Lack of Hyperbolicity in Asymptotic Erd\"os--Renyi Sparse Random Graphs

TL;DR

This paper proves that the giant component of sparse Erd"os--Renyi graphs is not Gromov hyperbolic, and as a consequence, it has zero spectral gap almost surely as the graph size grows large.

Contribution

It demonstrates the non-hyperbolicity of the giant component in sparse Erd"os--Renyi graphs and offers an alternative proof for the zero spectral gap property.

Findings

Giant component is not Gromov hyperbolic for c>1

Zero spectral gap almost surely as n→∞

Provides an alternative proof for spectral gap result

Abstract

In this work we prove that the giant component of the Erd\"os--Renyi random graph $G(n,c/n)$ for c a constant greater than 1 (sparse regime), is not Gromov $\delta$-hyperbolic for any positive $\delta$ with probability tending to one as $n\to\infty$. As a corollary we provide an alternative proof that the giant component of $G(n,c/n)$ when c>1 has zero spectral gap almost surely as $n\to\infty$.