The vertex degree distribution of random intersection graphs

Random intersection graphs are a model of random graphs in which each vertex is assigned a subset of a set of objects independently and two vertices are adjacent if their assigned subsets are not disjoint. The number of vertices is denoted by n and the number of objects is supposed to be ⌊ n α ⌋ for some α > 0. We determine the distribution of the degree of a typical vertex and show that it changes sharply between α > 1, α = 1, and α > 1.