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 e-prints
- 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
- E-Print:
- 42 pages, to appear in Probability Theory and Related Fields