The Voter model is a well-studied stochastic process that models the invasion of 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 by occasionally 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 the probability that eventually the whole population adopts trait $A$, i.e., the fixation probability. In more realistic settings, however, the invasion bias is not ubiquitous, but rather manifested only in parts of the network. For instance, when modeling the spread of a social trait, the invasion bias represents localized incentives. In this paper, we generalize the standard biased Voter model to the positional Voter model, in which the invasion bias is effectuated only on an arbitrary subset of the network nodes, called biased nodes. 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 a randomly occurring invasion. We show that the problem is NP-hard both for finite $\delta$ and when $\delta \rightarrow \infty$ (strong bias), while the objective function is not submodular in either setting, indicating strong computational 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. We complement our theoretical results with an experimental evaluation of some proposed heuristics.

翻译：选举者模型是一个被广泛研究的随机过程，用于模拟新特征A（例如新观点、社交模因、基因突变、磁自旋）在带有现有驻留特征B的个体网络（代理、人、基因、粒子）中的入侵。个体通过偶尔采样邻居的特征来改变自身特征，同时入侵偏倚δ≥0表示采用新特征A相对于驻留特征B的随机偏好。入侵强度通过最终整个群体采用特征A的概率（即固定概率）来度量。然而，在更现实的场景中，入侵偏倚并非普遍存在，而仅体现在网络的部分区域。例如，在建模社会特征传播时，入侵偏倚代表局部激励机制。本文我们将标准有偏选举者模型推广为定位选举者模型，其中入侵偏倚仅作用于网络节点的任意子集（称为偏倚节点）。我们研究由此产生的优化问题：给定预算k，选择k个偏倚节点以最大化随机发生入侵的固定概率。我们证明该问题在有限δ和δ→∞（强偏倚）情况下均为NP难问题，且目标函数在这两种情况下均非次模，表明其具有强计算复杂度。另一方面，当δ→0（弱偏倚）时，我们可以在O(n^{2ω})时间内获得紧近似，其中ω是矩阵乘法指数。我们通过若干启发式算法的实验评估补充了理论结果。