  Sparse exchangeable graphs on $\mathbb{R}_+$, and the associated graphex
framework for sparse graphs, generalize exchangeable graphs on $\mathbb{N}$,
and the associated graphon framework for dense graphs. We develop the graphex
framework as a tool for statistical network analysis by identifying the
sampling scheme that is naturally associated with the models of the framework,
and by introducing a general consistent estimator for the parameter (the
graphex) underlying these models. The sampling scheme is a modification of
independent vertex sampling that throws away vertices that are isolated in the
sampled subgraph. The estimator is a dilation of the empirical graphon
estimator, which is known to be a consistent estimator for dense exchangeable
graphs; both can be understood as graph analogues to the empirical distribution
in the i.i.d. sequence setting. Our results may be viewed as a generalization
of consistent estimation via the empirical graphon from the dense graph regime
to also include sparse graphs.
