*3.1. Entropy H*α

To test the sample size dependence of *H*<sup>α</sup>, we computed *H*α for the first *n* = 2*k* consecutive tokens, where *k* = 6, 7, ... , 27 for the 10 versions of our database (each with a di fferent random article order) and calculated averages. Figure 4A shows the convergence pattern for the five α-values in a superimposed scatter plot with connected dots where the colors of each *y*-axis correspond to one α-value (cf. the legend in Figure 4, the axes are log-scaled for improved visibility). For values of α < 1.00, there is no indication of convergence, while for *H*<sup>α</sup>=*1.50* and *H*<sup>α</sup>=*2.00*, it seems that *H*α converges rather quickly. To test the observed relationship between the sample size and *H*α for di fferent α-values, we calculated the Spearman correlation between the sample size and *H*α for di fferent minimum sample sizes. For example, a minimum sample size of *n* = 2<sup>17</sup> indicates that we restrict the calculation to sample sizes ranging between *n* = 2<sup>17</sup> and *n* = 227. For those 11 datapoints, we computed the Spearman correlation between the sample size and *H*α and ran the permutation test. Table 2 summarizes the results. For all α-values, except for α = 2.00, there is a clear indication for a significant (at *p* < 0.001) strong, positive, monotonic relationship between *H*α and the sample size for all the minimum sample sizes. Thus, while Figure 4A seems to indicate that *H*<sup>α</sup>=*1.50* converges rather quickly, the Spearman analysis reveals that the sample size dependence of *H*<sup>α</sup>=*1.50* persists for higher values of *k* with a minimum ρ of 0.80. Except for the last two minimum sample sizes, all the coe fficients pass the permutation test. For α = 2.00, *H*α starts to converge after *n* = 2<sup>14</sup> word tokens. None of the correlation coe fficients of higher minimum sample sizes passes the permutation test. In line with the results of [21,22], this sugges<sup>t</sup> α = 2.00 as a pragmatic choice when calculating *H*<sup>α</sup>. However, it is important to point out that for α = 2.00, the computation of *H*α is almost completely determined by the most frequent words (cf. Table 1). For lower values of α, the basic problem of sample size dependence (cf. Figure 1) persists. If it is the aim of a study to compare *H*α for databases with varying sizes, this has to be taken into account. Correspondingly, [23] reached similar conclusions for the convergence of <sup>R</sup>ény<sup>i</sup> entropy of order α = 2.00 for di fferent languages and for di fferent kinds of texts, both on the level of words and on the level of characters. In Appendix B, we have replicated the results of Table 2 based on <sup>R</sup>ényi's formulation of the entropy generalization. Table A5 shows that the results are almost identical, which is to be expected because the Havrda–Charvat–Lindhard–Nielsen–Aczél–Daróczy–Tsallis entropy is a monotone function of the <sup>R</sup>ény<sup>i</sup> entropy [20].

**Figure 4.** Generalized entropies *H*α and divergences *D*α as a function of the sample size. (**A**) *P*<sup>α</sup>, (**B**) *D*<sup>α</sup>.



\* An asterisk indicates that the corresponding correlation coefficient passed the permutation test at *p* < 0.001. For minimum sample sizes above 220, an exact permutation test is calculated.
