|
|
| Line 1: |
Line 1: |
| '''Evolutionary graph theory''' is an area of research lying at the intersection of [[graph theory]], [[probability theory]], and [[mathematical biology]]. Evolutionary graph theory is an approach to studying how [[topology]] affects [[evolution]] of a [[population]]. That the underlying topology can substantially affect the results of the evolutionary process is seen most clearly in a paper by [[Erez Lieberman]], Christoph Hauert and [[Martin Nowak]].<ref>{{cite doi|10.1038/nature03204}}</ref>
| | 23 year-old Electrician (Special Course ) James Vancamp from Nelson, has hobbies and interests such as beachcombing, [http://artshow.warnerbrosrecords.com/forum/information-new-singapore-property-gross-sales-launches property sale singapore] developers in singapore and walking. Has lately concluded a travel to Selous Game Reserve. |
| | |
| In evolutionary graph theory, individuals occupy [[vertex (graph theory)|vertices]] of a weighted [[directed graph]] and the weight w<sub>i j</sub> of an [[Edge (graph theory)|edge]] from vertex ''i'' to vertex ''j'' denotes the probability of ''i'' replacing ''j''. The weight corresponds to the biological notion of [[Fitness (biology)|fitness]] where fitter types propagate more readily.
| |
| One property studied on graphs with two types of individuals is the ''fixation probability'', which is defined as the probability that a single, randomly placed mutant of type A will replace a population of type B. According to the ''isothermal theorem'', a graph has the same fixation probability as the corresponding [[Moran process]] if and only if it is isothermal, thus the sum of all weights that lead into a vertex is the same for all vertices. This probability is
| |
| : <math> \begin{align} | |
| \rho_M = \frac{1-r^{-1}} { 1-r^{-N} }
| |
| \end{align}</math>
| |
| where ''r'' is the relative fitness of the invading type. Thus, a [[complete graph]] with equal weights describes a Moran process.
| |
| | |
| Graphs can be classified into amplifiers of selection and suppressors of selection. If the fixation probability of a single advantageous mutation <math> \rho_G </math> is higher than the fixation probability of the corresponding [[Moran process]] <math> \rho_M </math> then the graph is an amplifier, otherwise a suppressor of selection. One example of the suppressor of selection is a linear process where only vertex ''i-1'' can replace vertex ''i'' (but not the other way around). In this case the fixation probability is <math> \rho_G = 1/N </math> (where ''N'' is the number of vertices) since this is the probability that the mutation arises in the first vertex which will eventually replace all the other ones. Since <math> \rho_G < \rho_M </math> for all ''r'' greater than 1, this graph is by definition a suppressor of selection.
| |
| | |
| Evolutionary graph theory may also be studied in a dual formulation, as a [[coalescing random walk]].
| |
| | |
| Also [[Evolutionary game theory|evolutionary games]] can be studied on graphs where again an edge between ''i'' and ''j'' means that these two individuals will play a game against each other.
| |
| | |
| Closely related stochastic processes include the [[voter model]], which was introduced by Clifford and Sudbury (1973) and independently by Holley and Liggett (1975), and which has been studied extensively.
| |
| | |
| ==Bibliography==
| |
| * {{cite jstor|2959329}}
| |
| * {{cite book |author=Liggett, Thomas M. |title=Stochastic interacting systems: contact, voter, and exclusion processes |publisher=Springer |location=Berlin |year=1999 |pages= |isbn=3-540-65995-1 |oclc= |doi= |accessdate=}}
| |
| * {{cite doi|10.1093/biomet/60.3.581}}
| |
| * {{cite book |author=Martin A. Nowak |title=Evolutionary dynamics: exploring the equations of life |publisher=Belknap Press of Harvard University Press |location=Cambridge |year=2006 |pages= |isbn=0-674-02338-2 |oclc= |doi= |accessdate=}}
| |
| | |
| ==External links==
| |
| A virtual laboratory for studying evolution on graphs:[http://www.univie.ac.at/virtuallabs/Moran/]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Evolution]]
| |
| [[Category:Graph theory]]
| |
| [[Category:Evolutionary dynamics]]
| |
23 year-old Electrician (Special Course ) James Vancamp from Nelson, has hobbies and interests such as beachcombing, property sale singapore developers in singapore and walking. Has lately concluded a travel to Selous Game Reserve.