Figure 4. Pruning reveals the variable marginal return of words.

The Heaps scaling exponent b depends on the extent of the inclusion of the rarest words. For a given corpora and U_c value we make a scatter plot between N_w(t|U_c) and N_u(t|U_c) using words with u_i (t) ≥ U_c, using the same data color-U_c correspondence as in Fig. 3. (Panel Inset) We use OLS estimation to estimate the scaling exponent b(U_c) for the model N_w (t|U_c) ~ [N_u(t|U_c)]^b to show that b(U_c) increases from approximately 0.5 towards unity as we prune the corpora of extremely rare words. Our longitudinal language analysis provides insight into the structural importance of the most frequent words which are used more times per appearance and which play a crucial role in the usage of new and rare words.