## 3 Results  
The importance of high-degree nodes of ENBOSAP in the NTA of seismicity was first suggested in Reference [83]. It was observed that the inclusion of high-degree nodes in the geographical area in which we applied the NTA of seismicity was important for the clearer observation of \(\beta_{W,min}\) before strong EQs. In order to determine the important high-degree nodes that should have been included in the geographical areas where we studied seismicity in natural time, [PERSON] and [PERSON] [115] searched for \(k\)-cores in ENBOSAP. Recall that a \(k\)-core in a network is the largest subgraph whose vertices have a degree at least \(k\)[99]. Thus, in a \(k\)-core, each node has at least \(k\) nearest neighbors within that subgraph; see, e.g., the six-core depicted in red color in Figure 1. It is noteworthy that a network is hierarchically organized as a set of successfully enclosed \(k\)-cores [99], similar to the Russian nesting doll \"Matrioshka\". In other words, the five-core covered an area larger than the six-core and, as a network, it included the six-core itself; see Figure 1b,c of Reference [115]. The NTA of seismicity improved when we comparatively examined [115] the results from the areas N\({}_{25}^{50}\)E\({}_{7}^{46}\) (labeled \(A_{5}\)) and N\({}_{28}^{50}\)E\({}_{7}^{44}\) (labeled \(A_{6}\)) of the five-core and the six-core of the area N\({}_{25}^{50}\)E\({}_{5}^{46}\), respectively, in a fashion similar to that followed in [95]. More precisely, when focusing on the minima \(\beta_{W,min}\), which were simultaneously observed in both areas \(A_{5}\) and \(A_{6}\), we found that they preceded within six months, without any false alarm, all strong EQs with \(M\geq 7.1\) in the smaller region, i.e., \(A_{6}\) (see the red rectangle in Figure 1). The results for the variabilities \(\beta_{W}\) for both areas \(A_{5}\) and \(A_{6}\) for \(W=80\) and \(120\) EQs, corresponding to the number of EQs that occurred on average there for the periods of (approximately) two and three months, respectively, are shown in Figure 2 for the almost forty-year period from 1 January 1981 to 9 February 2021. A careful inspection of this figure showed that no additional minima were observed simultaneously since October 2018, which was the end of the period studied by [115]. In the same figure, with (red) open circles we marked the simultaneous \(\beta_{120,min}\) that preceded the strong EQs with \(M\geq 7.1\) in area \(A_{6}\), as mentioned. The dates corresponding to these minima are given in Table 1. In the same Table, we also inserted the date of another local minimum in both areas \(A_{5}\) and \(A_{6}\) which did not meet the criteria, \(\beta_{80}<\beta_{0}=0.253\) and \(r_{1}<r<r_{2}\), mentioned in the Materials and Methods to be considered as precursory, but it could be useful to provide an estimate of the epicenter of the Duzce EQ. Figure 3: **(a)** The cumulative distribution of the number \(n\) of EQs with \(M\geq 4.0\) that occurred between two EQs of magnitude \(M\geq 6.0\) in the area N\({}_{23}^{52}\)E\({}_{5}^{50}\) of Figure 1. The results came from 123 EQ cycles after using the Weibull model; see Table 1 of [123]. **(b)** The mean value and the standard deviation of the EPS estimated at the nearest to the epicenter point of a \(0.1^{\circ}\times 0.1^{\circ}\) square degrees grid at the time of the minima are depicted by the (red) open circles in Figure 2 as a function of the coarse grain radius \(R\). The (black) thick straight lines were designed as a guide for the eye. Since the aforementioned minima provided an estimate of the occurrence time (i.e., within the subsequent six months) of the \(M\geq 7.1\) EQs without any false alarm, we now proceeded to an estimate of the epicenter location of the impending EQ. The NTA already provided an answer to this question [31; 32] based on the simultaneous observation of \(\beta\) minima calculated in the large area and in various small areas. The latter was found to cluster around the epicenter location of the impending EQ.