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The seriation problem in the presence of a double Fiedler value

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Abstract

Seriation is a problem consisting of seeking the best enumeration order of a set of units whose interrelationship is described by a bipartite graph. An algorithm for spectral seriation based on the use of the Fiedler vector of the Laplacian matrix associated to the problem was developed by Atkins et al. under the assumption that the Fiedler value is simple. In this paper, we analyze the case in which the Fiedler value of the Laplacian is not simple, discuss its effect on the set of the admissible solutions, and study possible approaches to actually perform the computation. Examples and numerical experiments illustrate the effectiveness of the proposed methods.

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Data availability

Data sharing not applicable to this article. Only synthetic datasets were generated and analyzed during the current study. They can be generated by the computational code.

Code availability

The computational code is only prototypal, but it is available from the authors upon request.

References

  1. Hubert, L., Arabie, P., Meulman, J.: Graph-theoretic representations for proximity matrices through strongly-anti-Robinson or circular strongly-anti-Robinson matrices. Psychometrika 63(4), 341–358 (1998)

    Article  MATH  Google Scholar 

  2. Armstrong, S., Guzmán, C., Sing Long, C.A.: An optimal algorithm for strict circular seriation. SIAM J. Math. Data Sci. 3(4), 1223–1250 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Recanati, A., Kerdreux, T., d’Aspremont, A.: Reconstructing latent orderings by spectral clustering. arXiv:1807.07122 (2018)

  4. Coifman, R.R., Shkolnisky, Y., Sigworth, F.J., Singer, A.: Graph Laplacian tomography from unknown random projections. IEEE Trans. Image Process. 17(10), 1891–1899 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Evangelopoulos, X., Brockmeier, A.J., Mu, T., Goulermas, J.Y.: Circular object arrangement using spherical embeddings. Pattern Recognit. 103, 107192 (2020)

    Article  Google Scholar 

  6. Carmona, M., Chepoi, V., Naves, G., Préa, P.: A simple and optimal algorithm for strict circular seriation. arXiv:2205.04694 (2022)

  7. Petrie, W.M.F.: Sequences in prehistoric remains. J. R. Anthropol. Inst. 29, 295–301 (1899)

    Google Scholar 

  8. Brusco, M.J., Steinley, D.: Clustering, seriation, and subset extraction of confusion data. Psychol. Methods 11(3), 271–286 (2006)

    Article  Google Scholar 

  9. Eisen, M.B., Spellman, P.T., Brown, P.O., Botstein, D.: Cluster analysis and display of genome-wide expression patterns. P. Natl. Acad. Sci. U.S.A. 95, 14863–14868 (1998)

    Article  Google Scholar 

  10. Hodson, F.R., Kendall, D.G., Tautu, P: Mathematics in the Archaeological and Historical Sciences. University Press, Edinburgh (1971)

    MATH  Google Scholar 

  11. Mirkin, B.G., Rodin, S.N.: Graphs and Genes. Biomathematics, vol. 11. Springer (1984)

  12. Concas, A., Fenu, C., Rodriguez, G.: PQser: a Matlab package for spectral seriation. Numer. Algorithms 80(3), 879–902 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Barnard, S.T., Pothen, A., Simon, H.: A spectral algorithm for envelope reduction of sparse matrices. Numer. Linear Algebra Appl. 2(4), 317–334 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fogel, F., Jenatton, R., Bach, F., d’Aspremont, A.: Convex relaxations for permutation problems. In: Burges, C.J., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K.Q. (eds.) Advances in neural information processing systems vol. 26 (NIPS 2013) (2013)

  15. Recanati, A., Servant, N., Vert, J.-P., d’Aspremont, A.: Robust seriation and applications to cancer genomics. arXiv:1806.00664 (2018)

  16. Evangelopoulos, X., Brockmeier, A.J., Mu, T., Goulermas, J.Y.: Continuation methods for approximate large scale object sequencing. Mach. Learn. 108(4), 595–626 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lim, C.H., Wright, S.: A box-constrained approach for hard permutation problems. In: International conference on machine learning, pp. 2454–2463. PMLR (2016)

  18. Hahsler, M., Hornik, K., Buchta, C.: Getting things in order: an introduction to the R package seriation. J. Stat. Softw. 25(3), 1–34 (2008)

    Article  Google Scholar 

  19. Piana Agostinetti, P., Sommacal, M.: Il problema della seriazione in archeologia. Rivista di Scienze Preistoriche LV, 29–69 (2005)

    Google Scholar 

  20. Pardalos, P.M., Rendl, F., Wolkowicz, H.: The quadratic assignment problem: a survey and recent developments. In: Pardalos, P.M., Wolkowicz, H. (eds.) Quadratic assignment and related problems. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 16, pp. 1–42. American Mathematical Society (1994)

