On the total variation distance between the binomial random graph and the random intersection graph

TL;DR

This paper investigates the similarity between random intersection graphs and Erdős-Rényi graphs, proving that their total variation distance approaches zero when the set size grows faster than n^4.

Contribution

It extends previous results by showing the total variation distance tends to zero for larger set sizes, specifically when m grows faster than n^4.

Findings

Total variation distance tends to zero for m  n^4

Generalizes previous results to larger set sizes

Supports approximation of intersection graphs by Erdf6s-Re9nyi graphs

Abstract

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karo\'nski, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999), 131--159] introduced a random intersection graph by taking randomly assigned sets. The random intersection graph $G(n,m;p)$ has $n$ vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set $M$ of size $m$ where each element of $M$ belongs to each random subset with probability $p$, independently of all other elements in $M$. Fill, Scheinerman and Singer-Cohen [Random Struct Algorithms 16 (2000), 156--176] showed that the total variation distance between the random graph $G(n,m;p)$ and the Erd\"os-R\'enyi graph $G(n,\hat{p})$ tends to $0$ for any $0 \leq p=p(n) \leq 1$ if $m=n^{\alpha}$, $\alpha >6$, where $\hat{p}$ is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to $0$ for any $0 \leq p=p(n) \leq 1$ whenever $m \gg n^4$.