  Recently the presence of topologically protected edge-states in
Bi$_{14}$Rh$_3$I$_9$ was confirmed by scanning tunnelling microscopy
consolidating this compound as a weak 3D topological insulator (TI). Here, we
present a density-functional-theory-based study on a family of TIs derived from
the Bi$_{14}$Rh$_3$I$_9$ parent structure via substitution of Ru, Pd, Os, Ir
and Pt for Rh. Comparative analysis of the band-structures throughout the
entire series is done by means of a unified minimalistic tight-binding model
that evinces strong similarity between the quantum-spin-Hall (QSH) layer in
Bi$_{14}$Rh$_3$I$_9$ and graphene in terms of $p_z$-molecular orbitals.
Topologically non-trivial energy gaps are found for the Ir-, Rh-, Pt- and
Pd-based systems, whereas the Os- and Ru-systems remain trivial. Furthermore,
the energy position of the metal $d$-band centre is identified as the parameter
which governs the evolution of the topological character of the band structure
through the whole family of TIs. The $d$-band position is shown to correlate
with the chemical bonding within the QSH layers, thus revealing how the
chemical nature of the constituents affects the topological band character.
