Effect of the degree of an initial mutant in Moran processes in structured populations

We study effects of the mutant's degree on the fixation probability, extinction, and fixation times in Moran processes on Erdös-Rényi and Barabási-Albert graphs. We performed stochastic simulations and used mean-field-type approximations to obtain analytical formulas. We showed that the initial placement of a mutant has a significant impact on the fixation probability and extinction time, while it has no effect on the fixation time. In both types of graphs, an increase in the degree of an initial mutant results in a decreased fixation probability and a shorter time to extinction. Our results extend previous ones to arbitrary fitness values.

Figure 1

Fixation probability as a function of a degree of an initial mutant for three fitness values: 3, 2, and 1.5 in (a) ER and (b) BA graphs with the size of 500 and the average degree of 8.

Figure 2

Distribution of fixation probabilities of a mutant with respect to initial placements on ER and BA graphs with the size of 500 and the average degree of 8. The fitness is 2. Dash lines show the average fixation probability.

Figure 3

The distribution of temperatures in ER and BA graphs. The node temperature is simply defined as the inverse of the degree of all nodes leading to that node.

Figure 4

Fixation probability for various initial mutants with the same degree in ER and BA graphs. The size of the graph is 500 with the average degree of 8 and $r=2$.

Figure 5

Fixation time as a function of the degree of an initial mutant for three fitness values, 3, 2, and 1.5, in (a) ER and (b) BA graphs with the size of 500 and the average degree of 8.

Figure 6

Distribution of fixation times of a mutant with respect to initial placements on ER and BA graphs with the size of 500 and the average degree of 8. The fitness is 2. Dash lines show the average fixation time.

Figure 7

Extinction time as a function of a degree of an initial mutant for three fitness values, 3, 2, and 1.5, in (a) ER and (b) BA graphs with the size of 500 and the average degree of 8.

Figure 8

Distribution of extinction times of a mutant with respect to initial placements on ER and BA graphs with the size of 500 and the average degree of 8. The fitness is 2. The dash lines show the average extinction times.

Figure 9

Extinction time for various initial mutants with the same degree in both networks. The size of the networks is 500, the average degree 8, and $r=2$.

Figure 10

(a) The expected number of times the population started with one mutant passes through the state with $j$ mutants before fixation, $t_{1,j}^f$, Eq. (22) and (b) the expected time until the number of mutants reaches $j$ from 1, $T_j^f$, Eq. (17), for various values of $r$ in the ER graph with $N=500$ and average degree of 8.

Figure 11

(a) The expected number of times the population started with one mutant passes through the state with $j$ mutants before extinction, $t_{1,j}^e$, Eq. (23) and (b) the expected time until the number of mutants reaches $j$ from 1, $T_j^e$, Eq. (17), for various values of $r$ in the ER graph with $N=500$ and average degree of 8.

Figure 12

Fixation probability as a function of the degree of an initial mutant for various fitness values. The red line is our approximation given by (13), the blue dashed line is an approximation given by (B1), and the green dots show the result of simulations. In Eq. (B1) $s=1-1/r$.

Figure 13

The fitness effect on fixation probability for an initial mutant with degrees of 10 and 25. The red line is our approximation given by (13), the blue dashed line is an approximation given by (B1), and the green dots show the result of simulations. In Eq. (B1) $s=1-1/r$.