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The Moran process on graphs is a popular model to study the dynamics ofevolution in a spatially structured population. Exact analytical solutions forthe fixation probability and time of a new mutant have been found for only afew classes of graphs so far. Simulations are time-expensive and manyrealizations are necessary, as the variance of the fixation times is high. Wepresent an algorithm that numerically computes these quantities for arbitrarysmall graphs by an approach based on the transition matrix. The advantage oversimulations is that the calculation has to be executed only once. Building thetransition matrix is automated by our algorithm. This enables a fast andinteractive study of different graph structures and their effect on fixationprobability and time. We provide a fast implementation in C with this notehttps://github.com/hindersin/efficientFixation. Our code is very flexible, asit can handle two different update mechanisms (Birth-death or death-Birth), aswell as arbitrary directed or undirected graphs.

Exact numerical calculation of fixation probability and time on graphs