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Random sequential renormalization and agglomerative percolation in networks: application to Erdös-Rényi and
Golnoosh Bizhani1, Peter Grassberger, Maya Paczuski
1Complexity Science Group, University of Calgary, Calgary, Canada T2N 1N4.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 7, 2012
Summary
Random sequential renormalization (RSR) reveals a second-order phase transition in network models, characterized by agglomerative percolation and the emergence of a giant hub. This transition
Area of Science:
- Network science
- Statistical physics
- Complex systems analysis
Background:
- Network renormalization is crucial for understanding large-scale structures.
- Previous renormalization methods offered limited analytical depth.
- Understanding phase transitions in complex networks is an ongoing challenge.
Purpose of the Study:
- To investigate the statistical behavior of various network models under random sequential renormalization (RSR).
- To analyze the renormalization group (RG) flow with greater precision.
- To identify new features and phase transitions in network renormalization.
Main Methods:
- Application of random sequential renormalization (RSR) to Erdös-Rényi (ER) graphs, scale-free networks, and an annealed model.
- Local coarse-graining by randomly selecting nodes and connecting them to neighbors.
- Comparison with previous parallel renormalization techniques.
Main Results:
- All studied networks exhibit a second-order phase transition during RSR.
- This transition, termed agglomerative percolation, is linked to the formation of a giant hub.
- The transition occurs at different system sizes (N/N(0)) depending on the network type.
Conclusions:
- RSR provides a fine-grained analysis of RG flow, revealing novel network properties.
- Agglomerative percolation is a fundamental transition in graph renormalization.
- Critical exponents align with percolation theory and mean-field predictions for specific models.
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