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The Voter model is a well-studied stochastic process that models the invasionof a novel trait $A$ (e.g., a new opinion, social meme, genetic mutation,magnetic spin) in a network of individuals (agents, people, genes, particles)carrying an existing resident trait $B$. Individuals change traits byoccasionally sampling the trait of a neighbor, while an invasion bias$\delta\geq 0$ expresses the stochastic preference to adopt the novel trait $A$over the resident trait $B$. The strength of an invasion is measured by theprobability that eventually the whole population adopts trait $A$, i.e., thefixation probability. In more realistic settings, however, the invasion bias isnot ubiquitous, but rather manifested only in parts of the network. Forinstance, when modeling the spread of a social trait, the invasion biasrepresents localized incentives. In this paper, we generalize the standardbiased Voter model to the positional Voter model, in which the invasion bias iseffectuated only on an arbitrary subset of the network nodes, called biasednodes. We study the ensuing optimization problem, which is, given a budget $k$,to choose $k$ biased nodes so as to maximize the fixation probability of arandomly occurring invasion. We show that the problem is NP-hard both forfinite $\delta$ and when $\delta \rightarrow \infty$ (strong bias), while theobjective function is not submodular in either setting, indicating strongcomputational hardness. On the other hand, we show that, when$\delta\rightarrow 0$ (weak bias), we can obtain a tight approximation in$O(n^{2\omega})$ time, where $\omega$ is the matrix-multiplication exponent. Wecomplement our theoretical results with an experimental evaluation of someproposed heuristics.

Maximizing the Probability of Fixation in the Positional Voter Model