Maximal planar networks with large clustering coefficient and power-law degree distribution

Abstract

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