145 research outputs found
Critical States in a Dissipative Sandpile Model
A directed dissipative sandpile model is studied in the two-dimension.
Numerical results indicate that the long time steady states of this model are
critical when grains are dropped only at the top or, everywhere. The critical
behaviour is mean-field like. We discuss the role of infinite avalanches of
dissipative models in periodic systems in determining the critical behaviour of
same models in open systems.Comment: 4 pages (Revtex), 5 ps figures (included
Renormalization group approach to an Abelian sandpile model on planar lattices
One important step in the renormalization group (RG) approach to a lattice
sandpile model is the exact enumeration of all possible toppling processes of
sandpile dynamics inside a cell for RG transformations. Here we propose a
computer algorithm to carry out such exact enumeration for cells of planar
lattices in RG approach to Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett.
{\bf 59}, 381 (1987)] and consider both the reduced-high RG equations proposed
by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. {\bf 72}, 1690
(1994)] and the real-height RG equations proposed by Ivashkevich [Phys. Rev.
Lett. {\bf 76}, 3368 (1996)]. Using this algorithm we are able to carry out RG
transformations more quickly with large cell size, e.g. cell for
the square (sq) lattice in PVZ RG equations, which is the largest cell size at
the present, and find some mistakes in a previous paper [Phys. Rev. E {\bf 51},
1711 (1995)]. For sq and plane triangular (pt) lattices, we obtain the only
attractive fixed point for each lattice and calculate the avalanche exponent
and the dynamical exponent . Our results suggest that the increase of
the cell size in the PVZ RG transformation does not lead to more accurate
results. The implication of such result is discussed.Comment: 29 pages, 6 figure
About the fastest growth of Order Parameter in Models of Percolation
Can there be a `Litmus test' for determining the nature of transition in
models of percolation? In this paper we argue that the answer is in the
affirmative. All one needs to do is to measure the `growth exponent' of
the largest component at the percolation threshold; or
determines if the transition is continuous or discontinuous. We show that a
related exponent which describes how the average maximal jump
sizes in the Order Parameter decays on increasing the system size, is the
single exponent that describes the finite-size scaling of a number of
distributions related to the fastest growth of the Order Parameter in these
problems. Excellent quality scaling analysis are presented for the two single
peak distributions corresponding to the Order Parameters at the two ends of the
maximal jump, the bimodal distribution constructed by interpolation of these
distributions and for the distribution of the maximal jump in the Order
Parameter.Comment: 8 pages, 9 figure
On the scaling behavior of the abelian sandpile model
The abelian sandpile model in two dimensions does not show the type of
critical behavior familar from equilibrium systems. Rather, the properties of
the stationary state follow from the condition that an avalanche started at a
distance r from the system boundary has a probability proportional to 1/sqrt(r)
to reach the boundary. As a consequence, the scaling behavior of the model can
be obtained from evaluating dissipative avalanches alone, allowing not only to
determine the values of all exponents, but showing also the breakdown of
finite-size scaling.Comment: 4 pages, 5 figures; the new version takes into account that the
radius distribution of avalanches cannot become steeper than a certain power
la
Scale-free network on a vertical plane
A scale-free network is grown in the Euclidean space with a global
directional bias. On a vertical plane, nodes are introduced at unit rate at
randomly selected points and a node is allowed to be connected only to the
subset of nodes which are below it using the attachment probability: . Our numerical results indicate that the directed
scale-free network for belongs to a different universality class
compared to the isotropic scale-free network. For the
degree distribution is stretched exponential in general which takes a pure
exponential form in the limit of . The link length
distribution is calculated analytically for all values of .Comment: 4 pages, 4 figure
Precursors of catastrophe in the BTW, Manna and random fiber bundle models of failure
We have studied precursors of the global failure in some self-organised
critical models of sand-pile (in BTW and Manna models) and in the random fiber
bundle model (RFB). In both BTW and Manna model, as one adds a small but fixed
number of sand grains (heights) to any central site of the stable pile, the
local dynamics starts and continues for an average relaxation time (\tau) and
