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Fixation and consensus times on a network: a unified approach
G J Baxter1, R A Blythe, A J McKane
1School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand.
Physical Review Letters
|December 31, 2008
Summary
This study introduces a unified framework for stochastic models in biodiversity, genetics, language, and opinion dynamics. A key finding shows that the time to reach consensus is independent of network structure in language evolution models.
Area of Science:
- Complex Systems
- Mathematical Biology
- Network Science
Background:
- Stochastic models are crucial for understanding complex systems.
- Previous models often focused on specific phenomena like population genetics or opinion dynamics.
- A unified approach is needed to identify common principles across diverse fields.
Purpose of the Study:
- To develop a common stochastic framework applicable to biodiversity, population genetics, language evolution, and opinion dynamics.
- To derive a general expression for the mean time to fixation or consensus in networked systems.
- To investigate the impact of network structure on system dynamics.
Main Methods:
- Developed a generalized stochastic model with nodes having states between 0 and 1.
- Introduced interaction strengths m(ij) to define node interdependencies.
- Derived an approximate analytical expression for the mean time to reach fixation or consensus.
Main Results:
- A unified mathematical framework was established for diverse stochastic processes.
- An approximate expression for the mean time to fixation/consensus was derived.
- In language evolution models, this time was found to be independent of the underlying network structure.
Conclusions:
- The common framework simplifies the analysis of complex systems.
- Network topology does not influence the speed of consensus in language evolution.
- This finding has implications for understanding cultural and linguistic change.
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