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Here we introduce the Network Geometry with Flavor $s=-1,0,1$ (NGF)describing simplicial complexes defined in arbitrary dimension $d$ and evolvingby a non-equilibrium dynamics. The NGF can generate discrete geometries ofdifferent nature, ranging from chains and higher dimensional manifolds toscale-free networks with small-world properties, scale-free degree distributionand non-trivial community structure. The NGF admits as limiting cases both theBianconi-Barab\'asi model for complex networks the stochastic Apolloniannetwork, and the recently introduced model for Complex Quantum NetworkManifolds. The thermodynamic properties of NGF reveal that NGF obeys ageneralized area law opening a new scenario for formulating its coarse-grainedlimit. The structure of NGF is strongly dependent on the dimensionality $d$. Wealso show that NGF admits a quantum mechanical description in terms ofassociated quantum network states. Quantum network states are evolving by aMarkovian dynamics and a quantum network state at time $t$ encodes all possibleNGF evolutions up to time $t$. Interestingly the NGF remains fully classicalbut its statistical properties reveal the relation to its quantum mechanicaldescription. In fact the $\delta$-dimensional faces of the NGF have generalizeddegrees that follow either the Fermi-Dirac, Boltzmann or Bose-Einsteinstatistics depending on the flavor $s$ and the dimensions $d$ and $\delta$.

Network geometry with flavor: from complexity to quantum geometry