Evolutionary regime transitions in structured populations

The evolutionary dynamics of a finite population where resident individuals are replaced by mutant ones depends on its spatial structure. Usually, the population adopts the form of an undirected graph where the place occupied by each individual is represented by a vertex and it is bidirectionally linked to the places that can be occupied by its offspring. There are undirected graph structures that act as amplifiers of selection increasing the probability that the offspring of an advantageous mutant spreads through the graph reaching any vertex. But there also are undirected graph structures acting as suppressors of selection where this probability is less than that of the same individual placed in a homogeneous population. Here, firstly, we present the distribution of these evolutionary regimes for all undirected graphs with N ≤ 10 vertices. Some of them exhibit transitions between different regimes when the mutant fitness increases. In particular, as it has been already observed for small-order random graphs, we show that most graphs of order N ≤ 10 are amplifiers of selection. Secondly, we describe examples of amplifiers of order 7 that become suppressors from some critical value. In fact, for graphs of order N ≤ 7, we apply computer-aided techniques to symbolically compute their fixation probability and then their evolutionary regime, as well as the critical values for which they change their regime. Thirdly, the same technique is applied to some families of highly symmetrical graphs as a mean to explore methods of suppressing selection. The existence of suppression mechanisms that reverse an amplification regime when fitness increases could have a great interest in biology and network science. Finally, the analysis of all graphs from order 8 to order 10 reveals a complex and rich evolutionary dynamics, with multiple transitions between different regimes, which have not been examined in detail until now.

S1 Text: On some properties of Φ 0 (r) and its limit as N → ∞ In this text we prove some facts about the rational function Φ 0 (r) = r N−1 r N−1 + r N−2 + · · · + 1 stated in the main text of the paper.
Φ 0 (r) uniformly converges to its limit It is well known that the limit of Φ 0 (r) as N → ∞ is Result 1. This convergence is uniform, in fact, A direct computation gives the result. For r = 1 there is nothing to prove. For r < 1, Since r < 1, r j < r j−1 for any j ≥ 0. Therefore Since r > 1, r j > 1 for any j ≥ 1, hence r(r N−1 + r N−2 + · · · + 1) > N and therefore

First derivative convergence as N → ∞
It is stated in the paper that the sequence of functions Φ 0 (r) does not converge to It is interesting to see what happens to the limit of the derivatives at r = 1. A direct computation give us On the other hand, while there is no derivative of Φ ∞ (r) at r = 1, it is possible to compute the slope of the tangent from the right. In other words, the derivative of 1 − 1 r at r = 1, which is 1 (see Figure 1). The appearance of phase transitions in the sense of the paper is tied to the location of the inflection points of the functions Φ(r) of a given graph and Φ 0 (r) of the complete graph of the same order. Exploring in full generality the location of those points in all graphs is impossible. But just the behavior of the simplest one, the homogeneous population, give us a grasp of the complexities hidden around r = 1. Convexity and concavity can be computed from the sign of the second derivative. For small order, up to 5, it is possible to compute this derivative by hand to check that the function is in fact concave (see Figure 2). For greater orders the computation is just cumbersome, but it is possible to argue at some extent about it without the need of the direct computation.
The limit of Φ 0 (r) as r → ∞ is 1 and Φ 0 (r) 1 for all r > 0, therefore, at least for large enough r's, Φ 0 (r) should increase towards the asymptote 1, therefore, the second derivative should be negative for large enough r's. That is, the function is, for large values of its variable, concave.
But it is possible to compute the second derivative of the function for r = 1.
Substituting in (2) The sign of (3) is given by the sign of N 2 − 6N − 5. This quantity is 0 for N = 1 and N = 5. Between those points is negative, and it is positive everywhere else. Therefore, Result 2. For all orders greater or equal to 6, Φ 0 (r) is convex around 1, so there is, at least, one inflection point in (1, ∞).