# Resilience for stochastic systems interacting via a quasi-degenerate network

## TL;DR

This paper investigates how non-normality and quasi-degenerate network structures in stochastic reaction-diffusion systems can amplify noise, potentially leading to instability despite deterministic stability, thus challenging traditional resilience assessments.

## Contribution

It reveals that quasi-degenerate network spectra can cause noise amplification, affecting resilience evaluations in stochastic systems with non-normal interactions.

## Key findings

- Noise amplification increases with network size.
- Deterministically stable systems can become unstable due to network effects.
- Quasi-degenerate spectra influence stochastic stability.

## Abstract

A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasi degenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, may turn unstable, yielding seemingly regular patterns in the concentration amount. Non-normality and quasi-degenerate networks may therefore amplify the inherent stochasticity, and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.