an average number of topplings (\Delta) spread over a radial distance (\xi). We
find that these quantities all depend on the average height (h_{av}) of the
pile and they all diverge as (h_{av}) approaches the critical height (h_{c})
from below: (\Delta) (\sim (h_{c}-h_{av}))(^{-\delta}), (\tau \sim
(h_{c}-h_{av})^{-\gamma}) and (\xi) (\sim) ((h_{c}-h_{av})^{-\nu}). Numerically
we find (\delta \simeq 2.0), (\gamma \simeq 1.2) and (\nu \simeq 1.0) for both
BTW and Manna model in two dimensions. In the strained RFB model we find that
the breakdown susceptibility (\chi) (giving the differential increment of the
number of broken fibers due to increase in external load) and the relaxation
time (\tau), both diverge as the applied load or stress (\sigma) approaches the
network failure threshold (\sigma_{c}) from below: (\chi) (\sim) ((\sigma_{c})
(-)(\sigma)^{-1/2}) and (\tau) (\sim) ((\sigma_{c}) (-)(\sigma)^{-1/2}). These
self-organised dynamical models of failure therefore show some definite
precursors with robust power laws long before the failure point. Such
well-characterised precursors should help predicting the global failure point
of the systems in advance.Comment: 13 pages, 9 figures (eps
Clustering properties of a generalised critical Euclidean network
Many real-world networks exhibit scale-free feature, have a small diameter
and a high clustering tendency. We have studied the properties of a growing
network, which has all these features, in which an incoming node is connected
to its th predecessor of degree with a link of length using a
probability proportional to . For , the
network is scale free at with the degree distribution and as in the Barab\'asi-Albert model (). We find a phase boundary in the plane along which
the network is scale-free. Interestingly, we find scale-free behaviour even for
for where the existence of a new universality class
is indicated from the behaviour of the degree distribution and the clustering
coefficients. The network has a small diameter in the entire scale-free region.
The clustering coefficients emulate the behaviour of most real networks for
increasing negative values of on the phase boundary.Comment: 4 pages REVTEX, 4 figure
An optimal network for passenger traffic
The optimal solution of an inter-city passenger transport network has been
studied using Zipf's law for the city populations and the Gravity law
describing the fluxes of inter-city passenger traffic. Assuming a fixed value
for the cost of transport per person per kilometer we observe that while the
total traffic cost decreases, the total wiring cost increases with the density
of links. As a result the total cost to maintain the traffic distribution is
optimal at a certain link density which vanishes on increasing the network
size. At a finite link density the network is scale-free. Using this model the
air-route network of India has been generated and an one-to-one comparison of
the nodal degree values with the real network has been made.Comment: 5 pages, 4 figure
Local minimal energy landscapes in river networks
The existence and stability of the universality class associated to local
minimal energy landscapes is investigated. Using extensive numerical
simulations, we first study the dependence on a parameter of a partial
differential equation which was proposed to describe the evolution of a rugged
landscape toward a local minimum of the dissipated energy. We then compare the
results with those obtained by an evolution scheme based on a variational
principle (the optimal channel networks). It is found that both models yield
qualitatively similar river patterns and similar dependence on . The
aggregation mechanism is however strongly dependent on the value of . A
careful analysis suggests that scaling behaviors may weakly depend both on
and on initial condition, but in all cases it is within observational
data predictions. Consequences of our resultsComment: 12 pages, 13 figures, revtex+epsfig style, to appear in Phys. Rev. E
(Nov. 2000
The urban economy as a scale-free network
We present empirical evidence that land values are scale-free and introduce a
network model that reproduces the observations. The network approach to urban
modelling is based on the assumption that the market dynamics that generates
land values can be represented as a growing scale-free network. Our results
suggest that the network properties of trade between specialized activities
causes land values, and likely also other observables such as population, to be
power law distributed. In addition to being an attractive avenue for further
analytical inquiry, the network representation is also applicable to empirical
data and is thereby attractive for predictive modelling.Comment: Submitted to Phys. Rev. E. 7 pages, 3 figures. (Minor typos and
details fixed
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