We consider a dynamic ErdH{o}s-Renyi 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 $ntoinfty$. 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.
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