Figure 1

Heaps' law (a–e) and Zipf's law (f–l) in real datasets (a–d) and (f–i) and in the urn model with triggering (e,j).

Gutenberg⁴² (a), (f), Last.fm⁴³ (b), (g), Wikipedia⁴⁴ (c), (h), del.icio.us⁴⁵ (d), (i) datasets and the urn model with triggering (e), (j). Straight lines in the Heaps' law plots show functions of the form f(x) = ax^β, with the exponent β equal respectively to β = 0.45 (Gutenberg), β = 0.68 (Last.fm lyrics), β = 0.56 (Last.fm artist), β = 0.77 (Wikipedia) and β = 0.78 (del.icio.us) and to the ratio ν/ρ in the urn model with triggering, showing that the exponents for the Heaps' law of the model predicted by the analytic results are confirmed in the simulations. Straight lines in the Zipf's law plots show functions of the form f(x) = ax^−α, where the exponent α is equal to β⁻¹ for the different β's considered above. Note that the frequency-rank plots in real data deviate from a pure power-law behavior and the correspondence between the β and α exponents is valid only asymptotically (see discussion above and the Supplementary Information for a discussion about finite-size effects).