An Independent Process Approximation to Sparse Random Graphs with a Prescribed Number of Edges and Triangles

TL;DR

This paper establishes a bound on how closely a certain independent process approximates the uniform distribution of graphs with fixed numbers of edges and triangles, using Chen-Stein Poisson approximation.

Contribution

It introduces a pre-asymptotic bound on total variation distance between two graph distributions with fixed edge and triangle counts, providing explicit rates for finite parameters.

Findings

Total variation distance tends to zero when edge and triangle counts are small relative to n.

Explicit bounds are derived for all finite parameter values.

The method employs Chen-Stein Poisson approximation.

Abstract

We prove a $pre$-$asymptotic$ bound on the total variation distance between the uniform distribution over two types of undirected graphs with $n$ nodes. One distribution places a prescribed number of $k_T$ triangles and $k_S$ edges not involved in a triangle independently and uniformly over all possibilities, and the other is the uniform distribution over simple graphs with exactly $k_T$ triangles and $k_S$ edges not involved in a triangle. As a corollary, for $k_S = o(n)$ and $k_T = o(n)$ as $n$ tends to infinity, the total variation distance tends to $0$, at a rate that is given explicitly. Our main tool is Chen-Stein Poisson approximation, hence our bounds are explicit for all finite values of the parameters.