Systems with many dynamic units interacting through a complex environment (network) and driven by endogenous or external forces often evolve towards states with prominent cyclical fluctuations of the relevant variable. Intuitively, cycles can be seen as a consequence of dynamic nonlinearity in these systems, which enables energy accumulation, e.g. in sandpile models, and its release through avalanches of different sizes, uncorrelated with the strength of the applied force. Examples of such complex systems at different scales [1] range from physics laboratory samples, for example, the magnetization fluctuations on a hysteresis loop in antiferromagnetic bilayers [2], brain dynamics generative models [3], online social communications [4], and epidemic cycles [5] to solar activity and climate cycles [6].
Through a detailed study of two different cases $-$ simulated traffic jamming on complex networks [7] and data analysis of emotional communication in online social sites [8], we aim to demonstrate: firstly, the emergence of cyclical trends from the underlying natural mechanisms, interactions and constraints to the dynamics involved and, secondly, how the system$'$ s collective dynamic behavior modifies these cycles. Hence, the appearance of modulated cycles can signify a self-organized behavior of the dynamical units on a larger scale. Detrended multifractal analysis of the corresponding time series and associated singularity spectra adequately describe these modulated cycles (see [7] and references there). In the studied cases, we show how the shape of multifractal singularity spectra reflects the type of driving forces, constraints due to network structure, and the occurrence of long-range correlations in self-organized critical states.

[1] B. Tadic, R. Melnik, Self-organized critical dynamics as key to fundamental features of complexity in physical, biological and social networks, Dynamics 1, 181 (2021)
[2] S. Mijatovic, S. Graovac, D.Spasojevic, B. Tadic, Tuneable hysteresis loop and multifractal oscillations of ... ,Physica E: Low-dimensional systems and nanostructures 142, 115319 (2022)
[3] M. Ramezanian-Panahi, G. Abrevaya, J-C. Gagnon-Audet, V. Voleti, I. Rish, G. Dumas, Generative models of brain dynamics, review, Front. Art. Intell. 5, 807406 (2022)
[4] M. Mitrovic Dankulov, B. Tadic, R. Melnik, The dynamics of meaningful social interactions and the emergence of collective knowledge, Scientific Reports 5, 12197 (2015)
[5] M. Mitrovic Dankulov, B. Tadic, R. Melnik, Analysis of worldwide time-series data reveals some universal patterns of sars-cov-2 pandemic, Frontiers in Physics 10, 936618 (2022)
[6] J.L. Lean, Cycles and trends in solar irradiance and climate, WIREs Climate Change 1, 111 (2010)
[7] B, Tadic, Cyclical trends of network load fluctuations in traffic jamming, Dynamics 2,449 (2022)
[8] B. Tadic, M. Mitrovic Dankulov, R. Melnik, Evolving cycles and self-organized criticality in social dynamics, Chaos, Solitons and Fractals 171, 113459 (2023)