How to determine if a random graph with a fixed degree sequence has a giant component

TL;DR

This paper provides a criterion to determine the presence of a giant component in a random graph with a fixed degree sequence, based on the sum of degrees not equal to 2, with high probability results.

Contribution

It establishes a nearly condition-free method to predict giant components in random graphs with fixed degree sequences, requiring only a minimal degree sum condition.

Findings

High probability of giant component when sum of degrees not 2 exceeds a threshold

Bounded probabilities for having or not having a giant component otherwise

Minimal technical conditions on degree sequences for the results

Abstract

For a fixed degree sequence $\mathcal{D}=(d_1,...,d_n)$, let $G(\mathcal{D})$ be a uniformly chosen (simple) graph on $\{1,...,n\}$ where the vertex $i$ has degree $d_i$. In this paper we determine whether $G(\mathcal{D})$ has a giant component with high probability, essentially imposing no conditions on $\mathcal{D}$. We simply insist that the sum of the degrees in $\mathcal{D}$ which are not 2 is at least $\lambda(n)$ for some function $\lambda$ going to infinity with $n$. This is a relatively minor technical condition, and when $\mathcal{D}$ does not satisfy it, both the probability that $G(\mathcal{D})$ has a giant component and the probability that $G(\mathcal{D})$ has no giant component are bounded away from $1$.