Maximal Diversity and Zipf’s Law

Zipf’s law describes the empirical size distribution of the components of many systems in natural and social sciences and humanities. We show, by solving a statistical model, that Zipf’s law co-occurs with the maximization of the diversity of the component sizes. The law ruling the increase of such diversity with the total dimension of the system is derived and its relation with Heaps’s law is discussed. As an example, we show that our analytical results compare very well with linguistics and population datasets.

Figure 1

Pictorial representation of the problem. Power laws, $p(s)\sim s^{-\tau }$, are sketched with an increasing exponent $\tau$ (from bottom to top) alongside with relative typical realizations $\{s_n\}$. Entities of the same size are depicted as blocks of the same color and in all the cases they add up to $S$, the total length of the bar. For large values of $\tau$, most of the entities have small and similar sizes, resulting in a poor diversity $D$. In the other limit, small $\tau$, large sizes do get more probable but the total number of entities required to fill $S$ is smaller. Consequently, diversity is again small. The diversity is expected to be maximal for an intermediate value of $\tau$.

Figure 2

Average diversity $⟨D{⟩}_S$, obtained through numerical simulations for various sizes $S$ (see key), dashed lines are guides to the eye. Entities are extracted from the bare distribution Eq. (6) and the statistics is restricted over configurations respecting the global constraint. Results are averaged over ${10}^4-{10}^6$ configurations. Solid lines (shown only for the extreme sizes), are the analytical solutions given by Eq. (10). Inset: The exponent $\alpha (\tau )$, defined below Eq. (11), as a function of $\tau$. Solid line is the analytical result, dots are fits from the simulation data.

Figure 3

Top panel: diversity index $D$ evaluated from the data of Ref. [43]. Each green point is one of the more than 30 000 English books in the Project Gutenberg database (accessed July 2014), while the black squares are a running average over 20 points. The solid line is the result $⟨D{⟩}_S=2(S/\ln S{)}^{1/2}$, from Eq. (11) for $\tau =2$, which corresponds to maximal diversity. Bottom panel: diversity index using data from the GeoNames database [35] for cities. Each green point is a European country and the black squares are the corresponding running average. The solid line is Eq. (18) of the Supplemental Material [10], where the presence of a lower cutoff $s_L$ is taken into account. $s_L$ is estimated from the average of the smallest city in each country ($s_L\simeq 1313$), see Supplemental Material [10]. The dashed line is the behavior $⟨D{⟩}_S=2(S/\ln S{)}^{1/2}$, which is approached only asymptotically.