pm$ 0.09 & 4\\
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0.21 & 1 & 0.98 & 0.84 $\pm$ 0.02 & 0.79 $\pm$ 0.03 & 4.69 $\pm$ 0.1 & 4\\
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0.24 & 1 & 0.99 & 0.86 $\pm$ 0.02 & 0.79 $\pm$ 0.03 & 4.74 $\pm$ 0.11 & 4.01 $\pm$ 0.01\\
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\bottomrule
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\end{tabular}
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\end{table}
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\begin{table}[!h]
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\caption{\label{tab:per_model_table}\label{tab:per_model_rho}Quality metrics for $\rho$-colBiSBM}
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\centering
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\begin{tabular}[t]{rllllll}
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\toprule
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$\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$ & Recovered $Q_1$ & Recovered $Q_2$\\
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\midrule
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0.00 & 0 & 0 & 0 & 0 & 1 & 1\\
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0.03 & 0 & 0 & 0 & 0 & 1.01 $\pm$ 0.01 & 1.01 $\pm$ 0.01\\
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0.06 & 0.08 $\pm$ 0.01 & 0.08 $\pm$ 0.01 & 0.06 $\pm$ 0.01 & 0.07 $\pm$ 0.01 & 1.39 $\pm$ 0.05 & 1.6 $\pm$ 0.06\\
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0.09 & 0.72 $\pm$ 0.01 & 0.72 $\pm$ 0.01 & 0.53 $\pm$ 0.02 & 0.54 $\pm$ 0.02 & 3.39 $\pm$ 0.07 & 3.74 $\pm$ 0.07\\
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0.12 & 0.93 & 0.95 & 0.71 $\pm$ 0.03 & 0.75 $\pm$ 0.02 & 3.95 $\pm$ 0.02 & 4.5 $\pm$ 0.07\\
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\addlinespace
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0.15 & 0.97 & 0.99 & 0.78 $\pm$ 0.03 & 0.81 $\pm$ 0.02 & 4 & 4.49 $\pm$ 0.07\\
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0.18 & 0.98 & 1 & 0.76 $\pm$ 0.03 & 0.81 $\pm$ 0.02 & 4.01 $\pm$ 0.01 & 4.71 $\pm$ 0.09\\
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0.21 & 0.98 & 1 & 0.76 $\pm$ 0.03 & 0.81 $\pm$ 0.02 & 4.03 $\pm$ 0.02 & 4.72 $\pm$ 0.09\\
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0.24 & 0.98 & 1 & 0.74 $\pm$ 0.03 & 0.8 $\pm$ 0.02 & 4.06 $\pm$ 0.02 & 4.8 $\pm$ 0.1\\
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\bottomrule
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\end{tabular}
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\end{table}
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\begin{table}[!h]
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\caption{\label{tab:per_model_table}\label{tab:per_model_pirho}Quality metrics for $\pi\rho$-colBiSBM}
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\centering
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\begin{tabular}[t]{rllllll}
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\toprule
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$\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\text{ARI}_{1}$ & $\text{ARI}_{2}$ & Recovered $Q_1$ & Recovered $Q_2$\\
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\midrule
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0.00 & 0 & 0 & 0 & 0 & 1 & 1\\
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0.03 & 0 & 0 & 0 & 0 & 1.01 $\pm$ 0.01 & 1.01 $\pm$ 0.01\\
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0.06 & 0.07 $\pm$ 0.01 & 0.07 $\pm$ 0.01 & 0.07 $\pm$ 0.01 & 0.06 $\pm$ 0.01 & 1.48 $\pm$ 0.05 & 1.57 $\pm$ 0.06\\
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0.09 & 0.74 $\pm$ 0.01 & 0.73 $\pm$ 0.01 & 0.56 $\pm$ 0.03 & 0.55 $\pm$ 0.02 & 3.69 $\pm$ 0.06 & 3.66 $\pm$ 0.06\\
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0.12 & 0.96 $\pm$ 0.01 & 0.95 $\pm$ 0.01 & 0.73 $\pm$ 0.03 & 0.73 $\pm$ 0.03 & 4.31 $\pm$ 0.05 & 4.26 $\pm$ 0.05\\
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\addlinespace
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0.15 & 0.99 & 0.99 & 0.79 $\pm$ 0.02 & 0.78 $\pm$ 0.03 & 4.31 $\pm$ 0.05 & 4.35 $\pm$ 0.05\\
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0.18 & 1 & 1 & 0.83 $\pm$ 0.02 & 0.83 $\pm$ 0.02 & 4.31 $\pm$ 0.05 & 4.25 $\pm$ 0.04\\
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0.21 & 1 & 1 & 0.77 $\pm$ 0.03 & 0.77 $\pm$ 0.03 & 4.42 $\pm$ 0.05 & 4.34 $\pm$ 0.05\\
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0.24 & 1 & 1 & 0.82 $\pm$ 0.02 & 0.82 $\pm$ 0.02 & 4.25 $\pm$ 0.04 & 4.31 $\pm$ 0.05\\
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\bottomrule
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\end{tabular}
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\end{table}
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\paragraph{Results}
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For the model comparison, when \(\eps[\alpha]\) is small
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(\(\eps[\alpha]\in[0, .04]\)), the simulation model is close to the
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Erd\H{o}s-Reńyi network and it is very hard to find any structure beyond
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the one of a single block on each dimension.
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