Evolutionary graph theory is an area of research lying at the intersection of

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, and mathematical biology. Evolutionary graph theory is an approach to studying how

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

affects

evolution Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...

of a

population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...

. 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 Martin Andreas Nowak (born April 7, 1965) is an Austrian-born professor of mathematical biology, at Harvard University since 2003. He is one of the leading researchers in the field that studies the role of cooperation in evolution. Nowak has hel ...

. In evolutionary graph theory, individuals occupy vertices of a weighted

directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...

and the weight w_{i j} of an

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...

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 In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...

with equal weights describes a Moran process. The fixation probability is :

\begin 
\rho_M = \frac  
\end

where ''r'' is the relative fitness of the invading type. 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 < \rho_M

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, 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. Closely related stochastic processes include the

voter model In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975. One can imagine that there is a "voter" at each point on a connected graph, where the c ...

, which was introduced by Clifford and Sudbury (1973) and independently by Holley and Liggett (1975), and which has been studied extensively.

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