556 research outputs found
Synchronization is optimal in non-diagonalizable networks
We consider the problem of maximizing the synchronizability of oscillator
networks by assigning weights and directions to the links of a given
interaction topology. We first extend the well-known master stability formalism
to the case of non-diagonalizable networks. We then show that, unless some
oscillator is connected to all the others, networks of maximum
synchronizability are necessarily non-diagonalizable and can always be obtained
by imposing unidirectional information flow with normalized input strengths.
The extension makes the formalism applicable to all possible network
structures, while the maximization results provide insights into hierarchical
structures observed in complex networks in which synchronization plays a
significant role.Comment: 4 pages, 1 figure; minor revisio
(Non)Invariance of dynamical quantities for orbit equivalent flows
We study how dynamical quantities such as Lyapunov exponents, metric entropy,
topological pressure, recurrence rates, and dimension-like characteristics
change under a time reparameterization of a dynamical system. These quantities
are shown to either remain invariant, transform according to a multiplicative
factor or transform through a convoluted dependence that may take the form of
an integral over the initial local values. We discuss the significance of these
results for the apparent non-invariance of chaos in general relativity and
explore applications to the synchronization of equilibrium states and the
elimination of expansions
Dynamical and spectral properties of complex networks
Dynamical properties of complex networks are related to the spectral
properties of the Laplacian matrix that describes the pattern of connectivity
of the network. In particular we compute the synchronization time for different
types of networks and different dynamics. We show that the main dependence of
the synchronization time is on the smallest nonzero eigenvalue of the Laplacian
matrix, in contrast to other proposals in terms of the spectrum of the
adjacency matrix. Then, this topological property becomes the most relevant for
the dynamics.Comment: 14 pages, 5 figures, to be published in New Journal of Physic
How big is too big? Critical Shocks for Systemic Failure Cascades
External or internal shocks may lead to the collapse of a system consisting
of many agents. If the shock hits only one agent initially and causes it to
fail, this can induce a cascade of failures among neighoring agents. Several
critical constellations determine whether this cascade remains finite or
reaches the size of the system, i.e. leads to systemic risk. We investigate the
critical parameters for such cascades in a simple model, where agents are
characterized by an individual threshold \theta_i determining their capacity to
handle a load \alpha\theta_i with 1-\alpha being their safety margin. If agents
fail, they redistribute their load equally to K neighboring agents in a regular
network. For three different threshold distributions P(\theta), we derive
analytical results for the size of the cascade, X(t), which is regarded as a
measure of systemic risk, and the time when it stops. We focus on two different
regimes, (i) EEE, an external extreme event where the size of the shock is of
the order of the total capacity of the network, and (ii) RIE, a random internal
event where the size of the shock is of the order of the capacity of an agent.
We find that even for large extreme events that exceed the capacity of the
network finite cascades are still possible, if a power-law threshold
distribution is assumed. On the other hand, even small random fluctuations may
lead to full cascades if critical conditions are met. Most importantly, we
demonstrate that the size of the "big" shock is not the problem, as the
systemic risk only varies slightly for changes of 10 to 50 percent of the
external shock. Systemic risk depends much more on ingredients such as the
network topology, the safety margin and the threshold distribution, which gives
hints on how to reduce systemic risk.Comment: 23 pages, 7 Figure
Can aerosols be trapped in open flows?
The fate of aerosols in open flows is relevant in a variety of physical
contexts. Previous results are consistent with the assumption that such
finite-size particles always escape in open chaotic advection. Here we show
that a different behavior is possible. We analyze the dynamics of aerosols both
in the absence and presence of gravitational effects, and both when the
dynamics of the fluid particles is hyperbolic and nonhyperbolic. Permanent
trapping of aerosols much heavier than the advecting fluid is shown to occur in
all these cases. This phenomenon is determined by the occurrence of multiple
vortices in the flow and is predicted to happen for realistic particle-fluid
density ratios.Comment: Animation available at
http://www.pks.mpg.de/~rdvilela/leapfrogging.htm
Network Automata: Coupling structure and function in real-world networks
We introduce Network Automata, a framework which couples the topological
evolution of a network to its structure. It is useful for dealing with networks
in which the topology evolves according to some specified microscopic rules
and, simultaneously, there is a dynamic process taking place on the network
that both depends on its structure but is also capable of modifying it. It is a
generic framework for modeling systems in which network structure, dynamics,
and function are interrelated. At the practical level, this framework allows
for easy implementation of the microscopic rules involved in such systems. To
demonstrate the approach, we develop a class of simple biologically inspired
models of fungal growth.Comment: 7 pages, 5 figures, 1 tables. Revised content - surplus text and
figures remove
Enhancing complex-network synchronization
Heterogeneity in the degree (connectivity) distribution has been shown to
suppress synchronization in networks of symmetrically coupled oscillators with
uniform coupling strength (unweighted coupling). Here we uncover a condition
for enhanced synchronization in directed networks with weighted coupling. We
show that, in the optimum regime, synchronizability is solely determined by the
average degree and does not depend on the system size and the details of the
degree distribution. In scale-free networks, where the average degree may
increase with heterogeneity, synchronizability is drastically enhanced and may
become positively correlated with heterogeneity, while the overall cost
involved in the network coupling is significantly reduced as compared to the
case of unweighted coupling.Comment: 4 pages, 3 figure
Network Synchronization, Diffusion, and the Paradox of Heterogeneity
Many complex networks display strong heterogeneity in the degree
(connectivity) distribution. Heterogeneity in the degree distribution often
reduces the average distance between nodes but, paradoxically, may suppress
synchronization in networks of oscillators coupled symmetrically with uniform
coupling strength. Here we offer a solution to this apparent paradox. Our
analysis is partially based on the identification of a diffusive process
underlying the communication between oscillators and reveals a striking
relation between this process and the condition for the linear stability of the
synchronized states. We show that, for a given degree distribution, the maximum
synchronizability is achieved when the network of couplings is weighted and
directed, and the overall cost involved in the couplings is minimum. This
enhanced synchronizability is solely determined by the mean degree and does not
depend on the degree distribution and system size. Numerical verification of
the main results is provided for representative classes of small-world and
scale-free networks.Comment: Synchronization in Weighted Network
Critical Switching in Globally Attractive Chimeras
We report on a new type of chimera state that attracts almost all initial
conditions and exhibits power-law switching behavior in networks of coupled
oscillators. Such switching chimeras consist of two symmetric configurations,
which we refer to as subchimeras, in which one cluster is synchronized and the
other is incoherent. Despite each subchimera being linearly stable, switching
chimeras are extremely sensitive to noise: arbitrarily small noise triggers and
sustains persistent switching between the two symmetric subchimeras. The
average switching frequency scales as a power law with the noise intensity,
which is in contrast with the exponential scaling observed in typical
stochastic transitions. Rigorous numerical analysis reveals that the power-law
switching behavior originates from intermingled basins of attraction associated
with the two subchimeras, which in turn are induced by chaos and symmetry in
the system. The theoretical results are supported by experiments on coupled
optoelectronic oscillators, which demonstrate the generality and robustness of
switching chimeras
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