Invasion Dynamics in the Biased Voter Process

TL;DR

This paper investigates how to optimally select initial agents to maximize the spread of a trait in a biased voter model, revealing computational hardness and proposing approximation strategies.

Contribution

It proves NP-hardness of fixation probability maximization and shows submodularity for the case of bias towards invasion, enabling greedy approximation algorithms.

Findings

Fixation maximization is NP-hard for both bias directions.

Submodularity holds when bias favors invasion, allowing greedy algorithms.

Experimental results support the theoretical findings.

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 $r\in(0,\infty)$ 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 $r>1$ and $r<1$, while the latter case is also inapproximable within any multiplicative factor. 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.