Universality of fixation probabilities in randomly structured populations

Area of Science:

• Evolutionary biology
• Population genetics
• Network science

Background:

• Population structure influences evolutionary dynamics, but analytical results for random structures are limited.
• Existing theorems, like the isothermal theorem, often assume uniform interaction probabilities, which is unrealistic for biological systems.
• Biological populations exhibit variation and noise, necessitating models that account for complex interaction structures.

Purpose of the Study:

• To investigate the fixation probability of mutants in finite populations with general random graph structures under constant selection.
• To determine if evolutionary outcomes in structured populations converge to those in well-mixed populations.
• To extend the applicability of evolutionary theorems to more realistic, noisy population structures.

Main Methods:

• Modeling populations as weighted, directed random graphs where vertices are individuals and edges represent interactions.
• Establishing a robustness result for the isothermal theorem.
• Employing large deviation estimates to analyze the typical structure of random graphs.
• Simulating fixation probabilities on various network types (perturbed lattices, small-world, scale-free).

Main Results:

• The fixation probability of an invading mutant in a generalized Erdős-Rényi random graph model is approximately the same as in a well-mixed population with high probability.
• Simulations on perturbed lattices, small-world networks, and scale-free networks show similar behavior.
• A conjecture is proposed that the fixation probability in well-mixed populations is a universal predictor for many large random graph models.

Conclusions:

• The fixation probability in many structured populations, even with variation and noise, closely resembles that of well-mixed populations.
• This finding suggests a degree of universality in evolutionary dynamics across different population structures.
• The results have implications for understanding evolutionary processes in diverse biological systems with complex interaction networks.