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Timeliness criticality in complex systems


In complex systems, external parameters often determine the phase in which the system operates, that is, its macroscopic behaviour. For nearly a century, statistical physics has been used to extensively study systems’ transitions across phases, (universal) critical exponents and related dynamical properties. Here we consider the functionality of systems, particularly operations in socio-technical ones, production in economic ones and, more generally, any schedule-based system, where timing is of crucial importance. We introduce a stylized model of delay propagation on temporal networks, where the magnitude of the delay-mitigating buffer acts as a control parameter. The model exhibits timeliness criticality, a novel form of critical behaviour. We characterize fluctuations near criticality, commonly referred to as avalanches, and identify the corresponding critical exponents. The model exhibits timeliness criticality also when run on real-world temporal systems such as production networks. Additionally, we explore potential connections with the mode-coupling theory of glasses, depinning transition and directed polymer problem.

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Fig. 1: Schematic of our model in terms of temporal networks.
Fig. 2: Timeliness criticality.
Fig. 3: Characterizing timeliness criticality.
Fig. 4: Timeliness criticality on real-world temporal networks.

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The source data for all figures in the paper (including Supplementary Figs. 16) are linked from the ‘Data and codes for figures’ section in the Supplementary Information to the respective folders in the GitHub repository (

Code availability

The codes used to generate all the figures in the paper (including Supplementary Figs. 16) are linked from the ‘Data and codes for figures’ section in the Supplementary Information to the respective folders in the GitHub repository (


  1. Moran, J., Pijpers, F. P., Weitzel, U., Bouchaud, J.-P. & Panja, D. Temporal criticality in socio-technical systems. Preprint at (2023).

  2. Dekker, M. M. & Panja, D. Cascading dominates large-scale disruptions in transport over complex networks. PLoS ONE 16, e0246077 (2021).

    Article  Google Scholar 

  3. Giannikas, V. et al. A data-driven method to assess the causes and impact of delay propagation in air transportation systems. Transp. Res. Part C: Emerg. Technol. 143, 103862 (2022).

    Article  Google Scholar 

  4. Ledwoch, A., Brintrup, A., Mehnen, J. örn & Tiwari, A. Systemic risk assessment in complex supply networks. IEEE Syst. J. 12, 1826–1837 (2016).

    Article  ADS  Google Scholar 

  5. Brintrup, A. & Ledwoch, A. Supply network science: emergence of a new perspective on a classical field. Chaos 28, 033120 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  6. Colon, C. élian & Ghil, M. Economic networks: heterogeneity-induced vulnerability and loss of synchronization. Chaos 27, 126703 (2017).

    Article  ADS  Google Scholar 

  7. The Associated Press. Ship still stuck in Suez Canal as backlog grows to 150 other ships. CBC (25 March 2021).

  8. Colon, C., Hallegatte, S. & Rozenberg, J. Criticality analysis of a country’s transport network via an agent-based supply chain model. Nat. Sustain. 4, 209–215 (2020).

    Article  Google Scholar 

  9. Kosasih, E. E. & Brintrup, A. Reinforcement learning provides a flexible approach for realistic supply chain safety stock optimisation. IFAC-PapersOnLine 55, 1539–1544 (2022).

    Article  Google Scholar 

  10. Dessertaine, T., Moran, J., Benzaquen, M. & Bouchaud, J.-P. Out-of-equilibrium dynamics and excess volatility in firm networks. J. Econ. Dyn. Control 138, 104362 (2022).

    Article  MathSciNet  Google Scholar 

  11. Munoz, M. A. & Hwa, T. On nonlinear diffusion with multiplicative noise. Europhys. Lett. 41, 147 (1998).

    Article  ADS  Google Scholar 

  12. Hinrichsen, H. Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49, 815–958 (2000).

    Article  ADS  Google Scholar 

  13. Götze, W. Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory Vol. 143 (Oxford Univ. Press, 2009).

  14. Holme, P. & Saramäki, J. Temporal networks. Phys. Rep. 519., 97–125 (2012).

    Article  ADS  Google Scholar 

  15. Holme, P. and Saramäki, J. Temporal Networks (SpringerLink, 2013).

  16. Sellitto, M., Biroli, G. & Toninelli, C. Facilitated spin models on Bethe lattice: bootstrap percolation, mode-coupling transition and glassy dynamics. Europhys. Lett. 69, 496 (2005).

    Article  ADS  Google Scholar 

  17. Rosso, A., Le Doussal, P. & Wiese, K. J. Avalanche-size distribution at the depinning transition: a numerical test of the theory. Phys. Rev. B 80, 144204 (2009).

    Article  ADS  Google Scholar 

  18. Bouchaud, E. Scaling properties of cracks. J. Phys.: Condens. Matter 9, 4319 (1997).

    ADS  Google Scholar 

  19. Lin, J., Lerner, E., Rosso, A. & Wyart, M. Scaling description of the yielding transition in soft amorphous solids at zero temperature. Proc. Natl Acad. Sci. USA 111, 14382–14387 (2014).

    Article  ADS  Google Scholar 

  20. Ponson, L. & Pindra, N. Crack propagation through disordered materials as a depinning transition: a critical test of the theory. Phys. Rev. E 95, 053004 (2017).

