## 2 Materials and Methods
### Natural Time Analysis Background and Order Parameter Fluctuations for Seismicity  
When a time series comprises \(N\) EQs, we define natural time for the occurrence of the \(k\)-th EQ as \(\chi_{k}=k/N\). Considering that the quantity \(Q_{l}(l=1,2,\ldots,N)\) is proportional to the energy emitted during the \(l\)-th EQ, \(p_{k}=Q_{k}/\sum_{n=1}^{N}Q_{n}\) is the normalized energy of the \(k\)-th EQ and is estimated from the seismic catalog. Here, we used as \(Q_{l}\) the scalar seismic moment M\({}_{0}\) reported in GCMT, i.e., \(Q_{l}=\)(M\({}_{0}\))\({}_{l}\) (see References [1; 4; 13]). Alternatively, the quantity \(Q_{l}\) can be calculated directly from the moment magnitude \(M_{w}\)[94] or by converting the magnitude reported in the catalog to \(M_{w}\), see, e.g., Reference [95]. As already mentioned, the variance \(\kappa_{1}\) of natural time serves as an order parameter [4; 5; 19; 56] for seismicity:  
\[\kappa_{1}=\sum_{k=1}^{N}p_{k}\bigg{(}\frac{k}{N}\bigg{)}^{2}-\bigg{(}\sum_{k =1}^{N}p_{k}\frac{k}{N}\bigg{)}^{2}\equiv\langle\chi^{2}\rangle-\langle\chi \rangle^{2}. \tag{1}\]  
In order to calculate the fluctuation of \(\kappa_{1}\) within an excerpt of the EQ catalog comprising \(W\) consecutive EQs, we have to calculate a variety of \(\kappa_{1}\) values corresponding to this excerpt. Let us assume that the first EQ of the excerpt corresponds to energy \(Q_{1}\). We can then form sub-excerpts \(s_{j}=\Big{\{}Q_{j+k-1}\Big{\}}_{k=1,2,\ldots,N}\) of consecutive \(N=6\) EQs of energy \(Q_{j+k-1}\) and natural time \(\chi_{k}=k/N\) each (cf. at least six EQs are needed [5] for obtaining reliable \(\kappa_{1}\)). By substituting \(p_{k}=Q_{j+k-1}/\sum_{k=1}^{N}Q_{j+k-1}\) in Equation (1), we can calculate \(\kappa_{1}\) and by sliding \(s_{j}\) over the excerpt of \(W\) EQs setting \(j=1,2,\ldots,W-N+1(=W-5)\) a totality of \(W-5\) values of \(\kappa_{1}\) are estimated. We repeat this calculation for \(N=7,8,\ldots,W\), thus, obtaining an ensemble of \(W-5+W-6+\ldots+2+1=[(W-4)(W-5)]/2\) values of \(\kappa_{1}\). Then, we compute the average \(\mu(\kappa_{1})\) and the standard deviation \(\sigma(\kappa_{1})\) of the thus obtained ensemble of \([(W-4)(W-5)]/2\,\kappa_{1}\) values. The fluctuation of \(\kappa_{1}\) is quantified by the variability \(\beta_{W}\) defined to be:  
\[\beta_{W}\equiv\sigma(\kappa_{1})/\mu(\kappa_{1}). \tag{2}\]  
in order to follow the time evolution of \(\beta_{W}\), this value is assigned to the \((W+1)\)-th EQ in the EQ catalog. As stated in the Introduction, the number \(W\) is chosen [65] to correspond to the number of EQs that, on average, occur during the average lead time of an SES activity  
Figure 2: Frequency–magnitude diagram for earthquakes in the GCMT catalog. Estimated b-values with least squares (labeled ‘w LS’) and the maximum likelihood (labeled ‘max lik’) method are also shown at the bottom of each panel. (**a**) For the whole study period (1976–2022), (**b**) for the period 1976–2000, and (**c**) for the period 2001–2022, using the ZMAP software [93]. The number of earthquakes is counted in bins of 0.1-magnitude-unit width.  
that ranges from a few weeks to \(5\frac{1}{2}\) months (see Chapter 7 of Ref. [4]), i.e., a period of a few months.