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In evolutionary dynamics, the probability that a mutation spreads through thewhole population, having arisen in a single individual, is known as thefixation probability. In general, it is not possible to find the fixationprobability analytically given the mutant's fitness and the topologicalconstraints that govern the spread of the mutation, so one resorts tosimulations instead. Depending on the topology in use, a great number ofevolutionary steps may be needed in each of the simulation events, particularlyin those that end with the population containing mutants only. We introduce twotechniques to accelerate the determination of the fixation probability. Thefirst one skips all evolutionary steps in which the number of mutants does notchange and thereby reduces the number of steps per simulation eventconsiderably. This technique is computationally advantageous for some of theso-called layered networks. The second technique, which is not restricted tolayered networks, consists of aborting any simulation event in which the numberof mutants has grown beyond a certain threshold value, and counting that eventas having led to a total spread of the mutation. For large populations, andregardless of the network's topology, we demonstrate, both analytically and bymeans of simulations, that using a threshold of about 100 mutants leads to anestimate of the fixation probability that deviates in no significant way fromthat obtained from the full-fledged simulations. We have observed speedups oftwo orders of magnitude for layered networks with 10000 nodes.

Early appraisal of the fixation probability in directed networks