Last modified: 2014-04-01

#### Abstract

Spatial population structure can substantially change the stochastic dynamics of mutants invading a resident population. One way to introduce population structure is putting the individuals on the nodes of a network. Individuals can be replaced by offspring of their neighboring nodes, connected through undirected links. A quantity of interest is the probability that the mutant takes over the entire population, the fixation probability. Regular networks where every node has the same number of links (e.g. rings or lattices) are called isothermal and lead to the same fixation probability as a well-mixed population. A network that increases the fixation probability of advantageous mutants and decreases it for disadvantageous mutants compared to a well-mixed system, is called an amplifier of selection. The rate of evolution, i.e. the average waiting time for the first successful mutant to appear, is increased by amplifiers of selection. But besides leading to a higher fixation probability, the amplifiers can also increase the time until fixation of the advantageous mutants, slowing down evolution.

We analyze small graphs to explore the impact of network structure on the fixation time. In the simplest case of four nodes, there are six different connected networks. Among these, the fully connected network (corresponding to the well-mixed population) and the ring are isothermal. Surprisingly, the other four networks are amplifiers of selection and hence there are no suppressors of selection of size four. All these networks increase the fixation time of the mutants compared to the complete network. Intuitively, one might assume that the removal of a link from a given network would increase fixation time. But a simple counter-example shows that the fixation time can also decrease by removing a link.