53 research outputs found
Extreme events and event size fluctuations in biased random walks on networks
Random walk on discrete lattice models is important to understand various
types of transport processes. The extreme events, defined as exceedences of the
flux of walkers above a prescribed threshold, have been studied recently in the
context of complex networks. This was motivated by the occurrence of rare
events such as traffic jams, floods, and power black-outs which take place on
networks. In this work, we study extreme events in a generalized random walk
model in which the walk is preferentially biased by the network topology. The
walkers preferentially choose to hop toward the hubs or small degree nodes. In
this setting, we show that extremely large fluctuations in event-sizes are
possible on small degree nodes when the walkers are biased toward the hubs. In
particular, we obtain the distribution of event-sizes on the network. Further,
the probability for the occurrence of extreme events on any node in the network
depends on its 'generalized strength', a measure of the ability of a node to
attract walkers. The 'generalized strength' is a function of the degree of the
node and that of its nearest neighbors. We obtain analytical and simulation
results for the probability of occurrence of extreme events on the nodes of a
network using a generalized random walk model. The result reveals that the
nodes with a larger value of 'generalized strength', on average, display lower
probability for the occurrence of extreme events compared to the nodes with
lower values of 'generalized strength'
Spatial synchronization and extinction of species under external forcing
We study the interplay between synchronization and extinction of a species.
Using a general model we show that under a common external forcing, the species
with a quadratic saturation term in the population dynamics first undergoes
spatial synchronization and then extinction, thereby avoiding the rescue
effect. This is because the saturation term reduces the synchronization time
scale but not the extinction time scale. The effect can be observed even when
the external forcing acts only on some locations provided there is a
synchronizing term in the dynamics. Absence of the quadratic saturation term
can help the species to avoid extinction.Comment: 4 pages, 2 figure
Self-organized and driven phase synchronization in coupled maps
We study the phase synchronization and cluster formation in coupled maps on
different networks. We identify two different mechanisms of cluster formation;
(a) {\it Self-organized} phase synchronization which leads to clusters with
dominant intra-cluster couplings and (b) {\it driven} phase synchronization
which leads to clusters with dominant inter-cluster couplings. In the novel
driven synchronization the nodes of one cluster are driven by those of the
others. We also discuss the dynamical origin of these two mechanisms for small
networks with two and three nodes.Comment: 4 pages including 2 figure
Characterization and control of small-world networks
Recently Watts and Strogatz have given an interesting model of small-world
networks. Here we concretise the concept of a ``far away'' connection in a
network by defining a {\it far edge}. Our definition is algorithmic and
independent of underlying topology of the network. We show that it is possible
to control spread of an epidemic by using the knowledge of far edges. We also
suggest a model for better advertisement using the far edges. Our findings
indicate that the number of far edges can be a good intrinsic parameter to
characterize small-world phenomena.Comment: 9 pages and 6 figure
Kinks Dynamics in One-Dimensional Coupled Map Lattices
We examine the problem of the dynamics of interfaces in a one-dimensional
space-time discrete dynamical system. Two different regimes are studied : the
non-propagating and the propagating one. In the first case, after proving the
existence of such solutions, we show how they can be described using Taylor
expansions. The second situation deals with the assumption of a travelling wave
to follow the kink propagation. Then a comparison with the corresponding
continuous model is proposed. We find that these methods are useful in simple
dynamical situations but their application to complex dynamical behaviour is
not yet understood.Comment: 17pages, LaTex,3 fig available on cpt.univ-mrs.fr directory
pub/preprints/94/dynamical-systems/94-P.307
Random spread on the family of small-world networks
We present the analytical and numerical results of a random walk on the
family of small-world graphs. The average access time shows a crossover from
the regular to random behavior with increasing distance from the starting point
of the random walk. We introduce an {\em independent step approximation}, which
enables us to obtain analytic results for the average access time. We observe a
scaling relation for the average access time in the degree of the nodes. The
behavior of average access time as a function of , shows striking similarity
