Connectivity of the uniform random intersection graph

A uniform random intersection graph  G(n,m,k)G(n,m,k) is a random graph constructed as follows. Label each of nn nodes by a randomly chosen set of kk distinct colours taken from some finite set of possible colours of size mm. 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)G(n,m,k) when n→∞n→∞ in many situations. For example, when kk is a function of nn such that k≥2k≥2 and m=⌊nα⌋m=⌊nα⌋ for some fixed positive real number αα then G(n,m,k)G(n,m,k) is almost surely connected when lim infk2n/mlogn>1,lim infk2n/mlogn>1, and G(n,m,k)G(n,m,k) is almost surely disconnected when lim supk2n/mlogn<1.lim supk2n/mlogn<1.