771 research outputs found
In-Degree and PageRank of Web pages: Why do they follow similar power laws?
The PageRank is a popularity measure designed by Google to rank Web pages.
Experiments confirm that the PageRank obeys a `power law' with the same
exponent as the In-Degree. This paper presents a novel mathematical model that
explains this phenomenon. The relation between the PageRank and In-Degree is
modelled through a stochastic equation, which is inspired by the original
definition of the PageRank, and is analogous to the well-known distributional
identity for the busy period in the M/G/1 queue. Further, we employ the theory
of regular variation and Tauberian theorems to analytically prove that the tail
behavior of the PageRank and the In-Degree differ only by a multiplicative
factor, for which we derive a closed-form expression. Our analytical results
are in good agreement with experimental data.Comment: 20 pages, 3 figures; typos added; reference adde
The frustrated Brownian motion of nonlocal solitary waves
We investigate the evolution of solitary waves in a nonlocal medium in the
presence of disorder. By using a perturbational approach, we show that an
increasing degree of nonlocality may largely hamper the Brownian motion of
self-trapped wave-packets. The result is valid for any kind of nonlocality and
in the presence of non-paraxial effects. Analytical predictions are compared
with numerical simulations based on stochastic partial differential equationComment: 4 pages, 3 figures
Complex light: Dynamic phase transitions of a light beam in a nonlinear non-local disordered medium
The dynamics of several light filaments (spatial optical solitons)
propagating in an optically nonlinear and non-local random medium is
investigated using the paradigms of the physics of complexity. Cluster
formation is interpreted as a dynamic phase transition. A connection with the
random matrices approach for explaining the vibrational spectra of an ensemble
of solitons is pointed out. General arguments based on a Brownian dynamics
model are validated by the numerical simulation of a stochastic partial
differential equation system. The results are also relevant for Bose condensed
gases and plasma physics.Comment: 11 pages, 20 figures. Small revisions, added a referenc
Remarks on the Central Limit Theorem for Non-Convex Bodies
In this note, we study possible extensions of the Central Limit Theorem for
non-convex bodies. First, we prove a Berry-Esseen type theorem for a certain
class of unconditional bodies that are not necessarily convex. Then, we
consider a widely-known class of non-convex bodies, the so-called p-convex
bodies, and construct a counter-example for this class
PageRank in scale-free random graphs
We analyze the distribution of PageRank on a directed configuration model and
show that as the size of the graph grows to infinity it can be closely
approximated by the PageRank of the root node of an appropriately constructed
tree. This tree approximation is in turn related to the solution of a linear
stochastic fixed point equation that has been thoroughly studied in the recent
literature
Solitons in nonlocal nonlinear media: exact results
We investigate the propagation of one-dimensional bright and dark spatial
solitons in a nonlocal Kerr-like media, in which the nonlocality is of general
form. We find an exact analytical solution to the nonlinear propagation
equation in the case of weak nonlocality. We study the properties of these
solitons and show their stability.Comment: 9 figures, submitted to Phys. Rev.
Route to nonlocality and observation of accessible solitons
We develop a general theory of spatial solitons in a liquid crystalline
medium exhibiting a nonlinearity with an arbitrary degree of effective
nonlocality. The model accounts the observability of "accessible solitons" and
establishes an important link with parametric solitons.Comment: 4 pages, 2 figure
Learning the 3D fauna of the web
Learning 3D models of all animals in nature requires
massively scaling up existing solutions. With this ultimate
goal in mind, we develop 3D-Fauna, an approach that
learns a pan-category deformable 3D animal model for
more than 100 animal species jointly. One crucial bottleneck of modeling animals is the limited availability of training data, which we overcome by learning our model from
2D Internet images. We show that prior approaches, which
are category-specific, fail to generalize to rare species with
limited training images. We address this challenge by introducing the Semantic Bank of Skinned Models (SBSM),
which automatically discovers a small set of base animal
shapes by combining geometric inductive priors with semantic knowledge implicitly captured by an off-the-shelf
self-supervised feature extractor. To train such a model,
we also contribute a new large-scale dataset of diverse animal species. At inference time, given a single image of any
quadruped animal, our model reconstructs an articulated
3D mesh in a feed-forward manner in seconds
Almost-Euclidean subspaces of via tensor products: a simple approach to randomness reduction
It has been known since 1970's that the N-dimensional -space contains
nearly Euclidean subspaces whose dimension is . However, proofs of
existence of such subspaces were probabilistic, hence non-constructive, which
made the results not-quite-suitable for subsequently discovered applications to
high-dimensional nearest neighbor search, error-correcting codes over the
reals, compressive sensing and other computational problems. In this paper we
present a "low-tech" scheme which, for any , allows to exhibit nearly
Euclidean -dimensional subspaces of while using only
random bits. Our results extend and complement (particularly) recent work
by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1)
simplicity (we use only tensor products) and (2) yielding "almost Euclidean"
subspaces with arbitrarily small distortions.Comment: 11 pages; title change, abstract and references added, other minor
change
Smooth analysis of the condition number and the least singular value
Let \a be a complex random variable with mean zero and bounded variance.
Let be the random matrix of size whose entries are iid copies of
\a and be a fixed matrix of the same size. The goal of this paper is to
give a general estimate for the condition number and least singular value of
the matrix , generalizing an earlier result of Spielman and Teng for
the case when \a is gaussian.
Our investigation reveals an interesting fact that the "core" matrix does
play a role on tail bounds for the least singular value of . This
does not occur in Spielman-Teng studies when \a is gaussian.
Consequently, our general estimate involves the norm .
In the special case when is relatively small, this estimate is nearly
optimal and extends or refines existing results.Comment: 20 pages. An erratum to the published version has been adde
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