  From social interactions to the human brain, higher-order networks are key to
describe the underlying network geometry and topology of many complex systems.
While it is well known that network structure strongly affects its function,
the role that network topology and geometry has on the emerging dynamical
properties of higher-order networks is yet to be clarified. In this
perspective, the spectral dimension plays a key role since it determines the
effective dimension for diffusion processes on a network. Despite its
relevance, a theoretical understanding of which mechanisms lead to a finite
spectral dimension, and how this can be controlled, represents nowadays still a
challenge and is the object of intense research. Here we introduce two
non-equilibrium models of hyperbolic higher-order networks and we characterize
their network topology and geometry by investigating the interwined appearance
of small-world behavior, $\delta$-hyperbolicity and community structure. We
show that different topological moves determining the non-equilibrium growth of
the higher-order hyperbolic network models induce tunable values of the
spectral dimension, showing a rich phenomenology which is not displayed in
random graph ensembles. In particular, we observe that, if the topological
moves used to construct the higher-order network increase the area$/$volume
ratio, the spectral dimension continuously decreases, while the opposite effect
is observed if the topological moves decrease the area$/$volume ratio. Our work
reveals a new link between the geometry of a network and its diffusion
properties, contributing to a better understanding of the complex interplay
between network structure and dynamics.
