Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality

Area of Science:

• Evolutionary biology
• Mathematical modeling
• Theoretical ecology

Background:

• Evolutionary graph theory (EGT) analyzes evolutionary dynamics on graphs.
• Key goals include determining fixation probability and fixation time for mutants.
• Fixation time distributions remain less understood than fixation probabilities.

Purpose of the Study:

• To derive exact expressions for conditional characteristic functions (CCFs) related to fixation and extinction times.
• To provide a novel method for analyzing fixation time distributions in EGT.
• To enable exploration of parameter dependence on graph structure.

Main Methods:

• Identification of a product martingale for evolutionary graphs with multiple partitions.
• Derivation of CCFs for a proxy of fixation/extinction times based on mutant population size changes.
• Utilizing martingale properties applicable across various graph structures and partition numbers.

Main Results:

• First derivation of CCFs for fixation time proxies on graphs with any number of partitions.
• Explicit parameter dependence in derived CCFs allows for detailed analysis of graph structure effects.
• Demonstration of martingales as a powerful, generalizable tool for EGT problems.

Conclusions:

• The derived CCFs offer a significant advancement in understanding fixation time distributions in EGT.
• Martingales provide a robust framework for studying diverse graph families simultaneously.
• This work opens new avenues for analyzing evolutionary dynamics on complex networks.