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Geometric fractal growth model for scale-free networks
1Nonlinear and Complex Systems Laboratory, Department of Physics, Pohang University of Science and Technology, Pohang, Kyongbuk 790-784, Korea.
Summary
We present a deterministic model for scale-free networks, demonstrating that tuning a parameter (m) adjusts the power-law degree exponent (gamma) between 2 and 3. This model also reveals small-world properties in network structures.
Area of Science:
- Network Science
- Statistical Physics
- Complex Systems
Background:
- Scale-free networks are crucial in modeling real-world systems.
- Understanding their degree distribution and structural properties is key.
Purpose of the Study:
- Introduce a deterministic model for generating scale-free networks.
- Analyze the impact of vertex offspring generation on network properties.
- Investigate degree distribution and shortest-path distances.
Main Methods:
- Developed a deterministic model where vertex offspring number is proportional to its degree.
- Analyzed two network structures: tree (parent connection only) and loop (parent and grandparent connection).
- Derived analytical expressions for degree exponent and mean shortest-path distance.
Main Results:
- Both tree and loop models exhibit power-law degree distributions with exponent gamma = 1+ln(2m-1)/ln m.
- The degree exponent gamma can be tuned in the range (2, 3) by adjusting parameter m.
- The tree structure demonstrates small-world behavior with mean shortest-path distance d ~ ln N/ln K_macro.
- A constant offspring model results in an exponential decay in degree distribution.
Conclusions:
- The deterministic model effectively generates scale-free networks with tunable degree exponents.
- The model captures essential network properties like power-law distributions and small-world phenomena.
- The findings offer insights into the fundamental mechanisms governing complex network formation.