with that of the {\em characteristic length} of the graph. This observation may
have important applications in routing and switching in networks with large
number of nodes.Comment: RevTeX4 file with 6 figure
General mechanism for amplitude death in coupled systems
We introduce a general mechanism for amplitude death in coupled
synchronizable dynamical systems. It is known that when two systems are coupled
directly, they can synchronize under suitable conditions. When an indirect
feedback coupling through an environment or an external system is introduced in
them, it is found to induce a tendency for anti-synchronization. We show that,
for sufficient strengths, these two competing effects can lead to amplitude
death. We provide a general stability analysis that gives the threshold values
for onset of amplitude death. We study in detail the nature of the transition
to death in several specific cases and find that the transitions can be of two
types - continuous and discontinuous. By choosing a variety of dynamics for
example, periodic, chaotic, hyper chaotic, and time-delay systems, we
illustrate that this mechanism is quite general and works for different types
of direct coupling, such as diffusive, replacement, and synaptic couplings and
for different damped dynamics of the environment.Comment: 12 pages, 17 figure
Effect of noise on coupled chaotic systems
Effect of noise in inducing order on various chaotically evolving systems is
reviewed, with special emphasis on systems consisting of coupled chaotic
elements. In many situations it is observed that the uncoupled elements when
driven by identical noise, show synchronization phenomena where chaotic
trajectories exponentially converge towards a single noisy trajectory,
independent of the initial conditions. In a random neural network, with
infinite range coupling, chaos is suppressed due to noise and the system
evolves towards a fixed point. Spatiotemporal stochastic resonance phenomenon
has been observed in a square array of coupled threshold devices where a
temporal characteristic of the system resonates at a given noise strength. In a
chaotically evolving coupled map lattice with logistic map as local dynamics
and driven by identical noise at each site, we report that the number of
structures (a structure is a group of neighbouring lattice sites for whom
values of the variable follow certain predefined pattern) follow a power-law
decay with the length of the structure. An interesting phenomenon, which we
call stochastic coherence, is also reported in which the abundance and
lifetimes of these structures show characteristic peaks at some intermediate
noise strength.Comment: 21 page LaTeX file for text, 5 Postscript files for figure
Coupled Maps on Trees
We study coupled maps on a Cayley tree, with local (nearest-neighbor)
interactions, and with a variety of boundary conditions. The homogeneous state
(where every lattice site has the same value) and the node-synchronized state
(where sites of a given generation have the same value) are both shown to occur
for particular values of the parameters and coupling constants. We study the
stability of these states and their domains of attraction. As the number of
sites that become synchronized is much higher compared to that on a regular
lattice, control is easier to effect. A general procedure is given to deduce
the eigenvalue spectrum for these states. Perturbations of the synchronized
state lead to different spatio-temporal structures. We find that a mean-field
like treatment is valid on this (effectively infinite dimensional) lattice.Comment: latex file (25 pages), 4 figures included. To be published in Phys.
Rev.
Complex transitions to synchronization in delay-coupled networks of logistic maps
A network of delay-coupled logistic maps exhibits two different
synchronization regimes, depending on the distribution of the coupling delay
times. When the delays are homogeneous throughout the network, the network
synchronizes to a time-dependent state [Atay et al., Phys. Rev. Lett. 92,
144101 (2004)], which may be periodic or chaotic depending on the delay; when
the delays are sufficiently heterogeneous, the synchronization proceeds to a
steady-state, which is unstable for the uncoupled map [Masoller and Marti,
Phys. Rev. Lett. 94, 134102 (2005)]. Here we characterize the transition from
time-dependent to steady-state synchronization as the width of the delay
distribution increases. We also compare the two transitions to synchronization
as the coupling strength increases. We use transition probabilities calculated
via symbolic analysis and ordinal patterns. We find that, as the coupling
strength increases, before the onset of steady-state synchronization the
network splits into two clusters which are in anti-phase relation with each
other. On the other hand, with increasing delay heterogeneity, no cluster
formation is seen at the onset of steady-state synchronization; however, a
rather complex unsynchronized state is detected, revealed by a diversity of
transition probabilities in the network nodes
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