Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Batagelj, V., Mrvar, A.: Pajek - analysis and visualization of large networks. In: Jünger, M., Mutzel, P. (eds.) Graph Drawing Software. Mathematics and Visualization, pp. 77–103. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-642-18638-7_4
Carlsson, G., Collins, A., Guibas, L.J.: Persistence barcodes for shapes. Int. J. Shape Model. 11, 149–187 (2005)
Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76(1), 1–12 (2008)
Choi, H., Jin, K.H.: Predicting cognitive decline with deep learning of brain metabolism and amyloid imaging. Behav. Brain Res. 344, 103–109 (2018). https://doi.org/10.1016/j.bbr.2018.02.017. https://www.sciencedirect.com/science/article/pii/S0166432818301013
Chung, M.K., Bubenik, P., Kim, P.T.: Persistence diagrams of cortical surface data. In: Prince, J.L., Pham, D.L., Myers, K.J. (eds.) IPMI 2009. LNCS, vol. 5636, pp. 386–397. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02498-6_32
Chung, M.K., Villalta-Gil, V., Lee, H., Rathouz, P.J., Lahey, B.B., Zald, D.H.: Exact topological inference for paired brain networks via persistent homology. In: Niethammer, M., et al. (eds.) IPMI 2017. LNCS, vol. 10265, pp. 299–310. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59050-9_24
Chung, M.K., et al.: Topological brain network distances. arXiv:1809.03878 [stat.AP] (2018). https://arxiv.org/abs/1809.03878
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120 (2007)
Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Warner, F., Zucker, S.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. In: Proceedings of the National Academy of Sciences, pp. 7426–7431 (2005)
Edelsbrunner, H., Harer, J.: Persistent homology - a survey. Contemp. Math. 453, 257–282 (2008)
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society Press, New York (2009)
Friedman, J.: Computing Betti numbers via combinatorial Laplacians. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pp. 386–391 (1996)
Horak, D., Jost, J.: Spectra of combinatorial Laplace operators on simplicial complexes. Adv. Math. 244, 303–336 (2013)
Kim, Y.-J., Kook, W.: Harmonic cycles for graphs. Linear Multilinear Algebra, 1–11 (2018). https://doi.org/10.1080/03081087.2018.1440519
Lee, H., Chung, M.K., Kang, H., Choi, H., Kim, Y.K., Lee, D.S.: Abnormal hole detection in brain connectivity by kernel density of persistence diagram and Hodge Laplacian. In: 2018 IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018), pp. 20–23, April 2018. https://doi.org/10.1109/ISBI.2018.8363514
Lee, H., Chung, M.K., Kang, H., Kim, B.N., Lee, D.S.: Persistent brain network homology from the perspective of dendrogram. IEEE Trans. Med. Imaging 31, 2267–2277 (2012)
Lee, H., Chung, M.K., Kang, H., Lee, D.S.: Hole detection in metabolic connectivity of alzheimer’s disease using k–laplacian. In: Golland, P., Hata, N., Barillot, C., Hornegger, J., Howe, R. (eds.) MICCAI 2014, LNCS, vol. 8675, pp. 297–304. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10443-0_38
Lim, L.H.: Hodge Laplacians on graphs. Geometry and topology in statistical inference. In: Proceedings of Symposia in Applied Mathematics, vol. 73 (2015)
Petri, G., et al.: Homological scaffolds of brain functional networks. J. Roy. Soc. Interface 11(101), 20140873 (2014). https://doi.org/10.1098/rsif.2014.0873
Reininghaus, J., Huber, S., Bauer, U., Kwitt, R.: A stable multi-scale kernel for topological machine learning. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4741–4748, June 2015
Rolls, E.T., Joliot, M., Tzourio-Mazoyer, N.: Implementation of a new parcellation of the orbitofrontal cortex in the automated anatomical labeling atlas. Neuroimage 122, 1–5 (2015)
Sanabria-Diaz, G., Martìnez-Montes, E., Melie-Garcia, L., Alzheimer’s Disease Neuroimaging Initiative: Glucose metabolism during resting state reveals abnormal brain networks organization in the Alzheimer’s disease and mild cognitive impairment. PLOS ONE 8(7), 1–25 (2013). https://doi.org/10.1371/journal.pone.0068860
Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. J. Vis. 8, 1–18 (2008)
Sizemore, A., Giusti, C., Kahn, A., Vettel, J., Betzel, R., Bassett, D.: Cliques and cavities in the human connectome. J. Comput. Neurosci. 44, 115–145 (2018)
Solo, V., et al.: Connectivity in fMRI: blind spots and breakthroughs. IEEE Trans. Med. Imaging 37(7), 1537–1550 (2018). https://doi.org/10.1109/TMI.2018.2831261
Sporns, O., Tononi, G., Edelman, G.: Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices. Cereb. Cortex 10(2), 127–141 (2000). https://doi.org/10.1093/cercor/10.2.127
Sporns, O., Betzel, R.F.: Modular brain networks. Ann. Rev. Psychol. 67, 19.1–19.28 (2016)
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33, 249–274 (2005)
Acknowledgements
Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). 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 http://adni.loni.usc.edu. 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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Lee, H. et al. (2019). Harmonic Holes as the Submodules of Brain Network and Network Dissimilarity. In: Marfil, R., Calderón, M., Díaz del Río, F., Real, P., Bandera, A. (eds) Computational Topology in Image Context. CTIC 2019. Lecture Notes in Computer Science(), vol 11382. Springer, Cham. https://doi.org/10.1007/978-3-030-10828-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-10828-1_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-10827-4
Online ISBN: 978-3-030-10828-1
eBook Packages: Computer ScienceComputer Science (R0)