Type-II Apollonian network: More robust and more efficient Apollonian network

The family of planar graphs is a particularly important family and models many networks including the layout of printed circuits. The widely-known Apollonian packing process has been used as guideline to create the typical Apollonian network with planarity. In this paper, we propose a new principled framework based on the Apollonian packing process to generate model as complex network, and obtain a family of new networks called Type-II Apollonian network At. While our network and the typical Apollonian network are maximal planar, the former turns out to be Hamiltonian and Eulerian, however, the latter is not. Then, we in-depth study some fundamental structural properties on network At, and verify that network At is sparse, has scale-free feature and small-world property, and exhibits disassortative mixing structure. Next, we derive the asymptotic solution of the spanning tree entropy of network At by designing an effective algorithm, which suggests that Type-II Apollonian network is more robust to a random removal of edges than the typical Apollonian network. Additionally, we study trapping problem on network At, and use average trapping time as metric to show that Type-II Apollonian network At has more efficient underlying structure for fast information diffusion than the typical Apollonian network.