Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Higher-order organization of multivariate time series

Nature Physics volume 19, pages 221–229 (2023)Cite this article

Subjects

A Publisher Correction to this article was published on 25 January 2023

This article has been updated

Abstract

Time series analysis has proven to be a powerful method to characterize several phenomena in biology, neuroscience and economics, and to understand some of their underlying dynamical features. Several methods have been proposed for the analysis of multivariate time series, yet most of them neglect the effect of non-pairwise interactions on the emerging dynamics. Here, we propose a framework to characterize the temporal evolution of higher-order dependencies within multivariate time series. Using network analysis and topology, we show that our framework robustly differentiates various spatiotemporal regimes of coupled chaotic maps. This includes chaotic dynamical phases and various types of synchronization. Hence, using the higher-order co-fluctuation patterns in simulated dynamical processes as a guide, we highlight and quantify signatures of higher-order patterns in data from brain functional activity, financial markets and epidemics. Overall, our approach sheds light on the higher-order organization of multivariate time series, allowing a better characterization of dynamical group dependencies inherent to real-world data.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Higher-order structure of MTS: schematic representation.
Fig. 2: Global and local higher-order indicators distinguish the dynamical regimes of coupled chaotic maps.
Fig. 3: Higher-order approaches perform better in distinguishing the CML regimes.
Fig. 4: Higher-order indicators for real-world MTS.
Fig. 5: Projection of higher-order measures provides local spatiotemporal information.

Similar content being viewed by others

Data availability

All data needed to evaluate the conclusions of this paper are available on Zenodo (10.5281/zenodo.7210075).

Code availability

The code used in this work is available at: https://github.com/andresantoro/RHOSTS as well as a maintained version on E.A.’s GitHub page (https://github.com/eamico).

Change history

References

  1. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D.-U. Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  3. Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y. & Zhou, C. Synchronization in complex networks. Phy. Rep. 469, 93–153 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  4. Barrat, A., Barthelemy, M. & Vespignani, A. Dynamical Processes on Complex Networks (Cambridge Univ. Press, 2008).

    Book  MATH  Google Scholar 

  5. Watts, D. J. & Dodds, P. S. Influentials, networks, and public opinion formation. J. Consum. Res. 34, 441–458 (2007).

    Article  Google Scholar 

  6. Latora, V., Nicosia, V. & Russo, G. Complex Networks: Principles, Methods and Applications (Cambridge Univ. Press, 2017).

    Book  MATH  Google Scholar 

  7. Benson, A. R., Gleich, D. F. & Leskovec, J. Higher-order organization of complex networks. Science 353, 163–166 (2016).

    Article  ADS  Google Scholar 

  8. Petri, G. et al. Homological scaffolds of brain functional networks. J. R. Soc. Interface 10, 186–198 (2014).

    Google Scholar 

  9. Giusti, C., Pastalkova, E., Curto, C. & Itskov, V. Clique topology reveals intrinsic geometric structure in neural correlations. Proc. Natl Acad. Sci. USA 112, 13455–13460 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Sizemore, A. E. et al. Cliques and cavities in the human connectome. J. Comput. Neurosci. 44, 115–145 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  11. Grilli, J., Barabás, G., Michalska-Smith, M. J. & Allesina, S. Higher-order interactions stabilize dynamics in competitive network models. Nature 548, 210–213 (2017).

    Article  ADS  Google Scholar 

  12. Sanchez-Gorostiaga, A., Bajić, D., Osborne, M. L., Poyatos, J. F. & Sanchez, A. High-order interactions distort the functional landscape of microbial consortia. PLOS Biol. 17, 3000550 (2019).

    Article  Google Scholar 

  13. Battiston, F. et al. Networks beyond pairwise interactions: structure and dynamics. Phys. Rep. 874, 1–92 (2020).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Torres, L., Blevins, A. S., Bassett, D. & Eliassi-Rad, T. The why, how, and when of representations for complex systems. SIAM Rev. 63, 435–485 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  15. Battiston, F. et al. The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093–1098 (2021).

    Article  Google Scholar 

  16. Battiston, F. & Petri, G. Higher-Order Systems (Springer, 2022).

  17. Millán, A. P., Torres, J. J. & Bianconi, G. Synchronization in network geometries with finite spectral dimension. Phys. Rev. E 99, 022307 (2019).

