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In this article, we propose a simple rule that generates scale-free networkswith very large clustering coefficient and very small average distance. Thesenetworks are called {\bf Random Apollonian Networks}(RAN) as they can beconsidered as a variation of Apollonian networks. We obtain the analyticresults 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 thesimulation results. We prove that the increasing tendency of average distanceof 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, RANmay perform well in mimicking the reality. The RAN possess hierarchicalstructure as $C(k)\sim k^{-1}$ that in accord with the observations of manyreal-life networks. In addition, we prove that RAN are maximal planar networks,which are of particular practicability for layout of printed circuits and soon. The percolation and epidemic spreading process are also studies and thecomparison 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 numberof nodes) is relatively small(as $N\sim 10^4$), the performance of RAN underintentional attack is not sensitive to $N$, while that of BA networks is muchaffected by $N$. And the diseases spread slower in RAN than BA networks duringthe outbreaks, indicating that the large clustering coefficient may slower thespreading velocity especially in the outbreaks.Comment: 13 pages, 10 figure

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