140,293 research outputs found
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
Autocatalytic Sets and the Growth of Complexity in an Evolutionary Model
A model of interacting species is considered with two types of dynamical
variables. The fast variables are the populations of the species and slow
variables the links of a directed graph that defines the catalytic interactions
among them. The graph evolves via mutations of the least fit species. Starting
from a sparse random graph, we find that an autocatalytic set (ACS) inevitably
appears and triggers a cascade of exponentially increasing connectivity until
it spans the whole graph. The connectivity subsequently saturates in a
statistical steady state. The time scales for the appearance of an ACS in the
graph and its growth have a power law dependence on and the catalytic
probability. At the end of the growth period the network is highly non-random,
being localized on an exponentially small region of graph space for large .Comment: 13 pages REVTEX (including figures), 4 Postscript figure
Link analysis of the Web
Inlinks to a page of the web are often used as indicators of the authority of the page, because it is commonly felt that if a page is linked from many other pages it is worth looking at. According to a widely accepted model, the inlinks follow a power law. Outlinking analysis, on the contrary, has received less attention. Nevertheless, the outlink structure should be included in the description of the web graph and used for designing and testing the algorithms for the real web. The power law has been proposed also for the outlinks, but the experimental observation has shown that the distribution of the pages with a low number of outlinks is not well modeled by the power law. In this paper we propose a linking model based on a preferential attachment strategy and a uniform attachment strategy. Similar models have already been considered in the literature. The substantial novelty of our approach on the previous ones concerns the approximation technique applied to the steady state solution. With this technique the model appears to be well suited for describing both inlink and outlink distributions. The experimentation on two subsets of the real web shows that the two attachment strategies play a different role in the inlink and the outlink case
Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional Clifford circuits
Entanglement transitions in quantum dynamics present a novel class of phase transitions in nonequilibrium systems. When a many-body quantum system undergoes unitary evolution interspersed with monitored random measurements, the steady state can exhibit a phase transition between volume- and area-law entanglement. There is a correspondence between measurement-induced transitions in nonunitary quantum circuits in
d
spatial dimensions and classical statistical mechanical models in
d
+
1
dimensions. In certain limits these models map to percolation, but there is analytical and numerical evidence to suggest that away from these limits the universality class should generically be distinct from percolation. Intriguingly, despite these arguments, numerics on
1
+
1
D
qubit circuits give bulk exponents which are nonetheless close to those of 2D percolation, with some possible differences in surface behavior. In the first part of this work we explore the critical properties of
2
+
1
D
Clifford circuits. In the bulk, we find many properties suggested by the percolation picture, including several matching bulk exponents, and an inverse power law for the critical entanglement growth,
S
(
t
,
L
)
∼
L
(
1
−
a
/
t
)
, which saturates to an area law. We then utilize a graph-state-based algorithm to analyze in
1
+
1
D
and
2
+
1
D
the critical properties of entanglement clusters in the steady state. We show that in a model with a simple geometric map to percolation—the projective transverse field Ising model—these entanglement clusters are governed by percolation surface exponents. However, in the Clifford models we find large deviations in the cluster exponents from those of surface percolation, highlighting the breakdown of any possible geometric map to percolation. Given the evidence for deviations from the percolation universality class, our results raise the question of why nonetheless many bulk properties behave similarly to those of percolation
Power-law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species
We present determinant criteria for the preclusion of non-degenerate multiple
steady states in networks of interacting species. A network is modeled as a
system of ordinary differential equations in which the form of the species
formation rate function is restricted by the reactions of the network and how
the species influence each reaction. We characterize families of so-called
power-law kinetics for which the associated species formation rate function is
injective within each stoichiometric class and thus the network cannot exhibit
multistationarity. The criterion for power-law kinetics is derived from the
determinant of the Jacobian of the species formation rate function. Using this
characterization we further derive similar determinant criteria applicable to
general sets of kinetics. The criteria are conceptually simple, computationally
tractable and easily implemented. Our approach embraces and extends previous
work on multistationarity, such as work in relation to chemical reaction
networks with dynamics defined by mass-action or non-catalytic kinetics, and
also work based on graphical analysis of the interaction graph associated to
the system. Further, we interpret the criteria in terms of circuits in the
so-called DSR-graphComment: To appear in SIAM Journal on Applied Dynamical System
Localization transition, Lifschitz tails and rare-region effects in network models
Effects of heterogeneity in the suspected-infected-susceptible model on
networks are investigated using quenched mean-field theory. The emergence of
localization is described by the distributions of the inverse participation
ratio and compared with the rare-region effects appearing in simulations and in
the Lifschitz tails. The latter, in the linear approximation, is related to the
spectral density of the Laplacian matrix and to the time dependent order
parameter. I show that these approximations indicate correctly Griffiths Phases
both on regular one-dimensional lattices and on small world networks exhibiting
purely topological disorder. I discuss the localization transition that occurs
on scale-free networks at degree exponent.Comment: 9 pages, 9 figures, accepted version in PR
Free zero-range processes on networks
A free zero-range process (FRZP) is a simple stochastic process describing
the dynamics of a gas of particles hopping between neighboring nodes of a
network. We discuss three different cases of increasing complexity: (a) FZRP on
a rigid geometry where the network is fixed during the process, (b) FZRP on a
random graph chosen from a given ensemble of networks, (c) FZRP on a dynamical
network whose topology continuously changes during the process in a way which
depends on the current distribution of particles. The case (a) provides a very
simple realization of the phenomenon of condensation which manifests as the
appearance of a condensate of particles on the node with maximal degree. The
case (b) is very interesting since the averaging over typical ensembles of
graphs acts as a kind of homogenization of the system which makes all nodes
identical from the point of view of the FZRP. In the case (c), the distribution
of particles and the dynamics of network are coupled to each other. The
strength of this coupling depends on the ratio of two time scales: for changes
of the topology and of the FZRP. We will discuss a specific example of that
type of interaction and show that it leads to an interesting phase diagram.Comment: 11 pages, 4 figures, to appear in Proceedings of SPIE Symposium
"Fluctuations and Noise 2007", Florence, 20-24 May 200
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