Moran graphs: Difference between revisions
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::''Circulation Theorem:'' The Moran process on a graph results in the same fixation probability \(\rho_1\) of a single mutant as in an unstructured population if the graph is a circulation. | ::''Circulation Theorem:'' The Moran process on a graph results in the same fixation probability \(\rho_1\) of a single mutant as in an unstructured population if the graph is a circulation. | ||
An [[#Fixation probability on circulation graphs|illustrative sketch of the proof]] follows below. In contrast to unstructured (well-mixed) populations, the state of a structured population is not simply determined by just the number of mutants (or residents) but rather needs to take the distribution of mutants and residents into account. Rather surprisingly, however, it turns out that for circulations the spatial distribution of mutants and residents does not affect their fixation probabilities. The observation that fixation probabilities remain unchanged for diverse population structures forms the basis for the conjecture by [[#References|Maruyama (1970)]] and [[#References|Slatkin (1981)]] speculating that population structure does not affect fixation probabilities. Indeed this is true for structures as diverse as [https://en.wikipedia.org/wiki/Complete_graph complete graphs] (every vertex is connected to every other vertex), lattices or [https://en.wikipedia.org/wiki/Random_regular_graph random regular graphs] | An [[#Fixation probability on circulation graphs|illustrative sketch of the proof]] follows below. In contrast to unstructured (well-mixed) populations, the state of a structured population is not simply determined by just the number of mutants (or residents) but rather needs to take the distribution of mutants and residents into account. Rather surprisingly, however, it turns out that for circulations the spatial distribution of mutants and residents does not affect their fixation probabilities. The observation that fixation probabilities remain unchanged for diverse population structures forms the basis for the conjecture by [[#References|Maruyama (1970)]] and [[#References|Slatkin (1981)]] speculating that population structure does not affect fixation probabilities. Indeed this is true for structures as diverse as [https://en.wikipedia.org/wiki/Complete_graph complete graphs] (every vertex is connected to every other vertex), lattices or [https://en.wikipedia.org/wiki/Random_regular_graph random regular graphs]. | ||
Last but not least, the circulation theorem indirectly suggests that some population structures ''do'' affect fixation probabilities. Thus, it becomes an intriguing question what population structures may act as [[evolutionary suppressors]] that suppress selection and enhance random drift (\(\rho<\rho_1\) for \(r > 1\)) or, conversely, as [[evolutionary amplifiers]] that enhance selection and suppresses random drift (\(\rho>\rho_1\) for \(r > 1\)). | Last but not least, the circulation theorem indirectly suggests that some population structures ''do'' affect fixation probabilities. Thus, it becomes an intriguing question what population structures may act as [[evolutionary suppressors]] that suppress selection and enhance random drift (\(\rho<\rho_1\) for \(r > 1\)) or, conversely, as [[evolutionary amplifiers]] that enhance selection and suppresses random drift (\(\rho>\rho_1\) for \(r > 1\)). | ||
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<div class="lab_description Moran"> | <div class="lab_description Moran"> | ||
[[Image:1D lattice (N=150, k=2).png|left|200px|link=EvoLudoLab: Moran process on the | [[Image:1D lattice (N=150, k=2).png|left|200px|link=EvoLudoLab: Moran process on the cycle graph]] | ||
==== [[EvoLudoLab: Moran process on the | ==== [[EvoLudoLab: Moran process on the cycle graph|Evolutionary dynamics on the cycle graph]] ==== | ||
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<div class="lab_description Moran"> | <div class="lab_description Moran"> | ||
[[Image:1D lattice (N=150, k=2).png|left|200px|link=EvoLudoLab: Fixation probabilities on the | [[Image:1D lattice (N=150, k=2).png|left|200px|link=EvoLudoLab: Fixation probabilities on the cycle graph]] | ||
==== [[EvoLudoLab: Fixation probabilities on the | ==== [[EvoLudoLab: Fixation probabilities on the cycle graph|Fixation probabilities on the cycle graph]] ==== | ||
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<div class="lab_description Moran"> | <div class="lab_description Moran"> | ||
[[Image:1D lattice (N=150, k=2).png|left|200px|link=EvoLudoLab: Fixation times on the | [[Image:1D lattice (N=150, k=2).png|left|200px|link=EvoLudoLab: Fixation times on the cycle graph]] | ||
==== [[EvoLudoLab: Fixation times on the | ==== [[EvoLudoLab: Fixation times on the cycle graph|Fixation times on the cycle graph]] ==== | ||
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===References=== | ===References=== | ||
#Maruyama, T. (1974) A Simple Proof that Certain Quantities are Independent of the Geographical Structure of Population ''Theor. Pop. Biol.'' '''5''' 148-154. | #Maruyama, T. (1974) A Simple Proof that Certain Quantities are Independent of the Geographical Structure of Population ''Theor. Pop. Biol.'' '''5''' 148-154. | ||
#Slatkin, M. (1981) Fixation probabilities and fixation times in a subdivided population ''Evolution'' '''35''' 477-488. | #Slatkin, M. (1981) Fixation probabilities and fixation times in a subdivided population ''Evolution'' '''35''' 477-488. | ||
[[Category:Evolutionary graph theory]][[Category:Tutorial]] | [[Category:Evolutionary graph theory]][[Category:Tutorial]] | ||
