862
edits
Evolutionary graph theory: Difference between revisions
From EvoLudo
m
no edit summary
No edit summary |
mNo edit summary |
||
| Line 12: | Line 12: | ||
[[Image:Moran graph (lattice).png|thumb|300px|On lattices the [[spatial Moran process]] results in the same fixation probability of a single (or several) mutants as the original [[Moran process]] in unstructured populations.]] | [[Image:Moran graph (lattice).png|thumb|300px|On lattices the [[spatial Moran process]] results in the same fixation probability of a single (or several) mutants as the original [[Moran process]] in unstructured populations.]] | ||
Population structure can be modelled by arranging individuals on a graph such that each individual is represented by a vertex and its neighbourhood is defined through links connecting it to other individuals. Interestingly, for the [[spatial Moran process]] the fixation probability remains unaffected for a large class of population structures (graphs known as circulations), i.e. is the same as for the original [[Moran process]] in unstructured populations. For a single mutant this is | Population structure can be modelled by arranging individuals on a graph such that each individual is represented by a vertex and its neighbourhood is defined through links connecting it to other individuals. Interestingly, for the [[spatial Moran process]] the fixation probability remains unaffected for a large class of population structures (graphs known as circulations), i.e. is the same as for the original [[Moran process]] in unstructured populations. For a single mutant this is | ||
\ | \begin{align} | ||
\rho_1 = \frac{\displaystyle 1-\frac1r}{\displaystyle 1-\frac1{r^N}} | \rho_1 = \frac{\displaystyle 1-\frac1r}{\displaystyle 1-\frac1{r^N}} | ||
\ | \end{align} | ||
and is even the same regardless of the initial location of the mutant. This applies, for example, for the fixation probabilities of mutants on the complete graph where every vertex is connected to every other one, on cycles, on lattices, on random regular graphs, and many more. Interestingly, however, the same is not true for fixation times. In fact, fixation times sensitively depend on the details of the population structure. However, sufficiently [[Graph symmetries|symmetrical graphs]] can at least ensure that the fixation time does not depend on the initial location of a mutant. | and is even the same regardless of the initial location of the mutant. This applies, for example, for the fixation probabilities of mutants on the complete graph where every vertex is connected to every other one, on cycles, on lattices, on random regular graphs, and many more. Interestingly, however, the same is not true for fixation times. In fact, fixation times sensitively depend on the details of the population structure. However, sufficiently [[Graph symmetries|symmetrical graphs]] can at least ensure that the fixation time does not depend on the initial location of a mutant. | ||
{{-}} | {{-}} | ||
| Line 31: | Line 31: | ||
Interestingly, it is also possible to create population structures that amplify selection and suppress random drift. For example, on the star graph, where all nodes are connected to a central hub and vice versa, the fixation probability of a single, randomly placed mutant becomes | Interestingly, it is also possible to create population structures that amplify selection and suppress random drift. For example, on the star graph, where all nodes are connected to a central hub and vice versa, the fixation probability of a single, randomly placed mutant becomes | ||
\[ | \[ | ||
\rho_2 = \frac{\displaystyle 1-\frac1{r^2}}{\displaystyle 1-\frac1{r^{2N}}} | \rho_2 = \frac{\displaystyle 1-\frac1{r^2}}{\displaystyle 1-\frac1{r^{2N}}} | ||
\] | \] | ||
for large \(N\). Thus, any selective difference \(r\) is amplified to \(r^2\). Even more intriguingly, population structures exist that act as arbitrarily strong amplifiers of selection and suppressors of random drift. However, note that amplification has its price in that the average fixation time goes to infinity as the amplification increases. | for large \(N\). Thus, any selective difference \(r\) is amplified to \(r^2\). Even more intriguingly, population structures exist that act as arbitrarily strong amplifiers of selection and suppressors of random drift. However, note that amplification has its price in that the average fixation time goes to infinity as the amplification increases. | ||
