Dense percolation in large-scale mean-field random networks is provably "explosive"

Area of Science:

• Network science
• Statistical physics
• Complex systems

Background:

• Explosive percolation describes a sudden increase in network connectivity.
• This transition is typically continuous (second-order) in random network models.
• Dense percolation focuses on highly connected clusters within already connected networks.

Purpose of the Study:

• To investigate the nature of dense percolation transitions.
• To determine if dense percolation differs qualitatively from explosive percolation.
• To characterize the size of the largest dense cluster in mean-field random networks.

Main Methods:

• Analysis of the classical mean-field random network formation process.
• Mathematical proof of the transition order for dense percolation.
• Derivation of tight asymptotic bounds for dense cluster size.

Main Results:

• Dense percolation exhibits a discontinuous (first-order) phase transition.
• This is a qualitative difference from the continuous transition of explosive percolation.
• Explicit bounds for the largest dense cluster size were derived, extending clique size formulas.

Conclusions:

• Dense percolation represents a distinct phase transition phenomenon.
• The findings highlight qualitative differences in network connectivity transitions.
• Results have implications for understanding phase transitions in real-world networks.