scales). Following James et al. (2019), we used unbiased $\alpha$ estimates:

$\dfrac{1}{\alpha_{\mathrm{m}}}=\dfrac{1}{N-1}\sum\mathrm{ln}\left(\frac{X}{X_{\mathrm{min}}}\right).$
(11)

For $\alpha_{\mathrm{err}}$ we used the expression from the same work, despite
it being valid only for large $N$:

$\alpha_{\mathrm{err}}=\dfrac{\alpha_{\mathrm{m}}}{\sqrt{N-2}}.$ (12)

In the ideal case $a$ follows a normal distribution with a median of 0 and a
standard deviation $\sigma=1$. This means that $\alpha_{\mathrm{m}}$ provides
an unbiased estimate of $\and$, and $\alpha_{\mathrm{err}}$ has an intuitive
Gaussian meaning. To illustrate how closely $a$ obeys a normal distribution we
plot, in Fig. 22 (right sub-panel) , the percentiles at levels corresponding
to the median and its $\pm\sigma$, $\pm 2\sigma$, and $\pm 3\sigma$
counterparts. For the MLE method and relatively large sample sizes $N\gtrsim
100$, these levels are at $a=0,\pm 1,\pm 2,\pm 3$, indicating that $a$ very
closely resembles the normal distribution. For smaller sample sizes $a$ is
skewed towards negative values, and accurately described by a gamma
distribution (Eq. 2). We note that the median value of $a$ stays close to 0
for all sample sizes.

In stark contrast to this, the graphical method leads to a median value of $a$
on the order of 5 for the least squares error and 1.5 for the bootstrap error,
indicating that median $\alpha_{\mathrm{m}}$ tends to be larger than $\and$ by
a few $\alpha_{\mathrm{err}}$, regardless of sample size. The wide spread of
the percentile curves in Fig. 22 (left and middle sub-panel) demonstrates that
$\alpha_{\mathrm{err}}$ values are substantially underestimated when choosing
to use the graphical method .

We performed similar accuracy estimates for PLIs determined with the python
package powerlaw202020https://pypi.org/project/powerlaw/ (Alstott et al.,
2014). In the original code, $\alpha$ and $\alpha_{\mathrm{err}}$ are
calculated as

$\dfrac{1}{\alpha_{\mathrm{m}}}=\dfrac{1}{N}\sum\mathrm{ln}\left(\frac{X}{X_{\mathrm{min}}}\right),$
(13)

and

$\alpha_{\mathrm{err}}=\dfrac{\alpha_{\mathrm{m}}}{\sqrt{N}}.$ (14)

Fig. 23 (left panel) shows that this estimate performs worse than the version
modified according to Eqs. 11-12 (center panel).

In real-life observations $X_{\mathrm{min}}$ is often not known. The powerlaw
package offers fitting for the best $X_{\mathrm{min}}$ by minimizing the
Kolmogorov-Smirnov distance between the data and the theoretical power-law
fit. We have repeated the aforementioned simulations with fitting for
$X_{\mathrm{min}}$, restricting its range from $\mathrm{min}(X)$ to the third
largest value. In this case, the median value of $a$ is slightly more biased
towards small negative values and the distribution is somewhat wider (Fig.
23), but the MLE still provides a much better fit than the graphical method.

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