Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks

Area of Science:

• Graph theory
• Network science
• Complex systems

Background:

• Farey graphs are known for small-world, Hamiltonian, and perfect properties.
• However, Farey graphs lack scale-free characteristics, limiting their use in modeling some complex systems.
• This study introduces generalized Farey graphs as an extension.

Purpose of the Study:

• To analyze the topological characteristics of generalized Farey graphs.
• To investigate the growth mechanism and its impact on network properties.
• To determine if generalized Farey graphs exhibit scale-free properties.

Main Methods:

• Analysis of topological characteristics of generalized Farey graphs.
• Mathematical deduction based on the growth mechanism of these graphs.
• Comparison of degree distribution between Farey and generalized Farey graphs.

Main Results:

• Generalized Farey graphs exhibit small-world and highly clustered properties.
• These new models demonstrate scale-free characteristics.
• Exponential increase in node degrees during growth is identified as the cause of scale-free properties.

Conclusions:

• Generalized Farey graphs are a promising model for complex networks requiring scale-free properties.
• The growth mechanism, specifically exponential degree incrementation, is key to achieving scale-free networks.
• Unlike Farey graphs, generalized Farey graphs provide a better fit for certain empirical complex systems.