Geometric evolution of complex networks with degree correlations.

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Understanding network formation is crucial for modeling real-world systems.
• Geometric network models offer insights into spatial embedding and structure.
• Existing models often lack control over specific network properties like degree distributions.

Purpose of the Study:

• To present a general class of geometric network growth mechanisms.
• To demonstrate how to control network properties such as degree sequence and correlations.
• To investigate the influence of geometry on network structure and identify phase transitions.

Main Methods:

• Homogeneous attachment in a metric space with Fermi-Dirac connection probability.
• Utilizing a hidden variable framework to derive analytical expressions.
• Analyzing the average clustering coefficient to study geometric effects.

Main Results:

• An analytical expression for the degree sequence is derived.
• The model allows tuning of degree distributions, including scale-free networks.
• Network history influences degree-degree correlations, enabling assortative or disassortative tuning.
• A phase transition is identified between random and geometric regimes based on inverse temperature β.

Conclusions:

• The proposed geometric network growth mechanism offers significant control over network topology.
• The model successfully reproduces various degree distributions and correlation patterns.
• A clear geometric phase transition is observed, highlighting the role of spatial embedding.