Noise-induced desynchronization and stochastic escape from equilibrium in complex networks

Faster network disruption from layered oscillatory dynamics

Nonlinear complex network-coupled systems typically have multiple stable equilibrium states. Following perturbations or due to ambient noise, the system is pushed away from its initial equilibrium, and, depending on the direction and the amplitude of the excursion, it might undergo a transition to another equilibrium. It was recently demonstrated [M. Tyloo, J. Phys. Complex. 3 03LT01 (2022)] that layered complex networks may exhibit amplified fluctuations. Here, I investigate how noise with system-specific correlations impacts the first escape time of nonlinearly coupled oscillators. Interestingly, I show that, not only the strong amplification of the fluctuations is a threat to the good functioning of the network but also the spatial and temporal correlations of the noise along the lowest-lying eigenmodes of the Laplacian matrix. I analyze first escape times on synthetic networks and compare noise originating from layered dynamics to uncorrelated noise.

Reconstructing network structures from partial measurements

The dynamics of systems of interacting agents is determined by the structure of their coupling network. The knowledge of the latter is, therefore, highly desirable, for instance, to develop efficient control schemes, to accurately predict the dynamics, or to better understand inter-agent processes. In many important and interesting situations, the network structure is not known, however, and previous investigations have shown how it may be inferred from complete measurement time series on each and every agent. These methods implicitly presuppose that, even though the network is not known, all its nodes are. Here, we investigate the different problem of inferring network structures within the observed/measured agents. For symmetrically coupled dynamical systems close to a stable equilibrium, we establish analytically and illustrate numerically that velocity signal correlators encode not only direct couplings, but also geodesic distances in the coupling network within the subset of measurable agents. When dynamical data are accessible for all agents, our method is furthermore algorithmically more efficient than the traditional ones because it does not rely on matrix inversion.

Chaotic transients, riddled basins, and stochastic transitions in coupled periodic logistic maps

A system of two coupled map-based oscillators is studied. As units, we use identical logistic maps in two-periodic modes. In this system, increasing coupling strength significantly changes deterministic regimes of collective dynamics with coexisting periodic, quasiperiodic, and chaotic attractors. We study how random noise deforms these dynamical regimes in parameter zones of mono- and bistability, causes "order-chaos" transformations, and destroys regimes of in-phase and anti-phase synchronization. In the analytical study of these noise-induced phenomena, a stochastic sensitivity technique and a method of confidence domains for periodic and multi-band chaotic attractors are used. In this analysis, a key role of chaotic transients and geometry of "riddled" basins is revealed.

Minimal fatal shocks in multistable complex networks

Multistability is a common phenomenon which naturally occurs in complex networks. Often one of the coexisting stable states can be identified as being the desired one for a particular application. We present here a global approach to identify the minimal perturbation which will instantaneously kick the system out of the basin of attraction of its desired state and hence induce a critical or fatal transition we call shock-tipping. The corresponding Minimal Fatal Shock is a vector whose length can be used as a global stability measure and whose direction in state space allows us to draw conclusions on weaknesses of the network corresponding to critical network motifs. We demonstrate this approach in plant-pollinator networks and the power grid of Great Britain. In both system classes, tree-like substructures appear to be the most vulnerable with respect to the minimal shock perturbation.

Vulnerability in dynamically driven oscillatory networks and power grids

Vulnerability of networks has so far been quantified mainly for structural properties. In driven systems, however, vulnerability intrinsically relies on the collective response dynamics. As shown recently, dynamic response patterns emerging in driven oscillator networks and AC power grid models are highly heterogeneous and nontrivial, depending jointly on the driving frequency, the interaction topology of the network, and the node or nodes driven. Identifying which nodes are most susceptible to dynamic driving and may thus make the system as a whole vulnerable to external input signals, however, remains a challenge. Here, we propose an easy-to-compute Dynamic Vulnerability Index (DVI) for identifying those nodes that exhibit largest amplitude responses to dynamic driving signals with given power spectra and thus are most vulnerable. The DVI is based on linear response theory, as such generic, and enables robust predictions. It thus shows potential for a wide range of applications across dynamically driven networks, for instance, for identifying the vulnerable nodes in power grids driven by fluctuating inputs from renewable energy sources and fluctuating power output to consumers.

