  We introduce a family of complex networks that interpolates between the
Apollonian network and its binary version, recently introduced in [Phys. Rev. E
\textbf{107}, 024305 (2023)], 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 $2\pi / 3$ rotation symmetry as a tree-like 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.
