  Here we introduce the Network Geometry with Flavor $s=-1,0,1$ (NGF)
describing simplicial complexes defined in arbitrary dimension $d$ and evolving
by a non-equilibrium dynamics. The NGF can generate discrete geometries of
different nature, ranging from chains and higher dimensional manifolds to
scale-free networks with small-world properties, scale-free degree distribution
and non-trivial community structure. The NGF admits as limiting cases both the
Bianconi-Barab\'asi model for complex networks the stochastic Apollonian
network, and the recently introduced model for Complex Quantum Network
Manifolds. The thermodynamic properties of NGF reveal that NGF obeys a
generalized area law opening a new scenario for formulating its coarse-grained
limit. The structure of NGF is strongly dependent on the dimensionality $d$. We
also show that NGF admits a quantum mechanical description in terms of
associated quantum network states. Quantum network states are evolving by a
Markovian dynamics and a quantum network state at time $t$ encodes all possible
NGF evolutions up to time $t$. Interestingly the NGF remains fully classical
but its statistical properties reveal the relation to its quantum mechanical
description. In fact the $\delta$-dimensional faces of the NGF have generalized
degrees that follow either the Fermi-Dirac, Boltzmann or Bose-Einstein
statistics depending on the flavor $s$ and the dimensions $d$ and $\delta$.
