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Published on: October 13, 2023
Kleinberg navigation in fractal small-world networks
Mickey R Roberson1, Daniel ben-Avraham
1Department of Physics, Clarkson University, Potsdam, NY 13699-5820, USA.
Summary
Efficient navigation in fractal networks is achieved when long-range links follow a power-law distribution. The optimal exponent relates to the fractal dimension, with finite-size corrections observed.
Area of Science:
- Complex networks
- Network theory
- Fractal geometry
Background:
- The Kleinberg problem investigates efficient navigation strategies in complex networks.
- Small-world networks exhibit properties of both regular and random networks.
- Fractal lattices present unique topological characteristics impacting network dynamics.
Purpose of the Study:
- To investigate the Kleinberg navigation problem on fractal lattices.
- To determine the optimal distribution of long-range links for efficient navigation.
- To analyze the impact of fractal dimensions on navigation efficiency.
Main Methods:
- Extensive numerical simulations were conducted.
- Analysis of navigation efficiency based on link length distribution.
- Investigation of finite-size effects on network properties.
Main Results:
- Optimal navigation occurs when long-range link lengths follow P(r) ~ r^(-alpha).
- The exponent alpha is directly related to the fractal dimension of the lattice (alpha=d(f)).
- Finite-size corrections to the exponent alpha are proportional to 1/(ln N)^2.
Conclusions:
- The study confirms theoretical predictions for navigation on fractal networks.
- Fractal dimension is a key parameter governing efficient navigation strategies.
- Understanding finite-size effects is crucial for accurate modeling of large fractal networks.
