695 research outputs found
Growing Networks: Limit in-degree distribution for arbitrary out-degree one
We compute the stationary in-degree probability, , for a growing
network model with directed edges and arbitrary out-degree probability. In
particular, under preferential linking, we find that if the nodes have a light
tail (finite variance) out-degree distribution, then the corresponding
in-degree one behaves as . Moreover, for an out-degree distribution
with a scale invariant tail, , the corresponding
in-degree distribution has exactly the same asymptotic behavior only if
(infinite variance). Similar results are obtained when
attractiveness is included. We also present some results on descriptive
statistics measures %descriptive statistics such as the correlation between the
number of in-going links, , and outgoing links, , and the
conditional expectation of given , and we calculate these
measures for the WWW network. Finally, we present an application to the
scientific publications network. The results presented here can explain the
tail behavior of in/out-degree distribution observed in many real networks.Comment: 12 pages, 6 figures, v2 adds a section on descriptive statistics, an
analisis on www network, typos adde
Emergence of weight-topology correlations in complex scale-free networks
Different weighted scale-free networks show weights-topology correlations
indicated by the non linear scaling of the node strength with node
connectivity. In this paper we show that networks with and without
weight-topology correlations can emerge from the same simple growth dynamics of
the node connectivities and of the link weights. A weighted fitness network is
introduced in which both nodes and links are assigned intrinsic fitness. This
model can show a local dependence of the weight-topology correlations and can
undergo a phase transition to a state in which the network is dominated by few
links which acquire a finite fraction of the total weight of the network.Comment: (4 pages,3 figures
Geographical Coarsegraining of Complex Networks
We perform the renormalization-group-like numerical analysis of
geographically embedded complex networks on the two-dimensional square lattice.
At each step of coarsegraining procedure, the four vertices on each square box are merged to a single vertex, resulting in the coarsegrained
system of the smaller sizes. Repetition of the process leads to the observation
that the coarsegraining procedure does not alter the qualitative
characteristics of the original scale-free network, which opens the possibility
of subtracting a smaller network from the original network without destroying
the important structural properties. The implication of the result is also
suggested in the context of the recent study of the human brain functional
network.Comment: To appear in Phys. Rev. Let
Emergence of Clusters in Growing Networks with Aging
We study numerically a model of nonequilibrium networks where nodes and links
are added at each time step with aging of nodes and connectivity- and
age-dependent attachment of links. By varying the effects of age in the
attachment probability we find, with numerical simulations and scaling
arguments, that a giant cluster emerges at a first-order critical point and
that the problem is in the universality class of one dimensional percolation.
This transition is followed by a change in the giant cluster's topology from
tree-like to quasi-linear, as inferred from measurements of the average
shortest-path length, which scales logarithmically with system size in one
phase and linearly in the other.Comment: 8 pages, 6 figures, accepted for publication in JSTA
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
Scaling Behaviour of Developing and Decaying Networks
We find that a wide class of developing and decaying networks has scaling
properties similar to those that were recently observed by Barab\'{a}si and
Albert in the particular case of growing networks. The networks considered here
evolve according to the following rules: (i) Each instant a new site is added,
the probability of its connection to old sites is proportional to their
connectivities. (ii) In addition, (a) new links between some old sites appear
with probability proportional to the product of their connectivities or (b)
some links between old sites are removed with equal probability.Comment: 7 pages (revtex
Evolution of scale-free random graphs: Potts model formulation
We study the bond percolation problem in random graphs of weighted
vertices, where each vertex has a prescribed weight and an edge can
connect vertices and with rate . The problem is solved by the
limit of the -state Potts model with inhomogeneous interactions for
all pairs of spins. We apply this approach to the static model having
so that the resulting graph is scale-free with
the degree exponent . The number of loops as well as the giant
cluster size and the mean cluster size are obtained in the thermodynamic limit
as a function of the edge density, and their associated critical exponents are
also obtained. Finite-size scaling behaviors are derived using the largest
cluster size in the critical regime, which is calculated from the cluster size
distribution, and checked against numerical simulation results. We find that
the process of forming the giant cluster is qualitatively different between the
cases of and . While for the former, the giant
cluster forms abruptly at the percolation transition, for the latter, however,
the formation of the giant cluster is gradual and the mean cluster size for
finite shows double peaks.Comment: 34 pages, 9 figures, elsart.cls, final version appeared in NP
Exploring the assortativity-clustering space of a network's degree sequence
Nowadays there is a multitude of measures designed to capture different
aspects of network structure. To be able to say if the structure of certain
network is expected or not, one needs a reference model (null model). One
frequently used null model is the ensemble of graphs with the same set of
degrees as the original network. In this paper we argue that this ensemble can
be more than just a null model -- it also carries information about the
original network and factors that affect its evolution. By mapping out this
ensemble in the space of some low-level network structure -- in our case those
measured by the assortativity and clustering coefficients -- one can for
example study how close to the valid region of the parameter space the observed
networks are. Such analysis suggests which quantities are actively optimized
during the evolution of the network. We use four very different biological
networks to exemplify our method. Among other things, we find that high
clustering might be a force in the evolution of protein interaction networks.
We also find that all four networks are conspicuously robust to both random
errors and targeted attacks
Comment on "Breakdown of the Internet under Intentional Attack"
We obtain the exact position of the percolation threshold in intentionally
damaged scale-free networks.Comment: 1 page, to appear in Phys. Rev. Let
- …