Convergence rate of $H$-property for step-graphons

TL;DR

This paper investigates the convergence rate of the probability that a random graph sampled from a graphon has a cycle cover, revealing exponential and root n rates with rigorous proofs and numerical validation.

Contribution

It characterizes the convergence rates of the $H$-property in graphons, distinguishing between exponential and root n rates, and provides rigorous proofs and numerical validation.

Findings

Identifies two types of convergence rates: exponential and root n.

Provides rigorous mathematical proofs for the convergence rates.

Validates the theoretical results with numerical experiments.

Abstract

A graphon is said to have the $H$-property if a random undirected graph $G_n$ on $n$ nodes sampled from it has a node-wise disjoint cycle cover almost surely as $n\to\infty$. It has been shown in the earlier work that the $H$-property obeys the zero-one law, i.e., the probability that the random graph has a cycle cover tends to either one or zero. In this paper, we sharpen the result by characterizing the convergence rate of the probability. Specifically, we show that there are two different types of rates, with one being exponential and the other being root $n$. We provide a rigorous proof and numerical validation.