The Largest Component of Near-Critical Random Intersection Graph with Tunable Clustering

Abstract

In this paper, we study the largest component of the near-critical random intersection graph Gn,m,p with n nodes and m elements, where m=Θn which leads to the fact that the clustering is tunable. We prove that with high probability the size of the largest component in the weakly supercritical random intersection graph with tunable clustering on n vertices is of order nϵn, and it is of order ϵ−2nlognϵ3n in the weakly subcritical one, where ϵn⟶0 and n1/3ϵn⟶∞ as n⟶∞.