Emergent Hyperbolic Network Geometry.

Area of Science:

• Complex systems science
• Network theory
• Computational geometry

Background:

• Many complex systems involve interactions among more than two nodes, necessitating descriptions beyond simple networks.
• Simplicial complexes, composed of simplices like nodes, links, and triangles, offer a geometric framework for these systems.
• These structures are relevant in fields such as quantum gravity for spacetime discretization.

Purpose of the Study:

• To investigate the emergent geometry of complex networks by extending growing network models to simplicial complexes.
• To determine if this emergent geometry is hyperbolic.
• To analyze how dimensionality and network dynamics influence the properties of growing simplicial complexes.

Main Methods:

• Extending knowledge of growing complex networks to growing simplicial complexes.
• Utilizing purely combinatorial models for simplicial complex growth.
• Analyzing statistical and geometrical properties, including degree distribution, small-world effects, and community structure.

Main Results:

• A hyperbolic network geometry emerges spontaneously from combinatorial models of growing simplicial complexes.
• The statistical and geometrical properties are dependent on the dimensionality of the complex.
• The complexes exhibit universal properties of real-world networks, such as scale-free degree distribution, small-world phenomena, and community structures.
• Phase transitions occur with heterogeneous face fitness, impacting network geometry.

Conclusions:

• Growing simplicial complexes can spontaneously generate hyperbolic geometry.
• Dimensionality is a key factor influencing the emergent geometric and statistical properties.
• These models capture essential features of real complex networks and offer insights into their geometric underpinnings.