Consider an undirected graph G, representing a social network, where each node is blue or red, corresponding to positive or negative opinion on a topic. In the voter model, in discrete time rounds, each node picks a neighbour uniformly at random and adopts its colour. Despite its significant popularity, this model does not capture some fundamental real-world characteristics such as the difference in the strengths of individuals connections, individuals with neutral opinion on a topic, and individuals who are reluctant to update their opinion. To address these issues, we introduce and study a generalisation of the voter model. Motivating by campaigning strategies, we study the problem of selecting a set of seeds blue nodes to maximise the expected number of blue nodes after some rounds. We prove that the problem is NP- hard and provide a polynomial time approximation algorithm with the best possible approximation guarantee. Our experiments on real-world and synthetic graph data demonstrate that the proposed algorithm outperforms other algorithms. We also investigate the convergence properties of the model. We prove that the process could take an exponential number of rounds to converge. However, if we limit ourselves to strongly connected graphs, the convergence time is polynomial and the period (the number of states in convergence) divides the length of all cycles in the graph.

翻译：考虑一个表示社交网络的无向图G，其中每个节点为蓝色或红色，对应于对某个话题的正面或负面观点。在选民模型中，在离散时间轮次中，每个节点均匀随机地选择一个邻居并采纳其颜色。尽管该模型具有显著的影响力，但它未能捕捉某些现实世界的基本特征，例如个体连接强度的差异、对话题持中立观点的个体，以及不愿更新观点的个体。为解决这些问题，我们引入并研究了一种选民模型的推广形式。受竞选策略的启发，我们研究了选择一组种子蓝色节点以最大化若干轮后期望蓝色节点数量的问题。我们证明了该问题是NP难的，并提供了一个具有最佳可能近似保证的多项式时间近似算法。我们在真实世界和合成图数据上的实验表明，所提算法优于其他算法。我们还研究了该模型的收敛性质。我们证明了该过程可能需要指数级轮次才能收敛。然而，若将讨论范围限制在强连通图上，则收敛时间是多项式的，且周期（收敛状态的数量）可整除图中所有环的长度。