Synchronization dynamics in non-normal networks: the trade-off for optimality

TL;DR

This paper investigates how non-normality and directedness in complex networks influence synchronization robustness, revealing a trade-off that impacts the design of optimal, resilient systems.

Contribution

It demonstrates that standard stability analysis methods may fail for non-normal, directed networks and highlights the importance of considering non-normality in network design.

Findings

Synchronization robustness is affected by non-normality and directedness.

Standard techniques like MSF may not predict stability accurately in non-normal networks.

A trade-off exists between non-normality and directedness for optimal synchronization.

Abstract

Synchronization is an important behavior that characterizes many natural and human made systems composed by several interacting units. It can be found in a broad spectrum of applications, ranging from neuroscience to power-grids, to mention a few. Such systems synchronize because of the complex set of coupling they exhibit, the latter being modeled by complex networks. The dynamical behavior of the system and the topology of the underlying network are strongly intertwined, raising the question of the optimal architecture that makes synchronization robust. The Master Stability Function (MSF) has been proposed and extensively studied as a generic framework to tackle synchronization problems. Using this method, it has been shown that for a class of models, synchronization in strongly directed networks is robust to external perturbations. In this paper, our approach is to transform the non-autonomous system of coupled oscillators into an autonomous one, showing that previous results are model-independent. Recent findings indicate that many real-world networks are strongly directed, being potential candidates for optimal synchronization. Inspired by the fact that highly directed networks are also strongly non-normal, in this work, we address the matter of non-normality by pointing out that standard techniques, such as the MSF, may fail in predicting the stability of synchronized behavior. These results lead to a trade-off between non-normality and directedness that should be properly considered when designing an optimal network, enhancing the robustness of synchronization.