Study On Group Motion Behavior Of Self-adaptive Individual

The paper studied the group motion behavior of the self-adaptive individuals of human mobility patterns in the city marathons, mainly for the analyses on the velocity distributions of athletes in city marathons and the correlation between the transition of the velocity distribution in internal timing lengths and the intensity of the competition which is described by ranking changes. We collected the datasets of records of the finish times of athletes in the New York City marathon and American Chicago marathon for more than 10 years, and those in London marathon and Berlin marathon for several years, and the dataset of the finish times in every internal timing length of male runners in New York City marathon. From these datasets, we made empirical statistical analyses on the features of the velocity distributions of all finishers and those in the fastest group in full marathon course, and the features of the velocity distributions of all male finishers in different internal timing lengths of the New York City marathon. And we further made empirical statistical analyses on the rank changes of all male runners in every internal timing length in the New York City marathon and investigated the correlation between the transition of velocity distributions and the intensity of competitions which is described with the rank changes of runners in the internal timing lengths of the New York marathon. The main contents of this paper are summarized as follows:1. By making empirical statistical analyses on the velocity distributions of four famous city marathons, we found that all the velocity distributions of full marathon course in the four cities follow log-normal distribution, and the velocity distributions of the fastest age group of full marathon also follow log-normal distribution.2. The theoretical analysis shows that the logarithm values of average velocities and the standard deviations of the logarithmic velocities in different internal timing lengths in the New York Marathon are both approximately constant. These two conditions lead to the log-normal velocity distribution according to the maximum entropy principle. So the results of the velocity distributions from our empirical statistical analyses are in good accordance with those derived by the maximum entropy principle.3. We made empirical statistical analyses on the rank changes of all male runners in every internal timing length in the New York City marathon and investigated the correlation between the transition of velocity distributions and the intensity of competitions which is described with the rank changes of runners in the internal timing lengths of the New York marathon. We found the velocity distributions of all male runners in 5-kilometer-long internal timing lengths undergoes two transitions: from log-normal distribution in the initial lengths to the Gaussian distribution in the middle lengths, and to log-normal distribution in last lengths. The intensity of the competition, which is described by the root-mean-square rank changes of all runners, goes weaker from initial lengths to the middle lengths corresponding to the transition of the velocity distribution from log-normal distribution to Gaussian distribution, and when the competition gets stronger in the middle lengths, there will come a transition from Gaussian distribution to log-normal one in the last lengths.