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Self-avoiding walks and connective constants in small-world networks.
Carlos P Herrero1, Martha Saboyá
1Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones Científicas (CSIC), Campus de Cantoblanco, 28049 Madrid, Spain.
Summary
This study explores long-distance network properties using self-avoiding walks (SAWs) on rewired lattices. Network connectivity increases with disorder, showing distinct behaviors at low and high rewiring probabilities.
Area of Science:
- Network Science
- Statistical Physics
Background:
- Small-world networks exhibit unique long-distance characteristics.
- Self-avoiding walks (SAWs) are a key tool for studying these properties.
Purpose of the Study:
- To investigate how rewiring regular lattices affects network long-distance properties.
- To analyze the behavior of self-avoiding walks on these modified networks.
Main Methods:
- Generating networks by rewiring links in 1D and 2D regular lattices.
- Employing numerical simulations to calculate the number of SAWs (u(n)) over n steps.
- Analyzing the connective constant (μ) as a measure of long-distance behavior.
Main Results:
- The connective constant (μ) increases with disorder strength (rewiring probability, p).
- A linear relationship (μ = μ(0) + ap) is observed for small p.
- Network behavior approaches that of random graphs as p approaches 1.
Conclusions:
- Numerical simulations and analytical approaches agree for low rewiring probabilities.
- Discrepancies arise at high rewiring probabilities due to differing connectivity distributions.
- Rewiring probability is a critical factor in determining the long-distance characteristics of networks.