Abstract
An increasingly important class of networks is derived from physical systems that have a spatial basis. Specifically, nodes in the network have spatial coordinates associated with them, and conserved edges in two networks being aligned have correlated distance measures. An example of such a network is the human brain connectome – a network of co-activity of different regions of the brain, as observed in a functional MRI (fMRI). Here, the problem of identifying conserved patterns corresponds to the alignment of connectomes. In this context, one may structurally align the brains through co-registration to a common coordinate system. Alternately, one may align the networks, ignoring the structural basis of co-activity. In this paper, we formulate a novel problem – rigid graph alignment, which simultaneously aligns the network, as well as the underlying structure. We formally specify the problem and present a method based on expectation maximization, which alternately aligns the network and the structure via rigid body transformations. We demonstrate that our method significantly improves the quality of network alignment in synthetic graphs. We also apply rigid graph alignment to functional brain networks derived from 20 subjects drawn from the Human Connectome Project (HCP), and show over a two-fold increase in quality of alignment. Our results are broadly applicable to other applications and abstracted networks that can be embedded in metric spaces – e.g., through spectral embeddings.
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References
Bayati, M., Gleich, D.F., Saberi, A., Wang, Y.: Message-passing algorithms for sparse network alignment. ACM Trans. Knowl. Discov. Data 7(1), 3:1–3:31 (2013). http://doi.acm.org/10.1145/2435209.2435212
Berg, J., Lässig, M.: Local graph alignment and motif search in biological networks. Proc. Natl. Acad. Sci. 101(41), 14689–14694 (2004). https://www.pnas.org/content/101/41/14689
Berman, H.M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T.N., Weissig, H., Shindyalov, I.N., Bourne, P.E.: The protein data bank. Nucleic Acids Res. 28(1), 235–242 (2000)
Besl, P., McKay, H.: A method for registration of 3-D shapes. IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992)
Bouaziz, S., Tagliasacchi, A., Pauly, M.: Sparse iterative closest point. In: Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing, SGP 2013, pp. 113–123. Eurographics Association, Aire-la-Ville (2013). http://dx.doi.org/10.1111/cgf.12178
Ciriello, G., Mina, M., Guzzi, P.H., Cannataro, M., Guerra, C.: Alignnemo: a local network alignment method to integrate homology and topology. PLOS ONE 7(6), 1–14 (2012)
Conroy, B.R., Ramadge, P.J.: The grouped two-sided orthogonal procrustes problem. In: 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3688–3691, May 2011
Eggert, D., Lorusso, A., Fisher, R.: Estimating 3-D rigid body transformations: a comparison of four major algorithms. Mach. Vis. Appl. 9(5), 272–290 (1997)
Emmert-Streib, F., Dehmer, M., Shi, Y.: Fifty years of graph matching, network alignment and network comparison. Inf. Sci. 346(C), 180–197 (2016). https://doi.org/10.1016/j.ins.2016.01.074
Essen, D.C.V., Smith, S.M., Barch, D.M., Behrens, T.E., Yacoub, E., Ugurbil, K.: The WU-Minn human connectome project: an overview. NeuroImage 80, 62–79 (2013)
Finn, E.S., Shen, X., Scheinost, D., Rosenberg, M.D., Huang, J., Chun, M.M., Papademetris, X., Constable, R.T.: Functional connectome fingerprinting: identifying individuals using patterns of brain connectivity. Nature Neurosci. 18(11), 1664–1671 (2015)
Jenkinson, M., Bannister, P., Brady, M., Smith, S.: Improved optimization for the robust and accurate linear registration and motion correction of brain images. NeuroImage 17(2), 825–841 (2002)
Kabsch, W.: A solution for the best rotation to relate two sets of vectors. Acta Crystallogr. Sect. A 32(5), 922–923 (1976)
Kuchaiev, O., Milenković, T., Memišević, V., Hayes, W., Pržulj, N.: Topological network alignment uncovers biological function and phylogeny. J. Roy. Soc. Interface (2010). http://rsif.royalsocietypublishing.org/content/early/2010/03/24/rsif.2010.0063
Murphy, K., Birn, R.M., Handwerker, D.A., Jones, T.B., Bandettini, P.A.: The impact of global signal regression on resting state correlations: are anti-correlated networks introduced? NeuroImage 44(3), 893–905 (2009)
Patro, R., Kingsford, C.: Global network alignment using multiscale spectral signatures. Bioinformatics 28(23), 3105–3114 (2012)
Rusinkiewicz, S., Levoy, M.: Efficient variants of the ICP algorithm. In: Proceedings of the 3DIM 2001, October 2001
Sabata, B., Aggarwal, J.: Estimation of motion from a pair of range images: a review. CVGIP: Image Underst. 54(3), 309–324 (1991). http://www.sciencedirect.com/science/article/pii/104996609190032K
Schönemann, P.H.: A generalized solution of the orthogonal procrustes problem. Psychometrika 31(1), 1–10 (1966)
Singh, R., Xu, J., Berger, B.: Pairwise global alignment of protein interaction networks by matching neighborhood topology. In: Proceedings of the 11th Annual International Conference on Research in Computational Molecular Biology, RECOMB 2007, pp. 16–31. Springer, Heidelberg (2007)
Smith, S.M., Beckmann, C.F., Andersson, J., Auerbach, E.J., Bijsterbosch, J., Douaud, G., Duff, E., Feinberg, D.A., Griffanti, L., Harms, M.P., Kelly, M., Laumann, T., Miller, K.L., Moeller, S., Petersen, S., Power, J., Salimi-Khorshidi, G., Snyder, A.Z., Vu, A.T., Woolrich, M.W., Xu, J., Yacoub, E., Uğurbil, K., Essen, D.C.V., Glasser, M.F.: Resting-state fMRI in the human connectome project. NeuroImage 80, 144–168 (2013)
Smith, S.M.: Fast robust automated brain extraction. Hum. Brain Mapp. 17(3), 143–155 (2002)
Sussenguth, E.H.: A graph-theoretic algorithm for matching chemical structures. J. Chem. Documentation 5(1), 36–43 (1965)
Umeyama, S.: An eigendecomposition approach to weighted graph matching problems. IEEE Trans. Pattern Anal. Mach. Intell. 10(5), 695–703 (1988)
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The authors are supported by the National Science Foundation grants CCF-1149756 and IIS-1546488.
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Ravindra, V., Nassar, H., Gleich, D.F., Grama, A. (2020). Rigid Graph Alignment. In: Cherifi, H., Gaito, S., Mendes, J., Moro, E., Rocha, L. (eds) Complex Networks and Their Applications VIII. COMPLEX NETWORKS 2019. Studies in Computational Intelligence, vol 881. Springer, Cham. https://doi.org/10.1007/978-3-030-36687-2_52
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DOI: https://doi.org/10.1007/978-3-030-36687-2_52
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