- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Maximum likelihood estimation of burst-merging kernels for bursty time series

Tibebe Birhanu and Hang-Hyun Jo*

  • *Contact author: h2jo@catholic.ac.kr
PDF

Phys. Rev. E 111, 054317 – Published 27 May, 2025

DOI: https://doi.org/10.1103/PhysRevE.111.054317

Abstract

Various time series in natural and social processes have been found to be bursty. Events in the time series rapidly occur within short time periods, forming bursts, which are alternated with long inactive periods. As the timescale defining bursts increases, individual events are sequentially merged to become small bursts and then bigger ones, eventually leading to the single burst containing all events. Such a merging pattern has been depicted by a tree that fully reveals the hierarchical structure of bursts, thus called a burst tree. The burst-tree structure can be simply characterized by a burst-merging kernel that dictates which bursts are merged together as the timescale increases. In this work, we develop the maximum likelihood estimation method of the burst-merging kernel from time series, which is successfully tested against the time series generated using several model kernels. We also apply our method to some empirical time series from various backgrounds. Our method provides a useful tool to precisely characterize the time series data, hence enabling to study their underlying mechanisms more accurately.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (49)

  1. A.-L. Barabási and M. Pósfai, Network Science (Cambridge University Press, Cambridge, 2016).
  2. M. E. J. Newman, Networks, 2nd ed. (Oxford University Press, Oxford, UK, 2018).
  3. F. Menczer, S. Fortunato, and C. A. Davis, A First Course in Network Science (Cambridge University Press, Cambridge, 2020).
  4. S. N. Dorogovtsev and J. F. F. Mendes, The Nature of Complex Networks (Oxford University Press, Oxford, UK, 2022).
  5. D. Easley and J. Kleinberg, Networks, Crowds, and Markets: Reasoning About a Highly Connected World (Cambridge University Press, Cambridge, UK, 2010).
  6. A. Barrat, M. Barthélemy, and A. Vespignani, Dynamical Processes on Complex Networks, 1st ed. (Cambridge University Press, Cambridge, UK, 2008).
  7. P. Holme and J. Saramäki, Temporal networks, Phys. Rep. 519, 97 (2012).
  8. Edited by P. Holme and J. Saramäki, Temporal Network Theory, Computational social sciences (Springer International Publishing, Cham, 2019).
  9. N. Masuda and R. Lambiotte, A Guide to Temporal Networks, Series on complexity science (World Scientific, New Jersey, 2016).
  10. A.-L. Barabási, The origin of bursts and heavy tails in human dynamics, Nature (London) 435, 207 (2005).
  11. M. Karsai, H.-H. Jo, and K. Kaski, Bursty Human Dynamics (Springer International Publishing, Cham, 2018).
  12. H.-H. Jo and T. Hiraoka, Bursty time series analysis for temporal networks, in Temporal Network Theory, edited by P. Holme and J. Saramäki (Springer International Publishing, Cham, 2023), 2nd ed., pp. 165–183.
  13. P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality: An explanation of the 1/f noise, Phys. Rev. Lett. 59, 381 (1987).
  14. H. J. Jensen, Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems, 1st ed. (Cambridge University Press, Cambridge, UK, 1998).
  15. M. S. Wheatland, P. A. Sturrock, and J. M. McTiernan, The waiting-time distribution of solar flare hard x-ray bursts, ApJ 509, 448 (1998).
  16. Á. Corral, Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes, Phys. Rev. Lett. 92, 108501 (2004).
  17. J. M. Beggs and D. Plenz, Neuronal avalanches in neocortical circuits, J. Neurosci. 23, 11167 (2003).
  18. T. Petermann, T. C. Thiagarajan, M. A. Lebedev, M. A. L. Nicolelis, D. R. Chialvo, and D. Plenz, Spontaneous cortical activity in awake monkeys composed of neuronal avalanches, Proc. Natl. Acad. Sci. USA 106, 15921 (2009).
  19. T. Kemuriyama, H. Ohta, Y. Sato, S. Maruyama, M. Tandai-Hiruma, K. Kato, and Y. Nishida, A power-law distribution of inter-spike intervals in renal sympathetic nerve activity in salt-sensitive hypertension-induced chronic heart failure, BioSystems 101, 144 (2010).
