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 attentiondeficit hyperactivity disorder children, 26 autism spectrum disorder children and 11 pediatric control subjects.

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Abstract.The difference between networks has been often assessed by the difference of global topological measures such as the clustering coefficient, degree distribution and modularity. In this paper, we introduce a new framework for measuring the network difference using the GromovHausdorff (GH) distance, which is often used in shape analysis. In order to apply the GH distance, we define the shape of the brain network by piecing together the patches of locally connected nearest neighbors using the graph filtration. The shape of the network is then transformed to an algebraic form called the single linkage matrix. The single linkage matrix is subsequently used in measuring network differences using the GH distance. As an illustration, we apply the proposed framework to compare the FDG-PET based functional brain networks out of 24 attention deficit hyperactivity disorder (ADHD) children, 26 autism spectrum disorder (ASD) children and 11 pediatric control subjects.

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 .

We present a novel framework for characterizing paired brain networks using techniques in hyper-networks, sparse learning and persistent homology. The framework is general enough for dealing with any type of paired images such as twins, multimodal and longitudinal images. The exact nonparametric statistical inference procedure is derived on testing monotonic graph theory features that do not rely on time consuming permutation tests. The proposed method computes the exact probability in quadratic time while the permutation tests require exponential time. As illustrations, we apply the method to simulated networks and a twin fMRI study. In case of the latter, we determine the statistical significance of the heritability index of the large-scale reward network where every voxel is a network node.