Complex networks in the framework of nonassociative geometry

Area of Science:

• Complex systems science
• Network theory
• Nonassociative geometry

Background:

• Statistical analysis of complex networks often overlooks underlying geometric structures.
• Understanding the 'small-world' property in networks is crucial for explaining their behavior.
• Existing models struggle to fully account for the intricate connectance patterns observed in real-world networks like the Internet.

Purpose of the Study:

• To develop an effective model extending statistical treatments of complex networks by incorporating hidden geometry.
• To investigate the role of nonlocal curvature in controlling the small-world property of networks.
• To apply this novel approach to analyze the Internet as a complex network embedded in hyperbolic space.

Main Methods:

• Development of an effective model within a nonassociative geometry framework.
• Incorporation of nonlocal curvature to model the small-world property.
• Application and validation of the model using empirical data from the Internet.

Main Results:

• The model demonstrates remarkable agreement with available empirical data for Internet connectance.
• The proposed approach successfully explains features of Internet connectance that are not captured by other models.
• Nonlocal curvature in hyperbolic space is identified as a key factor in network small-world properties.

Conclusions:

• The nonassociative geometry framework provides a powerful tool for modeling complex networks with hidden geometry.
• The model offers a new perspective on understanding and analyzing diverse complex networks, including transportation, social, and biological systems.
• This approach advances the study of complex networks by integrating geometric principles with statistical analysis.