Curvature and temperature of complex networks

Area of Science:

• Complex networks
• Network topology
• Hyperbolic geometry

Background:

• Scale-free networks exhibit heterogeneous degree distributions.
• The origins of these distributions in complex systems remain an active area of research.
• Understanding network structure is crucial for applications like internet routing.

Purpose of the Study:

• To demonstrate how hidden hyperbolic space expansion can generate scale-free network topologies.
• To establish a physical interpretation for network link energies using Fermi-Dirac statistics.
• To explore the potential of hyperbolic embedding for practical network applications, such as routing.

Main Methods:

• Developed a hyperbolic model where network emergence is driven by exponential expansion.
• Utilized Fermi-Dirac statistics to link hyperbolic distances with energy.
• Embedded the internet topology into a hyperbolic plane for comparison with the model.

Main Results:

• Heterogeneous degree distributions in complex networks arise from hyperbolic space expansion.
• Hidden space curvature directly influences degree distribution heterogeneity.
• Network clustering is determined by temperature within the hyperbolic model.
• Empirical embedding of the internet showed strong agreement with the hyperbolic model.

Conclusions:

• The proposed hyperbolic model realistically explains the emergence of scale-free network topologies.
• The model provides a framework for understanding network properties like degree heterogeneity and clustering.
• Internet embedding validates the model and suggests novel routing strategies using local information, promising enhanced performance.