TDA

TDA analyzes topological features of brain networks using persistent homology on weighted brain graphs (weights from Pearson correlations) to quantify cycles with Betti plots and perform exact statistical inference of those topological features.

Key Features:

• Betti plots construction: Constructs Betti plots from weighted brain graphs derived from Pearson correlation matrices to represent topological features across filtration values.
• Persistent homology: Applies persistent homology to extract Betti numbers (zeroth and first) across filtrations.
• Exact statistical inference: Implements a framework for exact statistical inference on Betti plots to assess significance of topological features.
• Kolmogorov–Smirnov (KS) distance extension: Employs the KS distance, extending its application from zeroth Betti numbers to first Betti numbers to measure network similarity across filtrations.
• Heritability analysis: Applies the inference framework to twin imaging data from the Human Connectome Project to evaluate heritability of cycle counts in resting-state functional connectivity networks.
• Validation with simulations: Validates performance using random network simulations with known ground truths.

Scientific Applications:

• Brain network integration and connectivity: Uses cycle structure analysis to provide insights into the integration and connectivity strength of brain networks.
• Statistical significance testing in neuroimaging: Enables rigorous testing of topological features in neuroimaging studies via exact inference on Betti plots.
• Heritability assessment: Assesses whether the number of cycles in resting-state functional connectivity networks is a heritable trait using twin data from the Human Connectome Project.

Methodology:

Constructs Betti plots from weighted brain graphs with weights from Pearson correlations, applies persistent homology to compute zeroth and first Betti numbers across filtrations, uses the Kolmogorov–Smirnov (KS) distance extended to first Betti numbers for network comparison, performs exact statistical inference on Betti plots, and validates methods using random network simulations; applied to twin imaging data from the Human Connectome Project to evaluate heritability of cycles.