35 research outputs found
Fitness-based network growth with dynamic feedback
This article is a preprint of a paper that is currently under review with Physical Review E.We study a class of network growth models in which the choice of attachment by new nodes is governed by intrinsic attractiveness, or tness, of the existing nodes. The key feature of the models is a feedback mechanism whereby the distribution from which fitnesses of new nodes are drawn is dynamically updated to account for the evolving degree distribution. It is shown that in the case of linear mapping between fitnesses and degrees, the models lead to tunable stationary powerlaw degree distribution, while in the non-linear case the distributions converge to the stretched exponential form
Network growth model with intrinsic vertex fitness
© 2013 American Physical SocietyWe study a class of network growth models with attachment rules governed by intrinsic node fitness. Both the individual node degree distribution and the degree correlation properties of the network are obtained as functions of the network growth rules. We also find analytical solutions to the inverse, design, problems of matching the growth rules to the required (e.g., power-law) node degree distribution and more generally to the required degree correlation function. We find that the design problems do not always have solutions. Among the specific conditions on the existence of solutions to the design problems is the requirement that the node degree distribution has to be broader than a certain threshold and the fact that factorizability of the correlation functions requires singular distributions of the node fitnesses. More generally, the restrictions on the input distributions and correlations that ensure solvability of the design problems are expressed in terms of the analytical properties of their generating functions
Crossover Between Universality Classes in the Statistics of Rare Events in Disordered Conductors
The crossover from orthogonal to the unitary universality classes in the
distribution of the anomalously localized states (ALS) in two-dimensional
disordered conductors is traced as a function of magnetic field. We demonstrate
that the microscopic origin of the crossover is the change in the symmetry of
the underlying disorder configurations, that are responsible for ALS. These
disorder configurations are of weak magnitude (compared to the Fermi energy)
and of small size (compared to the mean free path). We find their shape
explicitly by means of the direct optimal fluctuation method.Comment: 7 pages including 2 figure
A New Type of Intensity Correlation in Random Media
A monochromatic point source, embedded in a three-dimensional disordered
medium, is considered. The resulting intensity pattern exhibits a new type of
long-range correlations. The range of these correlations is infinite and their
magnitude, normalized to the average intensity, is of order , where
and are the wave number and the mean free path respectively.Comment: RevTeX, 8 pages, 3 figures, Accepted to Phys. Rev. Let
Universality of Parametric Spectral Correlations: Local versus Extended Perturbing Potentials
We explore the influence of an arbitrary external potential perturbation V on
the spectral properties of a weakly disordered conductor. In the framework of a
statistical field theory of a nonlinear sigma-model type we find, depending on
the range and the profile of the external perturbation, two qualitatively
different universal regimes of parametric spectral statistics (i.e.
cross-correlations between the spectra of Hamiltonians H and H+V). We identify
the translational invariance of the correlations in the space of Hamiltonians
as the key indicator of universality, and find the connection between the
coordinate system in this space which makes the translational invariance
manifest, and the physically measurable properties of the system. In
particular, in the case of localized perturbations, the latter turn out to be
the eigenphases of the scattering matrix for scattering off the perturbing
potential V. They also have a purely statistical interpretation in terms of the
moments of the level velocity distribution. Finally, on the basis of this
analysis, a set of results obtained recently by the authors using random matrix
theory methods is shown to be applicable to a much wider class of disordered
and chaotic structures.Comment: 16 pages, 7 eps figures (minor changes and reference [17] added
Nucleation of superconducting pairing states at mesoscopic scales at zero temperature
We find the spin polarized disordered Fermi liquids are unstable to the
nucleation of superconducting pairing states at mesoscopic scales even when
magnetic fields which polarize the spins are substantially higher than the
critical one. We study the probability of finding superconducting pairing
states at mesoscopic scales in this limit. We find that the distribution
function depends only on the film conductance. The typical length scale at
which pairing takes place is universal, and decreases when the magnetic field
is increased. The number density of these states determines the strength of the
random exchange interactions between mesoscopic pairing states.Comment: 11 pages, no figure
Quest for Rare Events in three-dimensional Mesoscopic Disordered Metals
The study reports on the first large statistics numerical experiment
searching for rare eigenstates of anomalously high amplitudes in
three-dimensional diffusive metallic conductors. Only a small fraction of a
huge number of investigated eigenfunctions generates the far asymptotic tail of
their amplitude distribution function. The relevance of the relationship
between disorder and spectral averaging, as well as of the quantum transport
properties of the investigated mesoscopic samples, for the numerical
exploration of eigenstate statistics is divulged. The quest provides exact
results to serve as a reference point in understanding the limits of
approximations employed in different analytical predictions, and thereby the
physics (quantum vs semiclassical) behind large deviations from the universal
predictions of random matrix theory.Comment: 5 pages, 3 embedded EPS figures, figure 3 replaced with new findings
on spectral vs disorder averagin
Statistics of Rare Events in Disordered Conductors
Asymptotic behavior of distribution functions of local quantities in
disordered conductors is studied in the weak disorder limit by means of an
optimal fluctuation method. It is argued that this method is more appropriate
for the study of seldom occurring events than the approaches based on nonlinear
-models because it is capable of correctly handling fluctuations of the
random potential with large amplitude as well as the short-scale structure of
the corresponding solutions of the Schr\"{o}dinger equation. For two- and
three-dimensional conductors new asymptotics of the distribution functions are
obtained which in some cases differ significantly from previously established
results.Comment: 17 pages, REVTeX 3.0 and 1 Postscript figur
Eigenstate Structure in Graphs and Disordered Lattices
We study wave function structure for quantum graphs in the chaotic and
disordered regime, using measures such as the wave function intensity
distribution and the inverse participation ratio. The result is much less
ergodicity than expected from random matrix theory, even though the spectral
statistics are in agreement with random matrix predictions. Instead, analytical
calculations based on short-time semiclassical behavior correctly describe the
eigenstate structure.Comment: 4 pages, including 2 figure