  We investigate structural properties of large, sparse random graphs through
the lens of "sampling convergence" (Borgs et. al. (2017)). Sampling convergence
generalizes left convergence to sparse graphs, and describes the limit in terms
of a "graphex". We introduce a notion of sampling convergence for sequences of
multigraphs, and establish the graphex limit for the configuration model, a
preferential attachment model, the generalized random graph, and a bipartite
variant of the configuration model. The results for the configuration model,
preferential attachment model and bipartite configuration model provide
necessary and sufficient conditions for these random graph models to converge.
The limit for the configuration model and the preferential attachment model is
an augmented version of an exchangeable random graph model introduced by Caron
and Fox (2017).
