Rigid Graph Alignment

  • Vikram RavindraEmail author
  • Huda Nassar
  • David F. Gleich
  • Ananth Grama
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 881)


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.


Graph alignment Structural alignment 



The authors are supported by the National Science Foundation grants CCF-1149756 and IIS-1546488.


  1. 1.
    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). Scholar
  2. 2.
    Berg, J., Lässig, M.: Local graph alignment and motif search in biological networks. Proc. Natl. Acad. Sci. 101(41), 14689–14694 (2004). Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Besl, P., McKay, H.: A method for registration of 3-D shapes. IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992)CrossRefGoogle Scholar
  5. 5.
    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).
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    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 2011Google Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Emmert-Streib, F., Dehmer, M., Shi, Y.: Fifty years of graph matching, network alignment and network comparison. Inf. Sci. 346(C), 180–197 (2016). Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Kabsch, W.: A solution for the best rotation to relate two sets of vectors. Acta Crystallogr. Sect. A 32(5), 922–923 (1976)CrossRefGoogle Scholar
  14. 14.
    Kuchaiev, O., Milenković, T., Memišević, V., Hayes, W., Pržulj, N.: Topological network alignment uncovers biological function and phylogeny. J. Roy. Soc. Interface (2010).
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Patro, R., Kingsford, C.: Global network alignment using multiscale spectral signatures. Bioinformatics 28(23), 3105–3114 (2012)CrossRefGoogle Scholar
  17. 17.
    Rusinkiewicz, S., Levoy, M.: Efficient variants of the ICP algorithm. In: Proceedings of the 3DIM 2001, October 2001Google Scholar
  18. 18.
    Sabata, B., Aggarwal, J.: Estimation of motion from a pair of range images: a review. CVGIP: Image Underst. 54(3), 309–324 (1991). Scholar
  19. 19.
    Schönemann, P.H.: A generalized solution of the orthogonal procrustes problem. Psychometrika 31(1), 1–10 (1966)MathSciNetCrossRefGoogle Scholar
  20. 20.
    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)Google Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Smith, S.M.: Fast robust automated brain extraction. Hum. Brain Mapp. 17(3), 143–155 (2002)CrossRefGoogle Scholar
  23. 23.
    Sussenguth, E.H.: A graph-theoretic algorithm for matching chemical structures. J. Chem. Documentation 5(1), 36–43 (1965)CrossRefGoogle Scholar
  24. 24.
    Umeyama, S.: An eigendecomposition approach to weighted graph matching problems. IEEE Trans. Pattern Anal. Mach. Intell. 10(5), 695–703 (1988)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Vikram Ravindra
    • 1
    Email author
  • Huda Nassar
    • 1
  • David F. Gleich
    • 1
  • Ananth Grama
    • 1
  1. 1.Purdue UniversityWest LafayetteUSA

Personalised recommendations