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，用于模拟网络结构群体中新型性状的兴起与取代。在生物学背景下，一组适应度为$m\in(0,\infty)$的突变体入侵适应度为$1$的居民群体。每个个体以其适应度为比例进行繁殖，其后代随机替换网络中的邻居个体。该过程在突变体要么固定（取代整个群体）要么灭绝时结束。固定概率衡量入侵的成功程度。为考虑环境异质性，我们研究了标准过程的推广形式——异质莫兰过程。在此过程中，每个个体的适应度由其类型（居民/突变体）及其占据的节点共同决定。我们研究了种子选择的自然优化问题：给定预算$k$，应选择哪$k$个个体发起突变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。接着，我们聚焦于突变偏向网络——每个节点的突变体适应度至少与居民适应度相同。我们证明该问题仍是$\mathbf{NP}$-难的，但固定概率函数具有次模性，因此优化问题可接受贪心$(1-1/e)$-近似算法。基于实际数据集对贪心算法及多种启发式方法的实验评估验证了我们的结论。