  Topological data analysis (TDA) approaches are becoming increasingly popular
for studying the dependence patterns in multivariate time series data. In
particular, various dependence patterns in brain networks may be linked to
specific tasks and cognitive processes, which can be altered by various
neurological impairments such as epileptic seizures. Existing TDA approaches
rely on the notion of distance between data points that is symmetric by
definition for building graph filtrations. For brain dependence networks, this
is a major limitation that constrains practitioners to using only symmetric
dependence measures, such as correlations or coherence. However, it is known
that the brain dependence network may be very complex and can contain a
directed flow of information from one brain region to another. Such dependence
networks are usually captured by more advanced measures of dependence such as
partial directed coherence, which is a Granger causality based dependence
measure. These dependence measures will result in a non-symmetric distance
function, especially during epileptic seizures. In this paper we propose to
solve this limitation by decomposing the weighted connectivity network into its
symmetric and anti-symmetric components using matrix decomposition and
comparing the anti-symmetric component prior to and post seizure. Our analysis
of epileptic seizure EEG data shows promising results.
