  Statistical network modeling has focused on representing the graph as a
discrete structure, namely the adjacency matrix, and considering the
exchangeability of this array. In such cases, the Aldous-Hoover representation
theorem (Aldous, 1981;Hoover, 1979} applies and informs us that the graph is
necessarily either dense or empty. In this paper, we instead consider
representing the graph as a measure on $\mathbb{R}_+^2$. For the associated
definition of exchangeability in this continuous space, we rely on the
Kallenberg representation theorem (Kallenberg, 2005). We show that for certain
choices of such exchangeable random measures underlying our graph construction,
our network process is sparse with power-law degree distribution. In
particular, we build on the framework of completely random measures (CRMs) and
use the theory associated with such processes to derive important network
properties, such as an urn representation for our analysis and network
simulation. Our theoretical results are explored empirically and compared to
common network models. We then present a Hamiltonian Monte Carlo algorithm for
efficient exploration of the posterior distribution and demonstrate that we are
able to recover graphs ranging from dense to sparse--and perform associated
tests--based on our flexible CRM-based formulation. We explore network
properties in a range of real datasets, including Facebook social circles, a
political blogosphere, protein networks, citation networks, and world wide web
networks, including networks with hundreds of thousands of nodes and millions
of edges.
