The fixation probability is the probability that a new mutant introduced in a homogeneous population eventually takes over the entire population. The fixation probability is a fundamental quantity of natural selection, and known to depend on the population structure. Amplifiers of natural selection are population structures which increase the fixation probability of advantageous mutants, as compared to the baseline case of well-mixed populations. In this work we focus on symmetric population structures represented as undirected graphs. In the regime of undirected graphs, the strongest amplifier known has been the Star graph, and the existence of undirected graphs with stronger amplification properties has remained open for over a decade. In this work we present the Comet and Comet-swarm families of undirected graphs. We show that for a range of fitness values of the mutants, the Comet and Cometswarm graphs have fixation probability strictly larger than the fixation probability of the Star graph, for fixed population size and at the limit of large populations, respectively. Evolutionary dynamics study populations of reproducing individuals and the composition of the population over the course of time. A fundamental quantity is the fixation probability 1-10 , which characterizes the chances of an invading mutant to get fixed in a homogeneous population of residents. The most well-known mathematical model for studying evolutionary dynamics on finite populations is the birth-death Moran process 11 . Initially, a population of N individuals consists of two types: N − 1 residents, and 1 invading mutant. The residents are associated with a normalized fitness of 1, whereas the invading mutant has a fitness advantage r > 1, which is constant and independent of the composition of the population. The population size remains fixed over the course of time. At each time point, an individual is chosen for reproduction with probability proportional to its fitness, and its offspring replaces an individual chosen uniformly at random. In this setting, the population is well-mixed, as the reproducing individual may replace any other individual. The fixation probability is defined as the probability that the Moran process results in a population of N mutants (i.e., the mutants get fixed in the population). The fixation probability, ρ, for well-mixed populations is a function of r and N, and equals Published: xx xx xxxx OPEN www.nature.com/scientificreports/ 2 Scientific RepoRts | 7: 82 |
The work of REF introduced the Metastars as a family of unweighted undirected graphs with better amplification properties than Stars, for specific values of the fitness advantage r. Other aspects of the Moran process on graphs have also been studied in the literature.
Amplification on Undirected Population Structures: Comets Beat Stars
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