901 research outputs found
Degrees and distances in random and evolving Apollonian networks
This paper studies Random and Evolving Apollonian networks (RANs and EANs),
in d dimension for any d>=2, i.e. dynamically evolving random d dimensional
simplices looked as graphs inside an initial d-dimensional simplex. We
determine the limiting degree distribution in RANs and show that it follows a
power law tail with exponent tau=(2d-1)/(d-1). We further show that the degree
distribution in EANs converges to the same degree distribution if the
simplex-occupation parameter in the n-th step of the dynamics is q_n->0 and
sum_{n=0}^infty q_n =infty. This result gives a rigorous proof for the
conjecture of Zhang et al. that EANs tend to show similar behavior as RANs once
the occupation parameter q->0. We also determine the asymptotic behavior of
shortest paths in RANs and EANs for arbitrary d dimensions. For RANs we show
that the shortest path between two uniformly chosen vertices (typical
distance), the flooding time of a uniformly picked vertex and the diameter of
the graph after n steps all scale as constant times log n. We determine the
constants for all three cases and prove a central limit theorem for the typical
distances. We prove a similar CLT for typical distances in EANs
Coherence in scale-free networks of chaotic maps
We study fully synchronized states in scale-free networks of chaotic logistic
maps as a function of both dynamical and topological parameters. Three
different network topologies are considered: (i) random scale-free topology,
(ii) deterministic pseudo-fractal scale-free network, and (iii) Apollonian
network. For the random scale-free topology we find a coupling strength
threshold beyond which full synchronization is attained. This threshold scales
as , where is the outgoing connectivity and depends on the
local nonlinearity. For deterministic scale-free networks coherence is observed
only when the coupling strength is proportional to the neighbor connectivity.
We show that the transition to coherence is of first-order and study the role
of the most connected nodes in the collective dynamics of oscillators in
scale-free networks.Comment: 9 pages, 8 figure
Non-nequilibrium model on Apollonian networks
We investigate the Majority-Vote Model with two states () and a noise
on Apollonian networks. The main result found here is the presence of the
phase transition as a function of the noise parameter . We also studies de
effect of redirecting a fraction of the links of the network. By means of
Monte Carlo simulations, we obtained the exponent ratio ,
, and for several values of rewiring probability . The
critical noise was determined and also was calculated. The
effective dimensionality of the system was observed to be independent on ,
and the value is observed for these networks. Previous
results on the Ising model in Apollonian Networks have reported no presence of
a phase transition. Therefore, the results present here demonstrate that the
Majority-Vote Model belongs to a different universality class as the
equilibrium Ising Model on Apollonian Network.Comment: 5 pages, 5 figure
Synchronous and Asynchronous Recursive Random Scale-Free Nets
We investigate the differences between scale-free recursive nets constructed
by a synchronous, deterministic updating rule (e.g., Apollonian nets), versus
an asynchronous, random sequential updating rule (e.g., random Apollonian
nets). We show that the dramatic discrepancies observed recently for the degree
exponent in these two cases result from a biased choice of the units to be
updated sequentially in the asynchronous version
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