Persistent Brain Network Homology From the Perspective of Dendrogram

The brain network is usually constructed by estimating the connectivity matrix and thresholding it at an arbitrary level. The problem with this standard method is that we do not have any generally accepted criteria for determining a proper threshold. Thus, we propose a novel multiscale framework that models all brain networks generated over every possible threshold. Our approach is based on persistent homology and its various representations such as the Rips filtration, barcodes, and dendrograms. This new persistent homological framework enables us to quantify various persistent topological features at different scales in a coherent manner. The barcode is used to quantify and visualize the evolutionary changes of topological features such as the Betti numbers over different scales. By incorporating additional geometric information to the barcode, we obtain a single linkage dendrogram that shows the overall evolution of the network. The difference between the two networks is then measured by the Gromov-Hausdorff distance over the dendrograms. As an illustration, we modeled and differentiated the FDG-PET based functional brain networks of 24 attention-deficit hyperactivity disorder children, 26 autism spectrum disorder children, and 11 pediatric control subjects.