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Traversal times for random walks on small-world networks
1Consortium of the Americas for Interdisciplinary Science and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 31, 2005
Summary
We investigated random walks on small-world networks with differing step rates. A transition in traversal time was observed when edge steps were much faster than shortcut steps, revealing network percolation.
Area of Science:
- Network Science
- Statistical Physics
- Complex Systems
Background:
- Newman-Watts small-world networks are crucial models for understanding real-world systems.
- Previous studies on random walks on these networks often assumed uniform transition rates.
- Understanding traversal time is key to network efficiency and information flow.
Purpose of the Study:
- To investigate the mean traversal time of random walks on Newman-Watts small-world networks.
- To analyze the impact of differing transition rates for edge versus small-world connections.
- To identify and characterize novel transitions in traversal time behavior.
Main Methods:
- Analysis of random walks on Newman-Watts small-world networks.
- Introduction of distinct transition rates for local (edge) and long-range (small-world) steps.
- Development of a self-consistent effective-medium theory.
Main Results:
- A distinct transition in mean traversal time was observed when the rate of edge steps (f) significantly exceeded the rate of small-world steps (F).
- This transition is linked to the percolation of the random graph component of the network.
- Data collapsed onto a universal curve under these conditions, a phenomenon not seen in prior studies with equal transition rates.
- The effective-medium theory accurately describes traversal time across most parameter regimes, except near the transition point.
Conclusions:
- Varying transition rates between local and long-range steps fundamentally alters random walk dynamics on small-world networks.
- Network percolation plays a critical role in traversal time transitions under specific rate conditions.
- The developed effective-medium theory provides a valuable, albeit approximate, framework for analyzing such systems.
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