  In clinical neuroscience, epileptic seizures have been associated with the
sudden emergence of coupled activity across the brain. The resulting functional
networks - in which edges indicate strong enough coupling between brain regions
- are consistent with the notion of percolation, which is a phenomenon in
complex networks corresponding to the sudden emergence of a giant connected
component. 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 edge birth and edge death, as well as noise. This class is used
to understand the type of phase transitions undergone in a seizure, and in
particular, distinguishing between different percolation regimes in epileptic
seizures. We develop a hypothesis testing framework for inferring putative
percolation mechanisms. As a necessary precursor, we present an EM algorithm
for estimating parameters from a sequence of noisy networks only observed at a
longitudinal subsampling of time points. Our results suggest that different
types of percolation can occur in human seizures. The type inferred may suggest
tailored treatment strategies and provide new insights into the fundamental
science of epilepsy.
