Table 1.

Critical exponents

Dataset	${\alpha }_{r\prime }$	${\beta }_{r\prime }$	${\theta }_{r\prime }$	${\delta }_{r\prime }$	$\widetilde{{\beta }_{r\prime }}$	${\alpha }_r$	${\beta }_r$	${\theta }_r$	${\delta }_r$	$\widetilde{{\beta }_r}$
$\mathit{D}1$	${1.22}_{\pm 0.03}$	${0.89}_{\pm 0.04}$	${0.89}_{\pm 0.02}$	${0.15}_{\pm 0.02}$	${0.94}_{\pm 0.04}$	${2.02}_{\pm 0.08}$	${1.50}_{\pm 0.06}$	${0.88}_{\pm 0.02}$	${0.16}_{\pm 0.04}$	${1.61}_{\pm 0.09}$
$D2$	${1.28}_{\pm 0.07}$	${1.00}_{\pm 0.07}$	${0.94}_{\pm 0.02}$	${0.16}_{\pm 0.02}$	${1.04}_{\pm 0.08}$	${1.75}_{\pm 0.05}$	${1.35}_{\pm 0.03}$	${0.92}_{\pm 0.02}$	${0.17}_{\pm 0.03}$	${1.44}_{\pm 0.07}$
$D3$	${1.00}_{\pm 0.07}$	${0.64}_{\pm 0.03}$	${0.67}_{\pm 0.03}$	$-{0.07}_{\pm 0.15}$	${0.70}_{\pm 0.16}$	${1.80}_{\pm 0.14}$	${1.57}_{\pm 0.18}$	${0.83}_{\pm 0.04}$	${0.24}_{\pm 0.08}$	${1.25}_{\pm 0.16}$

We measured ${\alpha }_{r\prime }$, ${\beta }_{r\prime }$, ${\theta }_{r\prime }$, and ${\delta }_{r\prime }$ independently for each dataset by using rank as distance metric. We estimate the errors in our measurements based on 95% confidence level. We then compute $\widetilde{{\beta }_{r\prime }}={\alpha }_{r\prime }{\theta }_{r\prime }-{\delta }_{r\prime }$ using Eq. 8. The error of $\widetilde{{\beta }_{r\prime }}$, $\sigma \left(\widetilde{{\beta }_{r\prime }}\right)$, is calculated using error propagations $\sigma \left(\widetilde{{\beta }_{r\prime }}\right)=\sqrt{{\theta }_{r\prime }^2{\sigma }^2\left({\alpha }_{r\prime }\right)+{\alpha }_{r\prime }^2{\sigma }^2\left({\theta }_{r\prime }\right)+{\sigma }^2\left({\delta }_{r\prime }\right)}$. We find that $\widetilde{{\beta }_{r\prime }}$ largely agrees with ${\beta }_{r\prime }$ within uncertainties across all datasets. Similarly, we repeated the same measurements by using geodesic distance, obtaining ${\alpha }_r$, ${\beta }_r$, ${\theta }_r$, ${\delta }_r$, and their corresponding errors, allowing us to compute $\widetilde{{\beta }_r}$ and its error $\sigma \left(\widetilde{{\beta }_r}\right)$. We find $\widetilde{{\beta }_r}$ also well approximates ${\beta }_r$. The largest deviations are observed in $D3$, which is characterized by much larger uncertainties in estimations of all exponents. This is due to its much smaller data size. Because both our data size and noninteger nature of distance metrics prevent us from using standard fitting algorithms for power laws (57), we computed all our exponents by using the least-square method.