5,098 research outputs found
Self-avoiding walks on scale-free networks
Several kinds of walks on complex networks are currently used to analyze
search and navigation in different systems. Many analytical and computational
results are known for random walks on such networks. Self-avoiding walks (SAWs)
are expected to be more suitable than unrestricted random walks to explore
various kinds of real-life networks. Here we study long-range properties of
random SAWs on scale-free networks, characterized by a degree distribution
. In the limit of large networks (system size ), the average number of SAWs starting from a generic site
increases as , with . For finite ,
is reduced due to the presence of loops in the network, which causes the
emergence of attrition of the paths. For kinetic growth walks, the average
maximum length, , increases as a power of the system size: , with an exponent increasing as the parameter is
raised. We discuss the dependence of on the minimum allowed degree in
the network. A similar power-law dependence is found for the mean
self-intersection length of non-reversal random walks. Simulation results
support our approximate analytical calculations.Comment: 9 pages, 7 figure
Grassmann Integral Representation for Spanning Hyperforests
Given a hypergraph G, we introduce a Grassmann algebra over the vertex set,
and show that a class of Grassmann integrals permits an expansion in terms of
spanning hyperforests. Special cases provide the generating functions for
rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All
these results are generalizations of Kirchhoff's matrix-tree theorem.
Furthermore, we show that the class of integrals describing unrooted spanning
(hyper)forests is induced by a theory with an underlying OSP(1|2)
supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J.
Phys.
Fractal Characterizations of MAX Statistical Distribution in Genetic Association Studies
Two non-integer parameters are defined for MAX statistics, which are maxima
of simpler test statistics. The first parameter, , is the
fractional number of tests, representing the equivalent numbers of independent
tests in MAX. If the tests are dependent, . The second
parameter is the fractional degrees of freedom of the chi-square
distribution that fits the MAX null distribution. These two
parameters, and , can be independently defined, and can be
non-integer even if is an integer. We illustrate these two parameters
using the example of MAX2 and MAX3 statistics in genetic case-control studies.
We speculate that is related to the amount of ambiguity of the model
inferred by the test. In the case-control genetic association, tests with low
(e.g. ) are able to provide definitive information about the disease
model, as versus tests with high (e.g. ) that are completely uncertain
about the disease model. Similar to Heisenberg's uncertain principle, the
ability to infer disease model and the ability to detect significant
association may not be simultaneously optimized, and seems to measure the
level of their balance
Corrections to scaling in multicomponent polymer solutions
We calculate the correction-to-scaling exponent that characterizes
the approach to the scaling limit in multicomponent polymer solutions. A direct
Monte Carlo determination of in a system of interacting
self-avoiding walks gives . A field-theory analysis based
on five- and six-loop perturbative series leads to . We
also verify the renormalization-group predictions for the scaling behavior
close to the ideal-mixing point.Comment: 21 page
Consistent coarse-graining strategy for polymer solutions in the thermal crossover from Good to Theta solvent
We extend our previously developed coarse-graining strategy for linear
polymers with a tunable number n of effective atoms (blobs) per chain [D'Adamo
et al., J. Chem. Phys. 137, 4901 (2012)] to polymer systems in thermal
crossover between the good-solvent and the Theta regimes. We consider the
thermal crossover in the region in which tricritical effects can be neglected,
i.e. not too close to the Theta point, for a wide range of chain volume
fractions Phi=c/c* (c* is the overlap concentration), up to Phi=30. Scaling
crossover functions for global properties of the solution are obtained by
Monte-Carlo simulations of the Domb-Joyce model. They provide the input data to
develop a minimal coarse-grained model with four blobs per chain. As in the
good-solvent case, the coarse-grained model potentials are derived at zero
density, thus avoiding the inconsistencies related to the use of
state-dependent potentials. We find that the coarse-grained model reproduces
the properties of the underlying system up to some reduced density which
increases when lowering the temperature towards the Theta state. Close to the
lower-temperature crossover boundary, the tetramer model is accurate at least
up to Phi<10, while near the good-solvent regime reasonably accurate results
are obtained up to Phi<2. The density region in which the coarse-grained model
is predictive can be enlarged by developing coarse-grained models with more
blobs per chain. We extend the strategy used in the good-solvent case to the
crossover regime. This requires a proper treatment of the length rescalings as
before, but also a proper temperature redefinition as the number of blobs is
increased. The case n=10 is investigated. Comparison with full-monomer results
shows that the density region in which accurate predictions can be obtained is
significantly wider than that corresponding to the n=4 case.Comment: 21 pages, 14 figure
High-Precision Entropy Values for Spanning Trees in Lattices
Shrock and Wu have given numerical values for the exponential growth rate of
the number of spanning trees in Euclidean lattices. We give a new technique for
numerical evaluation that gives much more precise values, together with
rigorous bounds on the accuracy. In particular, the new values resolve one of
their questions.Comment: 7 pages. Revision mentions alternative approach. Title changed
slightly. 2nd revision corrects first displayed equatio
Monte Carlo Procedure for Protein Design
A new method for sequence optimization in protein models is presented. The
approach, which has inherited its basic philosophy from recent work by Deutsch
and Kurosky [Phys. Rev. Lett. 76, 323 (1996)] by maximizing conditional
probabilities rather than minimizing energy functions, is based upon a novel
and very efficient multisequence Monte Carlo scheme. By construction, the
method ensures that the designed sequences represent good folders
thermodynamically. A bootstrap procedure for the sequence space search is
devised making very large chains feasible. The algorithm is successfully
explored on the two-dimensional HP model with chain lengths N=16, 18 and 32.Comment: 7 pages LaTeX, 4 Postscript figures; minor change
Spanning Trees and bootstrap reliability estimation in correlation based networks
We introduce a new technique to associate a spanning tree to the average
linkage cluster analysis. We term this tree as the Average Linkage Minimum
Spanning Tree. We also introduce a technique to associate a value of
reliability to links of correlation based graphs by using bootstrap replicas of
data. Both techniques are applied to the portfolio of the 300 most capitalized
stocks traded at New York Stock Exchange during the time period 2001-2003. We
show that the Average Linkage Minimum Spanning Tree recognizes economic sectors
and sub-sectors as communities in the network slightly better than the Minimum
Spanning Tree does. We also show that the average reliability of links in the
Minimum Spanning Tree is slightly greater than the average reliability of links
in the Average Linkage Minimum Spanning Tree.Comment: 17 pages, 3 figure
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