Fixation Maximization in the Positional Moran Process

Abstract

The Moran process is a classic stochastic process that models invasion dynamics on graphs. A single "mutant" (e.g., a new opinion, strain, social trait etc.) invades a population of residents spread over the nodes of a graph. The mutant fitness advantage $\delta\geq 0$ determines how aggressively mutants propagate to their neighbors. The quantity of interest is the fixation probability, i.e., the probability that the initial mutant eventually takes over the whole population. However, in realistic settings, the invading mutant has an advantage only in certain locations. E.g., a bacterial mutation allowing for lactose metabolism only confers an advantage on places where dairy products are present. In this paper we introduce the positional Moran process, a natural generalization in which the mutant fitness advantage is only realized on specific nodes called active nodes. The associated optimization problem is fixation maximization: given a budget $k$, choose a set of $k$ active nodes that maximize the fixation probability of the invading mutant. We show that the problem is NP-hard, while the optimization function is not submodular, thus indicating strong computational hardness. Then we focus on two natural limits. In the limit of $\delta\to\infty$ (strong selection), although the problem remains NP-hard, the optimization function becomes submodular and thus admits a constant-factor approximation using a simple greedy algorithm. In the limit of $\delta\to 0$ (weak selection), we show that in $O(m^\omega)$ time we can obtain a tight approximation, where $m$ is the number of edges and $\omega$ is the matrix-multiplication exponent. Finally, we present an experimental evaluation of the new algorithms together with some proposed heuristics.