954 research outputs found
Random walks on the Apollonian network with a single trap
Explicit determination of the mean first-passage time (MFPT) for trapping
problem on complex media is a theoretical challenge. In this paper, we study
random walks on the Apollonian network with a trap fixed at a given hub node
(i.e. node with the highest degree), which are simultaneously scale-free and
small-world. We obtain the precise analytic expression for the MFPT that is
confirmed by direct numerical calculations. In the large system size limit, the
MFPT approximately grows as a power-law function of the number of nodes, with
the exponent much less than 1, which is significantly different from the
scaling for some regular networks or fractals, such as regular lattices,
Sierpinski fractals, T-graph, and complete graphs. The Apollonian network is
the most efficient configuration for transport by diffusion among all
previously studied structure.Comment: Definitive version accepted for publication in EPL (Europhysics
Letters
Planar growth generates scale free networks
In this paper we introduce a model of spatial network growth in which nodes
are placed at randomly selected locations on a unit square in ,
forming new connections to old nodes subject to the constraint that edges do
not cross. The resulting network has a power law degree distribution, high
clustering and the small world property. We argue that these characteristics
are a consequence of the two defining features of the network formation
procedure; growth and planarity conservation. We demonstrate that the model can
be understood as a variant of random Apollonian growth and further propose a
one parameter family of models with the Random Apollonian Network and the
Deterministic Apollonian Network as extreme cases and our model as a midpoint
between them. We then relax the planarity constraint by allowing edge crossings
with some probability and find a smooth crossover from power law to exponential
degree distributions when this probability is increased.Comment: 27 pages, 9 figure
Maximal planar networks with large clustering coefficient and power-law degree distribution
In this article, we propose a simple rule that generates scale-free networks
with very large clustering coefficient and very small average distance. These
networks are called {\bf Random Apollonian Networks}(RAN) as they can be
considered as a variation of Apollonian networks. We obtain the analytic
results of power-law exponent and clustering coefficient
, which agree very well with the
simulation results. We prove that the increasing tendency of average distance
of RAN is a little slower than the logarithm of the number of nodes in RAN.
Since most real-life networks are both scale-free and small-world networks, RAN
may perform well in mimicking the reality. The RAN possess hierarchical
structure as that in accord with the observations of many
real-life networks. In addition, we prove that RAN are maximal planar networks,
which are of particular practicability for layout of printed circuits and so
on. The percolation and epidemic spreading process are also studies and the
comparison between RAN and Barab\'{a}si-Albert(BA) as well as Newman-Watts(NW)
networks are shown. We find that, when the network order (the total number
of nodes) is relatively small(as ), the performance of RAN under
intentional attack is not sensitive to , while that of BA networks is much
affected by . And the diseases spread slower in RAN than BA networks during
the outbreaks, indicating that the large clustering coefficient may slower the
spreading velocity especially in the outbreaks.Comment: 13 pages, 10 figure
Energy landscapes, scale-free networks and Apollonian packings
We review recent results on the topological properties of two spatial
scale-free networks, the inherent structure and Apollonian networks. The
similarities between these two types of network suggest an explanation for the
scale-free character of the inherent structure networks. Namely, that the
energy landscape can be viewed as a fractal packing of basins of attraction.Comment: 10 pages, 8 figure
Self-similar disk packings as model spatial scale-free networks
The network of contacts in space-filling disk packings, such as the
Apollonian packing, are examined. These networks provide an interesting example
of spatial scale-free networks, where the topology reflects the broad
distribution of disk areas. A wide variety of topological and spatial
properties of these systems are characterized. Their potential as models for
networks of connected minima on energy landscapes is discussed.Comment: 13 pages, 12 figures; some bugs fixed and further discussion of
higher-dimensional packing
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