Information decay in networks generates two effects. First, it differentiates how well informed different players within the same component are, and therefore how attractive they are to sponsor links to. Second, players may prefer to sponsor links to players they are already connected to. By focusing on small decay we analyze the first effect in isolation. We characterize the set of Nash equilibrium networks in the two-way flow model of network formation with small decay for any increasing benefit function of the players. The results show that small decay is consistent with two well-known stylized facts, namely that (i) many real world networks have high diameters, and (ii) that the diameter of such networks is typically small relative to the population size. We show that even stochastically stable networks may have any diameter when the benefit function is linear or strictly concave. Finally we study implied stability relations. We find that if any non-empty minimal network is stable, then so is the periphery-sponsored star. With strictly convex benefit functions, we find that other stars tend to be stable for a larger range of parameters than larger diameter networks which satisfy our characterization. However, with strictly concave benefit functions the other stars are stable for a smaller range of parameters than the larger diameter networks.
