Distinguishing between different percolation regimes in noisy dynamic networks with an application to epileptic seizures

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. 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. Author summaryPrior work has shown that an explosive density increase (i.e., more edges) in the brain functional connectivity networks in epilepsy patients during seizure onset aligns with the notion of percolation-a phenomenon in complex networks corresponding to the sudden emergence of a giant connected component. Our work delves deeper to provide statistical methods to uncover the underlying network evolution behavior behind the density increase. We aim to answer the question: How can we distinguish between different percolation regimes in practice? We develop a class of random graph hidden Markov models (RG-HMMs) and the necessary inferential methodologies applicable to real-world context, for characterizing percolation regimes in dynamically evolving networks in the presence of edge birth, death and noise. We present an EM algorithm with particle filtering and data augmentation for estimating parameters, and a hypothesis testing framework using Bayes factor for inferring between the Erdos-Renyi (a classical type) and the product-rule (an explosive type) percolation regimes. Our application to real seizure data suggests that different types of percolation can occur at different stages of human seizures depending on the clinical seizure types.