62 lines
2.8 KiB
TeX
62 lines
2.8 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 of size $n^{m=1}_1 = n^{m=1}_2 = 40$ 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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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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\paragraph{Results}
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