In evolutionary graph theory, individuals occupy
vertices of a weighted
directed graph and the weight wi j of an
edge from vertex
i to vertex
j denotes the probability of
i replacing
j. The weight corresponds to the biological notion of
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. Thus, for example, a
complete graph with equal weights describes a Moran process. The fixation probability is : \begin{align} \rho_M = \frac{1-r^{-1}} { 1-r^{-N} } \end{align} where
r is the relative fitness of the invading type.
Selection amplifiers and suppressors Graphs can be classified into amplifiers of selection and suppressors of selection. If the fixation probability of a single advantageous mutation \rho_G is higher than the fixation probability of the corresponding
Moran process \rho_M 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 \rho_G = 1/N (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 \rho_G for all
r greater than 1, this graph is by definition a suppressor of selection.
Alternative formulations Evolutionary graph theory may also be studied in a dual formulation, as a
coalescing random walk, or as a
stochastic process. We may consider the mutant population on a graph as a
random walk between absorbing barriers representing mutant extinction and mutant fixation. For highly symmetric graphs, we can then use martingales to find the
fixation probability as illustrated by Monk (2018). Also
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. == Related concepts ==