  21. Atkins, J.E., Boman, E.G., Hendrickson, B.: A spectral algorithm for seriation and the consecutive ones problem. SIAM J. Comput. 28(1), 297–310 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fiedler, M.: Algebraic connectivity of graphs. Czech. Math. J. 23(2), 298–305 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fiedler, M.: A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czech. Math. J. 25(4), 619–633 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fiedler, M.: Laplacian of graphs and algebraic connectivity. Banach Center Publ. 25(1), 57–70 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. De Abreu, N.M.M.: Old and new results on algebraic connectivity of graphs. Linear Algebra Appl. 423(1), 53–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jamakovic, A, Mieghem, P.V: On the robustness of complex networks by using the algebraic connectivity. In: International conference on research in networking, pp. 183–194. Springer (2008)

  27. Jamakovic, A, Uhlig, S: On the relationship between the algebraic connectivity and graph’s robustness to node and link failures. In: 2007 Next generation internet networks, pp 96–102. IEEE (2007)

  28. Kendall, D.G.: A statistical approach to Flinders–Petries sequence-dating. Bull. Int. Stat. Inst. 40(2), 657–681 (1963)

    MathSciNet  Google Scholar 

  29. Kendall, D.G.: A mathematical approach to seriation. Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci. 269(1193), 125–134 (1970)

    Google Scholar 

  30. Brainerd, G.W.: The place of chronological ordering in archaeological analysis. Am. Antiq. 16(4), 301–313 (1951)

    Article  Google Scholar 

  31. Robinson, W.S.: A method for chronologically ordering archaeological deposits. Am. Antiq. 16(4), 293–301 (1951)

    Article  Google Scholar 

  32. Chepoi, V., Fichet, B.: Recognition of Robinsonian dissimilarities. J. Classif. 14(2), 311–325 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Laurent, M., Seminaroti, M.: A Lex-BFS-based recognition algorithm for Robinsonian matrices. Discret. Appl. Math. 222, 151–165 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Laurent, M., Seminaroti, M.: Similarity-first search: a new algorithm with application to Robinsonian matrix recognition. SIAM Discret Math. 31 (3), 1765–1800 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Préa, P., Fortin, D.: An optimal algorithm to recognize Robinsonian dissimilarities. J. Classif 31(3), 3–51 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Seston, M.: Dissimilarités de Robinson: algorithmes de reconnaissance et d’approximation. PhD thesis Aix Marseille vol. 2 (2008)

  37. Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wilkinson, J.H.: The Algebraic Eigenvalue Problem vol. 87. Clarendon Press, Oxford (1965)

    Google Scholar 

  39. Ortega, J.M.: On Sturm sequences for tridiagonal matrices. J. ACM 7(3), 260–263 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  40. Davis, P.J: Circulant Matrices. Wiley (1979)

  41. Coxeter, H.S.M.: Self-dual configurations and regular graphs. Bull. Amer. Math. Soc. 56(5), 413–455 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  42. Watkins, M.E.: A theorem on tait colorings with an application to the generalized Petersen graphs. J. Comb. Theory 6(2), 152–164 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kaveh, A., Rahami, H.: Block circulant matrices and applications in free vibration analysis of cyclically repetitive structures. Acta Mech. 217(1), 51–62 (2011)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors are indebted to the two reviewers for their care in reviewing the manuscript. Their remarks led to a substantial improvement in the content of the paper and suggested possible future research development.

Funding

Anna Concas, Caterina Fenu, and Giuseppe Rodriguez were partially supported by Regione Autonoma della Sardegna research project “Algorithms and Models for Imaging Science (AMIS)” (RASSR57257, intervento finanziato con risorse FSC 2014-2020 - Patto per lo Sviluppo della Regione Sardegna) and by the INdAM-GNCS research project “Tecniche numeriche per l’analisi delle reti complesse e lo studio dei problemi inversi.” Caterina Fenu also gratefully acknowledges Regione Autonoma della Sardegna for the financial support provided under the Operational Programme P.O.R. Sardegna F.S.E. (European Social Fund 2014-2020 - Axis III Education and Formation, Objective 10.5, Line of Activity 10.5.12). The research of Raf Vandebril was partially supported by the Research Council KU Leuven, project C16/21/002 (Manifactor: Factor Analysis for Maps into Manifolds).

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    Authors and Affiliations

    1. Department of Mathematics and Computer Science, University of Cagliari, Via Ospedale 72, Cagliari, 09124, Italy

      Anna Concas, Caterina Fenu & Giuseppe Rodriguez

    2. Department of Computer Science, KU Leuven, Celestijnenlaan 200A, Leuven, 3001, Belgium

      Raf Vandebril

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    1. Anna Concas
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    Correspondence to Caterina Fenu.

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    The author Giuseppe Rodriguez is a member of the editorial board of Numerical Algorithms. The authors declare no other conflict of interest.

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    Concas, A., Fenu, C., Rodriguez, G. et al. The seriation problem in the presence of a double Fiedler value. Numer Algor 92, 407–435 (2023). https://doi.org/10.1007/s11075-022-01461-1

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