Random sequential renormalization of networks: application to critical trees.

Area of Science:

• Network Science
• Statistical Physics
• Graph Theory

Background:

• Existing renormalization schemes for networks can be complex to implement.
• Understanding network evolution and coarse-graining is crucial in various scientific domains.

Purpose of the Study:

• Introduce a simplified graph renormalization procedure called random sequential renormalization (RSR).
• Analyze the behavior of critical trees under RSR and derive analytical results.
• Identify scaling behaviors with existing network processes like agglomerative percolation.

Main Methods:

• Developed the random sequential renormalization (RSR) procedure for arbitrary networks.
• Applied RSR to critical trees and derived analytical results.
• Utilized random walk, scaling theory, and mean-field theory for analysis.

Main Results:

• Identified three distinct regimes in the evolution of critical trees under RSR.
• Derived an exponent ν=1/2 and observed a transition to a power-law degree distribution (p(k) ~ 1/k^2).
• Observed hub formation and star configurations in later stages, with scaling behaviors linked to agglomerative percolation.

Conclusions:

• RSR provides a simpler and effective method for network renormalization and coarse-graining.
• The study elucidates the complex evolution of critical trees under renormalization, revealing distinct scaling regimes.
• The findings connect RSR to agglomerative percolation, offering insights into continuous phase transitions in network structures.