14,150 research outputs found
Deterministic Scale-Free Networks
Scale-free networks are abundant in nature and society, describing such
diverse systems as the world wide web, the web of human sexual contacts, or the
chemical network of a cell. All models used to generate a scale-free topology
are stochastic, that is they create networks in which the nodes appear to be
randomly connected to each other. Here we propose a simple model that generates
scale-free networks in a deterministic fashion. We solve exactly the model,
showing that the tail of the degree distribution follows a power law
Quasistatic Scale-free Networks
A network is formed using the sites of an one-dimensional lattice in the
shape of a ring as nodes and each node with the initial degree .
links are then introduced to this network, each link starts from a distinct
node, the other end being connected to any other node with degree randomly
selected with an attachment probability proportional to . Tuning
the control parameter we observe a transition where the average degree
of the largest node changes its variation from to
at a specific transition point of . The network is scale-free i.e.,
the nodal degree distribution has a power law decay for .Comment: 4 pages, 5 figure
Excitable Scale Free Networks
When a simple excitable system is continuously stimulated by a Poissonian
external source, the response function (mean activity versus stimulus rate)
generally shows a linear saturating shape. This is experimentally verified in
some classes of sensory neurons, which accordingly present a small dynamic
range (defined as the interval of stimulus intensity which can be appropriately
coded by the mean activity of the excitable element), usually about one or two
decades only. The brain, on the other hand, can handle a significantly broader
range of stimulus intensity, and a collective phenomenon involving the
interaction among excitable neurons has been suggested to account for the
enhancement of the dynamic range. Since the role of the pattern of such
interactions is still unclear, here we investigate the performance of a
scale-free (SF) network topology in this dynamic range problem. Specifically,
we study the transfer function of disordered SF networks of excitable
Greenberg-Hastings cellular automata. We observe that the dynamic range is
maximum when the coupling among the elements is critical, corroborating a
general reasoning recently proposed. Although the maximum dynamic range yielded
by general SF networks is slightly worse than that of random networks, for
special SF networks which lack loops the enhancement of the dynamic range can
be dramatic, reaching nearly five decades. In order to understand the role of
loops on the transfer function we propose a simple model in which the density
of loops in the network can be gradually increased, and show that this is
accompanied by a gradual decrease of dynamic range.Comment: 6 pages, 4 figure
Classification of scale-free networks
While the emergence of a power law degree distribution in complex networks is
intriguing, the degree exponent is not universal. Here we show that the
betweenness centrality displays a power-law distribution with an exponent \eta
which is robust and use it to classify the scale-free networks. We have
observed two universality classes with \eta \approx 2.2(1) and 2.0,
respectively. Real world networks for the former are the protein interaction
networks, the metabolic networks for eukaryotes and bacteria, and the
co-authorship network, and those for the latter one are the Internet, the
world-wide web, and the metabolic networks for archaea. Distinct features of
the mass-distance relation, generic topology of geodesics and resilience under
attack of the two classes are identified. Various model networks also belong to
either of the two classes while their degree exponents are tunable.Comment: 6 Pages, 6 Figures, 1 tabl
Scale-free networks without growth
In this letter, we proposed an ungrowing scale-free network model, wherein
the total number of nodes is fixed and the evolution of network structure is
driven by a rewiring process only. In spite of the idiographic form of , by
using a two-order master equation, we obtain the analytic solution of degree
distribution in stable state of the network evolution under the condition that
the selection probability in rewiring process only depends on nodes'
degrees. A particular kind of the present networks with linearly correlated
with degree is studied in detail. The analysis and simulations show that the
degree distributions of these networks can varying from the Possion form to the
power-law form with the decrease of a free parameter , indicating the
growth may not be a necessary condition of the self-organizaton of a network in
a scale-free structure.Comment: 4 pages and 3 figure
Nucleation in scale-free networks
We have studied nucleation dynamics of the Ising model in scale-free networks
with degree distribution by using forward flux sampling
method, focusing on how the network topology would influence the nucleation
rate and pathway. For homogeneous nucleation, the new phase clusters grow from
those nodes with smaller degree, while the cluster sizes follow a power-law
distribution. Interestingly, we find that the nucleation rate decays
exponentially with the network size , and accordingly the critical nucleus
size increases linearly with , implying that homogeneous nucleation is not
relevant in the thermodynamic limit. These observations are robust to the
change of and also present in random networks. In addition, we have
also studied the dynamics of heterogeneous nucleation, wherein impurities
are initially added, either to randomly selected nodes or to targeted ones with
largest degrees. We find that targeted impurities can enhance the nucleation
rate much more sharply than random ones. Moreover, scales as and for targeted and
random impurities, respectively. A simple mean field analysis is also present
to qualitatively illustrate above simulation results.Comment: 7 pages, 5 figure
Scale-Free Networks are Ultrasmall
We study the diameter, or the mean distance between sites, in a scale-free
network, having N sites and degree distribution p(k) ~ k^-a, i.e. the
probability of having k links outgoing from a site. In contrast to the diameter
of regular random networks or small world networks which is known to be d ~
lnN, we show, using analytical arguments, that scale free networks with 2<a<3
have a much smaller diameter, behaving as d ~ lnlnN. For a=3, our analysis
yields d ~ lnN/lnlnN, as obtained by Bollobas and Riordan, while for a>3, d ~
lnN. We also show that, for any a>2, one can construct a deterministic scale
free network with d ~ lnlnN, and this construction yields the lowest possible
diameter.Comment: Latex, 4 pages, 2 eps figures, small corrections, added explanation
Highly clustered scale-free networks
We propose a model for growing networks based on a finite memory of the
nodes. The model shows stylized features of real-world networks: power law
distribution of degree, linear preferential attachment of new links and a
negative correlation between the age of a node and its link attachment rate.
Notably, the degree distribution is conserved even though only the most
recently grown part of the network is considered. This feature is relevant
because real-world networks truncated in the same way exhibit a power-law
distribution in the degree. As the network grows, the clustering reaches an
asymptotic value larger than for regular lattices of the same average
connectivity. These high-clustering scale-free networks indicate that memory
effects could be crucial for a correct description of the dynamics of growing
networks.Comment: 6 pages, 4 figure
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