    Article  ADS  Google Scholar 

  18. Schaub, M. T., Benson, A. R., Horn, P., Lippner, G. & Jadbabaie, A. Random walks on simplicial complexes and the normalized Hodge 1-Laplacian. SIAM Rev. 62, 353–391 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  19. Carletti, T., Battiston, F., Cencetti, G. & Fanelli, D. Random walks on hypergraphs. Phys. Rev. E 101, 022308 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  20. Iacopini, I., Petri, G., Barrat, A. & Latora, V. Simplicial models of social contagion. Nat. Commun. 10, 1–9 (2019).

    Article  Google Scholar 

  21. de Arruda, G. F., Petri, G. & Moreno, Y. Social contagion models on hypergraphs. Phys. Rev. Res. 2, 023032 (2020).

    Article  Google Scholar 

  22. Sahasrabuddhe, R., Neuhäuser, L. & Lambiotte, R. Modelling non-linear consensus dynamics on hypergraphs. J. Phys. Complex. 2, 025006 (2021).

    Article  ADS  Google Scholar 

  23. Alvarez-Rodriguez, U. et al. Evolutionary dynamics of higher-order interactions in social networks. Nat. Hum. Behav. 5, 586–595 (2021).

    Article  Google Scholar 

  24. Deco, G., Jirsa, V. K. & McIntosh, A. R. Emerging concepts for the dynamical organization of resting-state activity in the brain. Nat. Rev. Neurosci. 12, 43–56 (2011).

    Article  Google Scholar 

  25. Avena-Koenigsberger, A., Misic, B. & Sporns, O. Communication dynamics in complex brain networks. Nat. Rev. Neurosci. 19, 17–33 (2018).

    Article  Google Scholar 

  26. Mantegna, R. N. & Stanley, H. E. Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge Univ. Press, 1999).

    Book  MATH  Google Scholar 

  27. Peron, T. K. D. & Rodrigues, F. A. Collective behavior in financial markets. Europhys. Lett. 96, 48004 (2011).

    Article  ADS  Google Scholar 

  28. Brugere, I., Gallagher, B. & Berger-Wolf, T. Y. Network structure inference, a survey: motivations, methods, and applications. ACM Comput. Surv. 51, 24–12439 (2018).

    Google Scholar 

  29. Young, J. G., Petri, G. & Peixoto, T. P. Hypergraph reconstruction from network data. Commun. Phys. 4, 1–11 (2021).

    Article  Google Scholar 

  30. Musciotto, F., Battiston, F. & Mantegna, R. N. Detecting informative higher-order interactions in statistically validated hypergraphs. Commun. Phys. 4, 1–9 (2021).

    Article  Google Scholar 

  31. Wang, H., Ma, C., Chen, H.-S., Lai, Y.-C. & Zhang, H.-F. Full reconstruction of simplicial complexes from binary contagion and ising data. Nat. Commun. 13, 3043 (2022).

    Article  ADS  Google Scholar 

  32. Lizotte, S., Young, J.-G., Allard, A. Hypergraph reconstruction from noisy pairwise observations. Preprint at https://arxiv.org/abs/2208.06503 (2022).

  33. Rosas, F. E. et al. Disentangling high-order mechanisms and high-order behaviours in complex systems. Nat. Phys. 18, 476–477 (2022).

    Article  Google Scholar 

  34. Faes, L. et al. A New Framework for the Time- and Frequency-Domain Assessment of High-Order Interactions in Networks of Random Processes. In IEEE Transactions on Signal Processing 1–12 (IEEE, 2022).

  35. Rosas, F. E., Mediano, P. A. M., Gastpar, M. & Jensen, H. J. Quantifying high-order interdependencies via multivariate extensions of the mutual information. Phys. Rev. E 100, 32305 (2019).

    Article  ADS  Google Scholar 

  36. Stramaglia, S., Scagliarini, T., Daniels, B. C. & Marinazzo, D. Quantifying dynamical high-order interdependencies from the O-information: an application to neural spiking dynamics. Front. Physiol. 11, 1–11 (2021).

