  We consider the empirical neighborhood distribution of marked sparse
Erd\"os-R\'enyi random graphs, obtained by decorating edges and vertices of a
sparse Erd\"os-R\'enyi random graph with i.i.d.\ random elements taking values
on Polish spaces. We prove that the empirical neighborhood distribution of this
model satisfies a large deviation principle in the framework of local weak
convergence. We rely on the concept of BC-entropy introduced by Delgosha and
Anantharam~(2019) which is inspired on the previous work by Bordenave and
Caputo~(2015). Our main technical contribution is an approximation result that
allows one to pass from graph with marks in discrete spaces to marks in general
Polish spaces. As an application of the results developed here, we prove a
large deviation principle for interacting diffusions driven by gradient
evolution and defined on top of sparse Erd\"os-R\'enyi random graphs. In
particular, our results apply for the stochastic Kuramoto model. We obtain
analogous results for the sparse uniform random graph with given number of
edges.
