Network geometry with flavor: from complexity to quantum geometry

Abstract

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 \(\delta\).