  Large graphs are sometimes studied through their degree sequences (power law
or regular graphs). We study graphs that are uniformly chosen with a given
degree sequence. Under mild conditions, it is shown that sequences of such
graphs have graph limits in the sense of Lov\'{a}sz and Szegedy with
identifiable limits. This allows simple determination of other features such as
the number of triangles. The argument proceeds by studying a natural
exponential model having the degree sequence as a sufficient statistic. The
maximum likelihood estimate (MLE) of the parameters is shown to be unique and
consistent with high probability. Thus $n$ parameters can be consistently
estimated based on a sample of size one. A fast, provably convergent, algorithm
for the MLE is derived. These ingredients combine to prove the graph limit
theorem. Along the way, a continuous version of the Erd\H{o}s--Gallai
characterization of degree sequences is derived.
