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

For a fixed degree sequence $${\mathcal {D}}=(d_1,\ldots ,d_n)$$ , let $$G({\mathcal {D}})$$ be a uniformly chosen (simple) graph on $$\{1,\ldots ,n\}$$ 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. [ABSTRACT FROM AUTHOR]