Fig 1.
Classifying the effects of graph structure on weak selection.
(A) The fixation probability ρG(r), for a mutation of fitness r on a graph G, can be expanded under weak selection as . The zeroth-order coefficient, ρ∘, is the fixation probability of a neutral mutation, while the first-order coefficient, ρ′, determines the effect of weak selection. (B) The effects of graph structure on weak selection can be classified using ρ∘ and ρ′. All isothermal graphs (black dot) have ρ∘ = 1/N and ρ′ = (N − 1)/(2N). We say a graph is an absolute amplifier if ρ′ > (N − 1)/(2N), and an absolute suppressor if ρ′ < (N − 1)/(2N). We say a graph is a relative amplifier if ρ′/ρ∘ > (N − 1)/2 and a relative suppressor if ρ′/ρ∘ < (N − 1)/2. The three possible combinations for temperature initialization are shown in the colored regions.
Fig 2.
Exhaustive analysis of fixation probabilities under weak selection for small graphs.
(A) The values of ρ∘ and ρ′ are plotted for all 11,716,571 connected unweighted graphs of size 10. Colors correspond to the classification of graphs as shown in Fig 1B. (B) Scatter plot of ρ′ versus Nρ∘ for all graphs up to size 10. Note that Nρ∘ ≤ 1 for all graphs, with equality if only if the graph is isothermal (or regular, in the context of unweighted graphs).
Table 1.
Classification of small graphs with temperature initialization.
Fig 3.
Small graphs with extreme effects for temperature initialization.
The graphs with the largest or smallest values of ρ′ (which characterizes the likelihood of selected mutations to become fixed), and ρ′/ρ∘ (which quantifies the balance of selection versus drift) are shown for sizes 7 to 10. The Star graph minimizes ρ′ but maximizes ρ′/ρ∘ for these sizes. The largest ρ′ values arise for graphs with multiple components joined by single edges, while the smallest ρ′/ρ∘ ratios occur for Detour graphs.
Fig 4.
Small graphs with extreme effects for uniform initialization.
The graphs with the largest or smallest values of ρ′ are shown for graphs of size 7 to 10. (Only ρ′ is shown because, for uniform initialization, ρ∘ = 1/N for every graph.) Star graphs have the largest ρ′, while the smallest ρ′ values appear for graphs with a highly connected component linked to a tail.
Table 2.
Classification of small graphs with uniform initialization.
Fig 5.
Discovering absolute amplifiers for temperature initialization.
We used a genetic algorithm to discover graphs with large weak-selection effect ρ′. The resulting graphs (middle column) have a central “hub” joined by single links to outlying “islands”. To formalize this structure, we introduce a family of “Cartwheel” graphs CWn,m,h, consisting of a hub of size h and n islands of m vertices each (rightmost column). We find that the optimal Cartwheel graph has ρ′ exceeding that found by the genetic algorithm, except for N = 12 for which the same graph was identified by both methods. All graphs found by both methods are absolute amplifiers of weak selection, meaning ρ′ > (N − 1)/(2N).
Fig 6.
Discovering relative suppressors for temperature initialization.
When seeking to minimize the ratio ρ′/ρ∘, the genetic algorithm, in all cases, found Detour graphs [10], consisting of a complete graph with one edge replaced by a path. For comparison, we calculated ρ′/ρ∘ for all Detour graphs of the given sizes. The results were identical to those of the genetic algorithm except in the case N = 14, for which the genetic algorithm found a Detour graph with a non-optimal number of ring vertices.
Fig 7.
Discovering suppressors for uniform initialization.
When seeking to minimize the ratio ρ′/ρ∘, the genetic algorithm produced structures consisting of a well-connected part and a tail. We compared these to Lollipop graphs (known for their random walk properties [51–53]), and two new families, which we call Balloons and Balloon-Stars. The minimal ρ′ from these families improved on the genetic algorithm results for N = 12, 13, 14 (albeit by less than 10−4 for N = 14), and matched the genetic algorithm results for N = 11 and N = 15.
Fig 8.
Fixation probability, ρ(r) is plotted against mutant fitness r, for the Star and the complete graph. Dotted lines show the linear approximation ρ(1+ δ) ≈ ρ∘ + δρ′, accurate for weak selection (δ ≪ 1). (A) For temperature initialization, the Star is an absolute suppressor of weak selection, ρ′ < (N − 1)/(2N), but a relative amplifier, ρ′/ρ∘ > (N − 1)/2. (B) For uniform initialization, the Star is an amplifier of weak selection, ρ′ > (N − 1)/(2N). Note that ρ(1) = ρ∘ = 1/N for both graphs, as is true for any graph under uniform initialization.
