Network geometry with flavor: from complexity to quantum geometry

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 $\delta$.