Childhood maltreatment may adversely affect brain development and consequently influence behavioral, emotional, and psychological patterns during adulthood. In this study, we propose an analytical pipeline for modeling the altered topological structure of brain white matter in maltreated and typically developing children. We perform topological data analysis (TDA) to assess the alteration in the global topology of the brain white matter structural covariance network among children. We use persistent homology, an algebraic technique in TDA, to analyze topological features in the brain covariance networks constructed from structural magnetic resonance imaging and diffusion tensor imaging. We develop a novel framework for statistical inference based on the Wasserstein distance to assess the significance of the observed topological differences. Using these methods in comparing maltreated children with a typically developing control group, we find that maltreatment may increase homogeneity in white matter structures and thus induce higher correlations in the structural covariance; this is reflected in the topological profile. Our findings strongly suggest that TDA can be a valuable framework to model altered topological structures of the brain. The MATLAB codes and processed data used in this study can be found at https://github.com/laplcebeltrami/maltreated . Author Summary We employ topological data analysis (TDA) to investigate altered topological structures in the white matter of children who have experienced maltreatment. Persistent homology in TDA is utilized to quantify topological differences between typically developing children and those subjected to maltreatment, using magnetic resonance imaging and diffusion tensor imaging data. The Wasserstein distance is computed between topological features to assess disparities in brain networks. Our findings demonstrate that persistent homology effectively characterizes the altered dynamics of white matter in children who have suffered maltreatment.

A cycle in a brain network is a subset of a connected component with redundant additional connections. If there are many cycles in a connected component, the connected component is more densely connected. Whereas the number of connected components represents the integration of the brain network, the number of cycles represents how strong the integration is. However, it is unclear how to perform statistical inference on the number of cycles in the brain network. In this study, we present a new statistical inference framework for determining the significance of the number of cycles through the Kolmogorov-Smirnov (KS) distance, which was recently introduced to measure the similarity between networks across different filtration values by using the zeroth Betti number. In this paper, we show how to extend the method to the first Betti number, which measures the number of cycles. The performance analysis was conducted using the random network simulations with ground truths. By using a twin imaging study, which provides biological ground truth, the methods are applied in determining if the number of cycles is a statistically significant heritable network feature in the resting-state functional connectivity in 217 twins obtained from the Human Connectome Project. The MATLAB codes as well as the connectivity matrices used in generating results are provided at http://www.stat.wisc.edu/∼mchung/TDA . Author Summary In this paper, we propose a new topological distance based on the Kolmogorov-Smirnov (KS) distance that is adapted for brain networks, and compare them against other topological network distances including the Gromov-Hausdorff (GH) distances. KS-distance is recently introduced to measure the similarity between networks across different filtration values by using the zeroth Betti number, which measures the number of connected components. In this paper, we show how to extend the method to the first Betti number, which measures the number of cycles. The performance analysis was conducted using random network simulations with ground truths. Using a twin imaging study, which provides biological ground truth (of network differences), we demonstrate that the KS distances on the zeroth and first Betti numbers have the ability to determine heritability.