Fig 9.
(A) The Cartwheel graph CWn,m,h contains n islands of m vertices each and h ≥ n hub vertices. Each island is connected to a distinct hub vertex by an edge of weight ϵ; vertices within the hub and within each island are joined by edges of weight 1. (B) The special case n = h and m = 2 has a “spider” structure; this graph has the largest ρ′ in the ϵ → 0 limit. (C) Plot of ρ∘ vs ρ′ for various Cartwheel graphs of size N = 30, with temperature initialization. Points are shown for each ϵ = 2k, where k varies from −5 to 9 in increments of 0.2. Larger points correspond to ϵ = 1 and the ϵ → 0 limit, as derived in Eq (19). Note that limϵ→0 ρ∘ = 1/N for all Cartwheel graphs. CW10,2,10 (the “spider” case) has by far the largest ρ′ in the ϵ → 0 limit; it also has the largest ρ/ρ∘, for all ϵ, among the graphs displayed. However, CW6,3,12 has the largest ρ′ for ϵ = 1 among Cartwheels of size 30. CW4,6,6 and CW2,10,10 both have h = m and therefore have the same fixation probability as a well-mixed population in the ϵ → 0 limit, according to Eq (18). CW3,8,6 has h < m and is therefore a suppressor of weak selection in the ϵ → 0 limit.
Table 3.
Unweighted Cartwheel graphs CWn,m,h that maximize ρ′.
Fig 10.
Nonweak selection on the Cartwheel graph.
In the limit as the hub-to-island weight ϵ goes to zero, the fixation probability for arbitrary mutant fitness r is expressed in closed form by Eq (17). Temperature and uniform initialization are equivalent in this limit. Dashed lines show the weak-selection approximation, with slope ρ′ given by Eq (19). (A) Fixation probability is plotted against r for two Cartwheels of size 12. CW4,2,4 has h > m and is therefore an amplifier of weak selection. CW2,5,2 has h < m and is therefore a suppressor of weak selection (but appears to amplify selection for r > 2.4219). Cartwheels with h = m have the same fixation probability as the well-mixed population, for all r, in the ϵ → 0 limit. (B) In the limits ϵ → 0 and n = h → ∞, with m = 2, the fixation probability jumps discontinuously from 0 to 1/3 as r crosses 1; the expression for r > 1 is given in Eq (20). For comparison we also show the upper bound ρ(r) ≤ 1 − (r + 1)−1, derived by Pavlogiannis et al. [8], which applies to all weighted graphs with no self-loops under temperature initialization.
Fig 11.
(A) The Detour graph Dc,d begins with a complete graph of size c, and replaces the edge between two vertices by a path with d interior vertices. (B) Values of ρ∘ and ρ′ are plotted as d varies, for N = 100. The Detour Dc,d is a relative suppressor of weak selection except for d = 1. The minimal ratio ρ′/ρ∘ is achieved for d = 15 (marked with a star). (C) The minimal ρ′/(Nρ∘) ratio, and the value of d achieving this ratio, is plotted for each N. Note that the minimizing d grows sub-linearly with N.
Fig 12.
(A) This bowtie-shaped graph is the smallest absolute amplifier of weak selection under temperature initialization. (B) The difference in fixation probabilities between this graph and the complete graph K6 is plotted against mutant fitness, r. The dashed line shows the linear approximation to this difference at r = 1, computed using our weak-selection methods. Although this difference is increasing at r = 1 (because the graph is an absolute amplifier of weak selection), the difference is negative for all values of r. Thus this graph does not amplify selection in the usual sense of for all r > 1.
Fig 13.
(A) The Fan Fn,m (or Windmill [56]) consists of one hub and n blades containing m vertices each. We consider a weighted version, with edge weights as shown. Pictured here is the case n = m = 3. (B,C) The neutral fixation probability ρ∘ and weak selection coefficient ρ′, plotted as ϵ varies from 0 to infinity, for all Fan graphs of sizes N = 13 and N = 101. As ϵ increases, the behavior changes from absolute and relative suppressor, to absolute and relative amplifier, to relative amplifier but absolute suppressor. For ϵ → ∞, the ρ∘ and ρ′ values approach those of the Star graph Snm (marked by a red star). (D) As n → ∞, there are three regimes of behavior, depending whether ϵ is held constant (blue line) or scales as n−1 (purple line), or as n−2 (green line). The maximal ρ′ for fixed m is ρ′ = (m + 1)/(2m), achieved for n → ∞ with .