The Moran Process on a Random Graph

We study the fixation probability for two versions of the Moran process on the random graph Gn,p$$ {G}_{n,p} $$ at the threshold for connectivity. The Moran process models the spread of a mutant population in a network. Throughout the process, there are vertices of two types, mutants, and non-mutants. Mutants have fitness s$$ s $$ and non-mutants have fitness 1. The process starts with a unique individual mutant located at the vertex v0$$ {v}_0 $$. In the Birth-Death version of the process a random vertex is chosen proportionally to its fitness and then changes the type of a random neighbor to its own. The process continues until the set of mutants X$$ X $$ is empty or [n]$$ \left[n\right] $$. In the Death-Birth version, a uniform random vertex is chosen and then takes the type of a random neighbor, chosen according to fitness. The process again continues until the set of mutants X$$ X $$ is empty or [n]$$ \left[n\right] $$. The fixation probability is the probability that the process ends with X=& empty;$$ X=\varnothing $$. We show that asymptotically correct estimates of the fixation probability depend only on the degree of v0$$ {v}_0 $$ and its neighbors. In some cases we can provide values for these estimates and in other places we can only provide non-linear recurrences that could be used to compute values.