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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, whereeach 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 arepositive and negative otherwise. A node set $\mathcal{S}$ is a Target Set (TS)whenever the following holds: if $\mathcal{S}$ is fully positive initially, allnodes in the graph become positive eventually. We focus on a generalization ofTS, called Timed TS (TTS), where it is permitted to assign a positive state toa node at any step of the process, rather than just at the beginning.  We provide graph structures for which the minimum TTS is significantlysmaller than the minimum TS, indicating that timing is an essential aspect ofsuccessful target selection strategies. Furthermore, we prove tight bounds onthe minimum size of a TTS in terms of the number of nodes and maximum degreewhen the thresholds are assigned based on the majority rule.  We show that the problem of determining the minimum size of a TTS is NP-hardand provide an Integer Linear Programming formulation and a greedy algorithm.We evaluate the performance of our algorithm by conducting experiments onvarious synthetic and real-world networks. We also present a linear-time exactalgorithm for trees.

Minimum Target Sets in Non-Progressive Threshold Models: When Timing Matters