  We consider a dynamic Erd\H{o}s-R\'enyi random graph (ERRG) on $n$ vertices
in which each edge switches on at rate $\lambda$ and switches off at rate
$\mu$, independently of other edges. The focus is on the analysis of the
evolution of the associated empirical graphon in the limit as $n\to\infty$. Our
main result is a large deviation principle (LDP) for the sample path of the
empirical graphon observed until a fixed time horizon. The rate is
$\binom{n}{2}$, the rate function is a specific action integral on the space of
graphon trajectories. We apply the LDP to identify (i) the most likely path
that starting from a constant graphon creates a graphon with an atypically
large density of $d$-regular subgraphs, and (ii) the mostly likely path between
two given graphons. It turns out that bifurcations may occur in the solutions
of associated variational problems.
