On small subgraphs in a random intersection digraph

Given a set of vertices V and a set of attributes W let each vertex v ∈ V include an attribute w ∈W into a set S−(v) with probability p− and let it include w into a set S(v) with probability p+ independently for each w ∈ W . The random binomial intersection digraph on the vertex set V is defined as follows: for each u, v ∈ V the arc uv is present if S−(u) and S(v) are not disjoint. For any h = 2, 3, . . . we determine the birth threshold of the complete digraph on h vertices and describe the configurations of intersecting sets that realise the threshold.