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

TL;DR

This paper explores the importance of timing in influence spread models on social networks, introducing the concept of Timed Target Sets (TTS) and demonstrating their advantages over traditional Target Sets (TS).

Contribution

It generalizes the target set problem to include timing, provides bounds, proves NP-hardness, and offers algorithms including an exact linear-time solution for trees.

Findings

Timed TTS can be significantly smaller than TS.

Provided tight bounds for TTS size under majority thresholds.

Developed algorithms including an ILP, greedy, and linear-time for trees.

Abstract

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.