    Article  Google Scholar 

  37. MacMahon, M. & Garlaschelli, D. Community detection for correlation matrices. Phys. Rev. X 5, 021006 (2015).

    Google Scholar 

  38. Tagliazucchi, E., Balenzuela, P., Fraiman, D. & Chialvo, D. Criticality in large-scale brain fMRI dynamics unveiled by a novel point process analysis. Front. Physiol. 3, 15 (2012).

    Article  Google Scholar 

  39. Liu, X. & Duyn, J. H. Time-varying functional network information extracted from brief instances of spontaneous brain activity. Proc. Natl Acad. Sci. USA 110, 4392–4397 (2013).

    Article  ADS  Google Scholar 

  40. Faskowitz, J., Esfahlani, F. Z., Jo, Y., Sporns, O. & Betzel, R. F. Edge-centric functional network representations of human cerebral cortex reveal overlapping system-level architecture. Nat. Neurosci. 23, 1644–1654 (2020).

    Article  Google Scholar 

  41. Esfahlani, F. Z. et al. High-amplitude cofluctuations in cortical activity drive functional connectivity. Proc. Natl Acad. Sci. USA 117, 28393–28401 (2020).

    Article  ADS  Google Scholar 

  42. Van De Ville, D., Farouj, Y., Preti, M. G., Liégeois, R. & Amico, E. When makes you unique: temporality of the human brain fingerprint. Sci. Adv. 7, 0751 (2021).

    ADS  Google Scholar 

  43. Petri, G., Scolamiero, M., Donato, I. & Vaccarino, F. Topological strata of weighted complex networks. PLoS ONE 8, e66506 (2013).

    Article  ADS  Google Scholar 

  44. Zomorodian, A. & Carlsson, G. Computing persistent homology. Discrete Comput. Geom. 33, 249–274 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  45. Ghrist, R. Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45, 61–75 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  46. Carrière, M., Cuturi, M., Oudot, S. Sliced Wasserstein kernel for persistence diagrams. In Proceedings of the 34th International Conference on Machine Learning 70, 664–673. Proceedings of Machine Learning Research (PMLR, 2017).

  47. Kaneko, K. Overview of coupled map lattices. Chaos 2, 279–282 (1992).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Kaneko, K. Pattern dynamics in spatiotemporal chaos. Physica D 34, 1–41 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Lacasa, L., Nicosia, V. & Latora, V. Network structure of multivariate time series. Sci. Rep. 5, 1–9 (2015).

    Article  Google Scholar 

  50. Shrout, P. E. & Fleiss, J. L. Intraclass correlations: uses in assessing rater reliability. Psychol. Bull. 86, 420–428 (1979).

    Article  Google Scholar 

  51. Gatica, M. et al. High-order interdependencies in the aging brain. Brain Connect. 11, 734–744 (2021).

  52. Van Essen, D. C. et al. The WU-Minn Human Connectome Project: an overview. NeuroImage 80, 62–79 (2013).

    Article  Google Scholar 

  53. Scarpino, S.V. & Petri, G. On the predictability of infectious disease outbreaks. Nat. Commun. 10, 1–8 (2019).

    Article  Google Scholar 

  54. Schaefer, A. et al. Local–global parcellation of the human cerebral cortex from intrinsic functional connectivity MRI. Cereb. Cortex 28, 3095–3114 (2018).

    Article  Google Scholar 

  55. Glasser, M. F. et al. The minimal preprocessing pipelines for the Human Connectome Project. NeuroImage 80, 105–124 (2013).

    Article  Google Scholar 

  56. Deco, G. & Kringelbach, M. L. Turbulent-like dynamics in the human brain. Cell Rep. 33, 108471 (2020).

    Article  Google Scholar 

  57. Deco, G., Kringelbach, M. L., Jirsa, V. K. & Ritter, P. The dynamics of resting fluctuations in the brain: metastability and its dynamical cortical core. Sci. Rep. 7, 3095 (2017).

    Article  ADS  Google Scholar 

  58. Chialvo, D. R. Emergent complex neural dynamics. Nat. Phys. 6, 744–750 (2010).

    Article  Google Scholar 

  59. Smith, S. M. et al. Correspondence of the brain’s functional architecture during activation and rest. Proc. Natl Acad. Sci. USA 106, 13040–13045 (2009).

