Patterns of non-normality in networked systems

TL;DR

This work investigates pattern formation in reaction-diffusion systems on networks, focusing on non-normal (directed) coupling as a mechanism to induce macroscopic patterns from stable homogeneous states. It analyzes a two-species RD model on directed networks, showing that non-normality can extend the domain of pattern formation beyond the classical Turing conditions by leveraging transient amplification, and introduces the Brusselator as a testbed to illustrate topology-driven and non-normal-induced patterns. A key contribution is the pseudo-dispersion relation, built on the pseudospectrum, which predicts instability precursors beyond conventional linear analysis via a continuation method that tracks eigenvalue evolution from the Jacobian to a perturbed operator. The findings demonstrate that non-normality enlarges the pattern-forming region, reduces the perturbation threshold, and yields pattern amplitudes comparable to standard Turing patterns, offering a generalized route to self-organization on complex networks with potential applications across physics, chemistry, and biology.

Abstract

Several mechanisms have been proposed to explain the spontaneous generation of self-organized patterns, hypothesised to play a role in the formation of many of the magnificent patterns observed in Nature. In several cases of interest, the system under scrutiny displays a homogeneous equilibrium, which is destabilized via a symmetry breaking instability which reflects the specificity of the problem being inspected. The Turing instability is among the most celebrated paradigms for pattern formation. In its original form, the diffusion constants of the two mobile species need to be quite different from each other for the instability to develop. Unfortunately, this condition limits the applicability of the theory. To overcome this impediment, and with the ambitious long term goal to eventually reconcile theory and experiments, we here propose an alternative mechanism for promoting the onset of patterns. To this end a multi-species reaction-diffusion system is studied on a discrete, network-like support: the instability is triggered by the non-normality of the embedding network. The non-normal character of the dynamics instigates a short time amplification of the imposed perturbation, thus making the system unstable for a choice of parameters that would yield stability under the conventional scenario. Importantly, non-normal networks are pervasively found, as we shall here briefly review.

Figures (9)

• Figure 1: Attraction landscape. A schematic layout to depict the landscape of a generic reaction-diffusion system, defined on a symmetric network (a): the basin of attraction associated with the homogeneous equilibrium, $x^*$ (light blue) is displayed; the latter extends considerably to eventually entwine a large fraction of orbits (as e.g. $A_1$). Only trajectories which are distant enough from the homogeneous fixed point (as e.g. $B_1$) can evolve towards a different, possibly non homogeneous, equilibrium $x^\prime$ (turquoise). Once the dynamics is made to flow on a non-normal support (panel b), the effective basin of attraction of the homogeneous fixed point shrinks considerably. This is the direct signature of the short time amplification of the imposed perturbation, as stimulated by the non-normal character of the underlying support. The amplification makes it possible for the system to overcome the barrier as displayed in (c) and eventually results in the dynamical landscape, that is pictorially exemplified in (b).
• Figure 2: Non-normal dynamics. Time evolution for the norm of $\textbf{x}$, solution of the stable linear system $\dot{\textbf{x}} = \textbf{Ax}$ for a normal (blue) and non-normal (red) system. In both cases the system is stable, meaning that the spectral abscissa is negative $\alpha(\textbf{A})\equiv\sup\Re\sigma(\textbf{A}) < 0$. The norm $\parallel \textbf{x}(t)\parallel$ can however undergo a short time amplification if the numerical abscissa is positive, $\omega(\textbf{A})\equiv\sup\sigma(\textbf{H})>0$ where $\textbf{H}=\left(\textbf{A}+\textbf{A}^*\right)/2$ is the Hermitian part of $\textbf{A}$. The norm of the solution is lower bounded by the Kreiss constant $\mathcal{K}(\textbf{A})\leq\sup_{t\geq 0}\parallel \textbf{x}(t)\parallel/\parallel \textbf{x}(0)\parallel$trefethen.
• Figure 3: Different domains in the parameter plane $(b,c)$ that yield pattern formation for the Brusselator model. Distinct domains of interest are isolated in the reference plane $(b,c)$ and depicted with different colour codes. In region $(iii)$ (shaded in green), a conventional Turing instability develops for the Brusselator model defined on a symmetric support. In region $(ii)$ (yellow), topology driven patterns emerge: the homogeneous fixed point is triggered unstable by the directed nature of the spatial support; here patterns can be stationary or wave-like. Finally, red dots define region $(i)$, where non-normality drives the onset of pattern formation, notwithstanding the prediction of the linear stability analysis that deems the homogeneous fixed point stable, and thus resilient to external perturbation. In white is the region where patterns cannot be established. The Turing region is bounded from below by the curve $c=b-1$ (dashed line) and from above by $c=\frac{D_v}{D_u}(b+1-2\sqrt{b})$ (solid curve). The underlying non-normal network (generated according to the procedure discussed in the main body of the paper) is made up of $N=100$ nodes. The model parameters are $D_u = 0.5$ and $D_v = 1.925$. the initial conditions for $u$ (resp. $v$) are uniformly drawn from an $N$-dimensional sphere of radius $\delta = 0.2$ centred $u^*$ (resp. $v^*$).
• Figure 4: The dispersion relation and the ensuing patterns in different regions of the parameter plane. Turing patterns $(b,c)=(26,61.1)$ (left column), topology-driven patterns $(b,c)=(24,61.1)$ (middle column), patterns of non-normality $(b,c)=(22.5,61.1)$ (right column, enclosed in the red box). The position of the selected working points is marked in Fig. \ref{['fig:NNPatt']}, with, respectively, a triangle, a square and a circle. In all cases, the dispersion relation is drawn for the non-normal (directed) network (red dots) and the symmetrised network (green dots). Recall that the instability is set once the dispersion relation is positive. The insets in the panels that form the first row, show the time evolution of the norm of the system for the non-normal network (red curves) and the symmetric settings (green curves). Here, the system is initialised $\delta$-close to the equilibrium (see caption of Fig. \ref{['fig:NNPatt']}), with $\delta=0.2$. The remaining parameters, as well as the network employed, are those used in Fig. \ref{['fig:NNPatt']}. The ensuing patterns are displayed for, respectively, the symmetric setting (second row), and the non-normal network. In particular, in the third row, the patterns are depicted when representing the network with a stylized lattice layout (as for the symmetric setting). In the fourth row, the edges of the networks are instead shown. Nodes displaying a low concentration are assigned to the central portion of the cluster.
• Figure 5: On the amplitude of the ensuing patterns. Main panel (centre): the average pattern amplitude $\langle A \rangle$ is plotted as a function of the parameter $b$, for $c=61.1$. Red circles refer to the system evolved on a non-normal network, while green squares stand for its symmetrised analogue. The grey region corresponds to parameters giving rise to Turing instability and the blue one to instability driven by the network directionality. Finally, the domain coloured in pink highlights the region where the homogeneous fixed point is stable according to a linear stability analysis, but where patterns can emerge due to non-normality. The insets show histograms of pattern amplitudes, for two specific values of $b$, namely $b\sim 20.5$ (A), $b\sim 22.5$ (B). Red symbols refer to data collected when the Brusselator model is evolves on a non-normal support. Green histograms are computed from simulations that assume symmetric support. The remaining parameters have been set to $D_u=0.5$, $D_v=1.925$, $\delta=0.2$. Simulations have been performed using the same networks as in Fig. \ref{['fig:NNPatt']}.