Zipf Plot Research Articles

Zipf plots and the size distribution of firms

We use a Zipf plot to demonstrate that the upper tail of the size distribution of firms is too thin relative to the log normal rather than too fat, as had previously been believed.

Correlations in binary sequences and a generalized Zipf analysis.

We investigate correlated binary sequences using an n-tuple Zipf analysis, where we define ``words'' as strings of length n, and calculate the normalized frequency of occurrence \ensuremath{\omega}(R) of ``words'' as a function of the word rank R. We analyze sequences with short-range Markovian correlations, as well as those with long-range correlations generated by three different methods: inverse Fourier transformation, L\'evy walks, and the expansion-modification system. We study the relation between the exponent \ensuremath{\alpha} characterizing long-range correlations and the exponent \ensuremath{\zeta} characterizing power-law behavior in the Zipf plot. We also introduce a function P(\ensuremath{\omega}), the frequency density, which is related to the inverse Zipf function R(\ensuremath{\omega}), and find a simple relationship between \ensuremath{\zeta} and \ensuremath{\psi}, where \ensuremath{\omega}(R)\ensuremath{\sim}${\mathit{R}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\zeta}}}$ and P(\ensuremath{\omega})\ensuremath{\sim}${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\psi}}}$. Further, for Markovian sequences, we derive an approximate form for P(\ensuremath{\omega}). Finally, we study the effect of a coarse-graining ``renormalization'' on sequences with Markovian and with long-range correlations.