Computing the shape of brain networks using graph filtration and Gromov-Hausdorff metric

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 Gromov-Hausdorff (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.