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

Harmonic holes as the submodules of brain network and network dissimilarity