  Traditional Topological Data Analysis (TDA) methods, such as Persistent
Homology (PH), rely on distance measures (e.g., cross-correlation, partial
correlation, coherence, and partial coherence) that are symmetric by
definition. While useful for studying topological patterns in functional brain
connectivity, the main limitation of these methods is their inability to
capture the directional dynamics - which is crucial for understanding effective
brain connectivity. We propose the Causality-Based Topological Ranking (CBTR)
method, which integrates Causal Inference (CI) to assess effective brain
connectivity with Hodge Decomposition (HD) to rank brain regions based on their
mutual influence. Our simulations confirm that the CBTR method accurately and
consistently identifies hierarchical structures in multivariate time series
data. Moreover, this method effectively identifies brain regions showing the
most significant interaction changes with other regions during seizures using
electroencephalogram (EEG) data. These results provide novel insights into the
brain's hierarchical organization and illuminate the impact of seizures on its
dynamics.
