208 research outputs found
Boolean networks with reliable dynamics
We investigated the properties of Boolean networks that follow a given
reliable trajectory in state space. A reliable trajectory is defined as a
sequence of states which is independent of the order in which the nodes are
updated. We explored numerically the topology, the update functions, and the
state space structure of these networks, which we constructed using a minimum
number of links and the simplest update functions. We found that the clustering
coefficient is larger than in random networks, and that the probability
distribution of three-node motifs is similar to that found in gene regulation
networks. Among the update functions, only a subset of all possible functions
occur, and they can be classified according to their probability. More
homogeneous functions occur more often, leading to a dominance of canalyzing
functions. Finally, we studied the entire state space of the networks. We
observed that with increasing systems size, fixed points become more dominant,
moving the networks close to the frozen phase.Comment: 11 Pages, 15 figure
Crossover from Percolation to Self-Organized Criticality
We include immunity against fire as a new parameter into the self-organized
critical forest-fire model. When the immunity assumes a critical value,
clusters of burnt trees are identical to percolation clusters of random bond
percolation. As long as the immunity is below its critical value, the
asymptotic critical exponents are those of the original self-organized critical
model, i.e. the system performs a crossover from percolation to self-organized
criticality. We present a scaling theory and computer simulation results.Comment: 4 pages Revtex, two figures included, to be published in PR
The phase diagram of random threshold networks
Threshold networks are used as models for neural or gene regulatory networks.
They show a rich dynamical behaviour with a transition between a frozen and a
chaotic phase. We investigate the phase diagram of randomly connected threshold
networks with real-valued thresholds h and a fixed number of inputs per node.
The nodes are updated according to the same rules as in a model of the
cell-cycle network of Saccharomyces cereviseae [PNAS 101, 4781 (2004)]. Using
the annealed approximation, we derive expressions for the time evolution of the
proportion of nodes in the "on" and "off" state, and for the sensitivity
. The results are compared with simulations of quenched networks. We
find that for integer values of h the simulations show marked deviations from
the annealed approximation even for large networks. This can be attributed to
the particular choice of the updating rule.Comment: 8 pages, 6 figure
Scaling laws and simulation results for the self--organized critical forest--fire model
We discuss the properties of a self--organized critical forest--fire model
which has been introduced recently. We derive scaling laws and define critical
exponents. The values of these critical exponents are determined by computer
simulations in 1 to 8 dimensions. The simulations suggest a critical dimension
above which the critical exponents assume their mean--field values.
Changing the lattice symmetry and allowing trees to be immune against fire, we
show that the critical exponents are universal.Comment: 12 pages, postscript uuencoded, figures included, to appear in Phys.
Rev.
Forest fires and other examples of self-organized criticality
We review the properties of the self-organized critical (SOC) forest-fire
model. The paradigm of self-organized criticality refers to the tendency of
certain large dissipative systems to drive themselves into a critical state
independent of the initial conditions and without fine-tuning of the
parameters. After an introduction, we define the rules of the model and discuss
various large-scale structures which may appear in this system. The origin of
the critical behavior is explained, critical exponents are introduced, and
scaling relations between the exponents are derived. Results of computer
simulations and analytical calculations are summarized. The existence of an
upper critical dimension and the universality of the critical behavior under
changes of lattice symmetry or the introduction of immunity are discussed. A
survey of interesting modifications of the forest-fire model is given. Finally,
several other important SOC models are briefly described.Comment: 37 pages RevTeX, 13 PostScript figures (Figs 1, 4, 13 are of reduced
quality to keep download times small
The self-organized critical forest-fire model on large scales
We discuss the scaling behavior of the self-organized critical forest-fire
model on large length scales. As indicated in earlier publications, the
forest-fire model does not show conventional critical scaling, but has two
qualitatively different types of fires that superimpose to give the effective
exponents typically measured in simulations. We show that this explains not
only why the exponent characterizing the fire-size distribution changes with
increasing correlation length, but allows also to predict its asymptotic value.
We support our arguments by computer simulations of a coarse-grained model, by
scaling arguments and by analyzing states that are created artificially by
superimposing the two types of fires.Comment: 26 pages, 7 figure
The complex scaling behavior of non--conserved self--organized critical systems
The Olami--Feder--Christensen earthquake model is often considered the
prototype dissipative self--organized critical model. It is shown that the size
distribution of events in this model results from a complex interplay of
several different phenomena, including limited floating--point precision.
Parallels between the dynamics of synchronized regions and those of a system
with periodic boundary conditions are pointed out, and the asymptotic avalanche
size distribution is conjectured to be dominated by avalanches of size one,
with the weight of larger avalanches converging towards zero as the system size
increases.Comment: 4 pages revtex4, 5 figure
The properties of attractors of canalyzing random Boolean networks
We study critical random Boolean networks with two inputs per node that
contain only canalyzing functions. We present a phenomenological theory that
explains how a frozen core of nodes that are frozen on all attractors arises.
This theory leads to an intuitive understanding of the system's dynamics as it
demonstrates the analogy between standard random Boolean networks and networks
with canalyzing functions only. It reproduces correctly the scaling of the
number of nonfrozen nodes with system size. We then investigate numerically
attractor lengths and numbers, and explain the findings in terms of the
properties of relevant components. In particular we show that canalyzing
networks can contain very long attractors, albeit they occur less often than in
standard networks.Comment: 9 pages, 8 figure
Dynamics of a passive sliding particle on a randomly fluctuating surface
We study the motion of a particle sliding under the action of an external
field on a stochastically fluctuating one-dimensional Edwards-Wilkinson
surface. Numerical simulations using the single-step model shows that the
mean-square displacement of the sliding particle shows distinct dynamic scaling
behavior, depending on whether the surface fluctuates faster or slower than the
motion of the particle. When the surface fluctuations occur on a time scale
much smaller than the particle motion, we find that the characteristic length
scale shows anomalous diffusion with , where from numerical data. On the other hand, when the particle moves faster
than the surface, its dynamics is controlled by the surface fluctuations and
. A self-consistent approximation predicts that the
anomalous diffusion exponent is , in good agreement with simulation
results. We also discuss the possibility of a slow cross-over towards
asymptotic diffusive behavior. The probability distribution of the displacement
has a Gaussian form in both the cases.Comment: 6 pages, 4 figures, error in reference corrected and new reference
added, submitted to Phys. Rev.
Passive Sliders on Growing Surfaces and (anti-)Advection in Burger's Flows
We study the fluctuations of particles sliding on a stochastically growing
surface. This problem can be mapped to motion of passive scalars in a randomly
stirred Burger's flow. Renormalization group studies, simulations, and scaling
arguments in one dimension, suggest a rich set of phenomena: If particles slide
with the avalanche of growth sites (advection with the fluid), they tend to
cluster and follow the surface dynamics. However, for particles sliding against
the avalanche (anti-advection), we find slower diffusion dynamics, and density
fluctuations with no simple relation to the underlying fluid, possibly with
continuously varying exponents.Comment: 4 pages revtex
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