Evolutionary Dynamics on Degree-Heterogeneous Graphs

The evolution of two species with different fitness is investigated on degree-heterogeneous graphs. The population evolves either by one individual dying and being replaced by the offspring of a random neighbor (voter model dynamics) or by an individual giving birth to an offspring that takes over a random neighbor node (invasion process dynamics). The fixation probability for one species to take over a population of $N$ individuals depends crucially on the dynamics and on the local environment. Starting with a single fitter mutant at a node of degree $k$, the fixation probability is proportional to $k$ for voter model dynamics and to $1/k$ for invasion process dynamics.

Figure 1

Update illustration for two specific nodes. Genotypes $\mathbf{0}$ and $\mathbf{1}$ are denoted by ○ and ●, respectively. Shown are the possible transitions and their respective relative rates due to the interaction of two nodes across a link for LD, VM, and IP dynamics.

Figure 2

Moments of the $\mathbf{1}$ density in the biased VM and biased IP on a network of ${10}^4$ nodes with a power-law degree distribution $n_k\sim k^{-\nu }$ ($\nu =2.5$) and no correlations between node degrees. Nodes with degree larger than the mean degree are initialized to $\mathbf{1}$. while all other nodes are $\mathbf{0}$. For the VM, $s=8.5\times {10}^{-4}$, while for the IP, $s={10}^{-4}$.

Figure 3

Fixation probability of a single mutant initially at a node of degree $k$ on an uncorrelated power-law degree distributed ($n_k\sim k^{-\nu }$, $\nu =2.5$) graph with $N={10}^3$ and ${\mu }_1=8$. The empty symbols correspond to IP dynamics with $s=0.004$ (□), $s=0.008$ (○), and $s=0.016$ (△); the solid symbols correspond to VM dynamics with $s=0.01$ (■), $s=0.02$ (●), and $s=0.08$ (▲). The solid lines, with slopes $+1$ and $-1$, correspond to the second of Eqs. (15, 20).