Suppressors of selection

Inspired by recent works on evolutionary graph theory, an area of growing interest in mathematical and computational biology, we present examples of undirected structures acting as suppressors of selection for any fitness value r > 1. This means that the average fixation probability of an advantageous mutant or invader individual placed at some node is strictly less than that of this individual placed in a well-mixed population. This leads the way to study more robust structures less prone to invasion, contrary to what happens with the amplifiers of selection where the fixation probability is increased on average for advantageous invader individuals. A few families of amplifiers are known, although some effort was required to prove it. Here, we use computer aided techniques to find an exact analytical expression of the fixation probability for some graphs of small order (equal to 6, 8 and 10) proving that selection is effectively reduced for r > 1. Some numerical experiments using Monte Carlo methods are also performed for larger graphs and some variants.

where r > 0 is the relative fitness for the invader mutant, d i is the degree of the node i, and w i j = 1/d i if (i, j) ∈ E and w i j = 0 otherwise. Notice that all the outgoing probabilities P S,S have the same denominator, the total reproductive weight of S w S = r ∑ i∈S ∑ j∈V w i j + ∑ i∈V \S ∑ j∈V w i j = r|S| + N − |S|.
Let Φ S (r) be the probability of all population will become mutant at some point starting with mutants in the vertices belonging to S, that is, the probability of reaching the absorbing state V ∈ S starting from S ∈ S . Hence, the average fixation probability is To compute this quantity it is enough to solve the system of linear equations with boundary conditions Φ / 0 (r) = 0 and Φ V (r) = 1. If P = (P S,S ) is the transition matrix, Eq 2 can be written as where Φ Φ Φ = (0,Ψ Ψ Ψ, 1) is the vector with coordinates Φ S (r), (1, b, 0) is the vector with coordinates P S, / 0 , and (0, c, 1) is the vector with coordinates P S,V . It can be rewritten as where I is the identity matrix. It is clear that the entries of I − Q and the coordinates of c belongs to Q[r]. Therefore, Cramer's rule implies that the entries of the fundamental matrix N = (I − Q) −1 and the coordinates of the solution Ψ Ψ Ψ of the linear system Eq 4 are rational functions on r with rational coefficients. Then, Lemma 1. The average fixation probability Φ(r) is a rational function on r with rational coefficients.
In order to bound the degree of the numerator and denominator of Φ(r), we must simplify Eq 4 as follows. If we multiply each equation by the reproductive weight w S of the corresponding state S, we obtain the simpler equivalent linear system where T is the diagonal matrix with entries w S associated to the states S ∈ S different from / 0 and V . Obviously, all the coordinates of the matrix T (I − Q) and the vector T · c are functions of the form a r + b with a and b ∈ Q. By Cramer's rule, the fixation probability of any state S is where [T (I − Q)] S is the matrix obtained from T (I − Q) by replacing the column associated to S with the vector T · c. If the size of both matrices is d × d, since each entry is a polynomial of degree at most 1, Φ S (r) is the quotient of two polynomial with rational coefficients of degree at most d. Hence, Lemma 2. The average fixation probability Φ(r) is a rational function with both numerator and denominator polynomials of degree equal to the number of states minus two.

Remark 3.
A priori, the degree of the average fixation probability function is 2 N − 2 where N is the size of the graph. However, using the symmetries of the graph it is possible to reduce the number of possible states, and hence the degree of Φ(r). For example, the average fixation probability function for the complete graph K N has degree, at most, N − 1.
For the complete bipartite graphs K n,m it would have degree ≤ (n − 1)(m − 1) − 2 (see Refs. 1-3 for the concrete expressions). As we explained in the paper, for the -graphs, the degree d = N(N+1) On the other hand, since the average fixation probability tends to 1 as r → ∞, both numerator and denominator have the same lead coefficient, so it can be assumed to be 1.

The SageMath program
The essential tool in order to prove that 6 , 8 and 10 are suppressor of selection for any fitness value r > 1 is a SageMath program that symbolically • computes the exact fixation probability Φ(r) for the graphs 6 , 8 and 10 when r ∈ {1, . . . , d + 1, 1/2, . . . , 1/d}, where Φ (r) = ∑ d i=0 a i r i and Φ (r) = ∑ d i=0 b i r i are the numerator and the denominator of the rational function Φ(r). The symmetries of the -graphs allow to reduce the degree to d = N(N+1) The code is available at https://bitbucket.org/snippets/alvarolozano/7jnka/. Next, we give the exact values of Φ = Φ /Φ and ∆ = ∆ /∆ for the graphs 6 , 8