  20. K.-I. Goh and A.-L. Barabási, Burstiness and memory in complex systems, EPL (Europhysics Letters) 81, 48002 (2008).
  21. R. Crane and D. Sornette, Robust dynamic classes revealed by measuring the response function of a social system, Proc. Natl. Acad. Sci. USA 105, 15649 (2008).
  22. D. Rybski, S. V. Buldyrev, S. Havlin, F. Liljeros, and H. A. Makse, Scaling laws of human interaction activity, Proc. Natl. Acad. Sci. USA 106, 12640 (2009).
  23. H.-H. Jo, T. Hiraoka, and M. Kivelä, Burst-tree decomposition of time series reveals the structure of temporal correlations, Sci. Rep. 10, 12202 (2020).
  24. J. Choi, T. Hiraoka, and H.-H. Jo, Individual-driven versus interaction-driven burstiness in human dynamics: The case of Wikipedia edit history, Phys. Rev. E 104, 014312 (2021).
  25. M. Karsai, K. Kaski, A.-L. Barabási, and J. Kertész, Universal features of correlated bursty behaviour, Sci. Rep. 2, 397 (2012).
  26. T. Birhanu and H.-H. Jo, Burst-tree structure and higher-order temporal correlations, Phys. Rev. E 111, 014308 (2025).
  27. T. Pham, P. Sheridan, and H. Shimodaira, PAFit: A statistical method for measuring preferential attachment in temporal complex networks, PLoS ONE 10, e0137796 (2015).
  28. A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science 286, 509 (1999).
  29. P. L. Krapivsky, S. Redner, and F. Leyvraz, Connectivity of growing random networks, Phys. Rev. Lett. 85, 4629 (2000).
  30. M. E. J. Newman, Clustering and preferential attachment in growing networks, Phys. Rev. E 64, 025102(R) (2001).
  31. H. Jeong, Z. Néda, and A. L. Barabási, Measuring preferential attachment in evolving networks, Europhysics Letters (EPL) 61, 567 (2003).
  32. P. Sheridan, Y. Yagahara, and H. Shimodaira, Measuring preferential attachment in growing networks with missing-timelines using Markov chain Monte Carlo, Physica A 391, 5031 (2012).
  33. J. Kunegis, M. Blattner, and C. Moser, Preferential attachment in online networks: Measurement and explanations, in Proceedings of the 5th Annual ACM Web Science Conference (ACM, Paris, France, 2013), pp. 205–214.
  34. G. Gan, C. Ma, and J. Wu, Data Clustering: Theory, Algorithms, and Applications (Society for Industrial and Applied Mathematics, Alexandria, VA, 2007).
  35. W. H. Stockmayer, Theory of molecular size distribution and gel formation in branched-chain polymers, J. Chem. Phys. 11, 45 (1943).
  36. A. Lushnikov, Evolution of coagulating systems, J. Colloid Interface Sci. 45, 549 (1973).
  37. W. H. White, On the form of steady-state solutions to the coagulation equations, J. Colloid Interface Sci. 87, 204 (1982).
  38. E. M. Hendriks, M. H. Ernst, and R. M. Ziff, Coagulation equations with gelation, J. Stat. Phys. 31, 519 (1983).
  39. D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernoulli 5, 3 (1999).
  40. M. H. Lee, A survey of numerical solutions to the coagulation equation, J. Phys. A: Math. Gen. 34, 10219 (2001).
  41. F. Leyvraz, Scaling theory and exactly solved models in the kinetics of irreversible aggregation, Phys. Rep. 383, 95 (2003).
  42. F. Leyvraz, Rigorous results in the scaling theory of irreversible aggregation kinetics, J. Nonlinear Math. Phys. 12, 449 (2005).
  43. J. A. Wattis, An introduction to mathematical models of coagulation–fragmentation processes: A discrete deterministic mean-field approach, Physica D 222, 1 (2006).
  44. T. Birhanu and H.-H. Jo, Codes for the maximum likelihood estimation method of burst-merging kernels, https://github.com/tibebe22/Kernel-estimate (2025).
  45. English Wikipedia, https://dumps.wikimedia.org/.
  46. J. Yang and J. Leskovec, Patterns of temporal variation in online media, in Proceedings of the Fourth ACM International Conference on Web Search and Data Mining (ACM Press, Hong Kong, China, 2011), pp. 177–186.
  47. PhysioBank, https://physionet.org/physiobank/.
  48. Japan University Network Earthquake Catalog, http://wwweic.eri.u-tokyo.ac.jp/CATALOG/junec/.
  49. D. R. Hunter and K. Lange, A tutorial on MM algorithms, Am. Stat. 58, 30 (2004).

Outline

Information

Phys. Rev. E 111, 054317– Published 27 May, 2025
  • Received 18 March 2025
  • Accepted 9 May 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation