Riccardo Muolo — Page 3

arXiv2020

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

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.

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arXiv2018

Patterns of non-normality in networked systems

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.

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