Let $G$ be a graph, which represents a social network, and suppose each node $v$ has a threshold value $\tau(v)$. Consider an initial configuration, where each node is either positive or negative. In each discrete time step, a node $v$ becomes/remains positive if at least $\tau(v)$ of its neighbors are positive and negative otherwise. A node set $\mathcal{S}$ is a Target Set (TS) whenever the following holds: if $\mathcal{S}$ is fully positive initially, all nodes in the graph become positive eventually. We focus on a generalization of TS, called Timed TS (TTS), where it is permitted to assign a positive state to a node at any step of the process, rather than just at the beginning. We provide graph structures for which the minimum TTS is significantly smaller than the minimum TS, indicating that timing is an essential aspect of successful target selection strategies. Furthermore, we prove tight bounds on the minimum size of a TTS in terms of the number of nodes and maximum degree when the thresholds are assigned based on the majority rule. We show that the problem of determining the minimum size of a TTS is NP-hard and provide an Integer Linear Programming formulation and a greedy algorithm. We evaluate the performance of our algorithm by conducting experiments on various synthetic and real-world networks. We also present a linear-time exact algorithm for trees.

翻译：设$G$为一个表示社交网络的图，并假设每个节点$v$具有阈值$\tau(v)$。考虑初始配置，其中每个节点要么为正要么为负。在每个离散时间步中，若节点$v$至少有$\tau(v)$个邻居为正，则该节点变为/保持为正，否则为负。节点集$\mathcal{S}$被称为目标集（TS），当满足以下条件时：若$\mathcal{S}$初始时全为正，则图中所有节点最终都将变为正。我们关注TS的一种推广形式，称为计时目标集（TTS），它允许在过程的任意步骤中为节点分配正状态，而不仅仅是在初始时刻。我们给出了使得最小TTS远小于最小TS的图结构，表明时机是成功目标选择策略的关键因素。此外，我们证明了在基于多数规则分配阈值时，以节点数和最大度表示的最小TTS大小的紧界。结果表明，确定最小TTS大小的问题是NP难的，并提出了整数线性规划公式和贪心算法。我们通过在多种合成网络和真实网络上进行实验来评估算法的性能，同时给出了针对树结构的线性时间精确算法。