On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase

TL;DR

This paper analyzes the size of the largest component in a random graph with a subpower-law degree sequence in a subcritical phase, establishing bounds that match conjectures and previous results for similar models.

Contribution

It proves that for degree sequences obeying a gamma subpower law with gamma greater than 3, the largest component size is bounded by a specific power of n, aligning with conjectured bounds.

Findings

Largest component size does not exceed n^{1/γ+ε_n} with high probability

The bound matches the best possible power for such graphs

Results align with conjectures for models with independent degrees

Abstract

A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed $n^{1/\gamma+\varepsilon_n}$, $\varepsilon_n=O(\ln\ln n/\ln n)$, $1/\gamma$ being the best power for this random graph. This is similar to the best possible $n^{1/(\gamma-1)}$ bound for a different model of the random graph, one with independent vertex degrees, conjectured by Durrett, and proved recently by Janson.