*2.3. Herdan–Heaps's Law*

This law accounts for the sublinear increase of the number of different words *V*, and can be measured in physical units (i.e., as a function of the time elapsed *T*, *V*(*T*) ∼ *Tγ*) or in symbolic units (i.e., as a function of the total number of words spoken *L*, *V*(*L*) ∼ *Lβ*). Results are reported in Figure 4, certifying that (i) this law holds both in Catalan and Spanish and (ii) both in symbolic (*β*) and physical (*γ*) units, (iii) with a scaling exponent *β* ≈ *γ*, in good agreemen<sup>t</sup> to previous results [4] found for English: *β* ≈ 0.63 (Spanish and English) and *β* ≈ 0.62 (Catalan). In fact, this does not come as a surprise, given that a number of works have derived an inverse relationship between Zipf's and Herdan's exponents using different assumptions (see [47] or [48] for a review).

**Figure 4.** Herdan–Heaps's law. Sublinear increase of number of different words *V* versus time elapsed *T* (blue circles) and versus total number of words spoken *L* (green diamonds) for Catalan (**left**) and Spanish (**right**). As we are leading with a multiauthor corpus, each line represents a different way of permuting the order of concatenating each speaker. In every case we find scaling laws *V*(*L*) ∼ *Lβ* and *V*(*T*) ∼ *Tγ* which holds for about three decades. The scaling exponents *β* and *γ* are estimated for each permutation using the least-squares method, and the average value of each of them over all permutations is shown in the figure. We find *β* ≈ *γ*, as previously justified in [4], while its numerical value is on agreemen<sup>t</sup> with the one found for English [4].
