Heaps’ law in the Urn Model with Triggering

In parallel with the previous model, we will consider here a slightly different variant, in which the reinforcement does not act when an element is chosen for the first time. Hence, point (ii) of the previous rules will be changed into: We will show that both versions of the urn model above predict a Heaps’ law for |$D(t)$| and a frequency-rank distribution |$f(R)$| with a fat-tail behaviour, and the exponents of both laws do not depend on whether we choose (ii) or (ii.a). Strictly speaking, Zipf’s law requires an inverse proportionality between the frequency and rank of the considered quantities (Zipf, 1949). In the following, however, we shall always refer instead to a generalised version of Zipf’s law, in which the dependence of the frequency on the rank is power-law-like in the tail of the distribution, i.e. at large ranks. We will here simple formulas for the Heaps’ law exponent and the exponent of the asymptotic power-law behaviour of the frequency-rank distribution in terms of the model parameters |$\rho$| and |$\nu$|⁠.