Abstract
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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Data availability
The source data for all figures in the paper (including Supplementary Figs. 1–6) are linked from the ‘Data and codes for figures’ section in the Supplementary Information to the respective folders in the GitHub repository (https://github.com/jose-moran/timeliness_criticality/tree/main/figures/).
Code availability
The codes used to generate all the figures in the paper (including Supplementary Figs. 1–6) are linked from the ‘Data and codes for figures’ section in the Supplementary Information to the respective folders in the GitHub repository (https://github.com/jose-moran/timeliness_criticality/tree/main/figures/).
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Contributions
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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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. 20, 1352–1358 (2024). https://doi.org/10.1038/s41567-024-02525-w
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DOI: https://doi.org/10.1038/s41567-024-02525-w
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