On Connectivity in a General Random Intersection Graph

Abstract There has been growing interest in studies of general random intersection graphs. In this paper, we consider a general random intersection graphG(n,−→a ,−→ Kn, Pn) defined on a set Vn comprising n vertices, where−→a is a probability vector (a1, a2, . . . , am) and −→ Kn is (K1,n,K2,n, . . . ,Km,n). This graph has been studied in the literature [10,11,20,29] including a most recent work by Yağan [20]. Suppose there is a pool Pn consisting of Pn distinct objects. The n vertices in Vn are divided into m groups A1,A2, . . . ,Am. Each vertex v is independently assigned to exactly a group according to the probability distribution with P[v ∈ Ai] = ai, where i = 1, 2, . . . ,m. Afterwards, each vertex in group Ai independently chooses Ki,n objects uniformly at random from the object pool Pn. Finally, an undirected edge is drawn between two vertices in Vn that share at least one object. This graph model G(n,−→a , −→ Kn, Pn) has applications in secure sensor networks and social networks. We investigate connectivity in this general random intersection graph G(n,−→a ,−→ Kn, Pn) and present a sharp zero-one law. Our result is also compared with the zero-one law established by Yağan [20].