We consider the random graph induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. We show the existence of a zero-one law for the absence of isolated nodes, and complement it by a Poisson convergence for the number of isolated nodes. Leveraging earlier resu lts and analogies with Erd\H{o}s-Renyi graphs, we explore similar results for the property of graph connectivity. Implications for secure connectivity are discussed.