Random Apollonian networks with tailored clustering coefficient.

We introduce a family of complex networks that interpolates between the Apollonian network and its binary version, recently introduced in E. M. K. Souza et al. [Phys. Rev. E 107, 024305 (2023)2470-004510.1103/PhysRevE.107.024305], via random removal of nodes. The dilution process allows the clustering coefficient to vary from C=0.828 to C=0 while maintaining the behavior of average path length and other relevant quantities as in the deterministic Apollonian network. Robustness against the random deletion of nodes is also reported on spectral quantities such as the ground-state localization degree and its energy gap to the first excited state. The loss of the 2π/3 rotation symmetry as a treelike network emerges is investigated in the light of the hub wavefunction amplitude. Our findings expose the interplay between the small-world property and other distinctive traits exhibited by Apollonian networks, as well as their resilience against random attacks.

Binary Apollonian networks.

There is a well-known relationship between the binary Pascal's triangle and the Sierpinski triangle, in which the latter is obtained from the former by successive modulo 2 additions starting from a corner. Inspired by that, we define a binary Apollonian network and obtain two structures featuring a kind of dendritic growth. They are found to inherit the small-world and scale-free properties from the original network but display no clustering. Other key network properties are explored as well. Our results reveal that the structure contained in the Apollonian network may be employed to model an even wider class of real-world systems.