Abstract
Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth–death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations.
Highlights
Most analytical results for evolutionary dynamics have been obtained for well-mixed populations, regular networks or deme-structures
Moran process corresponds to the complete network, where every node is adjacent to all other nodes, which implies that the probability of being replaced is equal for all individuals
To assess the total time the process spends before absorption, we look at the so-called unconditional fixation time
Summary
Most analytical results for evolutionary dynamics have been obtained for well-mixed populations, regular networks or deme-structures. It turns out that amplifiers of selection can at the same time slow down the time until fixation Such an approach would have the drawback that while the time until a mutant appears is decreased, the time until it takes over is increased [2,3]. We consider all six possible undirected networks in detail and calculate (i) the probability of fixation, (ii) the time to fixation, and (iii) the sojourn times depending on the fitness of the mutant. This approach illustrates that the time to fixation can increase when a link is added to the network. Our approach shows that the optimal starting point for a novel mutation in terms of its fixation time depends on the fitness of this mutation
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