  This paper concerns the large deviations of a system of interacting particles
on a random graph. There is no stochasticity, and the only sources of disorder
are the random graph connections, and the initial condition. The average number
of afferent edges on any particular vertex must diverge to infinity as $N\to
\infty$, but can do so at an arbitrarily slow rate. These results are thus
accurate for both sparse and dense random graphs. A particular application to
sparse Erdos-Renyi graphs is provided. The theorem is proved by pushing forward
a Large Deviation Principle for a `nested empirical measure' generated by the
initial conditions to the dynamics. The nested empirical measure can be thought
of as the density of the density of edge connections: the associated weak
topology is more coarse than the topology generated by the graph cut norm, and
thus there is a broader range of application.
