Asymptotic expression for the fixation probability of a mutant in star graphs

We consider the Moran process in a graph called the"star"and obtain the asymptotic expression for the fixation probability of a single mutant when the size of the graph is large. The expression obtained corrects the previously known expression announced in reference [E Lieberman, C Hauert, and MA Nowak. Evolutionary dynamics on graphs. Nature, 433(7023):312-316, 2005] and further studied in [M. Broom and J. Rychtar. An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc. R. Soc. A-Math. Phys. Eng. Sci., 464(2098):2609-2627, 2008]. We also show that the star graph is an accelerator of evolution, if the graph is large enough.


Introduction
The central question in the mathematical study of population genetics is to understand how gene frequencies vary in time. This topic has a long story, since at least the works of Wright and Fisher [1,2]. A simpler, but in a certain sense equivalent model, was introduced by Moran [3]: in that case a population of constant size n of two types evolves in discrete time steps. To each type we attribute a positive number, called fitness. At each time one individual is selected to reproduce with probability proportional to the fitness and one to die (possibly the same one), with probability 1/n. The so called Moran process continues until the entire population consists of individuals of a single type. We say that one type has fixated while the other was extinct. One particularly important question is the fixation probability of a given type A, i.e., the probability that after a sufficiently long time, type A reaches fixation; see [4,5] In [6], topology, in the form of a graph, was introduced in the study of population dynamics. Individuals were represented by vertices, while edges represented possible positions for the offspring of individuals in a given vertex. In particular, consider a generic graph where all vertices but one (selected at random) are occupied by a type with fitness 1 (the resident type). The remaining vertex is occupied by a mutant with fitness r > 0. After a certain fixed time, one of the individuals (resident or mutant) is selected to reproduce with probability proportional to the fitness, and its offspring replaces one of the individuals occupying an adjacent vertex, selected with equal probability. This is the so called Moran process on graphs [3,6]. The invasion probability ρ is given by the probability that after a sufficiently long time all vertices are occupied by the mutant type (i.e., the mutant was fixed and the resident was extinct). For a complete graph (all pair of vertices are connected) with n vertices, the invasion probability is given by ρ C (r, n) := 1−r −1 1−r −n if r = 1 and ρ C (1, n) := 1 n . This corresponds to the original problem studied in [3]. We say that a graph is an accelerator of evolution if ρ > ρ C if and only if r > 1. If one of these inequalities is reversed, we say that the graph is a suppressor of evolution. See [5] for a more detailed explanation of concepts used in this article and also for examples of accelerators and suppressors of evolution.
In this article, we study a specific graph, called "star". The star is a graph with n+1 vertices, labeled from 0 to n, where vertex number 0 is called center and all the others are the leaves. The only edges are between vertex 0 and all other vertices. These edges are bidirectional. See figure 1. In [6], the star graph was introduced as a particular case of family of graphs, called "superstar" and parameterized by two natural numbers: k, the number of layers and n, the number of vertices at layer k + 1, for any k < k, connected to any given vertex at layer k . The first layer has a single vertex, called center. The star corresponds to k = 2. According to the conjecture in [6], However, recently [7] showed that this conjecture is false for k ≥ 5.
We start this article from a previous work (see [8]), where an explicit formula for the invasion probability in stars was derived and study the associated asymptotic expression when n, the number of individuals, is large. The aim of this work is to show that the expression for the leading term in the asymptotic expression of the invasion probability is given by Note that this expression is different from the one conjectured in [6], discussed above, for the case k = 2.
In section 2 we discuss the results presented in [8]. In section 3, we rigorously derive Eq. (1) and in section 4, we prove that the star is an accelerator of evolution, if n is large enough. We study this problem numerically and present our conclusions in the final section 5.

Evolutionary dynamics in the star graph
The first attempt to prove the formula conjectured in [6] was made in [8], where the exact expression for the invasion probability of mutant with fitness r in a star graph with n leaves was found.
We agree with the formula found in [8] for ρ n (r), however the asymptotic expression found when n → ∞ seems to not be correct. In particular, the asymptotic expression n nr nr+1 + r n+r (n + 1) · 1 + n n+r n−1 j=1 when n → ∞ is wrong (see [8, page 2616]).

The star is an accelerator of evolution
Now, let us compare the Moran process in the star and in the complete graph. For the neutral evolution (r = 1), we have the following simple result: Lemma 1. ρ n (1) = 1 n+1 = ρ C (1, n + 1). Proof. See Eq. (2) with r = 1. Equivalently, this result can be proved from Eq. (4) in the limit r → 1.
Note that lim r→1 ρ(r, n) = 1 n−1 = 1 n+1 , showing that the large population limit n → ∞ and the weak selection limit r → 1 are not interchangeable.

Discussion
Despite the fact that the expression conjectured in [6] is wrong for k = 2 and k ≥ 5, it has been widely used (e.g., [9,10,11]). However, exact or asymptotic expressions for k > 2 have not been found and should be subject of further investigations. See also [12] for a recent review in evolutionary graph theory. Differences between expression (1) and the expression 1−r −2 1−r −2n given by [8] and [6] are hardly noticeable from the numerical point of view. In fact, for r > 1, the difference between both expressions are exponentially small, as r −n → 0 exponentially fast when n → ∞. However, for r < 1 these differences, despite small, cannot be neglected. See table 1 and figure 2 for numerical comparisons.   Table 1: Invasion probabilities for different values of r and n 1: ρ n indicates the exact value given by Eq. (2); in the third column, we compute the approximate value given by Eq. (1) and its associated error = |ρ(r, n) − ρ n (r)|/ρ n (r); the last column indicates the approximation in references [6,5,8] and its associated error. Note that for r > 1, the two approximations are essentially equivalent; for r < 1 the first approximation is clearly superior. All calculations were performed in Sage v. 5.13. Figure 2: (Color online) For different values of n (x axis), we plot ρ n (r) (blue dots), ρ(n, r) (red line) and 1−r −2 1−r −2n (green line). We use a logarithmic scale in the y axis. Note that the red line is consistently a better approximation than the green one with respect to the exact values.