Laplacian matrix

• Diffusion dynamics on multiplex networks

From the diffusion equation

$$ \frac{dx_i^K}{dt} = D_K \sum_{j=1}^{N} w_{ij}^K (x_j^K - x_i^K) + \sum_{L=1}^{M} D_{KL}(x_i^L - x_i^K), $$

$$ \mathcal{L} = \left( \begin{array}{c|c} D_1 L_1 + D_x I & -D_x I \\\hline -D_x I & D_2 L_2 + D_x I \end{array} \right)$$

where \( L_i \) is the Laplacian of layer \(i\), \(D_i\) is the diffusion coefficient of the layer.

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