Amplifiers for the Moran Process

Abstract

The Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. A formal definition is provided in the paper. Strong amplification is a rather surprising property -it means that in such graphs, the fixation probability of a uniformly-placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n). The name "strongly amplifying" comes from the fact that this selective advantage is "amplified". Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly-amplifying families -superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. In this paper, we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family "megastars". When the algorithm is run on an n-vertex graph in this family, starting with a uniformly-chosen mutant, the extinction probability is roughly n −1/2 (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n). Finally, we prove that our analysis of megastars is fairly tightthere is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counter-example which clarifies the literature concerning the isothermal theorem of Lieberman et al.

Limits on amplifiers of natural selection under death-Birth updating

“…As a consequence, there are no super amplifiers under dB updating. These results are in sharp contrast to the Bd Moran process, for which amplifiers and super amplifiers have been constructed repeatedly [16,27,30,32], and, in fact, can be abundant [34]. Our findings suggest that the existence of amplifiers is sensitive to specific mechanisms of the evolutionary process, and hence their biological realization depends on which process captures actual population dynamics more faithfully.…”

Mutants and Residents with Different Connection Graphs in the Moran Process

“…They introduced the notions of amplifiers and suppressors of selection, a categorization of graphs based on the comparison of their fixation probabilities with that of the complete graph. They also found a sufficient condition (in fact [4] corrects the claim in [10] that the condition is also necessary) for a digraph to have the fixation probability of the complete graph, but a necessary condition is yet to be found.…”

Phase transitions of the Moran process and algorithmic consequences

“…In Section 1.2, we alluded to the fact that the problem of finding the strongest possible amplifier had essentially been solved. Any directed graph has extinction probability Ω( n −1/2 ) , which is tight up to a polylogarithmic factor , and any undirected graph has extinction probability Ω( n −1/3 ), which is tight up to a constant factor . In fact, the results of generalize to sparse graphs; any m ‐edge undirected graph has extinction probability , which is also tight up to a constant factor.…”

“…It is natural to ask: how strong can an n ‐vertex amplifier or suppressor be? For amplifiers this problem has essentially been solved for both directed and undirected graphs ; see Section 1.4 for details. For suppressors, much less is known; to our knowledge, the strongest family of (both directed and undirected) suppressors in the literature is due to Giakkoupis , and has fixation probability .…”