Fig 1.
Long and fast fixation times on a four-column graph GN.
a, For N = 4k, a graph GN consists of four columns of k nodes each (here k = 4 and N = 16). The grey edges within the yellow region are two-way, the black edges going from the side columns to the corresponding vertices of the middle column are one-way. The fractions shown highlight the probability that a step of the Birth-death occurs between the start and end vertex on the corresponding edge given the individual at the start vertex is selected for birth. Initially mutants occupy the left half and residents occupy the right half. As mutants (red nodes) spread upward through the third column, they can propagate along only one edge (red), whereas residents (blue nodes) fight back along multiple edges (blue). b, The timescale to fixation crucially depends on the mutant fitness advantage r. When r = 1.1 and the initial configuration S is all of the nodes on the left half, the fixation time Tr(GN, S) is exponential in N, whereas when r = 100 it is polynomial. Each data point is an average over at least 103 simulations.
Fig 2.
Fixation time is not monotone in r.
a, In an (undirected) star graph S4 on 4 nodes, one node (center) is connected to three leaf nodes by two-way edges. When the initial mutant appears at a leaf v, the fixation time Tr(S4, {v}) increases as r increases from r = 1 to roughly r = 1.023. Then it starts to decrease. b, Normalized fixation time Tr(G, {v})/Tr = 1(G, {v}) as a function of r ∈ [1, 1.3], for all 83 strongly connected graphs G with 4 nodes, and all four possible mutant starting nodes v. As r increases, the fixation time goes up for 182 of the 4 ⋅ 83 = 332 possible initial conditions. The increase is most pronounced for the so-called lollipop graph L4 and a starting node u. In contrast, for the same lollipop graph and a different starting node w, the fixation time decreases the fastest.
Fig 3.
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 .
Fig 4.
Fixation time and fixation probability of a single mutant under uniform initialization for all 5048 graphs with N = 5 nodes, for a, r = 1.1 and b, r = 2. Each graph is represented as a colored dot. The undirected graphs (with all edges two-way) are labeled in blue. The oriented graphs (with no edges two-way) are labeled in green. All other directed graphs are labeled in orange. The slowest graph is the (undirected) Star graph S5. Among the oriented graphs, the slowest graph is the Fan graph F5.
Fig 5.
Fixation time on slow oriented graphs.
a, The Fan graph with k blades has N = 2k + 1 nodes and 3k one-way edges (here k = 5 which yields N = 11). The Vortex graph with batch size k has N = 2k + 2 nodes and 4k edges (here k = 3 which yields N = 8). b-c, For both the Fan graphs and the Vortex graphs the fixation time scales roughly as N2, both for r = 1.1 and r = 100. (Each data point is an average over 1000 simulations).