271 research outputs found
Nonlocal failures in complex supply networks by single link additions
How do local topological changes affect the global operation and stability of
complex supply networks? Studying supply networks on various levels of
abstraction, we demonstrate that and how adding new links may not only promote
but also degrade stable operation of a network. Intriguingly, the resulting
overloads may emerge remotely from where such a link is added, thus resulting
in nonlocal failure. We link this counter-intuitive phenomenon to Braess'
paradox originally discovered in traffic networks. We use elementary network
topologies to explain its underlying mechanism for different types of supply
networks and find that it generically occurs across these systems. As an
important consequence, upgrading supply networks such as communication
networks, biological supply networks or power grids requires particular care
because even adding only single connections may destabilize normal network
operation and induce disturbances remotely from the location of structural
change and even global cascades of failures.Comment: 12 pages, 10 figure
Revealing networks from dynamics: an introduction
What can we learn from the collective dynamics of a complex network about its
interaction topology? Taking the perspective from nonlinear dynamics, we
briefly review recent progress on how to infer structural connectivity (direct
interactions) from accessing the dynamics of the units. Potential applications
range from interaction networks in physics, to chemical and metabolic
reactions, protein and gene regulatory networks as well as neural circuits in
biology and electric power grids or wireless sensor networks in engineering.
Moreover, we briefly mention some standard ways of inferring effective or
functional connectivity.Comment: Topical review, 48 pages, 7 figure
Inferring Network Topology from Complex Dynamics
Inferring network topology from dynamical observations is a fundamental
problem pervading research on complex systems. Here, we present a simple,
direct method to infer the structural connection topology of a network, given
an observation of one collective dynamical trajectory. The general theoretical
framework is applicable to arbitrary network dynamical systems described by
ordinary differential equations. No interference (external driving) is required
and the type of dynamics is not restricted in any way. In particular, the
observed dynamics may be arbitrarily complex; stationary, invariant or
transient; synchronous or asynchronous and chaotic or periodic. Presupposing a
knowledge of the functional form of the dynamical units and of the coupling
functions between them, we present an analytical solution to the inverse
problem of finding the network topology. Robust reconstruction is achieved in
any sufficiently long generic observation of the system. We extend our method
to simultaneously reconstruct both the entire network topology and all
parameters appearing linear in the system's equations of motion. Reconstruction
of network topology and system parameters is viable even in the presence of
substantial external noise.Comment: 11 pages, 4 figure
Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed
Stabilizing unstable periodic orbits in a chaotic invariant set not only
reveals information about its structure but also leads to various interesting
applications. For the successful application of a chaos control scheme,
convergence speed is of crucial importance. Here we present a predictive
feedback chaos control method that adapts a control parameter online to yield
optimal asymptotic convergence speed. We study the adaptive control map both
analytically and numerically and prove that it converges at least linearly to a
value determined by the spectral radius of the control map at the periodic
orbit to be stabilized. The method is easy to implement algorithmically and may
find applications for adaptive online control of biological and engineering
systems.Comment: 21 pages, 6 figure
Controlling Chaos Faster
Predictive Feedback Control is an easy-to-implement method to stabilize
unknown unstable periodic orbits in chaotic dynamical systems. Predictive
Feedback Control is severely limited because asymptotic convergence speed
decreases with stronger instabilities which in turn are typical for larger
target periods, rendering it harder to effectively stabilize periodic orbits of
large period. Here, we study stalled chaos control, where the application of
control is stalled to make use of the chaotic, uncontrolled dynamics, and
introduce an adaptation paradigm to overcome this limitation and speed up
convergence. This modified control scheme is not only capable of stabilizing
more periodic orbits than the original Predictive Feedback Control but also
speeds up convergence for typical chaotic maps, as illustrated in both theory
and application. The proposed adaptation scheme provides a way to tune
parameters online, yielding a broadly applicable, fast chaos control that
converges reliably, even for periodic orbits of large period
Unstable attractors induce perpetual synchronization and desynchronization
Common experience suggests that attracting invariant sets in nonlinear
dynamical systems are generally stable. Contrary to this intuition, we present
a dynamical system, a network of pulse-coupled oscillators, in which
\textit{unstable attractors} arise naturally. From random initial conditions,
groups of synchronized oscillators (clusters) are formed that send pulses
alternately, resulting in a periodic dynamics of the network. Under the
influence of arbitrarily weak noise, this synchronization is followed by a
desynchronization of clusters, a phenomenon induced by attractors that are
unstable. Perpetual synchronization and desynchronization lead to a switching
among attractors. This is explained by the geometrical fact, that these
unstable attractors are surrounded by basins of attraction of other attractors,
whereas the full measure of their own basin is located remote from the
attractor. Unstable attractors do not only exist in these systems, but moreover
dominate the dynamics for large networks and a wide range of parameters.Comment: 14 pages, 12 figure
Cycle flows and multistabilty in oscillatory networks: an overview
The functions of many networked systems in physics, biology or engineering
rely on a coordinated or synchronized dynamics of its constituents. In power
grids for example, all generators must synchronize and run at the same
frequency and their phases need to appoximately lock to guarantee a steady
power flow. Here, we analyze the existence and multitude of such phase-locked
states. Focusing on edge and cycle flows instead of the nodal phases we derive
rigorous results on the existence and number of such states. Generally,
multiple phase-locked states coexist in networks with strong edges, long
elementary cycles and a homogeneous distribution of natural frequencies or
power injections, respectively. We offer an algorithm to systematically compute
multiple phase- locked states and demonstrate some surprising dynamical
consequences of multistability
Transition to Reconstructibility in Weakly Coupled Networks
Across scientific disciplines, thresholded pairwise measures of statistical
dependence between time series are taken as proxies for the interactions
between the dynamical units of a network. Yet such correlation measures often
fail to reflect the underlying physical interactions accurately. Here we
systematically study the problem of reconstructing direct physical interaction
networks from thresholding correlations. We explicate how local common cause
and relay structures, heterogeneous in-degrees and non-local structural
properties of the network generally hinder reconstructibility. However, in the
limit of weak coupling strengths we prove that stationary systems with dynamics
close to a given operating point transition to universal reconstructiblity
across all network topologies.Comment: 15 pages, 4 figures, supplementary material include
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