  Cooperative self-assembly can result in complex nano-networks with new
hyperbolic geometry. However, the relation between the hyperbolicity and
spectral and dynamical features of these structures remains unclear. Using the
model of aggregation of simplexes introduced in I [Sci. Rep., 8:1987, 2018],
here we study topological and spectral properties of a large class of
self-assembled structures consisting of monodisperse building blocks (cliques
of size $n=3,4,5,6$) which self-assemble via sharing the geometrical shapes of
a lower order. The size of the shared sub-structure is tunned by varying the
chemical affinity $\nu$ such that for significant positive $\nu$ sharing the
largest face is the most probable, while for $\nu < 0$, attaching via a single
node dominates. Our results reveal that, while the parameter of hyperbolicity
remains $\delta_{max}=1$ across the assemblies, their structure and spectral
dimension $d_s$ vary with the size of cliques $n$ and the affinity when $\nu
\geq 0$. In this range, we findthat $d_s >4$ can be reached for $n\geq 5$ and
sufficiently large $\nu$. For the aggregates of triangles and tetrahedra, the
spectral dimension remains in the range $d_s\in [2,4)$, as well as for the
higher cliques at vanishing affinity. On the other end, for $\nu < 0$, we find
$d_s\eqsim 1.57$ independently on $n$. Moreover, the spectral distribution of
the normalised Laplacian eigenvalues has a characteristic shape with peaks and
a pronounced minimum, representing the hierarchical architecture of the
simplicial complexes. These findings show how the structures compatible with
complex dynamical properties can be assembled by controlling the higher-order
connectivity among the building blocks.
