  We introduce an innovative, data-driven topological data analysis (TDA)
technique for estimating the state spaces of dynamically changing functional
human brain networks at rest. Our method utilizes the Wasserstein distance to
measure topological differences, enabling the clustering of brain networks into
distinct topological states. This technique outperforms the commonly used
k-means clustering in identifying brain network state spaces by effectively
incorporating the temporal dynamics of the data without the need for explicit
model specification. We further investigate the genetic underpinnings of these
topological features using a twin study design, examining the heritability of
such state changes. Our findings suggest that the topology of brain networks,
particularly in their dynamic state changes, may hold significant hidden
genetic information. MATLAB code for the method is available at
https://github.com/laplcebeltrami/PH-STAT.
