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Moran graphs: Difference between revisions
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Recall that the number of mutants in the population changes only if a replacement occurs along any link that connects residents with mutants or vice versa. Because the Moran process essentially selects links with a probability proportional to the link weight and the fitness of the vertex at its tail, it follows that for each subset, the probability that another mutant is added is simply given by \(r/(1 + r)\) and the complementary probability to remove one mutant is \(1/(1 + r)\). Since this holds for each subset, it also holds for the entire population and is independent of the number, size, shape and distribution of mutant subsets. This invariance applies only if the circulation theorem is satisfied. Consequentially, the fixation probability on circulation graphs reduces to the recursive equation derived for the original [[Moran process]]. | Recall that the number of mutants in the population changes only if a replacement occurs along any link that connects residents with mutants or vice versa. Because the Moran process essentially selects links with a probability proportional to the link weight and the fitness of the vertex at its tail, it follows that for each subset, the probability that another mutant is added is simply given by \(r/(1 + r)\) and the complementary probability to remove one mutant is \(1/(1 + r)\). Since this holds for each subset, it also holds for the entire population and is independent of the number, size, shape and distribution of mutant subsets. This invariance applies only if the circulation theorem is satisfied. Consequentially, the fixation probability on circulation graphs reduces to the recursive equation derived for the original [[Moran process]]. | ||
Note that even though the fixation probabilities remain unchanged on circulation graphs, the corresponding fixation times are very sensitive to the details of the population structure and pose a much harder problem. Moreover, fixation times generally depend on the location of the initial mutant. Only additional [[Graph symmetries|structural symmetries]] can ensure that the fixation times are independent of the location on one graph but not across different graphs. The circulation theorem only ensures that the ''ratio'' of the transition probabilities \( | Note that even though the fixation probabilities remain unchanged on circulation graphs, the corresponding fixation times are very sensitive to the details of the population structure and pose a much harder problem. Moreover, fixation times generally depend on the location of the initial mutant. Only additional [[Graph symmetries|structural symmetries]] can ensure that the fixation times are independent of the location on one graph but not across different graphs. The circulation theorem only ensures that the ''ratio'' of the transition probabilities \(T_i^+/T_i^− = r\) remains unchanged, i.e. independent of the number of mutants \(i\) and their distribution. However, even on circulation graphs both \(T_i^+\) and \(T_i^−\) depend not only on the number but also on the distribution of mutants but those effects cancel for their ratio. Generally, population structures tend to substantially increase the fixation times because the structure limits the possibilities for mutants to conquer new vertices. | ||
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