Assume the tail follows a continuous power law for x \ge x_{min}:
- PDF: p(x) = (\alpha-1)\,x_{min}^{\alpha-1}\,x^{-\alpha}
- CDF: F(x) = 1 - (x/x_{min})^{1-\alpha}
Maximum-likelihood estimate of the exponent using tail data x_i \ge x_{min}:
\alpha = 1 + \dfrac{n}{\sum_{i=1}^{n} \ln\left(\dfrac{x_i}{x_{min}}\right)}
We select x_{min} by scanning candidates and minimizing the Kolmogorov–Smirnov distance:
D = \max_x |F_{emp}(x) - F_{model}(x)|
The p-value is computed by generating synthetic power-law samples, refitting, and counting how often simulated KS distances exceed the empirical one.
- Collect a list of positive event sizes from your system.
- Paste values into the data box using any separators.
- Set a minimum tail size to avoid tiny tails.
- Choose bootstrap iterations; higher improves reliability.
- Click compute and review exponent, KS, and p-value.
- Use the CSV or PDF buttons to export results.
| Run | Event size (x) | Notes |
|---|---|---|
| 1 | 3 | Small burst |
| 2 | 5 | Moderate burst |
| 3 | 8 | Moderate burst |
| 4 | 13 | Larger event |
| 5 | 21 | Tail candidate |
| 6 | 34 | Tail candidate |
| 7 | 55 | Extreme-like |
| 8 | 89 | Extreme-like |
1) What does this test actually tell me?
It checks whether the tail of your event sizes is statistically consistent with a power law. A good p-value supports the model, but it does not prove self-organized criticality.
2) Why do we estimate an xmin threshold?
Real data rarely follows a power law at small sizes. The threshold isolates the scaling region, reducing bias from detection limits, discreteness, and noncritical mechanisms.
3) What p-value should I consider “good”?
A common practice is p ≥ 0.10 to avoid false rejections in small samples. Some studies use 0.05. Interpret p together with diagnostics and domain knowledge.
4) How many events do I need for reliable results?
More is better. Hundreds to thousands of events improve stability, especially for bootstrapping. With very small datasets, xmin selection and alpha estimates can vary widely.
5) Can discrete data use this calculator?
It can still provide a useful approximation, but discrete power-law fitting is more exact for integer counts. If your sizes are strictly integer, consider dedicated discrete models too.
6) Does a power law automatically mean SOC?
No. Many mechanisms produce heavy tails. SOC claims usually require additional signatures like finite-size scaling, avalanche shape collapse, time correlations, and robustness across conditions.
7) Why might the calculator reject a power law?
Rejection can come from mixed regimes, censoring, limited dynamic range, or alternative tails like lognormal or stretched exponential. Improve preprocessing and compare competing models.