In this article, we examine the role played by heterogeneity in the popular "connections model" of Jackson and Wolinsky (J Econ Theory 71(1):355-365, 1996). We prove that when heterogeneity with respect to the values of resources, or the information decay parameter, depends on the identity of the player who forms the link, and the player who receives the link, all networks can be supported as Nash. Moreover, we show that Nash networks may not always exist. Interestingly, in the absence of decay, neither result occurs in a model with value heterogeneity. Finally, we show that when reducing heterogeneity, all networks cannot be supported as Nash, and the non-existence problem also disappears.