Equivalence of the random intersection graph and G(n,p)

TL;DR

This paper proves that the random intersection graph G(n,m,p) with m >= n^3 has the same threshold functions as an independently generated graph, confirming a conjecture and providing sharper equivalence results.

Contribution

It confirms the conjecture by Fill, Scheinerman, and Singer-Cohen that these two graph models are equivalent in terms of threshold functions, with improved results under certain conditions.

Findings

Established the equivalence of threshold functions for G(n,m,p) and independent-edge graphs.

Proved the conjecture posed by Fill, Scheinerman, and Singer-Cohen.

Derived sharper equivalence results under additional assumptions.

Abstract

We solve the conjecture posed by Fill, Scheinerman and Singer-Cohen and show the equivalence of the sharp threshold functions of the random intersection graph G(n,m,p) with $m >= n^3$ and a graph in which each edge appears independently. Moreover we prove sharper equivalence results under some additional assumptions.