The Moran process is a classic stochastic process that models the rise and takeover of novel traits in network-structured populations. In biological terms, a set of mutants, each with fitness $m\in(0,\infty)$ invade a population of residents with fitness $1$. Each agent reproduces at a rate proportional to its fitness and each offspring replaces a random network neighbor. The process ends when the mutants either fixate (take over the whole population) or go extinct. The fixation probability measures the success of the invasion. To account for environmental heterogeneity, we study a generalization of the Standard process, called the Heterogeneous Moran process. Here, the fitness of each agent is determined both by its type (resident/mutant) and the node it occupies. We study the natural optimization problem of seed selection: given a budget $k$, which $k$ agents should initiate the mutant invasion to maximize the fixation probability? We show that the problem is strongly inapproximable: it is $\mathbf{NP}$-hard to distinguish between maximum fixation probability 0 and 1. We then focus on mutant-biased networks, where each node exhibits at least as large mutant fitness as resident fitness. We show that the problem remains $\mathbf{NP}$-hard, but the fixation probability becomes submodular, and thus the optimization problem admits a greedy $(1-1/e)$-approximation. An experimental evaluation of the greedy algorithm along with various heuristics on real-world data sets corroborates our results.

翻译：莫兰过程是一种经典的随机过程，用于模拟网络结构种群中新颖特征的兴起与取代。在生物学语境中，一组适应度为 $m\in(0,\infty)$ 的突变体入侵适应度为 $1$ 的野生型种群。每个个体以其适应度成比例的速率繁殖，其后代随机替换网络中的一个邻居。当突变体要么固定（占据整个种群）要么灭绝时，过程终止。固定概率衡量入侵成功的可能性。为考虑环境异质性，我们研究了标准过程的推广形式，称为异质莫兰过程。在此过程中，每个个体的适应度由其类型（野生型/突变型）及其占据的节点共同决定。我们研究了自然优化问题——种子选择：给定预算 $k$，应选择哪 $k$ 个个体启动突变体入侵以最大化固定概率？我们证明该问题具有强不可近似性：区分最大固定概率为 0 和 1 是 $\mathbf{NP}$-难的。随后聚焦于突变偏向网络，其中每个节点的突变适应度至少与野生型适应度相同。我们证明该问题仍为 $\mathbf{NP}$-难，但固定概率变为子模函数，因此优化问题可接受贪婪 $(1-1/e)$-近似算法。基于真实数据集对贪婪算法及多种启发式方法的实验评估验证了我们的结论。