# Patterns of non-normality in networked systems

## TL;DR

This paper introduces a novel mechanism for pattern formation in networked systems, leveraging the non-normality of the underlying network to induce instabilities and patterns, expanding beyond traditional diffusion-based theories.

## Contribution

It proposes that non-normality in network structures can trigger pattern formation in reaction-diffusion systems, offering an alternative to classical Turing instability conditions.

## Key findings

- Non-normal networks can induce pattern formation.
- The proposed mechanism broadens the applicability of pattern formation theory.
- Non-normality leads to short-term amplification of perturbations.

## 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.