690 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
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
Percolation Critical Exponents in Scale-Free Networks
We study the behavior of scale-free networks, having connectivity
distribution P(k) k^-a, close to the percolation threshold. We show that for
networks with 3<a<4, known to undergo a transition at a finite threshold of
dilution, the critical exponents are different than the expected mean-field
values of regular percolation in infinite dimensions. Networks with 2<a<3
possess only a percolative phase. Nevertheless, we show that in this case
percolation critical exponents are well defined, near the limit of extreme
dilution (where all sites are removed), and that also then the exponents bear a
strong a-dependence. The regular mean-field values are recovered only for a>4.Comment: Latex, 4 page
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
The urban economy as a scale-free network
We present empirical evidence that land values are scale-free and introduce a
network model that reproduces the observations. The network approach to urban
modelling is based on the assumption that the market dynamics that generates
land values can be represented as a growing scale-free network. Our results
suggest that the network properties of trade between specialized activities
causes land values, and likely also other observables such as population, to be
power law distributed. In addition to being an attractive avenue for further
analytical inquiry, the network representation is also applicable to empirical
data and is thereby attractive for predictive modelling.Comment: Submitted to Phys. Rev. E. 7 pages, 3 figures. (Minor typos and
details fixed
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
Smallest small-world network
Efficiency in passage times is an important issue in designing networks, such
as transportation or computer networks. The small-world networks have
structures that yield high efficiency, while keeping the network highly
clustered. We show that among all networks with the small-world structure, the
most efficient ones have a single ``center'', from which all shortcuts are
connected to uniformly distributed nodes over the network. The networks with
several centers and a connected subnetwork of shortcuts are shown to be
``almost'' as efficient. Genetic-algorithm simulations further support our
results.Comment: 5 pages, 6 figures, REVTeX
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
Percolation in Directed Scale-Free Networks
Many complex networks in nature have directed links, a property that affects
the network's navigability and large-scale topology. Here we study the
percolation properties of such directed scale-free networks with correlated in-
and out-degree distributions. We derive a phase diagram that indicates the
existence of three regimes, determined by the values of the degree exponents.
In the first regime we regain the known directed percolation mean field
exponents. In contrast, the second and third regimes are characterized by
anomalous exponents, which we calculate analytically. In the third regime the
network is resilient to random dilution, i.e., the percolation threshold is
p_c->1.Comment: Latex, 5 pages, 2 fig
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
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