In the study of random networks, percolation – the sudden emergence of a giant connected component (GCC) – is of fundamental interest. Traditionally, work has concentrated on noise-free percolation with a monotonic process of network growth, but real-world networks are more complex. We develop a class of random graph hidden Markov models (RG-HMMs) for characterizing percolation regimes in noisy, dynamically evolving networks in the presence of both edge birth and edge death. This class subsumes a variety of random graph models already used in studying noise-free percolation. We focus on parameter estimation and testing of putative percolation regimes. We present an Expectation-Maximization (EM) algorithm, incorporating data augmentation and particle filtering, for estimating parameters in the model with a given sequence of noisy networks observed only at a longitudinal subsampling of time points. This in turn facilitates development of hypothesis testing strategies aimed ultimately at inferring putative percolation mechanisms in epileptic seizures.