Maximizing the Spread of an Opinion when Tertium Datur Est

Opinion diffusion has been largely studied in the literature on settings where the opinion whose spread has to be maximized, say white, competes against one opinion only, say black. For instance, for diffusion mechanisms modeled in terms of best response dynamics over majority agents (who change their opinion as to conform it to the majority of their neighbors), it is known that the spread can be maximized via certain greedy dynamics that can be computed in polynomial time. This setting is precisely the one considered in the paper. However, differently from earlier literature, it is assumed that one further opinion, say gray, is available to the agents. Moving from the observation that, with the third alternative to hand, greedy dynamics can dramatically fail to maximize the spread of opinion white, the paper then embarks in thorough computational, algorithmic and experimental studies. The picture that emerges is totally different from what is known for the case when two opinions are available only. Indeed, it is shown that greedy dynamics can dramatically fail in maximizing the spread. In particular, deciding whether there exists a dynamics that can spread the opinion white to at least k agents or can reach a consensus is shown to be intractable, formally NP-hard. On the other hand, islands of tractability based on certain structural properties of the interaction graph are identified. Finally, experimental results are discussed, which shed lights on opinion diffusion in real social networks.