How to determine if a random graph with a fixed degree sequence has a giant component
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 $1$.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 DOI:
 10.48550/arXiv.1601.03714
 arXiv:
 arXiv:1601.03714
 Bibcode:
 2016arXiv160103714J
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 Mathematics  Probability
 EPrint:
 42 pages, to appear in Probability Theory and Related Fields