Harmonic holes as the submodules of brain network and network dissimilarity

Persistent homology has been applied to brain network analysis for finding the shape of brain networks across multiple thresholds. In the persistent homology, the shape of networks is often quantified by the sequence of $k$-dimensional holes and Betti numbers.The Betti numbers are more widely used than holes themselves in topological brain network analysis. However, the holes show the local connectivity of networks, and they can be very informative features in analysis. In this study, we propose a new method of measuring network differences based on the dissimilarity measure of harmonic holes (HHs). The HHs, which represent the substructure of brain networks, are extracted by the Hodge Laplacian of brain networks. We also find the most contributed HHs to the network difference based on the HH dissimilarity. We applied our proposed method to clustering the networks of 4 groups, normal control (NC), stable and progressive mild cognitive impairment (sMCI and pMCI), and Alzheimer's disease (AD). The results showed that the clustering performance of the proposed method was better than that of network distances based on only the global change of topology.