245 research outputs found
Effect of the accelerating growth of communications networks on their structure
Motivated by data on the evolution of the Internet and World Wide Web we
consider scenarios of self-organization of the nonlinearly growing networks
into free-scale structures. We find that the accelerating growth of the
networks establishes their structure. For the growing networks with
preferential linking and increasing density of links, two scenarios are
possible. In one of them, the value of the exponent of the
connectivity distribution is between 3/2 and 2. In the other, and
the distribution is necessarily non-stationary.Comment: 4 pages revtex, 3 figure
Broadening of a nonequilibrium phase transition by extended structural defects
We study the effects of quenched extended impurities on nonequilibrium phase
transitions in the directed percolation universality class. We show that these
impurities have a dramatic effect: they completely destroy the sharp phase
transition by smearing. This is caused by rare strongly coupled spatial regions
which can undergo the phase transition independently from the bulk system. We
use extremal statistics to determine the stationary state as well as the
dynamics in the tail of the smeared transition, and we illustrate the results
by computer simulations.Comment: 4 pages, 4 eps figures, final version as publishe
Generic scale of the "scale-free" growing networks
We show that the connectivity distributions of scale-free growing
networks ( is the network size) have the generic scale -- the cut-off at
. The scaling exponent is related to the exponent
of the connectivity distribution, . We propose the
simplest model of scale-free growing networks and obtain the exact form of its
connectivity distribution for any size of the network. We demonstrate that the
trace of the initial conditions -- a hump at --
may be found for any network size. We also show that there exists a natural
boundary for the observation of the scale-free networks and explain why so few
scale-free networks are observed in Nature.Comment: 4 pages revtex, 3 figure
Evolving Networks with Multi-species Nodes and Spread in the Number of Initial Links
We consider models for growing networks incorporating two effects not
previously considered: (i) different species of nodes, with each species having
different properties (such as different attachment probabilities to other node
species); and (ii) when a new node is born, its number of links to old nodes is
random with a given probability distribution. Our numerical simulations show
good agreement with analytic solutions. As an application of our model, we
investigate the movie-actor network with movies considered as nodes and actors
as links.Comment: 5 pages, 5 figures, submitted to PR
Evolving networks with disadvantaged long-range connections
We consider a growing network, whose growth algorithm is based on the
preferential attachment typical for scale-free constructions, but where the
long-range bonds are disadvantaged. Thus, the probability to get connected to a
site at distance is proportional to , where is a
tunable parameter of the model. We show that the properties of the networks
grown with are close to those of the genuine scale-free
construction, while for the structure of the network is vastly
different. Thus, in this regime, the node degree distribution is no more a
power law, and it is well-represented by a stretched exponential. On the other
hand, the small-world property of the growing networks is preserved at all
values of .Comment: REVTeX, 6 pages, 5 figure
Pseudofractal Scale-free Web
We find that scale-free random networks are excellently modeled by a
deterministic graph. This graph has a discrete degree distribution (degree is
the number of connections of a vertex) which is characterized by a power-law
with exponent . Properties of this simple structure are
surprisingly close to those of growing random scale-free networks with
in the most interesting region, between 2 and 3. We succeed to find exactly and
numerically with high precision all main characteristics of the graph. In
particular, we obtain the exact shortest-path-length distribution. For the
large network () the distribution tends to a Gaussian of width
centered at . We show that the
eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail
with exponent .Comment: 5 pages, 3 figure
The effect of aging on network structure
In network evolution, the effect of aging is universal: in scientific
collaboration network, scientists have a finite time span of being active; in
movie actors network, once popular stars are retiring from stage; devices on
the Internet may become outmoded with techniques developing so rapidly. Here we
find in citation networks that this effect can be represented by an exponential
decay factor, , where is the node age, while other
evolving networks (the Internet for instance) may have different types of
aging, for example, a power-law decay factor, which is also studied and
compared. It has been found that as soon as such a factor is introduced to the
Barabasi-Albert Scale-Free model, the network will be significantly
transformed. The network will be clustered even with infinitely large size, and
the clustering coefficient varies greatly with the intensity of the aging
effect, i.e. it increases linearly with for small values of
and decays exponentially for large values of . At the same time, the
aging effect may also result in a hierarchical structure and a disassortative
degree-degree correlation. Generally the aging effect will increase the average
distance between nodes, but the result depends on the type of the decay factor.
The network appears like a one-dimensional chain when exponential decay is
chosen, but with power-law decay, a transformation process is observed, i.e.,
from a small-world network to a hypercubic lattice, and to a one-dimensional
chain finally. The disparities observed for different choices of the decay
factor, in clustering, average node distance and probably other aspects not yet
identified, are believed to bear significant meaning on empirical data
acquisition.Comment: 8 pages, 9 figures,V2, accepted for publication in Phys. Rev.
Scaling exponents and clustering coefficients of a growing random network
The statistical property of a growing scale-free network is studied based on
an earlier model proposed by Krapivsky, Rodgers, and Redner [Phys. Rev. Lett.
86, 5401 (2001)], with the additional constraints of forbidden of
self-connection and multiple links of the same direction between any two nodes.
Scaling exponents in the range of 1-2 are obtained through Monte Carlo
simulations and various clustering coefficients are calculated, one of which,
, is of order , indicating the network resembles a
small-world. The out-degree distribution has an exponential cut-off for large
out-degree.Comment: six pages, including 5 figures, RevTex 4 forma
Tree Networks with Causal Structure
Geometry of networks endowed with a causal structure is discussed using the
conventional framework of equilibrium statistical mechanics. The popular
growing network models appear as particular causal models. We focus on a class
of tree graphs, an analytically solvable case. General formulae are derived,
describing the degree distribution, the ancestor-descendant correlation and the
probability a randomly chosen node lives at a given geodesic distance from the
root. It is shown that the Hausdorff dimension of the causal networks is
generically infinite, in contrast to the maximally random trees, where it is
generically finite.Comment: 9 pages, 2-column revtex format, 1 eps figure, misprints correcte
Principles of statistical mechanics of random networks
We develop a statistical mechanics approach for random networks with
uncorrelated vertices. We construct equilibrium statistical ensembles of such
networks and obtain their partition functions and main characteristics. We find
simple dynamical construction procedures that produce equilibrium uncorrelated
random graphs with an arbitrary degree distribution. In particular, we show
that in equilibrium uncorrelated networks, fat-tailed degree distributions may
exist only starting from some critical average number of connections of a
vertex, in a phase with a condensate of edges.Comment: 14 pages, an extended versio
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