Community and rich-club detection are a well-known method to extract functionally specialized subnetwork in brain connectivity analysis. They find densely connected subregions with large modularity or high degree in brain connectivity studies. However, densely connected nodes are not the only representation of network shape. In this study, we propose a new method to extract abnormal holes, which are another representation of network shape. While densely connected component characterizes network's efficiency, abnormal holes characterize inefficiency. The proposed method differs from the existing hole detection in two respects. One is to use Hodge Laplacian to obtain a harmonic hole in the linear combination of edges, rather than a subset of edges. The other is to use the kernel density estimation of persistence diagram of random networks to determine the significance of a hole, rather than using the persistence of a hole. We applied the proposed method to find the abnormality of metabolic connectivity in the FDG PET data of ADNI. We found that, as AD severely progressed, the brain network had more abnormal holes. The localized holes showed how inefficient the structure of brain network became as the disease progressed.
Keywords: Alzheimer’s disease; Brain connectivity; Hodge Laplacian; Hole; Kernel density estimation.