In many scenarios, networks emerge endogenously as cognitive agents establish links in order to exchange information. Network formation has been widely studied in economics, but only on the basis of simplistic models that assume that the value of each additional piece of information is constant. In this paper we present a first model and associated analysis for network formation under the much more realistic assumption that the value of each additional piece of information depends on the type of that piece of information and on the information already possessed: information may be complementary or redundant. We model the formation of a network as a non-cooperative game in which the actions are the formation of links and the benefit of forming a link is the value of the information exchanged minus the cost of forming the link. We characterize the topologies of the networks emerging at a Nash equilibrium (NE) of this game and compare the efficiency of equilibrium networks with the efficiency of centrally designed networks. To quantify the impact of information redundancy and linking cost on social information loss, we provide estimates for the Price of Anarchy (PoA); to quantify the impact on individual information loss we introduce and provide estimates for a measure we call Maximum Information Loss (MIL). Finally, we consider the setting in which agents are not endowed with information, but must produce it. We show that the validity of the well-known "law of the few" depends on how information aggregates; in particular, the "law of the few" fails when information displays complementarities.