Dense networks with scale-free feature.

Area of Science:

• Network Science
• Complex Systems Analysis
• Graph Theory

Background:

• Many real-world networks, such as social and urban networks, are observed to be dense, contrasting with the typically sparse nature of scale-free networks.
• Existing models often fail to capture the dense characteristics observed in these complex systems.

Purpose of the Study:

• To propose a novel framework for generating scale-free graphs with inherent dense features.
• To theoretically analyze the properties of networks generated by the proposed framework, including their density, diameter, and assortativity.
• To investigate the community structure within these generated networks.

Main Methods:

• Development of a framework employing first-order subdivision and line operations for graph generation.
• Theoretical analysis of the resulting network density, characterized by power-law exponent (1<γ≤2).
• Calculation of Pearson correlation coefficients to assess network assortativity.
• Analysis of community size distribution in relation to modularity maximization.

Main Results:

• Successful generation of scale-free graphs exhibiting dense features with a power-law exponent between 1 and 2.
• Demonstration of the framework's ability to create models with large diameters and high assortativity, reaching theoretical upper bounds for Pearson correlation coefficients.
• Observation that community sizes in the generated models follow a power-law distribution related to modularity maximization.

Conclusions:

• The proposed framework effectively generates scale-free networks with dense properties, addressing limitations of previous models.
• The generated networks possess significant assortative structures and predictable community size distributions, valuable for modeling complex systems.
• This work provides a versatile approach for creating diverse networked models with tunable characteristics.