Fixation time in evolutionary graphs: A mean-field approach

Using an analytical method we calculate average conditional fixation time of mutants in a general graph-structured population of two types of species. The method is based on Markov chains and uses a mean-field approximation to calculate the corresponding transition matrix. Analytical results are compared with the results of simulation of the Moran process on a number of network structures.

Figure 1

Markov chain corresponding to the Moran process. The state $i$ represents all configurations with $i$ mutants in the network. The chain transits from state $i$ to state $i+1$ with probability ${\lambda }_i$ and to state $i-1$ with probability ${\mu }_i$. States 0 and $N$ are absorbing states.

Figure 2

Fixation time vs network size for a cycle graph with three different fitness values.

Figure 3

Fixation time vs network size for a complete graph with three different fitness values.

Figure 4

Fixation time vs network size for an Erdös-Rényi network with three different fitness values. The connection probability is set to $p=0.16$.

Figure 5

Fixation time vs connection probability $p$ in an Erdös-Rényi graph with $N=100$ nodes and fitness $r=3$.

Figure 6

Fixation time versus network size in a regular graph with $m=6$.

Figure 7

Fixation time vs network size in a scale-free network with $m=6$ for three different fitness values.

Figure 8

Fixation time vs network size in Watts-Strogatz model with $\langle k\rangle =6$ and $P_{\mathrm{WS}}=0.1$.