Yagan, OsmanMakowski, Armand M.Extended version of a paper submitted to ISIT 2008, Toronto (Canada), July 2008.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.en-USWireless sensor networksKey predistributionSecure connectivityAbsence of isolated nodesZero-one lawsPoisson convergenceTechnical Report