Early appraisal of the fixation probability in directed networks

TL;DR

This paper introduces two techniques to efficiently estimate fixation probabilities in directed networks, significantly reducing computational time while maintaining accuracy, especially for large populations and layered network topologies.

Contribution

The authors propose novel acceleration methods for fixation probability simulations, including skipping unchanged mutant steps and using thresholds to approximate fixation outcomes.

Findings

Speedups of up to 100 times for layered networks with 10,000 nodes.

Threshold-based approximation yields accurate fixation probability estimates.

Techniques are effective across different network topologies.

Abstract

In evolutionary dynamics, the probability that a mutation spreads through the whole population, having arisen in a single individual, is known as the fixation probability. In general, it is not possible to find the fixation probability analytically given the mutant's fitness and the topological constraints that govern the spread of the mutation, so one resorts to simulations instead. Depending on the topology in use, a great number of evolutionary steps may be needed in each of the simulation events, particularly in those that end with the population containing mutants only. We introduce two techniques to accelerate the determination of the fixation probability. The first one skips all evolutionary steps in which the number of mutants does not change and thereby reduces the number of steps per simulation event considerably. This technique is computationally advantageous for some of the so-called layered networks. The second technique, which is not restricted to layered networks, consists of aborting any simulation event in which the number of mutants has grown beyond a certain threshold value, and counting that event as having led to a total spread of the mutation. For large populations, and regardless of the network's topology, we demonstrate, both analytically and by means of simulations, that using a threshold of about 100 mutants leads to an estimate of the fixation probability that deviates in no significant way from that obtained from the full-fledged simulations. We have observed speedups of two orders of magnitude for layered networks with 10000 nodes.