  Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let
$\mathcal{G}(n,W)$ be the random graph obtained by independently including each
edge $jk$ with probability $W_{jk}$. Given a degree sequence ${\bf
d}=(d_1,\ldots, d_n)$, let $\mathcal{G}(n,{\bf d})$ denote a uniformly random
graph with degree sequence ${\bf d}$. We couple $\mathcal{G}(n,W)$ and
$\mathcal{G}(n,{\bf d})$ together so that a.a.s. $\mathcal{G}(n,W)$ is a
subgraph of $\mathcal{G}(n,{\bf d})$, where $W$ is some function of ${\bf d}$.
Let $\Delta({\bf d})$ denote the maximum degree in ${\bf d}$. Our coupling
result is optimal when $\Delta({\bf d})^2\ll \|{\bf d}\|_1$, i.e.\ $W_{ij}$ is
asymptotic to $\mathbb{P}(ij\in \mathcal{G}(n,{\bf d}))$ for every $i,j\in
[n]$. We also have coupling results for ${\bf d}$ that are not constrained by
the condition $\Delta({\bf d})^2\ll \|{\bf d}\|_1$. For such ${\bf d}$ our
coupling result is still close to optimal, in the sense that $W_{ij}$ is
asymptotic to $\mathbb{P}(ij\in \mathcal{G}(n,{\bf d}))$ for most pairs $i,j\in
[n]$.
