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.