Turing patterns in multiplex networks

The theory of patterns formation for a reaction-diffusion system defined on a multiplex is developed by means of a perturbative approach. The interlayer diffusion constants act as a small parameter in the expansion and the unperturbed state coincides with the limiting setting where the multiplex layers are decoupled. The interaction between adjacent layers can seed the instability of a homogeneous fixed point, yielding self-organized patterns which are instead impeded in the limit of decoupled layers. Patterns on individual layers can also fade away due to cross-talking between layers. Analytical results are compared to direct simulations.

Figure 1

A schematic illustration of a two layer multiplex network.

Figure 2

Main: ${\lambda }^{\max }$ is plotted vs $D_v^{12}$, starting from a condition for which the instability cannot occur when $D_v^{12}=0$. Circles refer to a direct numerical computation of ${\lambda }^{\max }$. The dashed (respectively solid) line represents the analytical solution as obtained at the first (respectively second) perturbative order. Upper inset: the dispersion relation $\lambda$ is plotted versus the eigenvalues of the (single layer) Laplacian operators, $L^1$ and $L^2$. The circles (respectively red and blue online) stand for $D_u^{12}=D_v^{12}=0$, while the squares (green online) are analytically calculated from (5), at the second order, for $D_u^{12}=0$ and $D_v^{12}=0.5$. The two layers of the multiplex have been generated as Watts-Strogatz (WD) [23] networks with probability of rewiring $p$ respectively equal to $0.4$ and $0.6$. The parameters are $b=8,c=17,D_u^1=D_u^2=1,D_v^1=4,D_v^2=5$. Lower inset: asymptotic concentration of species $u$ as function of the nodes index $i$. The first (blue online) $\Omega =100$ nodes refer to the network with $p=0.4$, the other $\Omega$ (red online) to $p=0.6$.

Figure 3

Main: ${\lambda }^{\max }$ is plotted vs $D^{12}\equiv D_v^{12}=D_u^{12}$, starting from the value $D^{12}=0$ for which the instability can occur. Circles refer to a direct numerical computation of ${\lambda }^{\max }$. The dashed (respectively solid) line represents the analytical solution as obtained at the first (respectively second) perturbative order. Inset: the dispersion relation $\lambda$ is plotted vs the eigenvalues of the (single layer) Laplacian operators, $L^1$ and $L^2$. The circles (respectively red and blue online) stand for $D_u^{12}=D_v^{12}=0$, while the squares (green online) are analytically calculated from (5), at the second order, for $D_u^{12}=D_v^{12}=0.2$. The two layers of the multiplex have been generated as Watts-Strogatz (WD) networks with probability of rewiring $p$ respectively equal to $0.4$ and $0.6$. The parameters are $b=8,c=16.2,D_u^1=D_u^2=1,D_v^1=4,D_v^2=5$.