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Recent studies have found that the modular structure of functional brain network is disrupted during the progress of Alzheimer's is the most basic topological disease. The modular structure of network invariant in determining the shape of network in the view of algebraic topology. In this study, we propose a new method to find another higher order topological invariant, hole, based on persistent homology. If a hole exists in the network, the information can be inefficiently delivered between regions. If we can localize the hole in the network, we can infer the reason of network inefficiency. We propose to detect the persistent hole using the spectrum of kappa-Laplacian, which is the generalized version of graph Laplacian. The method is applied to the metabolic network based on FDG-PET data of Alzheimer disease (AD), mild cognitive impairment (MCI) and normal control (NC) groups. The experiments show that the persistence of hole can be used as a biological marker of disease progression to AD. The localized hole may help understand the brain network abnormality in AD, revealing that the limbic-temporo-parietal association regions disturb direct connections between other regions.

Hole Detection in Metabolic Connectivity of Alzheimer’s Disease Using k −Laplacian