Critical behavior in heterogeneous random key graphs

Recently, there has been growing interest in studies of heterogeneous random key graphs. In this paper, we consider a heterogeneous random key graph G (n, a, Kn, Pn) defined on a set Vn comprising n nodes, where a is a probability vector (a1, a2,..., am) and Kn is (K1, n, K2, n, ..., Km, n). Suppose there is a pool Pn consisting of Pn distinct items. The n nodes in Vn are divided into m groups A1, A2,..., Am. Each node v is independently assigned to exactly a group according to the probability distribution with P[v ϵ Ai] = ai, where i = 1, 2,..., m. Afterwards, each node in group Ai independently chooses Ki, n items uniformly at random from the item pool Pn. Finally, an undirected edge is drawn between two nodes in Vn that share at least one item. This graph model G (n, a, Kn, Pn) has applications in secure sensor networks and social networks. We investigate critical behavior for the absence of isolated node in this heterogeneous random key graph G(n, a, Kn, Pn) and present a sharp zero-one law.