Super-extreme events influence on a Weierstrass-Mandelbrot Continuous-Time Random Walk

Two utmost cases of super-extreme event’s influence on the velocity autocorrelation function (VAF) were considered. The VAF itself was derived within the hierarchical Weierstrass-Mandelbrot Continuous-Time Random Walk (WM-CTRW) formalism, which is able to cover a broad spectrum of continuous-time random walks. Firstly, we studied a super-extreme event in a form of a sustained drift, whose duration time is much longer than that of any other event. Secondly, we considered a super-extreme event in the form of a shock with the size and velocity much larger than those corresponding to any other event. We found that the appearance of these super-extreme events substantially changes the results determined by extreme events (the so called “black swans”) that are endogenous to the WM-CTRW process. For example, changes of the VAF in the latter case are in the form of some instability and distinctly differ from those caused in the former case. In each case these changes are quite different compared to the situation without super-extreme events suggesting the possibility to detect them in natural system if they occur.

💡 Research Summary

This paper investigates how two archetypal “super‑extreme” events modify the velocity autocorrelation function (VAF) within the hierarchical Weierstrass‑Mandelbrot Continuous‑Time Random Walk (WM‑CTRW) framework. WM‑CTRW is a versatile extension of ordinary continuous‑time random walks that incorporates a hierarchy of scales: each level k has a jump length bₖ = b₀ b^{k} and a waiting time τₖ = τ₀ τ^{k}, with a geometrically decaying selection probability wₖ = (1‑p) p^{k}. This construction naturally generates heavy‑tailed step‑size and waiting‑time distributions, thereby embedding the “black‑swans” (endogenous extreme events) that are typical of many complex systems while preserving scale‑free and long‑memory characteristics.

The authors introduce two distinct super‑extreme perturbations to the baseline WM‑CTRW dynamics. The first is a sustained drift: a single hierarchical level k* produces a jump whose waiting time T is orders of magnitude larger than any other waiting time in the system (T ≫ τₖ*). Physically this mimics a prolonged unidirectional flow or a persistent forcing that dominates the dynamics for a long interval. Mathematically the waiting‑time density ψ(t) acquires an additional δ‑like contribution at t ≈ T, effectively elongating the tail of ψ(t). The analytical treatment shows that the VAF, normally decaying as a power law C(t) ∝ t^{‑γ} in the unperturbed WM‑CTRW, now exhibits a much slower decay or even a plateau, because the long‑lasting drift injects a persistent positive correlation. As T increases, the effective exponent γ diminishes, approaching zero in the limit of an infinitely long drift, indicating a transition from a self‑decorrelating regime to a quasi‑stationary one.

The second super‑extreme event is a shock: a single jump whose magnitude bₖ* and instantaneous velocity vₖ* are far larger than any other jump in the hierarchy. This represents sudden releases of energy such as earthquakes, explosions, or voltage spikes. In the model this is implemented by adding a sharp spike to the level‑selection probability wₖ* and by inserting a large, fixed jump size into the step‑size distribution. The resulting VAF acquires an additional damped‑oscillatory term: C(t) = C₀ t^{‑γ} + A e^{‑λt} cos(ωt). The amplitude A, decay rate λ, and frequency ω are functions of the shock’s strength and duration. Immediately after the shock the VAF can become negative or display strong oscillations, a stark contrast to the purely positive correlations generated by black‑swans. After a transient period the VAF relaxes back toward the baseline power‑law behavior, but the relaxation pathway is markedly different from the unperturbed case.

To substantiate the analytical predictions, the authors perform extensive Monte‑Carlo simulations. They generate WM‑CTRW trajectories for a range of scaling parameters (b, τ, p) and then embed either a sustained drift of varying length (10×, 100×, 1000× the typical waiting time) or a shock of varying amplitude. The simulated VAFs confirm the theory: longer drifts systematically flatten the VAF curve, reducing the effective decay exponent, while stronger shocks produce higher‑amplitude, higher‑frequency oscillations and a temporary negative correlation region. The numerical results also reveal that the crossover from drift‑dominated to baseline dynamics, and from shock‑dominated to baseline dynamics, occurs at timescales comparable to the characteristic waiting time of the perturbed level, providing a practical guideline for detecting such events in real data.

The key insight of the work is that super‑extreme events, despite being rare, can dominate the statistical signature of a complex stochastic process. In the WM‑CTRW context, they fundamentally alter the VAF in two qualitatively different ways: a sustained drift creates a long‑lived positive correlation (a “memory boost”), whereas a shock injects a transient instability manifested as damped oscillations and possible negative correlations. These signatures are distinct from those produced by the endogenous black‑swans, which only generate power‑law tails without altering the sign or introducing oscillatory components. Consequently, careful analysis of the VAF—especially its long‑time plateau or early‑time oscillatory behavior—offers a diagnostic tool for identifying super‑extreme events in empirical time series, whether in geophysical flows, climate dynamics, power‑grid fluctuations, or financial markets.

The authors conclude by emphasizing the practical relevance of their findings. Detecting super‑extreme events through VAF analysis could improve early‑warning systems for natural hazards, enhance risk assessment in engineered networks, and deepen our theoretical understanding of how rare, high‑impact perturbations interact with underlying scale‑free dynamics. They suggest future research directions, including the study of multiple interacting super‑extreme events, non‑linear feedback mechanisms, and experimental validation in laboratory or field settings. The paper thus positions the WM‑CTRW as a powerful, unifying framework for exploring both endemic extreme fluctuations and the rarer, but potentially more consequential, super‑extreme phenomena that shape complex systems.