Model of Apollonian Networks with Aging Mechanism

In relation to the problem of Apollonian packing, the Apollonian network is introduced. It has interesting properties like being scale-free, small-world, space-filling, and may help to explain the properties of energy landscapes, and can be applied to porous media, road networks, electrical systems, and the layout of the planar network. For further research on the Apollonian network, we propose a model of Apollonian Networks with aging mechanism. Using mathematical analysis, we deduce that the degree distribution obeys power-law form, which indicates that the network is power-law. Also we obtain the expression of clustering coefficient and prove that the average path length increases in the form of logarithm of network size, which indicates that the network is small-world. And then, the simulation results are given out, which agrees with the theoretic calculations. At last, we calculate the distribution of age of nodes and prove that the age of a node negatively correlates to the probability of the node's getting new links. Therefore, the model proposed in this article performs better than the model of Random Apollonian Networks and is more effective to the practice of layout of planar networks, such as printed circuits.