Connectivity of the uniform random intersection graph

A uniform random intersection graph G(n,m, k) is a random graph constructed as follows. Label each of n nodes by a randomly chosen set of k distinct colours taken from some finite set of possible colours of size m. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks, in particular when modelling the network graph of the well known key predistribution technique due to Eschenauer and Gligor. The paper determines the threshold for connectivity of the graph G(n,m, k) when n → ∞ in many situations. For example, when k is a function of n such that k ≥ 2 and m = ⌊nα⌋ for some fixed positive real number α then G(n,m, k) is almost surely connected when lim inf kn/m logn > 1, and G(n,m, k) is almost surely disconnected when lim sup kn/m logn < 1.