2,421 research outputs found
Kleinberg Navigation in Fractal Small World Networks
We study the Kleinberg problem of navigation in Small World networks when the
underlying lattice is a fractal consisting of N>>1 nodes. Our extensive
numerical simulations confirm the prediction that most efficient navigation is
attained when the length r of long-range links is taken from the distribution
P(r)~r^{-alpha}, where alpha=d_f, the fractal dimension of the underlying
lattice. We find finite-size corrections to the exponent alpha, proportional to
1/(ln N)^2
Exact mean first-passage time on the T-graph
We consider a simple random walk on the T-fractal and we calculate the exact
mean time to first reach the central node . The mean is performed
over the set of possible walks from a given origin and over the set of starting
points uniformly distributed throughout the sites of the graph, except .
By means of analytic techniques based on decimation procedures, we find the
explicit expression for as a function of the generation and of the
volume of the underlying fractal. Our results agree with the asymptotic
ones already known for diffusion on the T-fractal and, more generally, they are
consistent with the standard laws describing diffusion on low-dimensional
structures.Comment: 6 page
Wigner Surmise For Domain Systems
In random matrix theory, the spacing distribution functions are
well fitted by the Wigner surmise and its generalizations. In this
approximation the spacing functions are completely described by the behavior of
the exact functions in the limits s->0 and s->infinity. Most non equilibrium
systems do not have analytical solutions for the spacing distribution and
correlation functions. Because of that, we explore the possibility to use the
Wigner surmise approximation in these systems. We found that this approximation
provides a first approach to the statistical behavior of complex systems, in
particular we use it to find an analytical approximation to the nearest
neighbor distribution of the annihilation random walk
Synchronous and Asynchronous Recursive Random Scale-Free Nets
We investigate the differences between scale-free recursive nets constructed
by a synchronous, deterministic updating rule (e.g., Apollonian nets), versus
an asynchronous, random sequential updating rule (e.g., random Apollonian
nets). We show that the dramatic discrepancies observed recently for the degree
exponent in these two cases result from a biased choice of the units to be
updated sequentially in the asynchronous version
Facilitated diffusion of proteins on chromatin
We present a theoretical model of facilitated diffusion of proteins in the
cell nucleus. This model, which takes into account the successive
binding/unbinding events of proteins to DNA, relies on a fractal description of
the chromatin which has been recently evidenced experimentally. Facilitated
diffusion is shown quantitatively to be favorable for a fast localization of a
target locus by a transcription factor, and even to enable the minimization of
the search time by tuning the affinity of the transcription factor with DNA.
This study shows the robustness of the facilitated diffusion mechanism, invoked
so far only for linear conformations of DNA.Comment: 4 pages, 4 figures, accepted versio
Exponents appearing in heterogeneous reaction-diffusion models in one dimension
We study the following 1D two-species reaction diffusion model : there is a
small concentration of B-particles with diffusion constant in an
homogenous background of W-particles with diffusion constant ; two
W-particles of the majority species either coagulate ()
or annihilate () with the respective
probabilities and ; a B-particle and a
W-particle annihilate () with probability 1. The
exponent describing the asymptotic time decay of
the minority B-species concentration can be viewed as a generalization of the
exponent of persistent spins in the zero-temperature Glauber dynamics of the 1D
-state Potts model starting from a random initial condition : the
W-particles represent domain walls, and the exponent
characterizes the time decay of the probability that a diffusive "spectator"
does not meet a domain wall up to time . We extend the methods introduced by
Derrida, Hakim and Pasquier ({\em Phys. Rev. Lett.} {\bf 75} 751 (1995); Saclay
preprint T96/013, to appear in {\em J. Stat. Phys.} (1996)) for the problem of
persistent spins, to compute the exponent in perturbation
at first order in for arbitrary and at first order in
for arbitrary .Comment: 29 pages. The three figures are not included, but are available upon
reques
Memory-induced anomalous dynamics: emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model
We present a random walk model that exhibits asymptotic subdiffusive,
diffusive, and superdiffusive behavior in different parameter regimes. This
appears to be the first instance of a single random walk model leading to all
three forms of behavior by simply changing parameter values. Furthermore, the
model offers the great advantage of analytic tractability. Our model is
non-Markovian in that the next jump of the walker is (probabilistically)
determined by the history of past jumps. It also has elements of intermittency
in that one possibility at each step is that the walker does not move at all.
This rich encompassing scenario arising from a single model provides useful
insights into the source of different types of asymptotic behavior
Diffusion in sparse networks: linear to semi-linear crossover
We consider random networks whose dynamics is described by a rate equation,
with transition rates that form a symmetric matrix. The long time
evolution of the system is characterized by a diffusion coefficient . In one
dimension it is well known that can display an abrupt percolation-like
transition from diffusion () to sub-diffusion (D=0). A question arises
whether such a transition happens in higher dimensions. Numerically can be
evaluated using a resistor network calculation, or optionally it can be deduced
from the spectral properties of the system. Contrary to a recent expectation
that is based on a renormalization-group analysis, we deduce that is
finite; suggest an "effective-range-hopping" procedure to evaluate it; and
contrast the results with the linear estimate. The same approach is useful for
the analysis of networks that are described by quasi-one-dimensional sparse
banded matrices.Comment: 13 pages, 4 figures, proofed as publishe
Anomalous biased diffusion in a randomly layered medium
We present analytical results for the biased diffusion of particles moving
under a constant force in a randomly layered medium. The influence of this
medium on the particle dynamics is modeled by a piecewise constant random
force. The long-time behavior of the particle position is studied in the frame
of a continuous-time random walk on a semi-infinite one-dimensional lattice. We
formulate the conditions for anomalous diffusion, derive the diffusion laws and
analyze their dependence on the particle mass and the distribution of the
random force.Comment: 19 pages, 1 figur
Exact calculations of first-passage quantities on recursive networks
We present general methods to exactly calculate mean-first passage quantities
on self-similar networks defined recursively. In particular, we calculate the
mean first-passage time and the splitting probabilities associated to a source
and one or several targets; averaged quantities over a given set of sources
(e.g., same-connectivity nodes) are also derived. The exact estimate of such
quantities highlights the dependency of first-passage processes with respect to
the source-target distance, which has recently revealed to be a key parameter
to characterize transport in complex media. We explicitly perform calculations
for different classes of recursive networks (finitely ramified fractals,
scale-free (trans)fractals, non-fractals, mixtures between fractals and
non-fractals, non-decimable hierarchical graphs) of arbitrary size. Our
approach unifies and significantly extends the available results in the field.Comment: 16 pages, 10 figure
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