99 lines
No EOL
4.5 KiB
TeX
99 lines
No EOL
4.5 KiB
TeX
\paragraph{Simulation settings} We want to compare the performance of retrieving
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the nodes blocks with missing edges (that are labeled as \texttt{NA} in the
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incidence matrix).
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For this purpose we generate collections of networks with the following
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parameters:
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\begin{align*}
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\bm{\pi}^m = \begin{cases}
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\bm{\pi} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-}colBiSBM \\
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\sigma_1^m(\bm{\pi}) & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
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\end{cases} \\
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\bm{\rho}^m =
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\begin{cases}
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\bm{\rho} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-}colBiSBM \\
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\sigma_2^m(\bm{\rho}) & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM,
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\end{cases}
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\end{align*}
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for the block proportions, and two different structures with the corresponding
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$\bm{\alpha}$,
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\begin{align*}
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\bm{\alpha}^{modular} = \begin{pmatrix}
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0.9 & 0.05 & 0.05 \\
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0.05 & 0.2 & 0.05 \\
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0.05 & 0.05 & 0.8
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\end{pmatrix}, &
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\bm{\alpha}^{nested} = \begin{pmatrix}
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0.9 & 0.25 & 0.1 \\
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0.3 & 0.15 & 0.05 \\
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0.1 & 0.05 & 0.05
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\end{pmatrix},
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\end{align*}
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where $\bm{\alpha}^{modular}$ represents networks where there are look-a-like
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communities, which tends to interact preferentially within the community and
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less with the other communities. And $\bm{\alpha}^{nested}$ represents a common
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structure detected in ecology with generalist and specialist species and a
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\enquote{nested} structure.
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The collections contain two networks ($M=2$) of size $n^{m=1}_1 =
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n^{m=1}_2 = 40$ and
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$n^{m=2}_1 = n^{m=2}_2 = 120$. One collection is generated for each $colBiSBM$
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model. And the nodes block memberships (i.e., the row and column blocks they
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belong to) are saved.
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Per $colBiSBM$ model, 10 collections are generated and their results are
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averaged.
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In the network $m=1$ (i.e., the smaller one) a proportion of the edges
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$p_{\texttt{NA}}$ see their values replaced by \texttt{NA}s, the
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\enquote{forgotten} values are stored.
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\paragraph{Test procedure} A LBM is fitted on the first network, and the
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predicted block memberships are saved, along with the predicted links using the
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inferred parameters. This will serve as a baseline to see if the use of the
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collection benefits the predictions.
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A $colBiSBM$ model is then fitted (with a model matching the dataset considered)
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and we store the same predictions.
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\paragraph{Quality metrics} To benchmark the performance we use the
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\emph{Area Under the Curve} (AUC) for predicted versus real link values and the
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ARI for predicted versus real block memberships.
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For the comparison we subtract the metric given by the LBM to the one
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given by $colBiSBM$ and denote it $\Delta\mbox{metric}$.
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\begin{figure}[ht]
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\centering
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\input{../tikz/simulations/na_robustness/auc-model}
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\caption{$\Delta\mbox{AUC}$ in function of $p_{\texttt{NA}}$. The dashed red
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lines indicate the value 0 for which
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$\mbox{AUC}_{LBM} = \mbox{AUC}_{colBiSBM}$}
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\label{fig:auc-plot}
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\end{figure}
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\begin{figure}[ht]
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\centering
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\input{../tikz/simulations/na_robustness/ari-dim-model}
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\caption{$\Delta\mbox{ARI}$ in function of $p_{\texttt{NA}}$. The dashed red
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lines indicate the value 0 for which
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$\mbox{ARI}_{LBM} = \mbox{ARI}_{colBiSBM}$}
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\label{fig:ari-dim-plot-na}
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\end{figure}
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\paragraph{Results}
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On figure~\ref{fig:auc-plot} one can see that overall the nested structure seems
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to be the one benefitting most from the collection model having generally
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slightly higher $\Delta$AUC than the modular one.
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But in general it seems that for $\epsilon\in[0.1,0.7]$ there are no clear
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differences between LBM and colBiSBM regarding link prediction. For $\epsilon
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\in[0.8,0.9]$ this is where the collection model seems to be most effective.
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For the ARI, figure~\ref{fig:ari-dim-plot-na} suggests that collection model
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does at least as well as LBM and improves nodes memberships recovery for modular
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structure starting from $\epsilon = 0.7$. Again, nested structure benefits
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of collection model for smaller $\epsilon$ values but those increase in
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$\Delta$ARI are also smaller than what can be observed for modular structure.
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\clearpage |