Harmonic Holes as the Submodules of Brain Network and Network Dissimilarity

  • Hyekyoung LeeEmail author
  • Moo K. Chung
  • Hongyoon Choi
  • Hyejin Kang
  • Seunggyun Ha
  • Yu Kyeong Kim
  • Dong Soo Lee
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11382)


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 controls (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.


Topological data analysis Brain network Alzheimer’s disease Harmonic holes Hodge Laplacian 



Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database ( As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at This work is supported by Basic Science Research Program through the National Research Foundation (NRF) (No. 2013R1A1A2064593 and No. 2016R1D1A1B03935463), NRF Grant funded by MSIP of Korea (No. 2015M3C7A1028926 and No. 2017M3C7A1048079), NRF grant funded by the Korean Government (No. 2016R1D1A1A02937497, No. 2017R1A5A1015626, and No. 2011-0030815), and NIH grant EB022856.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Hyekyoung Lee
    • 1
    Email author
  • Moo K. Chung
    • 4
  • Hongyoon Choi
    • 1
  • Hyejin Kang
    • 2
  • Seunggyun Ha
    • 1
  • Yu Kyeong Kim
    • 3
  • Dong Soo Lee
    • 1
    • 2
  1. 1.Seoul National University HospitalSeoulRepublic of Korea
  2. 2.Seoul National UniversitySeoulRepublic of Korea
  3. 3.SMG-SNU Boramae Medical CenterSeoulRepublic of Korea
  4. 4.University of Wisconsin-MadisonMadisonUSA

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