Large earthquakes follow highly unequal ones

It was conjectured for a long time that the tectonic plates are in a self-organized state of criticality and that the Gutenberg-Richter law is a manifestation of that. It was recently shown that for a system near criticality, the inequality of their responses due to external driving would sharply rise and show universal behavior that could indicate proximity of the system to a critical point. As a result, measures such as the Gini and Kolkata indices that quantify inequality, can also serve as indicators of imminent criticallity and that of diverging (system spanning) responses. In the context of earthquakes, such a large response would correspond to events of high magnitudes. In this work, we show with numerical simulations and seismic data analysis that large earthquake events have a tendency to follow events that are highly unequal, similar to the case of a system near a critical point. Therefore, a continuous monitoring of the inequality indices of the earthquake time series could be useful for hazard estimates. We have applied this framework to models of earthquakes as well as to the earthquake time series from various tectonically active regions, such as North America, Southern Japan, parts of South-East Asia and Indonesia. The findings also indicate a quantitative estimate of the distance from criticality, when the tectonic plates are viewed as a self-organized critical system.

💡 Research Summary

The manuscript “Large earthquakes follow highly unequal ones” investigates whether measures of inequality—specifically the Gini and Kolkata (k) indices—can serve as precursors to large seismic events when the Earth’s crust is modeled as a self‑organized critical (SOC) system. The authors begin by reviewing the long‑standing conjecture that tectonic plates operate near a critical point, a view supported by the Gutenberg‑Richter (GR) magnitude law and the Omori‑Utsu aftershock decay law. Recent theoretical work on driven disordered systems has shown that, as a system approaches criticality, the distribution of its responses becomes increasingly unequal; the Gini and Kolkata indices, originally devised to quantify wealth inequality, rise sharply and exhibit universal values (≈0.87) just before a system‑spanning event.

To test this idea in the context of earthquakes, the authors employ two well‑established SOC models and a comprehensive analysis of real seismic catalogs.

1. Models

• Sandpile‑like model: A two‑dimensional lattice (N=400 sites) is uniformly loaded at rate v₀. When a site’s stress exceeds a threshold S_c, a cluster of neighboring sites above S_c−D₁ is identified; all sites in the cluster relax by a fixed amount D_s = D₂ + D₃. The avalanche size n (number of relaxed sites) is converted to a synthetic magnitude via M = (3/2) log n. Parameter choices (v₀=1, S_c=10, D₁=5, D₂=10, D₃=0.1) ensure a GR‑like power‑law distribution for large avalanches.
• Train model: A one‑dimensional chain of blocks is pulled slowly by a rightmost “engine” block. Each block experiences elastic forces from its neighbors and a random pinning force drawn uniformly from (0,1). When the net force exceeds the pinning, the block hops forward, possibly triggering a cascade. The number of blocks that move between successive engine steps defines an avalanche. This model maps onto the quenched Edwards‑Wilkinson equation and also displays SOC scaling.

Both models reproduce the GR law and, importantly, exhibit a sharp increase in inequality measures before large avalanches.

2. Inequality indices
The authors compute the Gini (g) and Kolkata (k) indices for a sliding window of the preceding W=100 events. For each window, event energies (or model avalanche sizes) are sorted, a Lorenz curve L(p) is built, and g = 1 − 2∫₀¹ L(p) dp, while k solves 1 − k = L(k). The calculated g and k are then assigned to the next event, providing a measure of the inequality that preceded it.

3. Simulation results
In the sandpile model, plotting avalanche sizes against their associated g values shows a clear clustering of large avalanches around g≈0.87. Sorting the entire time series by g produces a visual concentration of high‑magnitude events at the high‑g tail. The same pattern appears for the Kolkata index. Similar analyses of the train model yield comparable trends: larger avalanches preferentially follow windows with elevated g and k. Time‑series of g and k also reveal spikes immediately before large events, confirming the predictive potential of these metrics.

4. Real seismic data
The authors analyze USGS earthquake catalogs from 1975 to October 2025 for four tectonically active regions: Southern Japan, Southeast Asia, western North America, and Indonesia. Events with magnitude M ≥ 4.5 are retained, and magnitudes are converted to seismic energy via E = 10^{1.5 M} (ignoring a constant prefactor). Using the same sliding‑window approach (W=100), g and k are computed for each event. When the series are reordered by g (or k), large earthquakes (M ≥ 7.5) appear disproportionately in the high‑g (≈0.85–0.90) segment. The authors also test a “dynamic reset” scheme: when a large event is encountered, the left edge of the window is reset to that event, allowing the window to grow adaptively. This method still shows elevated inequality preceding major quakes, indicating robustness against window‑size choices.

5. Interpretation and implications
The convergence of g and k toward ~0.87 before large events mirrors theoretical predictions for systems on the brink of a critical transition, suggesting that the Earth’s crust behaves as an SOC system whose proximity to criticality can be quantified by inequality measures. Because g and k are easy to compute in real time from publicly available seismic catalogs, they could be incorporated into operational hazard monitoring frameworks. Moreover, the authors argue that inequality indices complement other proposed precursors (e.g., b‑value reductions, Omori‑Utsu parameter changes) and could be combined in multivariate early‑warning models.

6. Conclusions
The study provides compelling evidence that large earthquakes are statistically more likely to follow periods of highly unequal energy release among preceding events. This relationship holds across two distinct SOC models and across diverse real‑world tectonic settings. The universal rise of Gini and Kolkata indices to values near 0.87 serves as a quantitative marker of the system’s distance from criticality and offers a novel, data‑driven tool for seismic hazard assessment. Future work is suggested to integrate inequality metrics with other seismicity indicators and to explore their predictive skill in real‑time forecasting environments.