Structural Transitions in Densifying Networks.

Area of Science:

• Network Science
• Statistical Physics
• Graph Theory

Background:

• Understanding network formation is crucial in various scientific fields.
• Previous models often lack the ability to capture emergent dense structures and complex behaviors.

Purpose of the Study:

• Introduce a minimal generative model for network densification.
• Analyze the structural properties of networks generated by this copying mechanism.
• Investigate phase transitions and anomalies in dense network regimes.

Main Methods:

• A generative model where new nodes attach to a random target and its neighbors with probability p.
• Analysis of network sparsity and density based on the parameter p.
• Examination of network realizations, clique counts, and their dependence on the number of nodes (N).

Main Results:

• Networks are sparse for p<1/2 and dense for p≥1/2, with average degree increasing with N.
• Dense networks exhibit non-self-averaging behavior and disparate realizations.
• An infinite sequence of structural anomalies and phase transitions occur at specific p values (e.g., 2/3, 3/4).
• Linking to second neighbors can lead to complete graphs with non-zero probability as N approaches infinity.

Conclusions:

• The minimal generative model effectively captures the transition from sparse to dense networks.
• The model reveals complex structural phenomena, including phase transitions and anomalies, in dense network regimes.
• The findings have implications for understanding the formation and properties of various real-world complex networks.