Spreading Social Influence with both Positive and Negative Opinions in Online Networks

Social networks are important media for spreading information, ideas, and influence among individuals. Most existing research focuses on understanding the characteristics of social networks, investigating how information is spread through the "word-of-mouth" effect of social networks, or exploring social influences among individuals and groups. However, most studies ignore negative influences among individuals and groups. Motivated by the goal of alleviating social problems, such as drinking, smoking, and gambling, and influence-spreading problems, such as promoting new products, we consider positive and negative influences, and propose a new optimization problem called the Minimum-sized Positive Influential Node Set (MPINS) selection problem to identify the minimum set of influential nodes such that every node in the network can be positively influenced by these selected nodes with no less than a threshold of $\theta$. Our contributions are threefold. First, we prove that, under the independent cascade model considering positive and negative influences, MPINS is APX-hard. Subsequently, we present a greedy approximation algorithm to address the MPINS selection problem. Finally, to validate the proposed greedy algorithm, we conduct extensive simulations and experiments on random graphs and seven different real-world data sets that represent small-, medium-, and large-scale networks.