On small subgraphs in a random intersection digraph

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