Hierarchical localization in disordered Apollonian networks

Abstract

We investigate localization properties of the Apollonian network (AN) in the presence of diagonal and off-diagonal disorder. By employing a site-resolved localization measure, we show that the localization degree is strongly dependent on the energy and tied to the hierarchical topology of the network. At the spectral edges, eigenstates are strongly localized on highly connected sites originating from previous generations, a behavior that persists under both disorder mechanisms. In contrast, around zero energy localization is associated with the lowest-degree sites. As disorder breaks the underlying C3 symmetry of the AN, it promotes spatial reconfiguration of these states while preserving their support on low-degree nodes. For diagonal disorder, localization is enhanced over a broad range of negative energies, whereas off-diagonal disorder induces weakening of localization in this region. Finally, we show that the hub dominates the spectral edges but has negligible contribution near the band center, indicating that its associated localized states are robust against disorder. These results highlight how topology and disorder jointly shape localization in complex networks.