Fixation probability in Moran-like Processes on graphs

The well-known Isothermal Theorem was introduced in a Nature Communications article in 2005 and has since contributed to the creation of the rich field of evolutionary graph theory. The theorem states under which conditions certain Moran-like processes on graphs ("spatial Moran Processes") have the same fixation probability as the classic one-dimensional Moran Process that was introduced by Moran in 1958. Unfortunately, the Isothermal Theorem has never been proven completely. The main argument, that the projection of the process on the graph dynamics onto a one-dimensional process is a Birth-and-Death-Process, is not true in general, as the projection does not need to be Markovian. The aim of this paper is to present a more general version of the Isothermal Theorem using martingale techniques and a generalised framework using matrix notation. We follow up with a short study of small population size that shows the set of spatial Moran Processes with Moran fixation probability is even richer than previously understood. We underline the role played by the initial condition, and how individuals of the population are chosen for procreation.