Fig 3. Types of balanced graphs.
The class of balanced graphs includes the following families of graphs studied in the context of Moran process in the existing literature. a, Superstars [33] are the first proposed strong amplifiers of selection. b, Complete multipartite graphs [65] are a rare example of high-dimensional graphs for which the fixation probability of advantageous mutants can be expressed using an explicit formula. c, A certain form of Fan graphs [66] (with weighted and undirected edges) constitutes the strongest currently known amplifiers of selection under death-Birth updating. Theorem 4 implies that the fixation time on all those graphs is fast for all r ≥ 1. d, Not all graphs are balanced. For example, here for the highlighted node v the left hand side is 1, whereas the right-hand side is .