  Random graph models have played a dominant role in the theoretical study of
networked systems. The Poisson random graph of Erdos and Renyi, in particular,
as well as the so-called configuration model, have served as the starting point
for numerous calculations. In this paper we describe another large class of
random graph models, which we call equitable random graphs and which are
flexible enough to represent networks with diverse degree distributions and
many nontrivial types of structure, including community structure, bipartite
structure, degree correlations, stratification, and others, yet are exactly
solvable for a wide range of properties in the limit of large graph size,
including percolation properties, complete spectral density, and the behavior
of homogeneous dynamical systems, such as coupled oscillators or epidemic
models.
