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Three mechanisms for power laws on the Cayley tree
Ted Brookings1, J M Carlson, John Doyle
1Department of Physics, University of California, Santa Barbara, California 93106, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 31, 2005
Summary
This study compares three mechanisms—preferential growth, critical phase transitions, and highly optimized tolerance (HOT)—for generating power laws in complex systems. Criticality arises from random processes, while preferential growth and HOT involve exponential and nonexponential functions, respectively.
Area of Science:
- Complexity Science
- Statistical Physics
- Network Theory
Background:
- Power laws are observed across diverse natural and technological systems.
- Mechanisms like preferential growth, criticality, and highly optimized tolerance (HOT) are proposed to explain power-law distributions.
- Understanding these mechanisms is crucial for modeling complex systems.
Purpose of the Study:
- To directly compare preferential growth, critical phase transitions, and highly optimized tolerance (HOT) as generators of power laws.
- To analyze these mechanisms within the tractable framework of lattice percolation and forest fire models on a Cayley tree.
- To elucidate the mathematical underpinnings of power-law generation in these models.
Main Methods:
- Comparative analysis of three distinct theoretical mechanisms.
- Application to well-defined models: lattice percolation and forest fire models.
- Utilizing the analytically tractable structure of the Cayley tree for direct comparison.
Main Results:
- Criticality is characterized as a random process where power laws emerge from scale-free fluctuations.
- Preferential growth power laws are explained by competing exponential growth and decay dynamics.
- Highly Optimized Tolerance (HOT) generalizes these mechanisms, incorporating nonexponential functions derived from optimization principles.
Conclusions:
- Each mechanism offers a distinct perspective on power-law generation in complex systems.
- Criticality, preferential growth, and HOT provide complementary explanations for observed scale-free phenomena.
- The study highlights the diverse origins of power laws, from inherent randomness to adaptive optimization.
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