Network geometry with flavor: From complexity to quantum geometry.

Area of Science:

• Network geometry
• Quantum gravity
• Complex networks

Background:

• Network geometry is crucial for data mining, internet routing, and quantum gravity.
• Simplicial complexes are key structures in network geometry for discretizing spaces.
• Existing models like Bianconi-Barabási and Apollonian networks have limitations.

Purpose of the Study:

• Introduce a novel model, Network Geometry with Flavor (NGF), for simplicial complexes.
• Explore NGF's ability to generate diverse discrete geometries and complex network properties.
• Investigate the relationship between NGF's classical structure and its quantum mechanical description.

Main Methods:

• Define NGF for simplicial complexes in arbitrary dimensions with nonequilibrium dynamics.
• Analyze NGF's thermodynamic properties and emergent network characteristics (e.g., scale-free, small-world).
• Develop a quantum mechanical description for NGF using quantum network states.

Main Results:

• NGF generates diverse geometries, including manifolds and scale-free networks.
• NGF obeys a generalized area law, offering a new coarse-grained limit formulation.
• Scale-free properties emerge differently based on dimension: preferential attachment is needed for d=1, but not for d>1.
• NGF has a quantum mechanical description, with classical statistical properties reflecting quantum behavior.

Conclusions:

• NGF provides a unified framework for various network models and discrete geometries.
• The model reveals dimension-dependent mechanisms for generating scale-free networks.
• NGF bridges classical network science and quantum mechanics, with potential applications in quantum gravity and information theory.