Sharp transitions in random key graphs

Random key graphs have received much interest and been used in diverse applications. A random key graph with n nodes is constructed as follows: each node selects a set of Kn different items uniformly at random from the same pool of Pn distinct items, and two nodes establish an undirected edge in between if and only if they share at least one item. For such graph denoted by G(n, Kn, Pn), we present the following results in this paper.