Rank-frequency relation for Chinese characters

The Zipf's law states that the ordered frequencies $f_1>f_2> ...$ of different words in a text hold $f_r\propto r^{-\gamma}$ with $\gamma\approx 1$ and rank $r$. The law applies to many languages with alphabetical writing systems, but was so far found to be absent for the rank-frequency relation of the Chinese characters, the main (and oldest) example of the logographic writing system. Here we show that the Zipf's law for Chinese characters perfectly holds for sufficiently short texts (few thousand different characters). The scenario of its validity is similar to the Zipf's law for words in short English texts. We focus on short texts, since for the sake of the rank-frequency analysis, long texts are just mixtures of shorter, thematically homogenous pieces. For long texts (or for mixtures of short texts), the Zipf's law holds for a relatively small range of ranks, but it is still important, since for all Chinese texts (we studied) it carries out $simeq 40$ of the overall frequency. The previous results on the invalidity of the Zipf's law for long texts are accounted for by showing that before the region of very rare characters (hapax legomena) there emerges a range of ranks, where the rank-frequency relation is approximately exponential. From comparative analysis of the rank-frequency relations for Chinese and English, we suggest that the characters play for Chinese writers the same role as the words for those writing within alphabetical systems. Sufficiently long Chinese texts display a two-layer, hierarchic structure: power-law rank-frequency characters (first layer) and the exponential ones (second layer).