310,688 research outputs found
Degree-distribution Stability of Growing Networks
In this paper, we abstract a kind of stochastic processes from evolving
processes of growing networks, this process is called growing network Markov
chains. Thus the existence and the formulas of degree distribution are
transformed to the corresponding problems of growing network Markov chains.
First we investigate the growing network Markov chains, and obtain the
condition in which the steady degree distribution exists and get its exact
formulas. Then we apply it to various growing networks. With this method, we
get a rigorous, exact and unified solution of the steady degree distribution
for growing networks.Comment: 12 page
Causal and homogeneous networks
Growing networks have a causal structure. We show that the causality strongly
influences the scaling and geometrical properties of the network. In particular
the average distance between nodes is smaller for causal networks than for
corresponding homogeneous networks. We explain the origin of this effect and
illustrate it using as an example a solvable model of random trees. We also
discuss the issue of stability of the scale-free node degree distribution. We
show that a surplus of links may lead to the emergence of a singular node with
the degree proportional to the total number of links. This effect is closely
related to the backgammon condensation known from the balls-in-boxes model.Comment: short review submitted to AIP proceedings, CNET2004 conference;
changes in the discussion of the distance distribution for growing trees,
Fig. 6-right change
Greedy optimization for growing spatially embedded oscillatory networks
The coupling of some types of oscillators requires the mediation of a
physical link between them, rendering the distance between oscillators a
critical factor to achieve synchronization. In this paper we propose and
explore a greedy algorithm to grow spatially embedded oscillator networks. The
algorithm is constructed in such a way that nodes are sequentially added
seeking to minimize the cost of the added links' length and optimize the linear
stability of the growing network. We show that, for appropriate parameters, the
stability of the resulting network, measured in terms of the dynamics of small
perturbations and the correlation length of the disturbances, can be
significantly improved with a minimal added length cost. In addition, we
analyze numerically the topological properties of the resulting networks and
find that, while being more stable, their degree distribution is approximately
exponential and independent of the algorithm parameters. Moreover, we find that
other topological parameters related with network resilience and efficiency are
also affected by the proposed algorithm. Finally, we extend our findings to
more general classes of networks with different sources of heterogeneity. Our
results are a first step in the development of algorithms for the directed
growth of oscillatory networks with desirable stability, dynamical and
topological properties.Comment: 13 pages, 9 figure
Stability as a natural selection mechanism on interacting networks
Biological networks of interacting agents exhibit similar topological
properties for a wide range of scales, from cellular to ecological levels,
suggesting the existence of a common evolutionary origin. A general
evolutionary mechanism based on global stability has been proposed recently [J
I Perotti, O V Billoni, F A Tamarit, D R Chialvo, S A Cannas, Phys. Rev. Lett.
103, 108701 (2009)]. This mechanism is incorporated into a model of a growing
network of interacting agents in which each new agent's membership in the
network is determined by the agent's effect on the network's global stability.
We show that, out of this stability constraint, several topological properties
observed in biological networks emerge in a self organized manner. The
influence of the stability selection mechanism on the dynamics associated to
the resulting network is analyzed as well.Comment: 10 pages, 9 figure
Network Synchronization, Diffusion, and the Paradox of Heterogeneity
Many complex networks display strong heterogeneity in the degree
(connectivity) distribution. Heterogeneity in the degree distribution often
reduces the average distance between nodes but, paradoxically, may suppress
synchronization in networks of oscillators coupled symmetrically with uniform
coupling strength. Here we offer a solution to this apparent paradox. Our
analysis is partially based on the identification of a diffusive process
underlying the communication between oscillators and reveals a striking
relation between this process and the condition for the linear stability of the
synchronized states. We show that, for a given degree distribution, the maximum
synchronizability is achieved when the network of couplings is weighted and
directed, and the overall cost involved in the couplings is minimum. This
enhanced synchronizability is solely determined by the mean degree and does not
depend on the degree distribution and system size. Numerical verification of
the main results is provided for representative classes of small-world and
scale-free networks.Comment: Synchronization in Weighted Network
Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics
Linearized catalytic reaction equations modeling e.g. the dynamics of genetic
regulatory networks under the constraint that expression levels, i.e. molecular
concentrations of nucleic material are positive, exhibit nontrivial dynamical
properties, which depend on the average connectivity of the reaction network.
In these systems the inflation of the edge of chaos and multi-stability have
been demonstrated to exist. The positivity constraint introduces a nonlinearity
which makes chaotic dynamics possible. Despite the simplicity of such minimally
nonlinear systems, their basic properties allow to understand fundamental
dynamical properties of complex biological reaction networks. We analyze the
Lyapunov spectrum, determine the probability to find stationary oscillating
solutions, demonstrate the effect of the nonlinearity on the effective in- and
out-degree of the active interaction network and study how the frequency
distributions of oscillatory modes of such system depend on the average
connectivity.Comment: 11 pages, 5 figure
Universality in the synchronization of weighted random networks
Realistic networks display not only a complex topological structure, but also
a heterogeneous distribution of weights in the connection strengths. Here we
study synchronization in weighted complex networks and show that the
synchronizability of random networks with large minimum degree is determined by
two leading parameters: the mean degree and the heterogeneity of the
distribution of node's intensity, where the intensity of a node, defined as the
total strength of input connections, is a natural combination of topology and
weights. Our results provide a possibility for the control of synchronization
in complex networks by the manipulation of few parameters.Comment: 4 pages, 3 figure
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