A large variety of interacting complex systems are characterized by
interactions occurring between more than two nodes.
These systems are described by simplicial complexes. Simplicial complexes
are formed by simplices (nodes, links, triangles, tetrahedra etc.)
that have a natural geometric interpretation. As such simplicial complexes
are widely used in quantum gravity approaches that involve a discretization of spacetime.
Here, by extending our knowledge of growing complex networks to growing
simplicial complexes we investigate the nature of the emergent
geometry of complex networks and explore whether this geometry is hyperbolic.
Specifically we show that an hyperbolic network geometry emerges spontaneously
from models of growing simplicial complexes that are purely combinatorial.
The statistical and geometrical properties of the growing simplicial complexes strongly depend on
their dimensionality and display the major universal properties of real complex networks
(scale-free degree distribution, small-world and communities) at the same time.
Interestingly, when the network dynamics includes an heterogeneous fitness of the faces,
the statistical properties of the faces of the simplicies are described by Fermi-Dirac, Boltzmann and
Bose-Einstein statitics. Additionally the simplicial complexes can undergo a
phase transitions that are reflected by relevant changes in the network geometry.