The European Physical Journal Special Topics

, Volume 226, Issue 3, pp 383–389 | Cite as

Canonical horizontal visibility graphs are uniquely determined by their degree sequence

Open Access
Regular Article
Part of the following topical collections:
  1. Nonlinearity, Nonequilibrium and Complexity: Questions and Perspectives in Statistical Physics

Abstract

Horizontal visibility graphs (HVGs) are graphs constructed in correspondence with number sequences that have been introduced and explored recently in the context of graph-theoretical time series analysis. In most of the cases simple measures based on the degree sequence (or functionals of these such as entropies over degree and joint degree distributions) appear to be highly informative features for automatic classification and provide nontrivial information on the associated dynamical process, working even better than more sophisticated topological metrics. It is thus an open question why these seemingly simple measures capture so much information. Here we prove that, under suitable conditions, there exist a bijection between the adjacency matrix of an HVG and its degree sequence, and we give an explicit construction of such bijection. As a consequence, under these conditions HVGs are unigraphs and the degree sequence fully encapsulates all the information of these graphs, thereby giving a plausible reason for its apparently unreasonable effectiveness.

References

  1. 1.
    L. Lacasa, B. Luque, F. Ballesteros, J. Luque, J.C. Nuño, From time series to complex networks: the visibility graph, Proc. Natl. Acad. Sci. USA 105, 4972 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    B. Luque, L. Lacasa, F. Ballesteros, J. Luque, Horizontal visibility graphs: Exact results for random time series, Phys. Rev. E 80, 046103 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    L. Lacasa, On the degree distribution of horizontal visibility graphs associated to Markov processes and dynamical systems: diagrammatic and variational approaches, Nonlinearity 27, 2063 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    S. Severini, G. Gutin, T. Mansour, A characterization of horizontal visibility graphs and combinatorics on words, Physica A 390, 2421 (2011)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Flajolet, M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math. 204, 203 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    B. Luque, L. Lacasa, F. Ballesteros, A. Robledo, Analytical properties of horizontal visibility graphs in the Feigenbaum scenario, Chaos 22, 013109 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    B. Luque, A. Núñez, F. Ballesteros, A. Robledo, Quasiperiodic Graphs: Structural Design, Scaling and Entropic Properties, J. Nonlin. Sci. 23, 335 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    A.M. Núñez, B. Luque, L. Lacasa, J.P. Gómez, A. Robledo, Horizontal Visibility graphs generated by type-I intermittency, Phys. Rev. E 87, 052801 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    A. Aragoneses, L. Carpi, N. Tarasov, D.V. Churkin, M.C. Torrent, C. Masoller, S.K. Turitsyn, Unveiling temporal correlations characteristic of a phase transition in the output intensity of a fiber laser, Phys. Rev. Lett. 116, 033902 (2016)ADSCrossRefGoogle Scholar
  10. 10.
    M. Murugesan, R.I. Sujitha, Combustion noise is scale-free: transition from scale-free to order at the onset of thermoacoustic instability, J. Fluid Mech. 772, 225 (2015)ADSCrossRefGoogle Scholar
  11. 11.
    A. Charakopoulos, T.E. Karakasidis, P.N. Papanicolaou, A. Liakopoulos, The application of complex network time series analysis in turbulent heated jets, Chaos 24, 024408 (2014)ADSCrossRefMATHGoogle Scholar
  12. 12.
    P. Manshour, M.R. Rahimi Tabar, J. Peinche, Fully developed turbulence in the view of horizontal visibility graphs, J. Stat. Mech. 2015, P08031 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    R.V. Donner, J.F. Donges, Visibility graph analysis of geophysical time series: Potentials and possible pitfalls, Acta Geophys. 60, 3 (2012)CrossRefGoogle Scholar
  14. 14.
    V. Suyal, A. Prasad, H.P. Singh, Visibility-graph analysis of the solar wind velocity, Solar Phys. 289, 379 (2014)ADSCrossRefGoogle Scholar
  15. 15.
    Y. Zou, R.V. Donner, N. Marwan, M. Small, J. Kurths, Long-term changes in the north-south asymmetry of solar activity: a nonlinear dynamics characterization using visibility graphs, Nonlin. Processes Geophys. 21, 1113 (2014)ADSCrossRefGoogle Scholar
  16. 16.
    J.F. Donges, R.V. Donner, J. Kurths, Testing time series irreversibility using complex network methods, EPL 102, 10004 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    S. Jiang, C. Bian, X. Ning, Q.D.Y. Ma, Visibility graph analysis on heartbeat dynamics of meditation training, Appl. Phys. Lett. 102, 253702 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    M. Ahmadlou, H Adeli, A Adeli, New diagnostic EEG markers of the Alzheimer’s disease using visibility graph, J. Neural Transm. 117, 9 (2010)CrossRefGoogle Scholar
  19. 19.
    R. Flanagan, L. Lacasa, Irreversibility of financial time series: a graph-theoretical approach, Phys. Lett. A 380, 1689 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    J. Iacovacci, L. Lacasa, Sequential visibility-graph motifs, Phys. Rev. E 93, 042309 (2016)ADSCrossRefGoogle Scholar
  21. 21.
    B. Bollobas, Modern Graph Theory (Springer, New York, 1998)Google Scholar
  22. 22.
    M. Newman, The structure and function of complex networks, SIAM Rev. 45, 167 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    R. Tyshkevich, A. Chernyak, Unigraphs, I, Vesti Akademii Navuk BSSR 5, 5 (1978)MathSciNetMATHGoogle Scholar
  24. 24.
    R. Tyshkevich, A. Chernyak, Unigraphs, II, Vesti Akademii Navuk BSSR 1, 5 (1979)MathSciNetMATHGoogle Scholar
  25. 25.
    R. Tyshkevich, A. Chernyak, Unigraphs, III, Vesti Akademii Navuk BSSR 2, 5 (1979)MathSciNetMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Applied MathematicsSchool of Aeronautics, Technical University of Madrid (UPM), Plaza Cardenal CisnerosMadridSpain
  2. 2.School of Mathematical Sciences, Queen Mary University of LondonLondon E14NSUK

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