Social Networks with Competing Products

We introduce a new threshold model of social networks, in which the nodes influenced by their neighbours can adopt one out of several alternatives. We characterize social networks for which adoption of a product by the whole network is possible (respectively necessary) and the ones for which a unique outcome is guaranteed. These characterizations directly yield polynomial time algorithms that allow us to determine whether a given social network satisfies one of the above properties. We also study algorithmic questions for networks without unique outcomes. We show that the problem of determining whether a final network exists in which all nodes adopted some product is NP-complete. In turn, we also resolve the complexity of the problems of determining whether a given node adopts some (respectively, a given) product in some (respectively, all) network(s). Further, we show that the problem of computing the minimum possible spread of a product is NP-hard to approximate with an approximation ratio better than Ω(n), in contrast to the maximum spread, which is efficiently computable. Finally, we clarify that some of the above problems can be solved in polynomial time when there are only two products.