On the Ranking Property and Underlying Dynamics of Complex Systems

Ranking procedures are widely used to describe the phenomena in many different fields of social and natural sciences, e.g., sociology, economics, linguistics, demography, physics, biology, etc. In this dissertation, we dedicated to study the ranking properties and underlying dynamics embedded in complex systems. In particular, we focused on the scores/prizes ranking in sports systems and the words/characters usage ranking in human languages. The aim is to understand the mechanisms behind these issues by using the methods of statistical physics, Bayesian statistics and agent-based modeling. The concrete results concern the following aspects. We took up an interesting topic on the scores/prizes ranking in sports systems, and analyzed 40 data samples in 12 different sports fields. We found the striking similarities in different sports, i.e., the distributions of scores/prizes follow the universal power laws. We also showed that the data yielded the Pareto principle extensively observed in many social systems: 20% of the players accumulate 80% of the scores and money. For the tennis head-to-head data, we revealed that when two players compete, the probability that the higher-ranked player will win is related to the rank difference of the two opponents. In order to understand the origins of the universal scaling, we proposed an agent-based model, which can simulate the competitions of players in different matches, and results from our simulations are consistent with the empirical findings. Extensive simulation studies indicate that the model is quite robust with respect to the modifications of some parameters. Zipf’s law is the major regularity of statistical linguistics that served as a prototype for the rank-frequency relations and scaling laws in natural sciences. We investigated several English texts, clarified the valid range of Zipf’s law, and found this valid range increases upon mixing different texts. Based on the latent semantic analysis, we proposed a probabilistic model, in which we assumed that the words are drawn into the text with random probabilities, while their apriori density relates, via Bayesian statistics, to the general features of mental lexicon of the author who produced the text. Our model explained the Zipf’s law together with the limits of its validity, its generalization to high and low frequencies and hapax legomena. In another work, we specified the rank-frequency relations for Chinese characters. iii te l-0 08 39 31 0, v er si on 1 27 J un 2 01 3 We chose to study the short texts first, since for the sake of the rank-frequency analysis, long texts are just mixtures of shorter, thematically homogenous pieces. Our results showed that the Zipf’s law for Chinese characters perfectly holds for sufficiently short texts (few thousand different characters), and the scenario of its validity is similar to that for short English texts. We argued long Chinese texts display a two-layer, hierarchic structure: power-law rank-frequency characters (first layer) and the exponential ones (second layer). The previous results on the invalidity of the Zipf’s law for long texts are accounted for by showing that in between of the Zipfian range and 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 rank-frequency relations for Chinese and English, we suggested the characters play for Chinese writers the same role as the words for those writing within alphabetical systems.