The Moran Process

I will survey the Moran process, which models the spread of genetic 
mutations through populations.  The population is modelled by the 
vertices of a graph.  At the start, one vertex is chosen uniformly at 
random to receive the mutation.  Then, at each step of the process, a 
vertex is chosen to reproduce, with probability proportional to its 
"fitness", a parameter of the model.  It copies its mutant/non-mutant 
status to a uniformly random neighbour.  With probability 1, the 
mutation either dies out ("extinction") or spreads to every vertex 
("fixation").

The key quantities of interest are the probability of fixation and the
expected number of steps before fixation/extinction, and how they
depend on the graph and the relative fitness of the mutation.  We have
shown that the expected number of steps is polynomial in the size of
the graph, and this allows for very efficient approximation algorithms
for fixation probability.

Surprisingly, when mutants have higher fitness than non-mutants, there
are classes of graphs on which the fixation probability tends to 1 as
the graphs get larger.  This happens despite the initial mutant being
overwhelmingly outnumbered by non-mutants.

The talk will be aimed at a general audience, with no specialist
knowledge assumed.