On the second largest component of random hyperbolic graphs

TL;DR

This paper analyzes the size of the second largest component in random hyperbolic graphs within a specific parameter range, providing precise asymptotic results and answering open questions in the field.

Contribution

It establishes the asymptotic size of the second largest component in the hyperbolic graph model for key parameter ranges, resolving prior open problems.

Findings

Second largest component size is Θ((log n)^{1/(1-α)}) for 1/2 < α < 1

Size is Θ(log n) with constant probability at α=1/2

Size is polynomial in n at α=1

Abstract

We show that in the random hyperbolic graph model as formalized by Gugelmann et al. in the most interesting range of $\frac12 < \alpha < 1$ the size of the second largest component is $\Theta((\log n)^{1/(1-\alpha)})$, thus answering a question of Bode et al. We also show that for $\alpha=\frac12$ with constant probability the corresponding size is $\Theta(\log n)$, whereas for $\alpha=1$ it is $\Omega(n^{b})$ for some $b > 0$.