A random intersection digraph: Indegree and outdegree distributions

Let S ( 1 ) , … , S ( n ) , T ( 1 ) , … , T ( n ) be random subsets of the set [ m ] = { 1 , … , m } . We consider the random digraph D on the vertex set [ n ] defined as follows: the arc i → j is present in D whenever S ( i ) ∩ T ( j ) ≠ 0̸ . Assuming that the pairs of sets ( S ( i ) , T ( i ) ) , 1 ≤ i ≤ n , are independent and identically distributed, we study the in- and outdegree distributions of a typical vertex of D as n , m → ∞ .