The Mixed Birth-death/death-Birth Moran Process

We study evolutionary dynamics on graphs in which each step consists of one birth and one death, also known as the Moran processes. There are two types of individuals: residents with fitness $1$ and mutants with fitness $r$. Two standard update rules are used in the literature. In Birth-death (Bd), a vertex is chosen to reproduce proportional to fitness, and one of its neighbors is selected uniformly at random to be replaced by the offspring. In death-Birth (dB), a vertex is chosen uniformly to die, and then one of its neighbors is chosen, proportional to fitness, to place an offspring into the vacancy. We formalize and study a unified model, the $\lambda$-mixed Moran process, in which each step is independently a Bd step with probability $\lambda \in [0,1]$ and a dB step otherwise. We analyze this mixed process for undirected, connected graphs. As an interesting special case, we show at $\lambda =1\mathrm{/}2$, for any graph that the fixation probability when $r=1$ with a single mutant initially on the graph is exactly $1\mathrm{/}n$, and also at $\lambda =1\mathrm{/}2$ that the absorption time for any $r$ is $O_r(n^4)$. We also show results for graphs that are “almost regular,” in a manner defined in the paper. We use this to show that for suitable random graphs from $G\sim G(n,p)$ and fixed $r>1$, with high probability over the choice of graph, the absorption time is $O_r(n^4)$, the fixation probability is ${\mathrm{\Omega }}_r(n^{-2})$, and we can approximate the fixation probability in polynomial time. Another special case is when the graph has only two distinct degree values $\{d_1,d_2\}$ with $d_1\le d_2$. For those graphs, we give exact formulas for fixation probabilities when $r=1$ and any $\lambda$, and establish an absorption time of $O_r(n^4{\alpha }^4)$ for all $\lambda$, where $\alpha =d_2\mathrm{/}{d}_1$. We also provide explicit formulas for the star and cycle under any $r$ or $\lambda$.