Network desynchronization by non-Gaussian fluctuations

Many networks must maintain synchrony despite the fact that they operate in noisy environments. Important examples are stochastic inertial oscillators, which are known to exhibit fluctuations with broad tails in many applications, including electric power networks with renewable energy sources. Such non-Gaussian fluctuations can result in rare network desynchronization. Here we build a general theory for inertial oscillator network desynchronization by non-Gaussian noise. We compute the rate of desynchronization and show that higher moments of noise enter at specific powers of coupling: either speeding up or slowing down the rate exponentially depending on how noise statistics match the statistics of a network's slowest mode. Finally, we use our theory to introduce a technique that drastically reduces the effective description of network desynchronization. Most interestingly, when instability is associated with a single edge, the reduction is to one stochastic oscillator.

Enhancing power grid synchronization and stability through time-delayed feedback control

We study the synchronization and stability of power grids within the Kuramoto phase oscillator model with inertia with a bimodal natural frequency distribution representing the generators and the loads. The Kuramoto model describes the dynamics of the ac voltage phase and allows for a comprehensive understanding of fundamental network properties capturing the essential dynamical features of a power grid on coarse scales. We identify critical nodes through solitary frequency deviations and Lyapunov vectors corresponding to unstable Lyapunov exponents. To cure dangerous deviations from synchronization we propose time-delayed feedback control, which is an efficient control concept in nonlinear dynamic systems. Different control strategies are tested and compared with respect to the minimum number of controlled nodes required to achieve synchronization and Lyapunov stability. As a proof of principle, this fast-acting control method is demonstrated for different networks (the German and the Italian power transmission grid), operating points, configurations, and models. In particular, an extended version of the Kuramoto model with inertia is considered that includes the voltage dynamics, thus taking into account the interplay of amplitude and phase typical of the electrodynamical behavior of a machine.

Low-frequency oscillations in coupled phase oscillators with inertia

This work considers a second-order Kuramoto oscillator network periodically driven at one node to model low-frequency forced oscillations in power grids. The phase fluctuation magnitude at each node and the disturbance propagation in the network are numerically analyzed. The coupling strengths in this work are sufficiently large to ensure the stability of equilibria in the unforced system. It is found that the phase fluctuation is primarily determined by the network structural properties and forcing parameters, not the parameters specific to individual nodes such as power and damping. A new "resonance" phenomenon is observed in which the phase fluctuation magnitudes peak at certain critical coupling strength in the forced system. In the cases of long chain and ring-shaped networks, the Kuramoto model yields an important but somehow counter-intuitive result that the fluctuation magnitude distribution does not necessarily follow a simple attenuating trend along the propagation path and the fluctuation at nodes far from the disturbance source could be stronger than that at the source. These findings are relevant to low-frequency forced oscillations in power grids and will help advance the understanding of their dynamics and mechanisms and improve the detection and mitigation techniques.

Global robustness versus local vulnerabilities in complex synchronous networks

In complex network-coupled dynamical systems, two questions of central importance are how to identify the most vulnerable components and how to devise a network making the overall system more robust to external perturbations. To address these two questions, we investigate the response of complex networks of coupled oscillators to local perturbations. We quantify the magnitude of the resulting excursion away from the unperturbed synchronous state through quadratic performance measures in the angle or frequency deviations. We find that the most fragile oscillators in a given network are identified by centralities constructed from network resistance distances. Further defining the global robustness of the system from the average response over ensembles of homogeneously distributed perturbations, we find that it is given by a family of topological indices known as generalized Kirchhoff indices. Both resistance centralities and Kirchhoff indices are obtained from a spectral decomposition of the stability matrix of the unperturbed dynamics and can be expressed in terms of resistance distances. We investigate the properties of these topological indices in small-world and regular networks. In the case of oscillators with homogeneous inertia and damping coefficients, we find that inertia only has small effects on robustness of coupled oscillators. Numerical results illustrate the validity of the theory.