Diameter in ultra-small scale-free random graphs: Extended version

TL;DR

This paper investigates the diameter of ultra-small scale-free random graphs with infinite variance degrees, establishing precise asymptotics and constants for the diameter in different models.

Contribution

It provides the exact order and constants for the diameter of such graphs, extending understanding of their ultra-small world properties.

Findings

Diameter is of order log log n when minimal forward degree d ≥ 2.

Exact constant for diameter includes typical distance plus 2/ log d.

Proof techniques are unified for different models despite their differences.

Abstract

It is well known that many random graphs with infinite variance degrees are ultrasmall. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least $k$ is approximately $k^{-(\tau-1)}$ with $\tau\in(2,3)$, typical distances between pairs of vertices in a graph of size $n$ are asymptotic to $\frac{2\log\log n}{|\log(\tau-2)|}$ and $\frac{4\log\log n}{|\log(\tau-2)|}$, respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order $\log\log n$ precisely when the minimal forward degree $d$ of vertices is at least $2$. We identify the exact constant, which equals that of the typical distances plus $2/\log d$. Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.