    Article  ADS  Google Scholar 

  60. Yeo, B. T. et al. The organization of the human cerebral cortex estimated by intrinsic functional connectivity. J. Neurophysiol. 106, 1125–1165 (2011).

    Article  Google Scholar 

  61. Raichle, M. E. et al. A default mode of brain function. Proc. Natl Acad. Sci. USA 98, 676–682 (2001).

    Article  ADS  Google Scholar 

  62. Perl, Y.S., Escrichs, A., Tagliazucchi, E., Kringelbach, M.L., Deco, G. On the edge of criticality: strength-dependent perturbation unveils delicate balance between fluctuation and oscillation in brain dynamics. Preprint at bioRxiv https://doi.org/10.1101/2021.09.23.461520 (2022).

  63. Deco, G. et al. Rare long-range cortical connections enhance human information processing. Curr. Biol. 31, 4436–44485 (2021).

    Article  MathSciNet  Google Scholar 

  64. Amico, E., Arenas, A. & Goñi, J. Centralized and distributed cognitive task processing in the human connectome. Netw. Neurosci. 3, 455–474 (2019).

    Article  Google Scholar 

  65. Kutner, R. et al. Econophysics and sociophysics: their milestones & challenges. Physica A 516, 240–253 (2019).

    Article  ADS  Google Scholar 

  66. Musciotto, F., Pillo, J. & Mantegna, R. N. High-frequency trading and networked markets. Proc. Natl Acad. Sci. USA 118, 2015573118 (2021).

    Article  MathSciNet  Google Scholar 

  67. Musmeci, N., Nicosia, V., Aste, T., Di Matteo, T. & Latora, V. The multiplex dependency structure of financial markets. Complexity 2017, 9586064 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  68. Squartini, T., van Lelyveld, I. & Garlaschelli, D. Early-warning signals of topological collapse in interbank networks. Sci. Rep. 3, 3357 (2013).

    Article  Google Scholar 

  69. Wei, W. W. Time Series Analysis (Addison Wesley, 2005).

    Google Scholar 

  70. Bullmore, E. & Sporns, O. Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10, 186–198 (2009).

    Article  Google Scholar 

  71. Sporns, O. Networks of the Brain (MIT Press, 2010).

    Book  MATH  Google Scholar 

  72. Wei, W. W. Multivariate Time Series Analysis and Applications (Wiley, 2019).

    Book  MATH  Google Scholar 

  73. Hatcher, A. Algebraic Topology (Cambridge Univ. Press, 2005).

    MATH  Google Scholar 

  74. Wasserman, L. Topological data analysis. Annu. Rev. Stat. Appl. 5, 501–532 (2018).

    Article  MathSciNet  Google Scholar 

  75. Lum, P. Y. et al. Extracting insights from the shape of complex data using topology. Sci. Rep. 3, 1236 (2013).

    Article  Google Scholar 

  76. Nicolau, M., Levine, A. J. & Carlsson, G. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proc. Natl Acad. Sci. USA 108, 7265–7270 (2011).

    Article  ADS  Google Scholar 

  77. Saggar, M. et al. Towards a new approach to reveal dynamical organization of the brain using topological data analysis. Nat. Commun. 9, 1399 (2018).

    Article  ADS  Google Scholar 

  78. Saggar, M., Shine, J. M., Liégeois, R., Dosenbach, N. U. F. & Fair, D. Precision dynamical mapping using topological data analysis reveals a hub-like transition state at rest. Nat. Commun. 13, 4791 (2022).

    Article  ADS  Google Scholar 

  79. Edelsbrunner, H., Letscher, D., Zomorodian, A. Topological persistence and simplification. In Proc. of 41st Annual Symposium on Foundations of Computer Science 454–463 (2000).

  80. Ghrist, R. Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45, 61–75 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  81. Carlsson, G., Ishkhanov, T., de Silva, V. & Zomorodian, A. On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76, 1–12 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  82. Lee, H., Chung, M.K., Kang, H., Kim, B.-N., Lee, D.S.: Discriminative persistent homology of brain networks. In Proc. of 2011 IEEE International Symposium on Biomedical Imaging: from Nano to Macro 841–844 (IEEE, 2011).

