67 lines
2.9 KiB
TeX
67 lines
2.9 KiB
TeX
\section{Network clustering of simulated networks}\label{sec:network-clustering-of-simulated-networks}
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\paragraph{Simulation settings}
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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
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\(Q_1 = Q_2 = 3\). For the simulations the proportions are the
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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*} and for all \(m = 2,\dots,9\) \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*} where \(\sigma^1_m\) and \(\sigma^2_m\) are permutations of
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\{1, 2, 3\} proper to network \(m\) and
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\(\sigma^1 (\pi)= {(\pi_{\sigma^1 (i)})}_{i=\{1,\dots,3\}}\) and
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\(\sigma^2 (\rho)= {(\rho_{\sigma^2 (i)})}_{i=\{1,\dots,3\}}\). The
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networks are divided into 3 sub-collections of 3 networks with
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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*} with \(\epsilon \in [.1, .4]\). \(\bm{\alpha}^{as}\)
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represents a classical assortative community structure, while
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\(\bm{\alpha}^{cp}\) is a layered core-periphery structure with block 2
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acting as a semi-core. Finally, \(\bm{\alpha}^{dis}\) is a
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disassortative community structure with stronger connections between
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blocks than within blocks. If \(\epsilon = 0\), the three matrices are
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equal and the 9 networks have the same connection structure. Increasing
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\(\epsilon\) differentiates the 3 sub-collections of networks.
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\begin{figure}
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\centering
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\includegraphics{./img/ca0adc96e26b9b41eb8dec4c472696309ebcf0fe.png}
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\caption{\label{}ARI of the partition obtained by clustering in function
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of \(\eps\)}
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\end{figure}
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\paragraph{Results}
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The evaluation of our method involves a comparison between the resulting
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partition of the network collection and the simulated partition using
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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
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nearly perfect in all setups of the \(colBiSBM\).
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