Invasion Dynamics in the Biased Voter Process

Abstract

The voter process is a classic stochastic process that models the invasion of a mutant trait A (e.g., a new opinion, belief, legend, genetic mutation, magnetic spin) in a population of agents (e.g., people, genes, particles) who share a resident trait B, spread over the nodes of a graph. An agent may adopt the trait of one of its neighbors at any time, while the invasion bias r quantifies the stochastic preference towards (r>1) or against (r<1) adopting A over B. Success is measured in terms of the fixation probability, i.e., the probability that eventually all agents have adopted the mutant trait A. In this paper we study the problem of fixation probability maximization under this model: given a budget k, find a set of k agents to initiate the invasion that maximizes the fixation probability. We show that the problem is NP-hard for both regimes r>1 and r<1, while the latter case is also inapproximable within any multiplicative factor that is independent of r. On the positive side, we show that when r>1, the optimization function is submodular and thus can be greedily approximated within a factor 1-1/e. An experimental evaluation of some proposed heuristics corroborates our results.