  83. Carstens, C. J. & Horadam, K. J. Persistent homology of collaboration networks. Math. Probl. Eng. 2013, 815035 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  84. Horak, D., Maletić, S. & Rajković, M. Persistent homology of complex networks. J. Stat. Mech. Theory Exp. 2009, 03034 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  85. Barrat, A., Barthelemy, M., Pastor-Satorras, R. & Vespignani, A. The architecture of complex weighted networks. Proc. Natl Acad. Sci. USA 101, 3747–3752 (2004).

    Article  ADS  MATH  Google Scholar 

  86. Barabási, A.-L. Network Science (Cambridge Univ. Press, 2016).

    MATH  Google Scholar 

  87. Newman, M. Networks (Oxford Univ. Press, Oxford, 2018).

    Book  MATH  Google Scholar 

  88. Aroussi, R., et al. Yahoo! Finance market data downloader. GitHub https://github.com/ranaroussi/yfinance (2022).

  89. van Panhuis, W. G. et al. Contagious diseases in the United States from 1888 to the present. N. Engl. J. Med. 369, 2152–2158 (2013).

  90. Van Essen, D. C. et al. The Human Connectome Project: a data acquisition perspective. NeuroImage 62, 2222–2231 (2012).

    Article  Google Scholar 

  91. Smith, S. M. et al. Resting-state fMRI in the Human Connectome Project. NeuroImage 80, 144–168 (2013).

    Article  Google Scholar 

  92. Karahanoğlu, F. I. & Van De Ville, D. Transient brain activity disentangles fMRI resting-state dynamics in terms of spatially and temporally overlapping networks. Nat. Commun. 6, 7751 (2015).

    Article  ADS  Google Scholar 

  93. Preti, M. G., Bolton, T. A. & Van De Ville, D. The dynamic functional connectome: state-of-the-art and perspectives. NeuroImage 160, 41–54 (2017).

    Article  Google Scholar 

  94. Liu, X., Zhang, N., Chang, C. & Duyn, J. H. Co-activation patterns in resting-state fMRI signals. NeuroImage 180, 485–494 (2018).

    Article  Google Scholar 

Download references

Acknowledgements

We thank L. Lacasa and J. Goñi for feedback on an earlier version of the manuscript. Data were provided (in part) by the HCP, the WU-Minn Consortium (principal investigators D. V. Essen and K. Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research and by the McDonnell Center for Systems Neuroscience at Washington University. E.A. acknowledges financial support from the SNSF Ambizione project ‘Fingerprinting the brain: network science to extract features of cognition, behaviour and dysfunction’ (grant no. PZ00P2_185716). A.S. and E.A. acknowledge support from the SNSF COST project ‘Mapping the higher-order dynamics of neurodegeneration in human brain networks’ (grant no. IZCOZ0_198144). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Author information

Authors and Affiliations

  1. Neuro-X Institute, École Polytechnique Fédérale de Lausanne (EPFL), Geneva, Switzerland

    Andrea Santoro & Enrico Amico

  2. Department of Network and Data Science, Central European University, Vienna, Austria

    Federico Battiston

  3. CENTAI, Turin, Italy

    Giovanni Petri

  4. Department of Radiology and Medical Informatics, University of Geneva, Geneva, Switzerland

    Enrico Amico

Authors
  1. Andrea Santoro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Federico Battiston
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giovanni Petri
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Enrico Amico
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.S., F.B., G.P. and E.A. conceptualized the study. A.S. performed the numerical analysis. A.S., F.B., G.P. and E.A. wrote the paper.

Corresponding author

Correspondence to Enrico Amico.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Giulio Cimini, Bratislav Misic and the other, anonymous, reviewer(s) for their contribution to the peer review of this work

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–17, discussion and Tables 1 and 2.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Santoro, A., Battiston, F., Petri, G. et al. Higher-order organization of multivariate time series. Nat. Phys. 19, 221–229 (2023). https://doi.org/10.1038/s41567-022-01852-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-022-01852-0

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing