89,057 research outputs found
A stochastic model for the evolution of the Web
Recently several authors have proposed stochastic models of the growth of the Web graph that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the "rich get richer" phenomenon. However, these models fail to explain several distributions arising from empirical results, due to the fact that the predicted exponent is not consistent with the data. To address this problem, we extend the evolutionary model of the Web graph by including a non-preferential component, and we view the stochastic process in terms of an urn transfer model. By making this extension, we can now explain a wider variety of empirically discovered power-law distributions provided the exponent is greater than two. These include: the distribution of incoming links, the distribution of outgoing links, the distribution of pages in a Web site and the distribution of visitors to a Web site. A by-product of our results is a formal proof of the convergence of the standard stochastic model (first proposed by Simon)
A stochastic model for the evolution of the web allowing link deletion
Recently several authors have proposed stochastic evolutionary models for the growth of the web graph and other networks that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the ``rich get richer'' phenomenon. We present a generalisation of the basic model by allowing deletion of individual links and show that it also gives rise to a power-law distribution. We derive the mean-field equations for this stochastic model and show that by examining a snapshot of the distribution at the steady state of the model, we are able to tell whether any link deletion has taken place and estimate the link deletion probability. Our model enables us to gain some insight into the distribution of inlinks in the web graph, in particular it suggests a power-law exponent of approximately 2.15 rather than the widely published exponent of 2.1
Laplacian growth as one-dimensional turbulence
A new model of Laplacian stochastic growth is formulated using conformal
mappings. The model describes two growth regimes, stable and turbulent,
separated by a sharp phase transition. The first few Fourier components of the
mapping define the web, an envelope of the cluster. The web is used to study
the transition and the dynamics of large-scale features of the cluster
characterized by evolution from macro- to micro-scales. Also, we derive scaling
laws for the cluster size.Comment: 4 pages, RevTex, 4 figure
Froth across the Universe Dynamics and Stochastic Geometry of the Cosmic Foam
A review on the properties and dynamical origin and nature of the cosmic
foam, the tenuous space-filling frothy network permeating the interior of the
Universe. We discuss the properties of this striking and intriguing pattern,
describing its observational appearance, and seeking to elucidate its dynamical
origin and nature. An extensive discussion on the gravitational formation and
dynamical evolution of weblike patterns puts particular emphasis on the
formative role of the generic anisotropy of the cosmic gravitational force
fields. These tidal fields play an essential role in shaping the pattern of the
large scale cosmic matter distribution. Special attention is put on a
geometrical and stochastic framework for a systematic evaluation of its fossil
information content on the cosmic structure formation process. Its distinct
geometric character and the stochastic nature provides the cosmic web with some
unique and at first unexpected properties. The implications for galaxy
clustering are discussed on the basis of its relevant branch of mathematics,
stochastic geometry. Central within this context are Voronoi tessellations,
which have been found to represent a surprisingly versatile model for spatial
cellular distributions.Comment: Invited review, Proceedings 2nd Hellenic Cosmology Workshop, eds. M.
Plionis, S. Cotsakis, I. Georgantopoulos, Kluwer, 153 pages, 56 figures. Full
resolution version available at
http://www.astro.rug.nl/~weygaert/tim1publication/weyhellas2001.ps.g
Little Ado about Everything: CDM, a Cosmological Model with Fluctuation-driven Acceleration at Late Times
[abridged] We propose a model of the Universe (dubbed CDM) featuring a
stochastic evolution of the cosmological quantities, that is meant to render
small deviations from homogeneity/isotropy on scales of Mpc at
late cosmic times, associated to the emergence of the cosmic web. Specifically,
we prescribe that the behavior of the matter/radiation energy densities in
different patches of the Universe with such a size can be effectively described
by a stochastic version of the mass-energy evolution equation. The latter
includes an appropriate noise term that statistically accounts for local
fluctuations due to inhomogeneities, anisotropic stresses and matter flows. The
evolution of the different patches as a function of cosmic time is rendered via
the diverse realizations of the noise term; meanwhile, at any given cosmic
time, sampling the ensemble of patches will originate a nontrivial spatial
distribution of the cosmological quantities. The overall behavior of the
Universe will be obtained by averaging over the patch ensemble. We assume a
physically reasonable parameterization of the noise term, gauging it against a
wealth of cosmological datasets. We find that, with respect to standard
CDM, the ensemble-averaged cosmic dynamics in the CDM model is
substantially altered in three main respects: (i) an accelerated expansion is
enforced at late cosmic times without the need for any additional exotic
component (e.g., dark energy); (ii) the spatial curvature can stay small even
in a low-density Universe; (iii) matter can acquire an effective negative
pressure at late times. We provide predictions for the variance of the
cosmological quantities among different patches of the Universe at late cosmic
times. Finally, we show that in CDM the Hubble-tension is solved, and the
cosmic coincidence problem is relieved without invoking the anthropic
principle.Comment: 28 pages, 7 figures, typos corrected. Accepted by Ap
A stochastic evolutionary model exhibiting power-law behaviour with an exponential cutoff
Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the “rich get richer” phenomenon. Despite the generality of the proposed stochastic models, there are still some unexplained phenomena, which may arise due to the limited size of networks such as protein and e-mail networks. Such networks may in fact exhibit an exponential cutoff in the power-law scaling, although this cutoff may only be observable in the tail of the distribution for extremely large networks. We propose a modification of the basic stochastic evolutionary model, so that after a node is chosen preferentially, say according to the number of its inlinks, there is a small probability that this node will be discarded. We show that as a result of this modification, by viewing the stochastic process in terms of an urn transfer model, we obtain a power-law distribution with an exponential cutoff. Unlike many other models, the current model can capture instances where the exponent of the distribution is less than or equal to two. As a proof of concept, we demonstrate the consistency of our model by analysing a yeast protein interaction network, the distribution of which is known to follow a power law with an exponential cutoff
The Fractal Geometry of the Cosmic Web and its Formation
The cosmic web structure is studied with the concepts and methods of fractal
geometry, employing the adhesion model of cosmological dynamics as a basic
reference. The structures of matter clusters and cosmic voids in cosmological
N-body simulations or the Sloan Digital Sky Survey are elucidated by means of
multifractal geometry. A non-lacunar multifractal geometry can encompass three
fundamental descriptions of the cosmic structure, namely, the web structure,
hierarchical clustering, and halo distributions. Furthermore, it explains our
present knowledge of cosmic voids. In this way, a unified theory of the
large-scale structure of the universe seems to emerge. The multifractal
spectrum that we obtain significantly differs from the one of the adhesion
model and conforms better to the laws of gravity. The formation of the cosmic
web is best modeled as a type of turbulent dynamics, generalizing the known
methods of Burgers turbulence.Comment: 35 pages, 8 figures; corrected typos, added references; further
discussion of cosmic voids; accepted by Advances in Astronom
A model for collaboration networks giving rise to a power law distribution with exponential cutoff
Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the ``rich get richer'' phenomenon. Despite the generality of the proposed stochastic models, there are still some unexplained phenomena, which may arise due to the limited size of networks such as protein, e-mail, actor and collaboration networks. Such networks may in fact exhibit an exponential cutoff in the power-law scaling, although this cutoff may only be observable in the tail of the distribution for extremely large networks. We propose a modification of the basic stochastic evolutionary model, so that after a node is chosen preferentially, say according to the number of its inlinks, there is a small probability that this node will become inactive. We show that as a result of this modification, by viewing the stochastic process in terms of an urn transfer model, we obtain a power-law distribution with an exponential cutoff. Unlike many other models, the current model can capture instances where the exponent of the distribution is less than or equal to two. As a proof of concept, we demonstrate the consistency of our model empirically by analysing the Mathematical Research collaboration network, the distribution of which is known to follow a power law with an exponential cutoff
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