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 $\mathbb{R}^2$, 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 $\gamma =3$ and clustering coefficient $C={46/3}-36\texttt{ln}{3/2}\approx 0.74$, 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 $C(k)\sim k^{-1}$ 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 $N$(the total number of nodes) is relatively small(as $N\sim 10^4$), the performance of RAN under intentional attack is not sensitive to $N$, while that of BA networks is much affected by $N$. 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

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