Phase transitions of the Moran process and algorithmic consequences

Abstract

The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches “fixation,” where all vertices are mutants, or “extinction,” where none are. Our main result is an almost‐tight upper bound on expected absorption time. For all ϵ>0, we show that the expected absorption time on an n‐vertex graph is o(n3+ϵ). Specifically, it is at most n3eOfalse(false(loglognfalse)3false), and there is a family of graphs where it is Ω(n3). In proving this, we establish a phase transition in the probability of fixation, depending on the mutants' fitness r. We show that no similar phase transition occurs for digraphs, where it is already known that the expected absorption time can be exponential. Finally, we give an improved fully polynomial randomized approximation scheme (FPRAS) for approximating the probability of fixation. On degree‐bounded graphs where some basic properties are given, its running time is independent of the number of vertices.

Fast and strong amplifiers of natural selection

“…Recent research has shown the existence of strong amplifiers that are undirected and unweighted 27 , 29 . The absence of both weights and directions on the edges is known to lead to timescale that is polynomial in N 45 , 46 . However, this polynomial timescale still remains considerably slower than the logarithmic timescale of the well-mixed population.…”

“…However, these structures operate on long timescales where mutants and residents coexist for many generations until the population reaches a homogeneous state. The search for faster strong amplifiers has lead to structures, such as the Dense Incubators D N 27 , 29 , with timescale that is polynomial in N 45 , 46 . Since well-mixed populations resolve in generations, strong amplification still comes at the cost of a substantial increase in the timescale.…”

An extension of the Moran process using type-specific connection graphs

“…The lower bound in the undirected case remained 1 n , but the upper bound was significantly improved by Mertzios et al [16] to 1 − Ω(n −3/4 ), when r is independent of n. It was again improved by Giakkoupis [13] to 1 − Ω r −5/3 n −1/3 log −4/3 n , and finally by Goldberg et al [15] to 1 − Ω(n −1/3 ) where they also found a graph which shows that this is tight. While the general belief was that there are no undirected strong suppressors, Giakkoupis [13] showed that there is a class of graphs with fixation probability O(r 2 n −1/4 log n) and Goldberg et al [17] found a stronger suppressor with fixation probability O(r 2 n −1/2 ), opening the way for a potentially optimal strong suppressor to be discovered.…”

“…In that paper they prove NP-hardness and #P-hardness on the computation of the fixation probabilities for the aforementioned settings respectively, and they also prove PSPACE inclusion for both. Regarding the problem of computing the fixation probability in the single-graph setting, [20] and [17] provide fully polynomial, randomized time approximation scemes (FPRAS) which have significantly improved running time compared to the previous algorithm of [3].…”

Coexistence times in the Moran process with environmental heterogeneity

“…For example, it is known that for Star graphs, the process terminates after roughly N2log⁡N steps [28]. More generally, when the underlying graph is undirected (that is, all edges are two-way) then the process terminates after a number of steps that is polynomial in N [29,30]. The trade-off between fixation probability and fixation time has also been studied [31,32].…”