In a network formation framework, where payoffs reflect an agent’s ability to access information from direct and indirect contacts, we integrate negative externalities due to connectivity associated with two types of effects: competition for the access to information, and rivalrous use of information. We consider two separate models to capture the first and the second situations, respectively. In the first model we assume that information is a non-rivalrous good but that there is competition for the access to information, for example because an agent with many contacts must share his time between them and thus has fewer opportunities to pass on information to each particular contact. In the second model we do not assume that there is competition for the access to information, but rather that the use of information is rivalrous. In this case, it is assumed that when people are closer to the sender than an agent, the harmful effect is greater than when others are at the same distance to the sender as that agent. In both models we analyze pairwise stability and examine if the stability of a structure is preserved when the number of agents becomes very large. This leads to a new concept that we call asymptotic pairwise stability. We show that there can exist a tension between asymptotic pairwise stability and efficiency. The results allow us to compare and contrast the effects of two kinds of competition for information.