    Article  ADS  Google Scholar 

  21. Newman, M. E. J. & Barkema, G. T. Monte Carlo Methods in Statistical Physics (Oxford Univ. Press, 1999).

  22. Mastrandrea, R., Fournet, J. & Barrat, A. Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys. PLoS ONE 10, e0136497 (2015).

    Article  Google Scholar 

  23. Génois, M. et al. Data on face-to-face contacts in an office building suggest a low-cost vaccination strategy based on community linkers. Network Sci. 3, 326–347 (2015).

    Article  Google Scholar 

  24. Dekker, M. M., Schram, R. D., Ou, J. & Panja, D. Hidden dependence of spreading vulnerability on topological complexity. Phys. Rev. E 105, 054301 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  25. Dekker, M. M. et al. Quantifying agent impacts on contact sequences in social interactions. Sci. Rep. 12, 3483 (2022).

    Article  ADS  Google Scholar 

  26. Roux, S. & Herrmann, H. J. Disorder induced nonlinear conductivity. Europhys. Lett. 4, 1227 (1987).

    Article  ADS  Google Scholar 

  27. Wichmann, P., Brintrup, A., Baker, S., Woodall, P. & McFarlane, D. Towards automatically generating supply chain maps from natural language text. IFAC-PapersOnLine 51, 1726–1731 (2018).

    Article  Google Scholar 

  28. Sterman, J. Modeling managerial behavior: misperceptions of feedback in a dynamic decision making experiment. Manage. Sci. 35, 321–339 (1989).

    Article  Google Scholar 

  29. Dekker, M. M. et al. A next step in disruption management: combining operations research and complexity science. Public Transp. 14, 5–26 (2021).

    Article  Google Scholar 

  30. Shughrue, C., Werner, B. & Seto, K. C. Global spread of local cyclone damages through urban trade networks. Nat. Sustain. 3, 606–613 (2020).

    Article  Google Scholar 

  31. Bak, P., Chen, K., Scheinkman, J. & Woodford, M. Aggregate fluctuations from independent sectoral shocks: self-organized criticality in a model of production and inventory dynamics. Ric. Econ. 47, 3–30 (1993).

    Article  Google Scholar 

  32. Moran, J. & Bouchaud, J.-P. May’s instability in large economies. Phys. Rev. E 100, 032307 (2019).

    Article  ADS  Google Scholar 

  33. Bernanke, B. S., Gertler, M. & Gilchrist, S. The financial accelerator and the flight to quality. Rev. Econ. Stat. 78, 1–15 (1996).

    Article  Google Scholar 

  34. Bernanke, B. S., Geithner, T. F. & Paulson Jr, H. M. Firefighting: The Financial Crisis and Its Lessons (Penguin, 2019).

  35. Gabaix, X. The granular origins of aggregate fluctuations. Econometrica 79, 733–772 (2011).

    Article  MathSciNet  Google Scholar 

  36. Carvalho, V. M. & Tahbaz, A. Production networks: a primer. Annu. Rev. Econ. 11, 635–663 (2019).

    Article  Google Scholar 

  37. Pichler, A. & Farmer, J. D. Simultaneous supply and demand constraints in input–output networks: the case of COVID-19 in Germany, Italy, and Spain. Econ. Syst. Res. 34, 273–293 (2021).

    Article  Google Scholar 

  38. Lafrogne-Joussier, R., Martin, J. & Mejean, I. Supply shocks in supply chains: evidence from the early lockdown in China. IMF Econ. Rev. 71, 170–215 (2022).

    Article  Google Scholar 

  39. Pichler, A. & Diem, C. et al. Building an alliance to map global supply networks. Science 382, 270–272 (2023).

    Article  ADS  Google Scholar 

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Authors and Affiliations



J.-P.B. and D.P. conceptualized the timeliness criticality. J.-P.B., P.L.D., J.M., D.P., F.P.P. and M.R. analysed the timeliness criticality. All authors contributed to the paper.

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Correspondence to Debabrata Panja.

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The authors declare no competing interests.

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Nature Physics thanks Yuichi Ikeda and Jari Saramäki for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 An example of how an avalanche is defined.

The mean delay per node for the mean-field case is plotted (blue), along with an orange line corresponding to mean delay equals B (see text for the parameter values used for this plot). For the avalanche shown, we define tstart = 4, 760, 908 as the time when the mean delay per node exceeds B, and tend = 4, 785, 490 as the time when for the first time after tstart the mean delay per node falls below B again. Then the mean delay per node in the interval \(\left[{t}_{{{{\rm{start}}}}},{t}_{{{{\rm{end}}}}}\right)\) constitutes a delay avalanche, from which the persistence time and the size of the avalanche are obtained respectively as tp = tend − tstart = 24, 582, and as the area between the blue curve and the orange line over the interval \(\left[{t}_{{{{\rm{start}}}}},{t}_{{{{\rm{end}}}}}\right)\).

Supplementary information

Supplementary Information

Supplementary Sections A–E, Figs. 1–6 and Tables 1 and 2.

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Moran, J., Romeijnders, M., Doussal, P.L. et al. Timeliness criticality in complex systems. Nat. Phys. (2024).

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