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Diffusion, annihilation, and chemical reactions in complex networks with spatial constraints
Thorsten Emmerich1, Armin Bunde, Shlomo Havlin
1Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 11, 2012
Summary
We studied diffusion and reaction dynamics on complex Erdős-Rényi networks. The system
Area of Science:
- Statistical Physics
- Complex Networks
- Dynamical Processes
Background:
- Erdős-Rényi networks are fundamental models in network science.
- Previous studies explored network dimension (d) as a function of link-length exponent (δ).
- Dynamics on networks are influenced by their structural properties.
Purpose of the Study:
- Investigate diffusion, annihilation, and chemical reaction dynamics on spatially embedded Erdős-Rényi networks.
- Determine how network dimension (d) and random walk dimension (dw) control these processes.
- Analyze the impact of long-range links on dynamical processes.
Main Methods:
- Simulated random walkers on networks embedded in 1D and 2D Euclidean spaces.
- Analyzed average distance traveled (
) and return probability (P0(t)) over time (t). - Calculated network dimension (d) and random walk dimension (dw) as functions of δ and embedding dimension (de).
Main Results:
- Network dynamics are controlled by the system's dimension (d).
- The ratio d/dw governs the survival fraction in annihilation (A+A→0) and reaction (A+B→0) processes.
- Established relationships for ordered/disordered lattices hold for complex networks with long-range links.
Conclusions:
- The dimension (d) and random walk dimension (dw) are key parameters for understanding dynamics on complex networks.
- The d/dw ratio provides a universal descriptor for survival probabilities in reaction-diffusion systems.
- Findings extend the validity of established lattice theories to complex, long-range connected network structures.
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