60 lines
No EOL
3.4 KiB
TeX
60 lines
No EOL
3.4 KiB
TeX
\section{Network clustering of simulated networks}\label{sec:network-clustering-of-simulated-networks}
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\paragraph{Simulation settings} For all models we simulate $M = 9$ networks with
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$\forall m \in \{ 1 \dots M \} , n^m_1 = n^m_2 = 75$ with $Q_1 = Q_2 = 3$. For
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the simulations the proportions are the following:
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\begin{align*}
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\bm{\pi}^1 = \left( 0.2, 0.3, 0.5 \right) & & \bm{\rho}^1 = \left( 0.2, 0.3, 0.5 \right)
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\end{align*}
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and for all $m = 2,\dots,9$
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\begin{align*}
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\bm{\pi}^m = \begin{cases}
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\bm{\pi}^1 & \text{for } iid\text{-}colBiSBM \\
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\sigma^1_m(\bm{\pi}^1) & \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}^1 & \text{for } iid\text{-}colBiSBM \\
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\sigma^2_m(\bm{\rho}^1) & \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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where $\sigma^1_m$ and $\sigma^2_m$ are permutations of {1, 2, 3} proper to network $m$ and
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$\sigma^1 (\pi)= {(\pi_{\sigma^1 (i)})}_{i=\{1,\dots,3\}}$
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and $\sigma^2 (\rho)= {(\rho_{\sigma^2 (i)})}_{i=\{1,\dots,3\}}$.
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The networks are divided into 3 sub-collections of 3
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networks with connectivity parameters as follows:
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\begin{align*}
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\bm{\alpha}^{as} = .3 + \begin{pmatrix}
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\epsilon & - \frac{\epsilon}{2} & - \frac{\epsilon}{2} \\
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- \frac{\epsilon}{2} & \epsilon & - \frac{\epsilon}{2} \\
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- \frac{\epsilon}{2} & - \frac{\epsilon}{2} & \epsilon
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\end{pmatrix}, & &
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\bm{\alpha}^{cp} = .3 + \begin{pmatrix}
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\frac{3 \epsilon}{2} & \epsilon & \frac{\epsilon}{2} \\
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\epsilon & \frac{\epsilon}{2} & 0 \\
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\frac{\epsilon}{2} & 0 & - \frac{\epsilon}{2}
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\end{pmatrix}, & &
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\bm{\alpha}^{dis} = .3 + \begin{pmatrix}
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- \frac{\epsilon}{2} & \epsilon & \epsilon \\
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\epsilon & - \frac{\epsilon}{2} & \epsilon \\
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\epsilon & \epsilon & - \frac{\epsilon}{2}
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\end{pmatrix},
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\end{align*}
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with $\epsilon \in [.1, .4]$. $\bm{\alpha}^{as}$ represents a classical
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assortative community structure,
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while $\bm{\alpha}^{cp}$ is a layered core-periphery structure with block 2
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acting as a semi-core. Finally, $\bm{\alpha}^{dis}$ is a dis-assortative
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community structure with stronger
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connections between blocks than within blocks. If $\epsilon = 0$, the three
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matrices are equal and the 9 networks have the same connection structure.
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Increasing $\epsilon$ differentiates the 3 sub-collections of networks.
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% ARI boxplot
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\paragraph{Results} The evaluation of our method involves a comparison between
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the resulting partition of the network collection and the simulated partition
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using the ARI index. As the value of $\epsilon$ increases, our ability to
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distinguish between the networks improves, and this distinction becomes nearly
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perfect in all setups of the $colBiSBM$. |