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 $λ$-mixed Moran process, in which each step is independently a Bd step with probability $λ\in [0,1]$ and a dB step otherwise. We analyze this mixed process for undirected, connected graphs. As an interesting special case, we show at $λ=1/2$, for any graph that the fixation probability when $r=1$ with a single mutant initially on the graph is exactly $1/n$, and also at $λ=1/2$ 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 $Ω_r(n^{-2})$, 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 $d_1 \leq d_2$. For those graphs, we give exact formulas for fixation probabilities when $r = 1$ and any $λ$, and establish an absorption time of $O_r(n^4 α^4)$ for all $λ$, where $α= d_2 / d_1$. We also provide explicit formulas for the star and cycle under any $r$ or $λ$.

翻译：我们研究图上演化动力学，其中每一步包含一次出生和一次死亡，亦称为Moran过程。存在两类个体：适应度为$1$的定居者与适应度为$r$的突变体。文献中采用两种标准更新规则：在生死（Bd）规则中，按适应度比例选择一个顶点进行繁殖，并随机均匀选择其一个邻居被子代替代；在死生（dB）规则中，随机均匀选择一个顶点死亡，随后按适应度比例选择其一个邻居将子代置于空缺位置。我们形式化并研究了一个统一模型——$λ$混合Moran过程，其中每一步以概率$λ\in [0,1]$独立执行Bd步骤，否则执行dB步骤。我们针对无向连通图分析此混合过程。作为一个有趣的特殊情形，我们证明当$λ=1/2$时，对于任意图，初始单个突变体在$r=1$时的固定概率恰为$1/n$，且当$λ=1/2$时任意$r$的吸收时间为$O_r(n^4)$。我们还对论文中定义的“近似正则”图展示了相关结果。借此证明：对于来自$G \sim G(n,p)$的合适随机图及固定$r>1$，以高概率（基于图选择）吸收时间为$O_r(n^4)$，固定概率为$Ω_r(n^{-2})$，且可在多项式时间内近似计算固定概率。另一特殊情形是图仅含两个不同度值$\{d_1, d_2\}$（满足$d_1 \leq d_2$）。对此类图，我们给出了$r = 1$且任意$λ$时的精确固定概率公式，并建立了所有$λ$下吸收时间为$O_r(n^4 α^4)$的结论，其中$α= d_2 / d_1$。我们还针对星形图与环状图在任意$r$或$λ$下提供了显式公式。