222 research outputs found
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
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
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.
Renormalization group approach to the critical behavior of the forest fire model
We introduce a Renormalization scheme for the one and two dimensional
Forest-Fire models in order to characterize the nature of the critical state
and its scale invariant dynamics. We show the existence of a relevant scaling
field associated with a repulsive fixed point. This model is therefore critical
in the usual sense because the control parameter has to be tuned to its
critical value in order to get criticality. It turns out that this is not just
the condition for a time scale separation. The critical exponents are computed
analytically and we obtain , and ,
respectively for the one and two dimensional case, in very good agreement with
numerical simulations.Comment: 4 pages, 3 uuencoded Postcript figure
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
Exact Results for the One-Dimensional Self-Organized Critical Forest-Fire Model
We present the analytic solution of the self-organized critical (SOC)
forest-fire model in one dimension proving SOC in systems without conservation
laws by analytic means. Under the condition that the system is in the steady
state and very close to the critical point, we calculate the probability that a
string of neighboring sites is occupied by a given configuration of trees.
The critical exponent describing the size distribution of forest clusters is
exactly and does not change under certain changes of the model
rules. Computer simulations confirm the analytic results.Comment: 12 pages REVTEX, 2 figures upon request, dro/93/
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
Boolean networks with robust and reliable trajectories
We construct and investigate Boolean networks that follow a given reliable
trajectory in state space, which is insensitive to fluctuations in the updating
schedule, and which is also robust against noise. Robustness is quantified as
the probability that the dynamics return to the reliable trajectory after a
perturbation of the state of a single node. In order to achieve high
robustness, we navigate through the space of possible update functions by using
an evolutionary algorithm. We constrain the networks to having the minimum
number of connections required to obtain the reliable trajectory. Surprisingly,
we find that robustness always reaches values close to 100 percent during the
evolutionary optimization process. The set of update functions can be evolved
such that it differs only slightly from that of networks that were not
optimized with respect to robustness. The state space of the optimized networks
is dominated by the basin of attraction of the reliable trajectory.Comment: 12 pages, 9 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
Noise in random Boolean networks
We investigate the effect of noise on Random Boolean Networks. Noise is
implemented as a probability that a node does not obey its deterministic
update rule. We define two order parameters, the long-time average of the
Hamming distance between a network with and without noise, and the average
frozenness, which is a measure of the extent to which a node prefers one of the
two Boolean states. We evaluate both order parameters as function of the noise
strength, finding a smooth transition from deterministic () to fully
stochastic () dynamics for networks with , and a first order
transition at for . Most of the results obtained by computer
simulation are also derived analytically. The average Hamming distance can be
evaluated using the annealed approximation. In order to obtain the distribution
of frozenness as function of the noise strength, more sophisticated
self-consistent calculations had to be performed. This distribution is a
collection of delta peaks for K=1, and it has a fractal sructure for ,
approaching a continuous distribution in the limit .Comment: 9 pages, 8 figures, 1 tabl
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