24 research outputs found
Global Robustness vs. 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.Comment: 11 pages, 9 figure
Evolution of robustness in growing random networks
Networks are widely used to model the interaction between individual
dynamical systems. In many instances, the total number of units as well as the
interaction coupling are not fixed in time, but rather constantly evolve. In
terms of networks, this means that the number of nodes and edges change in
time. Various properties of coupled dynamical systems essentially depend on the
structure of the interaction network, such as their robustness to noise. It is
therefore of interest to predict how these properties are affected when the
network grows and what is their relation to the growing mechanism. Here, we
focus on the time-evolution of the network's Kirchhoff index. We derive closed
form expressions for its variation in various scenarios including both the
addition of edges and nodes. For the latter case, we investigate the evolution
where a single node with one and two edges connecting to existing nodes are
added recursively to a network. In both cases we derive relations between the
properties of the nodes to which the new one connects, and the global evolution
of the network robustness. In particular, we show how different scalings of the
Kirchhoff index as a function of the number of nodes are obtained. We
illustrate and confirm the theory with numerical simulations of growing random
networks.Comment: 11 pages, 5 figure
Layered Complex Networks as Fluctuation Amplifiers
In complex networked systems theory, an important question is how to evaluate
the system robustness to external perturbations. With this task in mind, I
investigate the propagation of noise in a multi-layer networked systems. I find
that, for a two layer network, noise originally injected in one layer can be
strongly amplified in the other layer, depending on how well-connected are the
complex networks in each layer and on how much the eigenmodes of their
Laplacian matrices overlap. These results allow to predict potentially harmful
conditions for the system and its sub-networks, where the level of fluctuations
is important, and how to avoid them.Comment: 4 pages, 2 figure
Assessing the impact of Byzantine attacks on coupled phase oscillators
For many coupled dynamical systems, the interaction is the outcome of the
measurement that each unit has of the others or of physical flows e.g. modern
inverter-based power grids, autonomous vehicular platoons or swarms of drones.
Synchronization among all the components of these systems is of primal
importance to avoid failures. The overall operational state of these systems
therefore crucially depends on the correct and reliable functioning of the
individual elements as well as the information they transmit through the
network. Here we investigate the effect of Byzantine attacks where one unit
does not behave as expected, but is controlled by an external attacker. For
such attacks, we assess the impact on the global collective behavior of
nonlinearly coupled phase oscillators. We relate the synchronization error
induced by the input signal to the properties of the attacked node. This allows
to anticipate the potential of an attacker and identify which network
components to secure.Comment: 6 pages, 5 figure
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, 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.Comment: 6 pages, 4 figure
Noise-Induced Desynchronization and Stochastic Escape from Equilibrium in Complex Networks
Complex physical systems are unavoidably subjected to external environments
not accounted for in the set of differential equations that models them. The
resulting perturbations are standardly represented by noise terms. We derive
conditions under which such noise terms perturb the dynamics strongly enough
that they lead to stochastic escape from the initial basin of attraction of an
initial stable equilibrium state of the unperturbed system. Focusing on
Kuramoto-like models we find in particular that, quite counterintuitively,
systems with inertia leave their initial basin faster than or at the same time
as systems without inertia, except for strong white-noise perturbations.Comment: Main text: 5 pages, 4 figures. Supplemental material: 6 pages, 7
figure
Robustness of Synchrony in Complex Networks and Generalized Kirchhoff Indices
In network theory, a question of prime importance is how to assess network
vulnerability in a fast and reliable manner. With this issue in mind, we
investigate the response to parameter changes of coupled dynamical systems on
complex networks. We find that for specific, non-averaged perturbations, the
response of synchronous states critically depends on the overlap between the
perturbation vector and the eigenmodes of the stability matrix of the
unperturbed dynamics. Once averaged over properly defined ensembles of such
perturbations, the response is given by new graph topological indices, which we
introduce as generalized Kirchhoff indices. These findings allow for a fast and
reliable method for assessing the specific or average vulnerability of a
network against changing operational conditions, faults or external attacks.Comment: 5 pages + supplemental material Fina
Network Inference using Sinusoidal Probing
The aim of this manuscript is to present a non-invasive method to recover the
network structure of a dynamical system. We propose to use a controlled probing
input and to measure the response of the network, in the spirit of what is done
to determine oscillation modes in large electrical networks. For a large class
of dynamical systems, we show that this approach is analytically tractable and
we confirm our findings by numerical simulations of networks of Kuramoto
oscillators. Our approach also allows us to determine the number of agents in
the network by probing and measuring a single one of them.Comment: 5 pages, 4 figure