The Moran process, as studied by Lieberman et al. [L05], is a stochastic process modeling the spread of genetic mutations in populations. In this process, agents of a two-type population (i.e. mutants and residents) are associated with the vertices of a graph. Initially, only one vertex chosen u.a.r. is a mutant, with fitness $r > 0$, while all other individuals are residents, with fitness $1$. In every step, an individual is chosen with probability proportional to its fitness, and its state (mutant or resident) is passed on to a neighbor which is chosen u.a.r. In this paper, we introduce and study for the first time a generalization of the model of [L05] by assuming that different types of individuals perceive the population through different graphs, namely $G_R(V,E_R)$ for residents and $G_M(V,E_M)$ for mutants. In this model, we study the fixation probability, i.e. the probability that eventually only mutants remain in the population, for various pairs of graphs. First, we transfer known results from the original single-graph model of [L05] to our 2-graph model. Among them, we provide a generalization of the Isothermal Theorem of [L05], that gives sufficient conditions for a pair of graphs to have the same fixation probability as a pair of cliques. Next, we give a 2-player strategic game view of the process where player payoffs correspond to fixation and/or extinction probabilities. In this setting, we attempt to identify best responses for each player and give evidence that the clique is the most beneficial graph for both players. Finally, we examine the possibility of efficient approximation of the fixation probability. We show that the fixation probability in the general case of an arbitrary pair of graphs cannot be approximated via a method similar to [D14]. Nevertheless, we provide a FPRAS for the special case where the mutant graph is complete.

中文翻译：

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Moran过程中具有不同连接图的突变体和居民

Lieberman 等人研究的 Moran 过程。[L05]，是一个随机过程，模拟基因突变在人群中的传播。在这个过程中，两种类型的种群（即突变体和居民）的代理与图的顶点相关联。最初，只有一个顶点选择 uar 是突变体，适应度 $r > 0$，而所有其他个体都是居民，适应度 $1$。在每一步中，选择一个个体的概率与其适应度成正比，并将其状态（突变或常驻）传递给被选择的邻居 uar 在本文中，我们首次介绍并研究了模型的泛化[L05] 通过假设不同类型的个体通过不同的图来感知种群，即居民的 $G_R(V,E_R)$ 和突变的 $G_M(V,E_M)$。在这个模型中，我们研究了固定概率，即最终只有突变体留在种群中的概率，对于不同的图形对。首先，我们将已知结果从 [L05] 的原始单图模型转移到我们的 2 图模型。其中，我们提供了 [L05] 的等温定理的推广，它为一对图具有与一对集团相同的固定概率提供了充分条件。接下来，我们给出了一个 2 人策略游戏视图，其中玩家收益对应于注视和/或灭绝概率。在这种情况下，我们尝试为每个玩家确定最佳响应，并提供证据证明集团是对两个玩家最有利的图表。最后，我们检查了固定概率的有效近似的可能性。我们表明，在任意一对图的一般情况下，固定概率不能通过类似于 [D14] 的方法来近似。尽管如此，我们为突变图完整的特殊情况提供了 FPRAS。